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                           THE PSYCHOLOGY OF
                               ARITHMETIC


                                   BY
                          EDWARD L. THORNDIKE

                       TEACHERS COLLEGE, COLUMBIA
                               UNIVERSITY


                                New York
                         THE MACMILLAN COMPANY
                                  1929

                         _All rights reserved_



                            COPYRIGHT, 1922,
                       BY THE MACMILLAN COMPANY.

      Set up and electrotyped. Published January, 1922. Reprinted
         October, 1924; May, 1926; August, 1927; October, 1929.


              · PRINTED IN THE UNITED STATES OF AMERICA ·




PREFACE


Within recent years there have been three lines of advance in psychology
which are of notable significance for teaching. The first is the new
point of view concerning the general process of learning. We now
understand that learning is essentially the formation of connections or
bonds between situations and responses, that the satisfyingness of the
result is the chief force that forms them, and that habit rules in the
realm of thought as truly and as fully as in the realm of action.

The second is the great increase in knowledge of the amount, rate, and
conditions of improvement in those organized groups or hierarchies of
habits which we call abilities, such as ability to add or ability to
read. Practice and improvement are no longer vague generalities, but
concern changes which are definable and measurable by standard tests and
scales.

The third is the better understanding of the so-called "higher
processes" of analysis, abstraction, the formation of general notions,
and reasoning. The older view of a mental chemistry whereby sensations
were compounded into percepts, percepts were duplicated by images,
percepts and images were amalgamated into abstractions and concepts, and
these were manipulated by reasoning, has given way to the understanding
of the laws of response to elements or aspects of situations and to many
situations or elements thereof in combination. James' view of reasoning
as "selection of essentials" and "thinking things together" in a
revised and clarified form has important applications in the teaching of
all the school subjects.

This book presents the applications of this newer dynamic psychology to
the teaching of arithmetic. Its contents are substantially what have
been included in a course of lectures on the psychology of the
elementary school subjects given by the author for some years to
students of elementary education at Teachers College. Many of these
former students, now in supervisory charge of elementary schools, have
urged that these lectures be made available to teachers in general. So
they are now published in spite of the author's desire to clarify and
reinforce certain matters by further researches.

A word of explanation is necessary concerning the exercises and problems
cited to illustrate various matters, especially erroneous pedagogy.
These are all genuine, having their source in actual textbooks, courses
of study, state examinations, and the like. To avoid any possibility of
invidious comparisons they are not quotations, but equivalent problems
such as represent accurately the spirit and intent of the originals.

I take pleasure in acknowledging the courtesy of Mr. S. A. Courtis, Ginn
and Company, D. C. Heath and Company, The Macmillan Company, The Oxford
University Press, Rand, McNally and Company, Dr. C. W. Stone, The
Teachers College Bureau of Publications, and The World Book Company, in
permitting various quotations.

                                                EDWARD L. THORNDIKE.

   TEACHERS COLLEGE
  COLUMBIA UNIVERSITY
     April 1, 1920




CONTENTS


 CHAPTER                                                          PAGE

    INTRODUCTION: THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS  xi

 I. THE NATURE OF ARITHMETICAL ABILITIES                             1

        Knowledge of the Meanings of Numbers
        Arithmetical Language
        Problem Solving
        Arithmetical Reasoning
        Summary
        The Sociology of Arithmetic

 II. THE MEASUREMENT OF ARITHMETICAL ABILITIES                      27

        A Sample Measurement of an Arithmetical Ability
        Ability to Add Integers
        Measurements of Ability in Computation
        Measurements of Ability in Applied Arithmetic:
          the Solution of Problems

 III. THE CONSTITUTION OF ARITHMETICAL ABILITIES                    51

        The Elementary Functions of Arithmetical Learning
        Knowledge of the Meaning of a Fraction
        Learning the Processes of Computation

 IV. THE CONSTITUTION OF ARITHMETICAL ABILITIES (_continued_)       70

        The Selection of the Bonds to Be Formed
        The Importance of Habit Formation
        Desirable Bonds Now Often Neglected
        Wasteful and Harmful Bonds
        Guiding Principles

 V. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS   102

        The Need of Stronger Elementary Bonds
        Early Mastery
        The Strength of Bonds for Temporary Service
        The Strength of Bonds with Technical Facts and Terms
        The Strength of Bonds Concerning the Reasons for
          Arithmetical Processes
        Propædeutic Bonds

 VI. THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF
          PRACTICE AND THE ORGANIZATION OF ABILITIES               122

        The Amount of Practice
        Under-learning and Over-learning
        The Organization of Abilities

 VII. THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS      141

        Conventional _versus_ Effective Orders
        Decreasing Interference and Increasing Facilitation
        Interest
        General Principles

 VIII. THE DISTRIBUTION OF PRACTICE                                156

        The Problem
        Sample Distributions
        Possible Improvements

 IX. THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL
          NOTIONS IN ARITHMETIC                                    169

        Responses to Elements and Classes
        Facilitating the Analysis of Elements
        Systematic and Opportunistic Stimuli to Analysis
        Adaptations to Elementary-school Pupils

 X. THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC            185

        The Essentials of Arithmetical Reasoning
        Reasoning as the Coöperation of Organized Habits

 XI. ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL            195

        The Utilization of Instinctive Interests
        The Order of Development of Original Tendencies
        Inventories of Arithmetical Knowledge and Skill
        The Perception of Number and Quantity
        The Early Awareness of Number

 XII. INTEREST IN ARITHMETIC                                       209

        Censuses of Pupils' Interests
        Relieving Eye Strain
        Significance for Related Activities
        Intrinsic Interest in Arithmetical Learning

 XIII. THE CONDITIONS OF LEARNING                                  227

        External Conditions
        The Hygiene of the Eyes in Arithmetic
        The Use of Concrete Objects in Arithmetic
        Oral, Mental, and Written Arithmetic

 XIV. THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE             266

        Illustrative Cases
        General Principles
        Difficulty and Success as Stimuli
        False Inferences

 XV. INDIVIDUAL DIFFERENCES                                        285

        Nature and Amount
        Differences within One Class
        The Causes of Individual Differences
        The Interrelations of Individual Differences

    BIBLIOGRAPHY OF REFERENCES                                     302

    INDEX                                                          311




GENERAL INTRODUCTION

THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS


The psychology of the elementary school subjects is concerned with the
connections whereby a child is able to respond to the sight of printed
words by thoughts of their meanings, to the thought of "six and eight"
by thinking "fourteen," to certain sorts of stories, poems, songs, and
pictures by appreciation thereof, to certain situations by acts of
skill, to certain others by acts of courtesy and justice, and so on and
on through the series of situations and responses which are provided by
the systematic training of the school subjects and the less systematic
training of school life during their study. The aims of elementary
education, when fully defined, will be found to be the production of
changes in human nature represented by an almost countless list of
connections or bonds whereby the pupil thinks or feels or acts in
certain ways in response to the situations the school has organized and
is influenced to think and feel and act similarly to similar situations
when life outside of school confronts him with them.

We are not at present able to define the work of the elementary school
in detail as the formation of such and such bonds between certain
detached situations and certain specified responses. As elsewhere in
human learning, we are at present forced to think somewhat vaguely in
terms of mental functions, like "ability to read the vernacular,"
"ability to spell common words," "ability to add, subtract, multiply,
and divide with integers," "knowledge of the history of the United
States," "honesty in examinations," and "appreciation of good music,"
defined by some general results obtained rather than by the elementary
bonds which constitute them.

The psychology of the school subjects begins where our common sense
knowledge of these functions leaves off and tries to define the
knowledge, interest, power, skill, or ideal in question more adequately,
to measure improvement in it, to analyze it into its constituent bonds,
to decide what bonds need to be formed and in what order as means to the
most economical attainment of the desired improvement, to survey the
original tendencies and the tendencies already acquired before entrance
to school which help or hinder progress in the elementary school
subjects, to examine the motives that are or may be used to make the
desired connections satisfying, to examine any other special conditions
of improvement, and to note any facts concerning individual differences
that are of special importance to the conduct of elementary school work.

Put in terms of problems, the task of the psychology of the elementary
school subjects is, in each case:--

(1) _What is the function?_ For example, just what is "ability to read"?
Just what does "the understanding of decimal notation" mean? Just what
are "the moral effects to be sought from the teaching of literature"?

(2) _How are degrees of ability or attainment, and degrees of progress
or improvement in the function or a part of the function measured?_ For
example, how can we determine how well a pupil should write, or how hard
words we expect him to spell, or what good taste we expect him to show?
How can we define to ourselves what knowledge of the meaning of a
fraction we shall try to secure in grade 4?

(3) _What can be done toward reducing the function to terms of
particular situation-response connections, whose formation can be more
surely and easily controlled?_ For example, how far does ability to
spell involve the formation one by one of bonds between the thought of
almost every word in the language and the thought of that word's letters
in their correct order; and how far does, say, the bond leading from the
situation of the sound of _ceive_ in _receive_ and _deceive_ to their
correct spelling insure the correct spelling of that part of _perceive_?
Does "ability to add" involve special bonds leading from "27 and 4" to
"31," from "27 and 5" to "32," and "27 and 6" to "33"; or will the bonds
leading from "7 and 4" to "11," "7 and 5" to "12" and "7 and 6" to "13"
(each plus a simple inference) serve as well? What are the situations
and responses that represent in actual behavior the quality that we call
school patriotism?

(4) _In almost every case a certain desired change of knowledge or skill
or power can be attained by any one of several sets of bonds. Which of
them is the best? What are the advantages of each?_ For example,
learning to add may include the bonds "0 and 0 are 0," "0 and 1 are 1,"
"0 and 2 are 2," "1 and 0 are 1," "2 and 0 are 2," etc.; or these may be
all left unformed, the pupil being taught the habits of entering 0 as
the sum of a column that is composed of zeros and otherwise neglecting 0
in addition. Are the rules of usage worth teaching as a means toward
correct speech, or is the time better spent in detailed practice in
correct speech itself?

(5) _A bond to be formed may be formed in any one of many degrees of
strength. Which of these is, at any given stage of learning the subject,
the most desirable, all things considered?_ For example, shall the dates
of all the early settlements of North America be learned so that the
exact year will be remembered for ten years, or so that the exact date
will be remembered for ten minutes and the date with an error plus or
minus of ten years will be remembered for a year or two? Shall the
tables of inches, feet, and yards, and pints, quarts, and gallons be
learned at their first appearance so as to be remembered for a year, or
shall they be learned only well enough to be usable in the work of that
week, which in turn fixes them to last for a month or so? Should a pupil
in the first year of study of French have such perfect connections
between the sounds of French words and their meanings that he can
understand simple sentences containing them spoken at an ordinary rate
of speaking? Or is slow speech permissible, and even imperative, on the
part of the teacher, with gradual increase of rate?

(6) _In almost every case, any set of bonds may produce the desired
change when presented in any one of several orders. Which is the best
order? What are the advantages of each?_ Certain systems for teaching
handwriting perfect the elementary movements one at a time and then
teach their combination in words and sentences. Others begin and
continue with the complex movement-series that actual words require.
What do the latter lose and gain? The bonds constituting knowledge of
the metric system are now formed late in the pupil's course. Would it be
better if they were formed early as a means of facilitating knowledge of
decimal fractions?

(7) _What are the original tendencies and pre-school acquisitions upon
which the connection-forming of the elementary school may be based or
which it has to counteract?_ For example, if a pupil knows the meaning
of a heard word, he may read it understandingly from getting its sound,
as by phonic reconstruction. What words does the average beginner so
know? What are the individual differences in this respect? What do the
instincts of gregariousness, attention-getting, approval, and
helpfulness recommend concerning group-work _versus_ individual-work,
and concerning the size of a group that is most desirable? The original
tendency of the eyes is certainly not to move along a line from left to
right of a page, then back in one sweep and along the next line. What is
their original tendency when confronted with the printed page, and what
must we do with it in teaching reading?

(8) _What armament of satisfiers and annoyers, of positive and negative
interests and motives, stands ready for use in the formation of the
intrinsically uninteresting connections between black marks and
meanings, numerical exercises and their answers, words and their
spelling, and the like?_ School practice has tried, more or less at
random, incentives and deterrents from quasi-physical pain to the most
sentimental fondling, from sheer cajolery to philosophical argument,
from appeals to assumed savage and primitive traits to appeals to the
interest in automobiles, flying-machines, and wireless telegraphy. Can
not psychology give some rules for guidance, or at least limit
experimentation to its more hopeful fields?

(9) _The general conditions of efficient learning are described in
manuals of educational psychology. How do these apply in the case of
each task of the elementary school?_ For example, the arrangement of
school drills in addition and in short division in the form of practice
experiments has been found very effective in producing interest in the
work and in improvement at it. In what other arithmetical functions may
we expect the same?

(10) _Beside the general principles concerning the nature and causation
of individual differences, there must obviously be, in existence or
obtainable as a possible result of proper investigation, a great fund
of knowledge of special differences relevant to the learning of reading,
spelling, geography, arithmetic, and the like. What are the facts as far
as known? What are the means of learning more of them?_ Courtis finds
that a child may be specially strong in addition and yet be specially
weak in subtraction in comparison with others of his age and grade. It
even seems that such subtle and intricate tendencies are inherited. How
far is such specialization the rule? Is it, for example, the case that a
child may have a special gift for spelling certain sorts of words, for
drawing faces rather than flowers, for learning ancient history rather
than modern?

       *       *       *       *       *

Such are our problems: this volume discusses them in the case of
arithmetic. The student who wishes to relate the discussion to the
general pedagogy of arithmetic may profitably read, in connection with
this volume: The Teaching of Elementary Mathematics, by D. E. Smith
['01], The Teaching of Primary Arithmetic, by H. Suzzallo ['11], How to
Teach Arithmetic, by J. C. Brown and L. D. Coffman ['14], The Teaching
of Arithmetic, by Paul Klapper ['16], and The New Methods in Arithmetic,
by the author ['21].




THE PSYCHOLOGY OF ARITHMETIC




CHAPTER I

THE NATURE OF ARITHMETICAL ABILITIES


According to common sense, the task of the elementary school is to
teach:--(1) the meanings of numbers, (2) the nature of our system
of decimal notation, (3) the meanings of addition, subtraction,
multiplication, and division, and (4) the nature and relations of
certain common measures; to secure (5) the ability to add, subtract,
multiply, and divide with integers, common and decimal fractions, and
denominate numbers, (6) the ability to apply the knowledge and power
represented by (1) to (5) in solving problems, and (7) certain specific
abilities to solve problems concerning percentage, interest, and other
common occurrences in business life.

This statement of the functions to be developed and improved is sound
and useful so far as it goes, but it does not go far enough to make the
task entirely clear. If teachers had nothing but the statement above as
a guide to what changes they were to make in their pupils, they would
often leave out important features of arithmetical training, and put in
forms of training that a wise educational plan would not tolerate. It is
also the case that different leaders in arithmetical teaching, though
they might all subscribe to the general statement of the previous
paragraph, certainly do not in practice have identical notions of what
arithmetic should be for the elementary school pupil.

The ordinary view of the nature of arithmetical learning is obscure or
inadequate in four respects. It does not define what 'knowledge of the
meanings of numbers' is; it does not take account of the very large
amount of teaching of _language_ which is done and should be done as a
part of the teaching of arithmetic; it does not distinguish between the
ability to meet certain quantitative problems as life offers them and
the ability to meet the problems provided by textbooks and courses of
study; it leaves 'the ability to apply arithmetical knowledge and power'
as a rather mystical general faculty to be improved by some educational
magic. The four necessary amendments may be discussed briefly.


KNOWLEDGE OF THE MEANINGS OF NUMBERS

Knowledge of the meanings of the numbers from one to ten may mean
knowledge that 'one' means a single thing of the sort named, that two
means one more than one, that three means one more than two, and so on.
This we may call the _series_ meaning. To know the meaning of 'six' in
this sense is to know that it is one more than five and one less than
seven--that it is between five and seven in the number series. Or we may
mean by knowledge of the meanings of numbers, knowledge that two fits a
collection of two units, that three fits a collection of three units,
and so on, each number being a name for a certain sized collection of
discrete things, such as apples, pennies, boys, balls, fingers, and the
other customary objects of enumeration in the primary school. This we
may call the _collection_ meaning. To know the meaning of six in this
sense is to be able to name correctly any collection of six separate,
easily distinguishable individual objects. In the third place, knowledge
of the numbers from one to ten may mean knowledge that two is twice
whatever is called one, that three is three times whatever is one, and
so on. This is, of course, the _ratio_ meaning. To know the meaning of
six in this sense is to know that if ___________ is one, a line half a
foot long is six, that if [___] is one, [____________] is about six,
while if [__] is one, [______] is about six, and the like. In the fourth
place, the meaning of a number may be a smaller or larger fraction of
its _implications_--its numerical relations, facts about it. To know six
in this sense is to know that it is more than five or four, less than
seven or eight, twice three, three times two, the sum of five and one,
or of four and two, or of three and three, two less than eight--that
with four it makes ten, that it is half of twelve, and the like. This we
may call the '_nucleus of facts_' or _relational_ meaning of a number.

Ordinary school practice has commonly accepted the second meaning as
that which it is the task of the school to teach beginners, but each of
the other meanings has been alleged to be the essential one--the series
idea by Phillips ['97], the ratio idea by McLellan and Dewey ['95] and
Speer ['97], and the relational idea by Grube and his followers.

This diversity of views concerning what the function is that is to be
improved in the case of learning the meanings of the numbers one to ten
is not a trifling matter of definition, but produces very great
differences in school practice. Consider, for example, the predominant
value assigned to counting by Phillips in the passage quoted below, and
the samples of the sort of work at which children were kept employed
for months by too ardent followers of Speer and Grube.

THE SERIES IDEA OVEREMPHASIZED

  "This is essentially the counting period, and any words that can be
  arranged into a series furnish all that is necessary. Counting is
  fundamental, and counting that is spontaneous, free from sensible
  observation, and from the strain of reason. A study of these original
  methods shows that multiplication was developed out of counting, and
  not from addition as nearly all textbooks treat it. Multiplication is
  counting. When children count by 4's, etc., they accent the same as
  counting gymnastics or music. When a child now counts on its fingers
  it simply reproduces a stage in the growth of the civilization of all
  nations.

  I would emphasize again that during the counting period there is a
  somewhat spontaneous development of the number series-idea which
  Preyer has discussed in his Arithmogenesis; that an immense momentum
  is given by a systematic series of names; and that these names are
  generally first learned and applied to objects later. A lady teacher
  told me that the Superintendent did not wish the teachers to allow the
  children to count on their fingers, but she failed to see why counting
  with horse-chestnuts was any better. Her children could hardly avoid
  using their fingers in counting other objects yet they followed the
  series to 100 without hesitation or reference to their fingers. This
  spontaneous counting period, or naming and following the series,
  should precede its application to objects." [D.E. Phillips, '97,
  p. 238.]

THE RATIO IDEA OVEREMPHASIZED

    [Illustration: FIG. 1.]

  "Ratios.--1. Select solids having the relation, or ratio, of _a_, _b_,
     _c_, _d_, _o_, _e_.

  2. Name the solids, _a_, _b_, _c_, _d_, _o_, _e_.

     The means of expressing must be as freely supplied as the means of
     discovery. The pupil is not expected to invent terms.

  3. Tell all you can about the relation of these units.

  4. Unite units and tell what the sum equals.

  5. Make statements like this: _o_ less _e_ equals _b_.

  6. _c_ can be separated into how many _d_'s? into how many _b_'s?

  7. _c_ can be separated into how many _b_'s? What is the name of the
     largest unit that can be found in both _c_ and _d_ an exact number
     of times?

  8. Each of the other units equals what part of _c_?

  9. If _b_ is 1, what is each of the other units?

 10. If _a_ is 1, what is each of the other units?

 11. If _b_ is 1, how many 1's are there in each of the other units?

 12. If _d_ is 1, how many 1's and parts of 1 in each of the other
     units?

 13. 2 is the relation of what units?

 14. 3 is the relation of what units?

 15. 1/2 is the relation of what units?

 16. 2/3 is the relation of what units?

 17. Which units have the relation 3/2?

 18. Which unit is 3 times as large as 1/2 of _b_?

 19. _c_ equals 6 times 1/3 of what unit?

 20. 1/3 of what unit equals 1/6 of _c_?

 21. What equals 1/2 of _c_? _d_ equals how many sixths of _c_?

 22. _o_ equals 5 times 1/3 of what unit?

 23. 1/3 of what unit equals 1/5 of _o_?

 24. 2/3 of _d_ equals what unit? _b_ equals how many thirds of _d_?

 25. 2 is the ratio of _d_ to 1/3 of what unit? 3 is the ratio of _d_
     to 1/2 of what unit?

 26. _d_ equals 3/4 of what unit? 3/4 is the ratio of what units?"
     [Speer, '97, p. 9f.]

THE RELATIONAL IDEA OVEREMPHASIZED

  An inspection of books of the eighties which followed the "Grube
  method" (for example, the _New Elementary Arithmetic_ by E.E. White
  ['83]) will show undue emphasis on the relational ideas. There will be
  over a hundred and fifty successive tasks all, or nearly all, on +7
  and -7. There will be much written work of the sort shown below:

  _Add:_
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    4    4
           4    1    2
          --   --   --

  which must have sorely tried the eyes of all concerned. Pupils are
  taught to "give the analysis and synthesis of each of the nine
  digits." Yet the author states that he does not carry the principle
  of the Grube method "to the extreme of useless repetition and
  mechanism."

It should be obvious that all four meanings have claims upon the
attention of the elementary school. Four is the thing between three and
five in the number series; it is the name for a certain sized collection
of discrete objects; it is also the name for a continuous magnitude
equal to four units--for four quarts of milk in a gallon pail as truly
as for four separate quart-pails of milk; it is also, if we know it
well, the thing got by adding one to three or subtracting six from ten
or taking two two's or half of eight. To know the meaning of a number
means to know somewhat about it in all of these respects. The difficulty
has been the narrow vision of the extremists. A child must not be left
interminably counting; in fact the one-more-ness of the number series
can almost be had as a by-product. A child must not be restricted to
exercises with collections objectified as in Fig. 2 or stated in words
as so many apples, oranges, hats, pens, etc., when work with measurement
of continuous quantities with varying units--inches, feet, yards,
glassfuls, pints, quarts, seconds, minutes, hours, and the like--is
so easy and so significant. On the other hand, the elaboration of
artificial problems with fictitious units of measure just to have
relative magnitudes as in the exercises on page 5 is a wasteful
sacrifice. Similarly, special drills emphasizing the fact that eighteen
is eleven and seven, twelve and six, three less than twenty-one, and the
like, are simply idolatrous; these facts about eighteen, so far as they
are needed, are better learned in the course of actual column-addition
and -subtraction.

    [Illustration: FIG. 2.]


ARITHMETICAL LANGUAGE

The second improvement to be made in the ordinary notion of what the
functions to be improved are in the case of arithmetic is to include
among these functions the knowledge of certain words. The understanding
of such words as _both_, _all_, _in all_, _together_, _less_,
_difference_, _sum_, _whole_, _part_, _equal_, _buy_, _sell_, _have
left_, _measure_, _is contained in_, and the like, is necessary in
arithmetic as truly as is the understanding of numbers themselves. It
must be provided for by the school; for pre-school and extra-school
training does not furnish it, or furnishes it too late. It can be
provided for much better in connection with the teaching of arithmetic
than in connection with the teaching of English.

It has not been provided for. An examination of the first fifty pages of
eight recent textbooks for beginners in arithmetic reveals very slight
attention to this matter at the best and no attention at all in some
cases. Three of the books do not even use the word _sum_, and one uses
it only once in the fifty pages. In all the four hundred pages the word
_difference_ occurs only twenty times. When the words are used, no great
ingenuity or care appears in the means of making sure that their
meanings are understood.

The chief reason why it has not been provided for is precisely that the
common notion of what the functions are that arithmetic is to develop
has left out of account this function of intelligent response to
quantitative terms, other than the names of the numbers and processes.

Knowledge of language over a much wider range is a necessary element in
arithmetical ability in so far as the latter includes ability to solve
verbally stated problems. As arithmetic is now taught, it does include
that ability, and a large part of the time of wise teaching is given to
improving the function 'knowing what a problem states and what it asks
for.' Since, however, this understanding of verbally stated problems may
not be an absolutely necessary element of arithmetic, it is best to
defer its consideration until we have seen what the general function of
problem-solving is.


PROBLEM-SOLVING

The third respect in which the function, 'ability in arithmetic,' needs
clearer definition, is this 'problem-solving.' The aim of the elementary
school is to provide for correct and economical response to genuine
problems, such as knowing the total due for certain real quantities at
certain real prices, knowing the correct change to give or get, keeping
household accounts, calculating wages due, computing areas, percentages,
and discounts, estimating quantities needed of certain materials to make
certain household or shop products, and the like. Life brings these
problems usually either with a real situation (as when one buys and
counts the cost and his change), or with a situation that one imagines
or describes to himself (as when one figures out how much money he must
save per week to be able to buy a forty-dollar bicycle before a certain
date). Sometimes, however, the problem is described in words to the
person who must solve it by another person (as when a life insurance
agent says, 'You pay only 25 cents a week from now till--and you get
$250 then'; or when an employer says, 'Your wages would be 9 dollars a
week, with luncheon furnished and bonuses of such and such amounts').
Sometimes also the problem is described in printed or written words to
the person who must solve it (as in an advertisement or in the letter of
a customer asking for an estimate on this or that). The problem may be
in part real, in part imagined or described to oneself, and in part
described to one orally or in printed or written words (as when the
proposed articles for purchase lie before one, the amount of money one
has in the bank is imagined, the shopkeeper offers 10 percent discount,
and the printed price list is there to be read).

To fit pupils to solve these real, personally imagined, or
self-described problems, and 'described-by-another' problems, schools
have relied almost exclusively on training with problems of the last
sort only. The following page taken almost at random from one of the
best recent textbooks could be paralleled by thousands of others; and
the oral problems put by teachers have, as a rule, no real situation
supporting them.

  1. At 70 cents per 100 pounds, what will be the amount of duty on an
     invoice of 3622 steel rails, each rail being 27 feet long and
     weighing 60 pounds to the yard?

  2. A man had property valued at $6500. What will be his taxes at the
     rate of $10.80 per $1000?

  3. Multiply seventy thousand fourteen hundred-thousandths by one
     hundred nine millionths, and divide the product by five hundred
     forty-five.

  4. What number multiplied by 43-3/4 will produce 265-5/8?

  5. What decimal of a bushel is 3 quarts?

  6. A man sells 5/8 of an acre of land for $93.75. What would be the
     value of his farm of 150-3/4 acres at the same rate?

  7. A coal dealer buys 375 tons coal at $4.25 per ton of 2240 pounds.
     He sells it at $4.50 per ton of 2000 pounds. What is his profit?

  8. Bought 60 yards of cloth at the rate of 2 yards for $5, and 80
     yards more at the rate of 4 yards for $9. I immediately sold the
     whole of it at the rate of 5 yards for $12. How much did I gain?

  9. A man purchased 40 bushels of apples at $1.50 per bushel.
     Twenty-five hundredths of them were damaged, and he sold them
     at 20 cents per peck. He sold the remainder at 50 cents per
     peck. How much did he gain or lose?

 10. If oranges are 37-1/2 cents per dozen, how many boxes, each
     containing 480, can be bought for $60?

 11. A man can do a piece of work in 18-3/4 days. What part of it can
     he do in 6-2/3 days?

 12. How old to-day is a boy that was born Oct. 29, 1896?
     [Walsh, '06, Part I, p. 165.]

As a result, teachers and textbook writers have come to think of the
functions of solving arithmetical problems as identical with the
function of solving the described problems which they give in school in
books, examination papers, and the like. If they do not think explicitly
that this is so, they still act in training and in testing pupils as if
it were so.

It is not. Problems should be solved in school to the end that pupils
may solve the problems which life offers. To know what change one should
receive after a given real purchase, to keep one's accounts accurately,
to adapt a recipe for six so as to make enough of the article for four
persons, to estimate the amount of seed required for a plot of a given
size from the statement of the amount required per acre, to make with
surety the applications that the household, small stores, and ordinary
trades require--such is the ability that the elementary school should
develop. Other things being equal, the school should set problems in
arithmetic which life then and later will set, should favor the
situations which life itself offers and the responses which life itself
demands.

Other things are not always equal. The same amount of time and effort
will often be more productive toward the final end if directed during
school to 'made-up' problems. The keeping of personal financial accounts
as a school exercise is usually impracticable, partly because some of
the children have no earnings or allowance--no accounts to keep, and
partly because the task of supervising work when each child has a
different problem is too great for the teacher. The use of real
household and shop problems will be easy only when the school program
includes the household arts and industrial education, and when these
subjects themselves are taught so as to improve the functions used by
real life. Very often the most efficient course is to make sure that
arithmetical procedures are applied to the real and personally initiated
problems which they fit, by having a certain number of such problems
arise and be solved; then to make sure that the similarity between these
real problems and certain described problems of the textbook or
teacher's giving is appreciated; and then to give the needed drill work
with described problems. In many cases the school practice is fairly
well justified in assuming that solving described problems will prepare
the pupil to solve the corresponding real problems actually much better
than the same amount of time spent on the real problems themselves.

All this is true, yet the general principle remains that, other things
being equal, the school should favor real situations, should present
issues as life will present them.

Where other things make the use of verbally described problems of the
ordinary type desirable, these should be chosen so as to give a maximum
of preparation for the real applications of arithmetic in life. We
should not, for example, carelessly use any problem that comes to mind
in applying a certain principle, but should stop to consider just what
the situations of life really require and show clearly the application
of that principle. For example, contrast these two problems applying
cancellation:--

  A. A man sold 24 lambs at $18 apiece on each of six days, and
     bought 8 pounds of metal with the proceeds. How much did he
     pay per ounce for the metal?

  B. How tall must a rectangular tank 16" long by 8" wide be to
     hold as much as a rectangular tank 24" by 18" by 6"?

The first problem not only presents a situation that would rarely or
never occur, but also takes a way to find the answer that would not, in
that situation, be taken since the price set by another would determine
the amount.

Much thought and ingenuity should in the future be expended in
eliminating problems whose solution does not improve the real function
to be improved by applied arithmetic, or improves it at too great cost,
and in devising problems which prepare directly for life's demands and
still can fit into a curriculum that can be administered by one teacher
in charge of thirty or forty pupils, under the limitations of school
life.

The following illustrations will to some extent show concretely what the
ability to apply the knowledge and power represented by abstract or pure
arithmetic--the so-called fundamentals--in solving problems should mean
and what it should not mean.

  _Samples of Desirable Applications of Arithmetic in Problems where
    the Situation is Actually Present to Sense in Whole or in Part_

Keeping the scores and deciding which side beat and by how much in
appropriate classroom games, spelling matches, and the like.

Computing costs, making and inspecting change, taking inventories, and
the like with a real or play store.

Mapping the school garden, dividing it into allotments, planning for the
purchase of seeds, and the like.

Measuring one's own achievement and progress in tests of word-knowledge,
spelling, addition, subtraction, speed of writing, and the like.
Measuring the rate of improvement per hour of practice or per week of
school life, and the like.

Estimating costs of food cooked in the school kitchen, articles made in
the school shops, and the like.

Computing the cost of telegrams, postage, expressage, for a real message
or package, from the published tariffs.

Computing costs from mail order catalogues and the like.

  _Samples of Desirable Applications of Arithmetic where the Situation
    is Not Present to Sense_

The samples given here all concern the subtraction of fractions.
Samples concerning any other arithmetical principle may be found in the
appropriate pages of any text which contains problem-material selected
with consideration of life's needs.

A

  1. Dora is making jelly. The recipe calls for 24 cups of sugar and
     she has only 21-1/2. She has no time to go to the store so she
     has to borrow the sugar from a neighbor. How much must she get?

      _Subtract_
          24        _Think "1/2 and 1/2 = 1." Write 1/2._
          21-1/2    _Think "2 and 2 = 4." Write the 2._
        --------
           2-1/2

  2. A box full of soap weighs 29-1/2 lb. The empty box weighs 3-1/2 lb.
     How much does the soap alone weigh?

  3. On July 1, Mr. Lewis bought a 50-lb. bag of ice-cream salt. On July
     15 there were just 11-1/2 lb. left. How much had he used in the two
     weeks?

  4. Grace promised to pick 30 qt. blueberries for her mother. So far
     she has picked 18-1/2 qt. How many more quarts must she pick?

B

  This table of numbers tells                Weight of Mary Adams
  what Nell's baby sister Mary            When born        7-3/8 lb.
  weighed every two months from           2 months old    11-1/4 lb.
  the time she was born till she          4 months old    14-1/8 lb.
  was a year old.                         6 months old    15-3/4 lb.
                                          8 months old    17-5/8 lb.
                                         10 months old    19-1/2 lb.
                                         12 months old    21-3/8 lb.

  1. How much did the Adams baby gain in the first two months?
  2. How much did the Adams baby gain in the second two months?
  3. In the third two months?
  4. In the fourth two months?
  5. From the time it was 8 months old till it was 10 months old?
  6. In the last two months?
  7. From the time it was born till it was 6 months old?

C

  1. Helen's exact average for December was 87-1/3. Kate's was 84-1/2.
     How much higher was Helen's than Kate's?

      87-1/3   How do you think of 1/2 and 1/3?
      84-1/2   How do you think of 1-2/6?
      ------
               How do you change the 4?

  2. Find the exact average for each girl in the following list. Write
     the answers clearly so that you can see them easily. You will use
     them in solving problems 3, 4, 5, 6, 7, and 8.

                 Alice  Dora  Emma  Grace  Louise  Mary  Nell  Rebecca
      Reading      91    87    83     81     79     77    76     73
      Language     88    78    82     79     73     78    73     75
      Arithmetic   89    85    79     75     84     87    89     80
      Spelling     90    79    75     80     82     91    68     81
      Geography    91    87    83     75     78     85    73     79
      Writing      90    88    75     72     93     92    95     78

  3. Which girl had the highest average?

  4. How much higher was her average than the next highest?

  5. How much difference was there between the highest and the lowest
     girl?

  6. Was Emma's average higher or lower than Louise's? How much?

  7. How much difference was there between Alice's average and Dora's?

  8. How much difference was there between Mary's average and Nell's?

  9. Write five other problems about these averages, and solve each of
     them.

_Samples of Undesirable Applications of Arithmetic_[1]

  Will has XXI marbles, XII jackstones, XXXVI pieces of string. How many
  things had he?

  George's kite rose CDXXXV feet and Tom's went LXIII feet higher. How
  high did Tom's kite rise?

  If from DCIV we take CCIV the result will be a number IV times as
  large as the number of dollars Mr. Dane paid for his horse. How much
  did he pay for his horse?

  Hannah has 5/8 of a dollar, Susie 7/25, Nellie 3/4, Norah 13/16. How
  much money have they all together?

  A man saves 3-17/80 dollars a week. How much does he save in a year?

  A tree fell and was broken into 4 pieces, 13-1/6 feet, 10-3/7 feet,
  8-1/2 feet, and 7-16/21 feet long. How tall was the tree?

  Annie's father gave her 20 apples to divide among her friends. She
  gave each one 2-2/9 apples apiece. How many playmates had she?

  John had 17-2/5 apples. He divided his whole apples into fifths. How
  many pieces had he in all?

  A landlady has 3-3/7 pies to be divided among her 8 boarders. How much
  will each boarder receive?

  There are twenty columns of spelling words in Mary's lesson and 16
  words in each column. How many words are in her lesson?

  There are 9 nuts in a pint. How many pints in a pile of 5,888,673
  nuts?

  The Adams school contains eight rooms; each room contains 48 pupils;
  if each pupil has eight cents, how much have they together?

  A pile of wood in the form of a cube contains 15-1/2 cords. What are
  the dimensions to the nearest inch?

  A man 6 ft. high weighs 175 lb. How tall is his wife who is of similar
  build, and weighs 125 lb.?

  A stick of timber is in the shape of the frustum of a square pyramid,
  the lower base being 22 in. square and the upper 14 in. square. How
  many cubic feet in the log, if it is 22 ft. long?

  Mr. Ames, being asked his age, replied: "If you cube one half of my
  age and add 41,472 to the result, the sum will be one half the cube of
  my age. How old am I?"

  [1] The following and later problems are taken from actual textbooks
      or courses of study or state examinations; to avoid invidious
      comparisons, they are not exact quotations, but are equivalents
      in principle and form, as stated in the preface.

These samples, just given, of the kind of problem-solving that should
not be emphasized in school training refer in some cases to books of
forty years back, but the following represent the results of a
collection made in 1910 from books then in excellent repute. It required
only about an hour to collect them; and I am confident that a thousand
such problems describing situations that the pupil will never encounter
in real life, or putting questions that he will never be asked in real
life, could easily be found in any ten textbooks of the decade from 1900
to 1910.

  If there are 250 kernels of corn on one ear, how many are there on 24
  ears of corn the same size?

  Maud is four times as old as her sister, who is 4 years old. What is
  the sum of their ages?

  If the first century began with the year 1, with what year does it
  end?

  Every spider has 8 compound eyes. How many eyes have 21 spiders?

  A nail 4 inches long is driven through a board so that it projects
  1.695 inches on one side and 1.428 on the other. How thick is the
  board?

  Find the perimeter of an envelope 5 in. by 3-1/4 in.

  How many minutes in 5/9 of 9/4 of an hour?

  Mrs. Knox is 3/4 as old as Mr. Knox, who is 48 years old. Their son
  Edward is 4/9 as old as his mother. How old is Edward?

  Suppose a pie to be exactly round and 10-1/2 miles in diameter. If it
  were cut into 6 equal pieces, how long would the curved edge of each
  piece be?

  8-1/3% of a class of 36 boys were absent on a rainy day. 33-1/3% of
  those present went out of the room to the school yard. How many were
  left in the room?

  Just after a ton of hay was weighed in market, a horse ate one pound
  of it. What was the ratio of what he ate to what was left?

  If a fan having 15 rays opens out so that the outer rays form a
  straight line, how many degrees are there between any two adjacent
  rays?

  One half of the distance between St. Louis and New Orleans is 280
  miles more than 1/10 of the distance; what is the distance between
  these places?

  If the pressure of the atmosphere is 14.7 lb. per square inch what is
  the pressure on the top of a table 1-1/4 yd. long and 2/3 yd. wide?

  13/28 of the total acreage of barley in 1900 was 100,000 acres; what
  was the total acreage?

  What is the least number of bananas that a mother can exactly divide
  between her 2 sons, or among her 4 daughters, or among all her
  children?

  If Alice were two years older than four times her actual age she would
  be as old as her aunt, who is 38 years old. How old is Alice?

  Three men walk around a circular island, the circumference of which is
  360 miles. A walks 15 miles a day, B 18 miles a day, and C 24 miles a
  day. If they start together and walk in the same direction, how many
  days will elapse before they will be together again?

With only thirty or forty dollars a year to spend on a pupil's
education, of which perhaps eight dollars are spent on improving his
arithmetical abilities, the immediate guidance of his responses to real
situations and personally initiated problems has to be supplemented
largely by guidance of his responses to problems described in words,
diagrams, pictures, and the like. Of these latter, words will be used
most often. As a consequence the understanding of the words used in
these descriptions becomes a part of the ability required in arithmetic.
Such word knowledge is also required in so far as the problems to be
solved in real life are at times described, as in advertisements,
business letters, and the like.

This is recognized by everybody in the case of words like _remainder_,
_profit_, _loss_, _gain_, _interest_, _cubic capacity_, _gross_, _net_,
and _discount_, but holds equally of _let_, _suppose_, _balance_,
_average_, _total_, _borrowed_, _retained_, and many such semi-technical
words, and may hold also of hundreds of other words unless the textbook
and teacher are careful to use only words and sentence structures which
daily life and the class work in English have made well known to the
pupils. To apply arithmetic to a problem a pupil must understand what
the problem is; problem-solving depends on problem-reading. In actual
school practice training in problem-reading will be less and less
necessary as we get rid of problems to be solved simply for the sake of
solving them, unnecessarily unreal problems, and clumsy descriptions,
but it will remain to some extent as an important joint task for the
'arithmetic' and 'reading' of the elementary school.


ARITHMETICAL REASONING

The last respect in which the nature of arithmetical abilities requires
definition concerns arithmetical reasoning. An adequate treatment of the
reasoning that may be expected of pupils in the elementary school and of
the most efficient ways to encourage and improve it cannot be given
until we have studied the formation of habits. For reasoning is
essentially the organization and control of habits of thought. Certain
matters may, however, be decided here. The first concerns the use of
computation and problems merely for discipline,--that is, the emphasis
on training in reasoning regardless of whether the problem is otherwise
worth reasoning about. It used to be thought that the mind was a set of
faculties or abilities or powers which grew strong and competent by
being exercised in a certain way, no matter on what they were exercised.
Problems that could not occur in life, and were entirely devoid of any
worthy interest, save the intellectual interest in solving them, were
supposed to be nearly or quite as useful in training the mind to reason
as the genuine problems of the home, shop, or trade. Anything that gave
the mind a chance to reason would do; and pupils labored to find when
the minute hand and hour hand would be together, or how many sheep a
shepherd had if half of what he had plus ten was one third of twice what
he had!

We now know that the training depends largely on the particular data
used, so that efficient discipline in reasoning requires that the pupil
reason about matters of real importance. There is no magic essence or
faculty of reasoning that works in general and irrespective of the
particular facts and relations reasoned about. So we should try to find
problems which not only stimulate the pupil to reason, but also direct
his reasoning in useful channels and reward it by results that are of
real significance. We should replace the purely disciplinary problems by
problems that are also valuable as special training for important
particular situations of life. Reasoning sought for reasoning's sake
alone is too wasteful an expenditure of time and is also likely to be
inferior as reasoning.

The second matter concerns the relative merits of 'catch' problems,
where the pupil has to go against some customary habit of thinking, and
what we may call 'routine' problems, where the regular ways of thinking
that have served him in the past will, except for some blunder, guide
him rightly.

Consider, for example, these four problems:

  1. "A man bought ten dozen eggs for $2.50 and sold them for 30 cents
     a dozen. How many cents did he lose?"

  2. "I went into Smith's store at 9 A.M. and remained until 10 A.M.
     I bought six yards of gingham at 40 cents a yard and three yards
     of muslin at 20 cents a yard and gave a $5.00 bill. How long was
     I in the store?"

  3. "What must you divide 48 by to get half of twice 6?"

  4. "What must you add to 19 to get 30?"

The 'catch' problem is now in disrepute, the wise teacher feeling by a
sort of intuition that to willfully require a pupil to reason to a
result sharply contrary to that to which previous habits lead him is
risky. The four illustrations just given show, however, that mere
'catchiness' or 'contra-previous-habit-ness' in a problem is not enough
to condemn it. The fourth problem is a catch problem, but so useful a
one that it has been adopted in many modern books as a routine drill!
The first problem, on the contrary, all, save those who demand no higher
criterion for a problem than that it make the pupil 'think,' would
reject. It demands the reversal of fixed habits _to no valid purpose_;
for in life the question in such case would never (or almost never) be
'How many cents did he lose?' but 'What was the result?' or simply 'What
of it?' This problem weakens without excuse the child's confidence in
the training he has had. Problems like (2) are given by teachers of
excellent reputation, but probably do more harm than good. If a pupil
should interrupt his teacher during the recitation in arithmetic by
saying, "I got up at 7 o'clock to multiply 9 by 2-3/4 and got 24-3/4 for
my answer; was that the right time to get up?" the teacher would not
thank fortune for the stimulus to thought but would think the child a
fool. Such catch questions may be fairly useful as an object lesson on
the value of search for the essential element in a situation if a great
variety of them are given one after another with routine problems
intermixed and with warning of the general nature of the exercise at the
beginning. Even so, it should be remembered that reasoning should be
chiefly a force organizing habits, not opposing them; and also that
there are enough bad habits to be opposed to give all necessary
training. Fabricated puzzle situations wherein a peculiar hidden element
of the situation makes the good habits called up by the situation
misleading are useful therefore rather as a relief and amusing variation
in arithmetical work than as stimuli to thought.

Problems like the third quoted above we might call puzzling rather than
'catch' problems. They have value as drills in analysis of a situation
into its elements that will amuse the gifted children, and as tests of
certain abilities. They also require that of many confusing habits, the
right one be chosen, rather than that ordinary habits be set aside by
some hidden element in the situation. Not enough is known about their
effect to enable us to decide whether or not the elementary school
should include special facility with them as one of the arithmetical
functions that it specially trains.

The fourth 'catch' quoted above, which all would admit is a good
problem, is good because it opposes a good habit for the sake of another
good habit, forces the analysis of an element whose analysis life very
much requires, and does it with no obvious waste. It is not safe to
leave a child with the one habit of responding to 'add, 19, 30' by 49,
for in life the 'have 19, must get .... to have 30' situation is very
frequent and important.

On the whole, the ordinary problems which ordinary life proffers seem to
be the sort that should be reasoned out, though the elementary school
may include the less noxious forms of pure mental gymnastics for those
pupils who like them.


SUMMARY

These discussions of the meanings of numbers, the linguistic demands of
arithmetic, the distinction between scholastic and real applications of
arithmetic, and the possible restrictions of training in reasoning,--may
serve as illustrations of the significance of the question, "What are
the functions that the elementary school tries to improve in its
teaching of arithmetic?" Other matters might well be considered in this
connection, but the main outline of the work of the elementary school is
now fairly clear. The arithmetical functions or abilities which it seeks
to improve are, we may say:--

(1) Working knowledge of the meanings of numbers as names for certain
sized collections, for certain relative magnitudes, the magnitude of
unity being known, and for certain centers or nuclei of relations to
other numbers.

(2) Working knowledge of the system of decimal notation.

(3) Working knowledge of the meanings of addition, subtraction,
multiplication, and division.

(4) Working knowledge of the nature and relations of certain common
measures.

(5) Working ability to add, subtract, multiply, and divide with
integers, common and decimal fractions, and denominate numbers, all
being real positive numbers.

(6) Working knowledge of words, symbols, diagrams, and the like as
required by life's simpler arithmetical demands or by economical
preparation therefor.

(7) The ability to apply all the above as required by life's simpler
arithmetical demands or by economical preparation therefor, including
(7 _a_) certain specific abilities to solve problems concerning areas
of rectangles, volumes of rectangular solids, percents, interest, and
certain other common occurrences in household, factory, and business
life.


THE SOCIOLOGY OF ARITHMETIC

  The phrase 'life's simpler arithmetical demands' is necessarily left
  vague. Just what use is being made of arithmetic in this country in
  1920 by each person therein, we know only very roughly. What may be
  called a 'sociology' of arithmetic is very much needed to investigate
  this matter. For rare or difficult demands the elementary school
  should not prepare; there are too many other desirable abilities that
  it should improve.

  A most interesting beginning at such an inventory of the actual uses
  of arithmetic has been made by Wilson ['19] and Mitchell.[2] Although
  their studies need to be much extended and checked by other methods of
  inquiry, two main facts seem fairly certain.

  First, the great majority of people in the great majority of their
  doings use only very elementary arithmetical processes. In 1737 cases
  of addition reported by Wilson, seven eighths were of five numbers or
  less. Over half of the multipliers reported were one-figure numbers.
  Over 95 per cent of the fractions operated with were included in this
  list: 1/2 1/4 3/4 1/3 2/3 1/8 3/8 1/5 2/5 4/5. Three fourths of all
  the cases reported were simple one-step computations with integers or
  United States money.

  Second, they often use these very elementary processes, not because
  such are the quickest and most convenient, but because they have lost,
  or maybe never had, mastery of the more advanced processes which would
  do the work better. The 5 and 10 cent stores, the counter with
  "Anything on this counter for 25¢," and the arrangements for payments
  on the installment plan are familiar instances of human avoidance of
  arithmetic. Wilson found very slight use of decimals; and Mitchell
  found men computing with 49ths as common fractions when the use of
  decimals would have been more efficient. If given 120 seconds to
  do a test like that shown below, leading lawyers, physicians,
  manufacturers, and business men and their wives will, according to my
  experience, get only about half the work right. Many women, finding on
  their meat bill "7-3/8 lb. roast beef $2.36," will spend time and
  money to telephone the butcher asking how much roast beef was per
  pound, because they have no sure power in dividing by a mixed number.

  [2] The work of Mitchell has not been published, but the author has
      had the privilege of examining it.

  Test

  Perform the operations indicated. Express all fractions in answers in
  lowest terms.

      _Add:_

      3/4 + 1/6 + .25       4 yr. 6 mo.
                            1 yr. 2 mo.
                            6 yr. 9 mo.
                            3 yr. 6 mo.
                            4 yr. 5 mo.
                            -----------

      _Subtract:_

      8.6 - 6.05007     7/8 - 2/3 =    5-7/16 - 2-3/16 =


      _Multiply:_

      29 ft. 6 in.     7 × 8 × 4-1/2 =
             8
      ------------

    _Divide:_

    4-1/2 ÷ 7 =

  It seems probable that the school training in arithmetic of the past
  has not given enough attention to perfecting the more elementary
  abilities. And we shall later find further evidence of this. On the
  other hand, the fact that people in general do not at present use a
  process may not mean that they ought not to use it.

  Life's simpler arithmetical demands certainly do not include matters
  like the rules for finding cube root or true discount, which no
  sensible person uses. They should not include matters like computing
  the lateral surface or volume of pyramids and cones, or knowing the
  customs of plasterers and paper hangers, which are used only by highly
  specialized trades. They should not include matters like interest on
  call loans, usury, exact interest, and the rediscounting of notes,
  which concern only brokers, bank clerks, and rich men. They should not
  include the technique of customs which are vanishing from efficient
  practice, such as simple interest on amount for times longer than a
  year, days of grace, or extremes and means in proportions. They should
  not include any elaborate practice with very large numbers, or
  decimals beyond thousandths, or the addition and subtraction of
  fractions which not one person in a hundred has to add or subtract
  oftener than once a year.

  When we have an adequate sociology of arithmetic, stating accurately
  just who should use each arithmetical ability and how often, we shall
  be able to define the task of the elementary school in this respect.
  For the present, we may proceed by common sense, guided by two
  limiting rules. The first is,--"It is no more desirable for the
  elementary school to teach all the facts of arithmetic than to teach
  all the words in the English language, or all the topography of the
  globe, or all the details of human physiology." The second is,--"It is
  not desirable to eliminate any element of arithmetical training until
  you have something better to put in its place."




CHAPTER II

THE MEASUREMENT OF ARITHMETICAL ABILITIES


One of the best ways to clear up notions of what the functions are which
schools should develop and improve is to get measures of them. If any
given knowledge or skill or power or ideal exists, it exists in some
amount. A series of amounts of it, varying from less to more, defines
the ability itself in a way that no general verbal description can do.
Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb., etc., helps to
tell us what we mean by weight. By finding a series of words like
_only_, _smoke_, _another_, _pretty_, _answer_, _tailor_, _circus_,
_telephone_, _saucy_, and _beginning_, which are spelled correctly by
known and decreasing percentages of children of the same age, or of the
same school grade, we know better what we mean by 'spelling-difficulty.'
Indeed, until we can measure the efficiency and improvement of a
function, we are likely to be vague and loose in our ideas of what the
function is.


A SAMPLE MEASUREMENT OF AN ARITHMETICAL ABILITY: THE ABILITY TO ADD
INTEGERS

Consider first, as a sample, the measurement of ability to add integers.

The following were the examples used in the measurements made by Stone
['08]:

           596     4695
           428      872
  2375      94     7948
  4052      75     6786
  6354     304      567
   260     645      858
  5041     984     9447
  1543     897     7499
  ----     ---     ----

The scoring was as follows: Credit of 1 for each column added correctly.
Stone combined measures of other abilities with this in a total score
for amount done correctly in 12 minutes. Stone also scored the
correctness of the additions in certain work in multiplication.

Courtis uses a sheet of twenty-four tasks or 'examples,' each consisting
of the addition of nine three-place numbers as shown below. Eight
minutes is allowed. He scores the amount done by the number of examples,
and also scores the number of examples done correctly, but does not
suggest any combination of these two into a general-efficiency score.

  927
  379
  756
  837
  924
  110
  854
  965
  344
  ---

The author long ago proposed that pupils be measured also with series
like _a_ to _g_ shown below, in which the difficulty increases step by
step.

  _a._           3  2  2  3  2  2  1  2
                 2  3  1  2  4  5  5  1
                 4  2  3  3  3  2  2  2
                 -  -  -  -  -  -  -  -

  _b._    21    32    12    24    34    34    22    12
          23    12    52    31    33    12    23    13
          24    25    15    14    32    23    43    61
          --    --    --    --    --    --    --    --

  _c._    22     3     4    35    32    83    22     3
           3    31     3     2    33    11     3    21
          38    45    52    52     2     4    33    64
          --    --    --    --    --    --    --    --

  _d._    30    20    10    22    10    20    52    12
          20    50    40    43    30     4     6    22
          40    17    24    13    40    23    30    44
          --    --    --    --    --    --    --    --

  _e._           4     5    20    12    12    20    10
          20    30     3    40     4    11    20    20
          10    30    20     4     1    23     7     2
          20     2    40    23    40    11    10    30
          20    20    10    11    20    22    30    25
          --    --    --    --    --    --    --    --

  _f._                19     9                 9
          14     2    19    24     9     4    13
           9    14    13    12    13    13     9    14
          17    23    13    15    15    34    12    25
          26    29    18    19    25    28    18    39
          --    --    --    --    --    --    --    --

  _g._                                  13
                13           9    14    12     9
                 9          13    12     9    14    24
          23    19    19    29     9     9    13    21
          28    26    26    14     8     8    29    23
          29    16    15    19    17    19    19    22
          --    --    --    --    --    --    --    --

Woody ['16] has constructed his well-known tests on this principle,
though he uses only one example at each step of difficulty instead of
eight or ten as suggested above. His test, so far as addition of
integers goes, is:--

SERIES A. ADDITION SCALE (in part)

By Clifford Woody

  (1)   (2)    (3)    (4)   (5)   (6)   (7)       (8)           (9)
   2     2     17     53    72    60   3 + 1 =   2 + 5 + 1 =    20
   3     4      2     45    26    37                            10
  --     3     --     --    --    --                             2
        --                                                      30
                                                                25
                                                                --

  (10) (11)   (12)   (13)    (14)        (15)   (16)  (17)    (18)
   21   32     43     23    25 + 42 =    100      9   199     2563
   33   59      1     25                  33     24   194     1387
   35   17      2     16                  45     12   295     4954
   --   --     13     --                 201     15   156     2065
               --                         46     19   ---     ----
                                         ---     --

            (19)        (20)            (21)       (22)
           $ .75       $12.50          $8.00        547
            1.25        16.75           5.75        197
             .49        15.75           2.33        685
           -----       ------           4.16        678
                                         .94        456
                                        6.32        393
                                       -----        525
                                                    240
                                                    152
                                                    ---

In his original report, Woody gives no scheme for scoring an individual,
wisely assuming that, with so few samples at each degree of difficulty,
a pupil's score would be too unreliable for individual diagnosis. The
test is reliable for a class; and for a class Woody used the degree of
difficulty such that a stated fraction of the class can do the work
correctly, if twenty minutes is allowed for the thirty-eight examples of
the entire test.

The measurement of even so simple a matter as the efficiency of a
pupil's responses to these tests in adding integers is really rather
complex. There is first of all the problem of combining speed and
accuracy into some single estimate. Stone gives no credit for a column
unless it is correctly added. Courtis evades the difficulty by reporting
both number done and number correct. The author's scheme, which gives
specified weights to speed and accuracy at each step of the series,
involves a rather intricate computation.

This difficulty of equating speed and accuracy in adding means precisely
that we have inadequate notions of what the ability is that the
elementary school should improve. Until, for example, we have decided
whether, for a given group of pupils, fifteen Courtis attempts with ten
right, is or is not a better achievement than ten Courtis attempts with
nine right, we have not decided just what the business of the teacher of
addition is, in the case of that group of pupils.

There is also the difficulty of comparing results when short and long
columns are used. Correctness with a short column, say of five figures,
testifies to knowledge of the process and to the power to do four
successive single additions without error. Correctness with a long
column, say of ten digits, testifies to knowledge of the process and to
the power to do nine successive single additions without error. Now if a
pupil's precision was such that on the average he made one mistake in
eight single additions, he would get about half of his five-digit
columns right and almost none of his ten-digit columns right. (He would
do this, that is, if he added in the customary way. If he were taught to
check results by repeated addition, by adding in half-columns and the
like, his percentages of accurate answers might be greatly increased in
both cases and be made approximately equal.) Length of column in a test
of addition under ordinary conditions thus automatically overweights
precision in the single additions as compared with knowledge of the
process, and ability at carrying.

Further, in the case of a column of whatever size, the result as
ordinarily scored does not distinguish between one, two, three, or more
(up to the limit) errors in the single additions. Yet, obviously, a
pupil who, adding with ten-digit columns, has half of his answer-figures
wrong, probably often makes two or more errors within a column, whereas
a pupil who has only one column-answer in ten wrong, probably almost
never makes more than one error within a column. A short-column test is
then advisable as a means of interpreting the results of a long-column
test.

Finally, the choice of a short-column or of a long-column test is
indicative of the measurer's notion of the kind of efficiency the world
properly demands of the school. Twenty years ago the author would have
been readier to accept a long-column test than he now is. In the world
at large, long-column addition is being more and more done by machine,
though it persists still in great frequency in the bookkeeping of weekly
and monthly accounts in local groceries, butcher shops, and the like.

The search for a measure of ability to add thus puts the problem of
speed _versus_ precision, and of short-column _versus_ long-column
additions clearly before us. The latter problem has hardly been
realized at all by the ordinary definitions of ability to add.

It may be said further that the measurement of ability to add gives the
scientific student a shock by the lack of precision found everywhere in
schools. Of what value is it to a graduate of the elementary school to
be able to add with examples like those of the Courtis test, getting
only eight out of ten right? Nobody would pay a computer for that
ability. The pupil could not keep his own accounts with it. The supposed
disciplinary value of habits of precision runs the risk of turning
negative in such a case. It appears, at least to the author, imperative
that checking should be taught and required until a pupil can add single
columns of ten digits with not over one wrong answer in twenty columns.
Speed is useful, especially indirectly as an indication of control of
the separate higher-decade additions, but the social demand for addition
below a certain standard of precision is _nil_, and its disciplinary
value is _nil_ or negative. This will be made a matter of further study
later.


MEASUREMENTS OF ABILITIES IN COMPUTATION

Measurements of these abilities may be of two sorts--(1) of the speed
and accuracy shown in doing one same sort of task, as illustrated by the
Courtis test for addition shown on page 28; and (2) of how hard a task
can be done perfectly (or with some specified precision) within a
certain assigned time or less, as illustrated by the author's rough test
for addition shown on pages 28 and 29, and by the Woody tests, when
extended to include alternative forms.

The Courtis tests, originated as an improvement on the Stone tests and
since elaborated by the persistent devotion of their author, are a
standard instrument of the first sort for measuring the so-called
'fundamental' arithmetical abilities with integers. They are shown on
this and the following page.

Tests of the second sort are the Woody tests, which include operations
with integers, common and decimal fractions, and denominate numbers, the
Ballou test for common fractions ['16], and the "Ladder" exercises of
the Thorndike arithmetics. Some of these are shown on pages 36 to 41.


Courtis Test

Arithmetic. Test No. 1. Addition

Series B

  You will be given eight minutes to find the answers to as many
  of these addition examples as possible. Write the answers on this
  paper directly underneath the examples. You are not expected
  to be able to do them all. You will be marked for both speed and
  accuracy, but it is more important to have your answers right than
  to try a great many examples.

    927    297    136    486    384    176    277    837
    379    925    340    765    477    783    445    882
    756    473    988    524    881    697    682    959
    837    983    386    140    266    200    594    603
    924    315    353    812    679    366    481    118
    110    661    904    466    241    851    778    781
    854    794    547    355    796    535    849    756
    965    177    192    834    850    323    157    222
    344    124    439    567    733    229    953    525
    ---    ---    ---    ---    ---    ---    ---    ---

  and sixteen more addition examples of nine three-place numbers.


Courtis Test

Arithmetic. Test No. 2. Subtraction

Series B

  You will be given four minutes to find the answers to as many
  of these subtraction examples as possible. Write the answers
  on this paper directly underneath the examples. You are not
  expected to be able to do them all. You will be marked for both
  speed and accuracy, but it is more important to have your answers
  right than to try a great many examples.

    107795491      75088824      91500053      87939983
     77197029      57406394      19901563      72207316
    ---------      --------      --------      --------

  and twenty more tasks of the same sort.


Courtis Test

Arithmetic. Test No. 3. Multiplication

Series B

  You will be given six minutes to work as many of these multiplication
  examples as possible. You are not expected to be able to do them all.
  Do your work directly on this paper; use no other. You will be marked
  for both speed and accuracy, but it is more important to get correct
  answers than to try a large number of examples.

    8246      7843      4837      3478      6482
      29       702        83        15        46
    ----      ----      ----      ----      ----

  and twenty more multiplication examples of the same sort.


Courtis Test

Arithmetic. Test No. 4. Division

Series B

  You will be given eight minutes to work as many of these division
  examples as possible. You are not expected to be able to do them all.
  Do your work directly on this paper; use no other. You will be marked
  for both speed and accuracy, but it is more important to get correct
  answers than to try a large number of examples.
      _____        ______        _____        ______
    25)6775      94)85352      37)9990      86)80066

  and twenty more division examples of the same sort.


SERIES B. MULTIPLICATION SCALE

By Clifford Woody

   (1)               (3)               (4)               (5)
   3 × 7 =           2 × 3 =           4 × 8 =            23
                                                           3
                                                          --

   (8)               (9)              (11)              (12)
    50               254              1036              5096
     3                 6                 8                 6
    --               ---              ----              ----

  (13)              (16)              (18)              (20)
  8754              7898                24               287
     8                 9               234               .05
  ----              ----               ---               ---

  (24)              (26)              (27)              (29)
    16              9742              6.25             1/8 × 2 =
     2-5/8            59               3.2
    ---             ----              ----

  (33)                (35)              (37)               (38)
  2-1/2 × 3-1/2 =     987-3/4    2-1/4 × 4-1/2 × 1-1/2 =   .0963-1/8
                       25                                  .084
                     ----                                  -----


SERIES B. DIVISION  SCALE

By Clifford Woody

  (1)               (2)              (7)               (8)
   __               ___                                 ___
  3)6              9)27             4 ÷ 2 =            9)0

  (11)              (14)              (15)              (17)
   ___              _____
  2)13             8)5856          1/4 of 128 =        50 ÷ 7 =

  (19)              (23)              (27)              (28)
                    ____                                 ______
  248 ÷ 7 =       23)469          7/8 of 624 =       .003).0936

      (30)                (34)                (36)
                                            ______________
   3/4 ÷ 5 =         62.50 ÷ 1-1/4 =       9)69 lbs. 9 oz.


Ballou Test

Addition of Fractions

            _Test 1_                                _Test 2_
  (1)  1/4            (2) 3/14             (1) 1/3            (2) 2/7
       1/4                1/14                 1/6                3/14
       ---                ----                 ---                ----


           _Test 3_                                 _Test 4_
  (1)  3/5            (2) 5/6              (1) 1/7            (2) 7/9
      11/15               1/2                  9/10               1/4
      -----               ---                  ----               ---


           _Test 5_                                 _Test 6_
  (1)  1/10           (2) 4/9              (1) 1/6            (2) 5/6
       1/6                5/12                 9/10               3/8
       ----               ----                 ----               ---


An Addition Ladder [Thorndike, '17, III, 5]

Begin at the bottom of the ladder. See if you can climb to the top
without making a mistake. Be sure to copy the numbers correctly.

  #Step 6.#
    _a._ Add 1-1/3 yd., 7/8 yd., 1-1/4 yd., 3/4 yd., 7/8 yd.,
          and 1-1/2 yd.
    _b._ Add 62-1/2¢, 66-2/3¢, 56-1/4¢, 60¢, and 62-1/2¢.
    _c._ Add 1-5/16, 1-9/32, 1-3/8, 1-11/32, and 1-7/16.
    _d._ Add 1-1/3 yd., 1-1/4 yd., 1-1/2 yd., 2 yd., 3/4 yd.,
          and 2/3 yd.

  #Step 5.#
    _a._ Add 4 ft. 6-1/2 in., 53-1/4 in., 5 ft. 1/2 in., 56-3/4 in.,
          and 5 ft.
    _b._ Add 7 lb., 6 lb. 11 oz., 7-1/2 lb., 6 lb. 4-1/2 oz.,
          and 8-1/2 lb.
    _c._ Add 1 hr. 6 min. 20 sec., 58 min. 15 sec., 1 hr. 4 min.,
          and 55 min.
    _d._ Add 7 dollars, 13 half dollars, 21 quarters, 17 dimes,
          and 19 nickels.

  #Step 4.#
    _a._ Add .05-1/2, .06, .04-3/4, .02-3/4, and .05-1/4.
    _b._ Add .33-1/3, .12-1/2, .18, .16-2/3, .08-1/3 and .15.
    _c._ Add .08-1/3, .06-1/4, .21, .03-3/4, and .16-2/3.
    _d._ Add .62, .64-1/2, .66-2/3, .10-1/4, and .68.

  #Step 3.#
    _a._ Add 7-1/4, 6-1/2, 8-3/8, 5-3/4, 9-5/8 and 3-7/8.
    _b._ Add 4-5/8, 12, 7-1/2, 8-3/4, 6 and 5-1/4.
    _c._ Add 9-3/4, 5-7/8, 4-1/8, 6-1/2, 7, 3-5/8.
    _d._ Add 12, 8-1/2, 7-1/3, 5, 6-2/3, and 9-1/2.

  #Step 2.#
    _a._ Add 12.04, .96, 4.7, 9.625, 3.25, and 20.
    _b._ Add .58, 6.03, .079, 4.206, 2.75, and 10.4.
    _c._ Add 52, 29.8, 41.07, 1.913, 2.6, and 110.
    _d._ Add 29.7, 315, 26.75, 19.004, 8.793, and 20.05.

  #Step 1.#
    _a._ Add 10-3/5, 11-1/5, 10-4/5, 11, 11-2/5, 10-3/5, and 11.
    _b._ Add 7-3/8, 6-5/8, 8, 9-1/8, 7-7/8, 5-3/8, and 8-1/8.
    _c._ Add 21-1/2, 18-3/4, 31-1/2, 19-1/4, 17-1/4, 22, and 16-1/2.
    _d._ Add 14-5/12, 12-7/12, 9-11/12, 6-1/12, and 5.


A Subtraction Ladder [Thorndike, '17, III, 11]

  #Step 9.#
    _a._  2.16 mi. - 1-3/4 mi.
    _b._  5.72 ft. - 5 ft. 3 in.
    _c._  2 min. 10-1/2 sec. - 93.4 sec.
    _d._  30.28 A. - 10-1/5 A.
    _e._  10 gal. 2-1/2 qt. - 4.623 gal.

  #Step 8.#
     _a_             _b_            _c_            _d_            _e_
    25-7/12         10-1/4          9-5/16         5-7/16         4-2/3
    12-3/4           7-1/3          6-3/8          2-3/4          1-3/4
    -------         ------          ------         ------         -----

  #Step 7.#
     _a_             _b_            _c_            _d_            _e_
    28-3/4          40-1/2         10-1/4         24-1/3         37-1/2
    16-1/8          14-3/8          6-1/2         11-1/2         14-3/4
    ------          ------         ------         ------         ------

  #Step 6.#
     _a_             _b_            _c_            _d_            _e_
    10-1/3           7-1/4         15-1/8         12-1/5          4-1/16
     4-2/3           2-3/4          6-3/8         11-4/5          2-7/16
    ------           -----         ------         ------          ------

  #Step 5.#
     _a_             _b_            _c_            _d_            _e_
    58-4/5          66-2/3         28-7/8         62-1/2          9-7/12
    52-1/5          33-1/3          7-5/8         37-1/2          4-5/12
    ------          ------         ------         ------          ------

  #Step 4.#
    _a._  4 hr. - 2 hr. 17 min.
    _b._  4 lb. 7 oz. - 2 lb. 11 oz.
    _c._  1 lb. 5 oz. - 13 oz.
    _d._  7 ft. - 2 ft. 8 in.
    _e._  1 bu. - 1 pk.

  #Step 3.#
      _a_             _b_         _c_        _d_            _e_
    92    mi.       6735 mi.    $3 - 89¢    28.4  mi.     $508.40
    84.15 mi.       6689 mi.                18.04 mi.      208.62
    ---------       --------    --------    ---------     -------

  #Step 2.#
      _a_        _b_        _c_        _d_            _e_
    $25.00     $100.00    $750.00   6124 sq. mi.   7846 sq. mi.
      9.36       71.28     736.50   2494 sq. mi.   2789 sq. mi.
    ------     -------    -------   ------------   ------------

  #Step 1.#
      _a_        _b_         _c_        _d_         _e_
    $18.64     $25.39      $56.70     819.4 mi.    67.55 mi.
      7.40      13.37       45.60     209.2 mi.    36.14 mi.
    ------     ------      ------     ---------    ---------


An Average Ladder [Thorndike, '17, III, 132]

Find the average of the quantities on each line. Begin with #Step 1#.
Climb to the top without making a mistake. Be sure to copy the numbers
correctly. Extend the division to two decimal places if necessary.

  #Step 6.#
    _a_. 2-2/3, 1-7/8, 2-3/4, 4-1/4, 3-5/8, 3-1/2
    _b_. 62-1/2¢, 66-2/3¢, 40¢, 83-1/3¢, $1.75, $2.25
    _c_. 3-11/16, 3-9/32, 3-3/8, 3-17/32, 3-7/16
    _d_. .17, 19, .16-2/3, .15-1/2, .23-1/4, .18

  #Step 5.#
    _a_. 5 ft. 3-1/2 in., 61-1/4 in., 58-3/4 in., 4 ft. 11 in.
    _b_. 6 lb. 9 oz., 6 lb. 11 oz., 7-1/4 lb., 7-3/8 lb.
    _c_. 1 hr. 4 min. 40 sec., 58 min. 35 sec., 1-1/4 hr.
    _d_. 2.8 miles, 3-1/2 miles, 2.72 miles

  #Step 4.#
    _a._ .03-1/2, .06, .04-3/4, .05-1/2, .05-1/4
    _b._ .043, .045, .049, .047, .046, .045
    _c._ 2.20, .87-1/2, 1.18, .93-3/4, 1.2925, .80
    _d._ .14-1/2, .12-1/2, .33-1/3, .16-2/3, .15, .17

  #Step 3.#
    _a._ 5-1/4, 4-1/2, 8-3/8, 7-3/4, 6-5/8, 9-3/8
    _b._ 9-5/8, 12, 8-1/2, 8-3/4, 6, 5-1/4, 9
    _c._ 9-3/8, 5-3/4, 4-1/8, 7-1/2, 6
    _d._ 11, 9-1/2, 10-1/3, 13, 16-2/3, 9-1/2

  #Step 2.#
    _a._ 13.05, .97, 4.8, 10.625, 3.37
    _b._ 1.48, 7.02, .93, 5.307, 4.1, 7, 10.4
    _c._ 68, 71.4, 59.8, 112, 96.1, 79.8
    _d._ 2.079, 3.908, 4.165, 2.74

  #Step 1.#
    _a._ 4, 9-1/2, 6, 5, 7-1/2, 8, 10, 9
    _b._ 6, 5, 3.9, 7.1, 8
    _c._ 1086, 1141, 1059, 1302, 1284
    _d._ $100.82, $206.49, $317.25, $244.73

As such tests are widened to cover the whole task of the elementary
school in respect to arithmetic, and accepted by competent authorities
as adequate measures of achievement in computing, they will give, as has
been said, a working definition of the task. The reader will observe,
for example, that work such as the following, though still found in many
textbooks and classrooms, does not, in general, appear in the modern
tests and scales.

Reduce the following improper fractions to mixed numbers:--

          19/13    43/21   176/25   198/14

Reduce to integral or mixed numbers:--

          61381/37   2134/67   413/413   697/225

Simplify:--

          3/4 of 8/9 of 3/5 of 15/22

Reduce to lowest terms:--

  357/527  264/312  492/779  418/874  854/1769  30/735  44/242

  77/847 18/243  96/224

Find differences:--

     6-2/7         8-5/11        8-4/13        5-1/4          7-1/8
     3-1/14        5-1/7         3-7/13        2-11/14        2-1/7
     ------        ------        ------        -------        ------

Square:--

  2/3  4/5  5/7  6/9  10/11  12/13  2/7  15/16  19/20  17/18

  25/30  41/53

Multiply:--

     2/11 × 33    32 × 3/14    39 × 2/13    60 × 11/28    77 × 4/11

     63 × 2/27    54 × 8/45    65 × 3/13     344-16/21   432-2/7


MEASUREMENTS OF ABILITY IN APPLIED ARITHMETIC: THE SOLUTION OF PROBLEMS

Stone ['08] measured achievement with the following problems, fifteen
minutes being the time allowed.

"Solve as many of the following problems as you have time for; work them
in order as numbered:

  1. If you buy 2 tablets at 7 cents each and a book for 65 cents, how
     much change should you receive from a two-dollar bill?

  2. John sold 4 Saturday Evening Posts at 5 cents each. He kept 1/2
     the money and with the other 1/2 he bought Sunday papers at 2 cents
     each. How many did he buy?

  3. If James had 4 times as much money as George, he would have $16.
     How much money has George?

  4. How many pencils can you buy for 50 cents at the rate of 2 for 5
     cents?

  5. The uniforms for a baseball nine cost $2.50 each. The shoes cost
     $2 a pair. What was the total cost of uniforms and shoes for the
     nine?

  6. In the schools of a certain city there are 2200 pupils; 1/2 are
     in the primary grades, 1/4 in the grammar grades, 1/8 in the high
     school, and the rest in the night school. How many pupils are there
     in the night school?

  7. If 3-1/2 tons of coal cost $21, what will 5-1/2 tons cost?

  8. A news dealer bought some magazines for $1. He sold them for
     $1.20, gaining 5 cents on each magazine. How many magazines were
     there?

  9. A girl spent 1/8 of her money for car fare, and three times as
     much for clothes. Half of what she had left was 80 cents. How much
     money did she have at first?

 10. Two girls receive $2.10 for making buttonholes. One makes 42,
     the other 28. How shall they divide the money?

 11. Mr. Brown paid one third of the cost of a building; Mr. Johnson
     paid 1/2 the cost. Mr. Johnson received $500 more annual rent than
     Mr. Brown. How much did each receive?

 12. A freight train left Albany for New York at 6 o'clock. An
     express left on the same track at 8 o'clock. It went at the rate of
     40 miles an hour. At what time of day will it overtake the freight
     train if the freight train stops after it has gone 56 miles?"

The criteria he had in mind in selecting the problems were as follows:--

"The main purpose of the reasoning test is the determination of the
ability of VI A children to reason in arithmetic. To this end, the
problems, as selected and arranged, are meant to embody the following
conditions:--

  1. Situations equally concrete to all VI A children.

  2. Graduated difficulties.
      _a._ As to arithmetical thinking.
      _b._ As to familiarity with the situation presented.

  3. The omission of
      _a._ Large numbers.
      _b._ Particular memory requirements.
      _c._ Catch problems.
      _d._ All subject matter except whole numbers, fractions, and
             United States money.

The test is purposely so long that only very rarely did any pupil fully
complete it in the fifteen minute limit."

Credits were given of 1, for each of the first five problems, 1.4, 1.2,
and 1.6 respectively for problems 6, 7, and 8, and of 2 for each of the
others.

Courtis sought to improve the Stone test of problem-solving, replacing
it by the two tests reproduced below.


ARITHMETIC--Test No. 6. Speed Test--Reasoning

#Do not work# the following examples. Read each example through, make
up your mind what operation you would use if you were going to work it,
then write the name of the operation selected in the blank space after
the example. Use the following abbreviations:--"Add." for addition,
"Sub." for subtraction, "Mul." for multiplication, and "Div." for
division.

                                                      +-----------+----+
                                                      | OPERATION |    |
                                                      |-----------+----|
  1. A girl brought a collection of 37 colored postal |           |    |
  cards to school one day, and gave away 19 cards to  |           |    |
  her friends. How many cards did she have left to    |           |    |
  take home?                                          |           |    |
                                                      |-----------+----|
  2. Five boys played marbles. When the game was      |           |    |
  over, each boy had the same number of marbles. If   |           |    |
  there were 45 marbles altogether, how many did each |           |    |
  boy have?                                           |           |    |
                                                      |-----------+----|
  3. A girl, watching from a window, saw 27           |           |    |
  automobiles pass the school the first hour, and     |           |    |
  33 the second. How many autos passed by the         |           |    |
  school in the two hours?                            |           |    |
                                                      |-----------+----|
  4. In a certain school there were eight rooms and   |           |    |
  each room had seats for 50 children. When all the   |           |    |
  places were taken, how many children were there in  |           |    |
  the school?                                         |           |    |
                                                      |-----------+----|
  5. A club of boys sent their treasurer to buy       |           |    |
  baseballs. They gave him $3.15 to spend. How many   |           |    |
  balls did they expect him to buy, if the balls cost |           |    |
  45¢. apiece?                                        |           |    |
                                                      |-----------+----|
  6. A teacher weighed all the girls in a certain     |           |    |
  grade. If one girl weighed 79 pounds and another    |           |    |
  110 pounds, how many pounds heavier was one girl    |           |    |
  than the other?                                     |           |    |
                                                      |-----------+----|
  7. A girl wanted to buy a 5-pound box of candy to   |           |    |
  give as a present to a friend. She decided to get   |           |    |
  the kind worth 35¢. a pound. What did she pay for   |           |    |
  the present?                                        |           |    |
                                                      |-----------+----|
  8. One day in vacation a boy went on a fishing trip |           |    |
  and caught 12 fish in the morning, and 7 in the     |           |    |
  afternoon. How many fish did he catch altogether?   |           |    |
                                                      |-----------+----|
  9. A boy lived 15 blocks east of a school; his chum |           |    |
  lived on the same street, but 11 blocks west of the |           |    |
  school. How many blocks apart were the two boys'    |           |    |
  houses?                                             |           |    |
                                                      |-----------+----|
  10. A girl was 5 times as strong as her small       |           |    |
  sister. If the little girl could lift a weight of   |           |    |
  20 pounds, how large a weight could the older girl  |           |    |
  lift?                                               |           |    |
                                                      |-----------+----|
  11. The children of a school gave a sleigh-ride     |           |    |
  party. There were 270 children to go on the ride    |           |    |
  and each sleigh held 30 children. How many sleighs  |           |    |
  were needed?                                        |           |    |
                                                      |-----------+----|
  12. In September there were 43 children in the      |           |    |
  eighth grade of a certain school; by June there     |           |    |
  were 59. How many children entered the grade        |           |    |
  during the year?                                    |           |    |
                                                      |-----------+----|
  13. A girl who lived 17 blocks away walked to       |           |    |
  school and back twice a day. What was the total     |           |    |
  number of blocks the girl walked each day in        |           |    |
  going to and from school?                           |           |    |
                                                      |-----------+----|
  14. A boy who made 67¢. a day carrying papers, was  |           |    |
  hired to run on a long errand for which he received |           |    |
  50¢. What was the total amount the boy earned that  |           |    |
  day?                                                |           |    |
                                                      |-----------+----|
                                          Total Right |           |    |
                                                      +-----------+----+

(Two more similar problems follow.)

Test 6 and Test 8 are from the Courtis Standard Test. Used by permission
of S. A. Courtis.


ARITHMETIC--Test No. 8. Reasoning

In the blank space below, work as many of the following examples as
possible in the time allowed. Work them in order as numbered, entering
each answer in the "answer" column before commencing a new example. Do
not work on any other paper.

                                                            +--------+-+
                                                            | ANSWER | |
                                                            |--------+-|
  1. The children in a certain school gave a Christmas      |        | |
  party. One of the presents was a box of candy. In filling |        | |
  the boxes, one grade used 16 pounds of candy, another 17  |        | |
  pounds, a third 12 pounds, and a fourth 13 pounds. What   |        | |
  did the candy cost at 26¢. a pound?                       |        | |
                                                            |--------+-|
  2. A school in a certain city used 2516 pieces of chalk   |        | |
  in 37 school days. Three new rooms were opened, each      |        | |
  room holding 50 children, and the school was then found   |        | |
  to use 84 sticks of chalk per day. How many more sticks   |        | |
  of chalk were used per day than at first?                 |        | |
                                                            |--------+-|
  3. Several boys went on a bicycle trip of 1500 miles.     |        | |
  The first week they rode 374 miles, the second week 264   |        | |
  miles, the third 423 miles, the fourth 401 miles. They    |        | |
  finished the trip the next week. How many miles did they  |        | |
  ride the last week?                                       |        | |
                                                            |--------+-|
  4. Forty-five boys were hired to pick apples from 15      |        | |
  trees in an apple orchard. In 50 minutes each boy had     |        | |
  picked 48 choice apples. If all the apples picked were    |        | |
  packed away carefully in 8 boxes of equal size, how many  |        | |
  apples were put in each box?                              |        | |
                                                            |--------+-|
  5. In a certain school 216 children gave a sleigh-ride    |        | |
  party. They rented 7 sleighs at a cost of $30.00 and paid |        | |
  $24.00 for the refreshments. The party travelled 15 miles |        | |
  in 2-1/2 hours and had a very pleasant time. What was     |        | |
  each child's share of the expense?                        |        | |
                                                            |--------+-|
  6. A girl found, by careful counting, that there were     |        | |
  2400 letters on one page of her history, and only 2295    |        | |
  letters on a page of her reader. How many more letters    |        | |
  had she read in one book than in the other if she had     |        | |
  read 47 pages in each of the books?                       |        | |
                                                            |--------+-|
  7. Each of 59 rooms in the schools of a certain city      |        | |
  contributed 25 presents to a Christmas entertainment for  |        | |
  poor children. The stores of the city gave 1986 other     |        | |
  articles for presents. What was the total number of       |        | |
  presents given away at the entertainment?                 |        | |
                                                            |--------+-|
  8. Forty-eight children from a certain school paid 10¢.   |        | |
  apiece to ride 7 miles on the cars to a woods. There in a |        | |
  few hours they gathered 2765 nuts. 605 of these were bad, |        | |
  but the rest were shared equally among the children. How  |        | |
  many good nuts did each one get?                          |        | |
                                                            |--------+-|
                                                      Total |        | |
                                                            +--------+-+

These proposed measures of ability to apply arithmetic illustrate very
nicely the differences of opinion concerning what applied arithmetic and
arithmetical reasoning should be. The thinker who emphasizes the fact
that in life out of school the situation demanding quantitative
treatment is usually real rather than described, will condemn a test all
of whose constituents are _described_ problems. Unless we are
excessively hopeful concerning the transfer of ideas of method and
procedure from one mental function to another we shall protest against
the artificiality of No. 3 of the Stone series, and of the entire
Courtis Test 8 except No. 4. The Courtis speed-reasoning test (No. 6) is
a striking example of the mixture of ability to understand quantitative
relations with the ability to understand words. Consider these five, for
example, in comparison with the revised versions attached.[3]

  [3] The form of Test 6 quoted here is that given by Courtis ['11-'12,
      p. 20]. This differs a little from the other series of Test 6,
      shown on pages 43 and 44.

  1. The children of a school gave a sleigh-ride party. There were 9
  sleighs, and each sleigh held 30 children. How many children were
  there in the party?

  REVISION. _If one sleigh holds 30 children, 9 sleighs hold ....
  children._

  2. Two school-girls played a number-game. The score of the girl
  that lost was 57 points and she was beaten by 16 points. What was
  the score of the girl that won?

  REVISION. _Mary and Nell played a game. Mary had a score of 57.
  Nell beat Mary by 16. Nell had a score of ...._

  3. A girl counted the automobiles that passed a school. The total
  was 60 in two hours. If the girl saw 27 pass the first hour how
  many did she see the second?

  REVISION. _In two hours a girl saw 60 automobiles. She saw 27 the
  first hour. She saw .... the second hour._

  4. On a playground there were five equal groups of children each
  playing a different game. If there were 75 children all together,
  how many were there in each group?

  REVISION. _75 pounds of salt just filled five boxes. The boxes were
  exactly alike. There were .... pounds in a box._

  5. A teacher weighed all the children in a certain grade. One girl
  weighed 70 pounds. Her older sister was 49 pounds heavier. How many
  pounds did the sister weigh?

  REVISION. _Mary weighs 70 lb. Jane weighs 49 pounds more than Mary.
  Jane weighs .... pounds._

The distinction between a problem described as clearly and simply as
possible and the same problem put awkwardly or in ill-known words or
willfully obscured should be regarded; and as a rule measurements of
ability to apply arithmetic should eschew all needless obscurity or
purely linguistic difficulty. For example,

  _A boy bought a two-cent stamp. He gave the man in the store 10
  cents. The right change was .... cents._

is better as a test than

  _If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in
  payment, what change should he be expected to receive in return?_

The distinction between the description of a _bona fide_ problem that a
human being might be called on to solve out of school and the
description of imaginary possibilities or puzzles should also be
considered. Nos. 3 and 9 of Stone are bad because to frame the problems
one must first know the answers, so that in reality there could never be
any point in solving them. It is probably safe to say that nobody in the
world ever did or ever will or ever should find the number of apples in
a box by the task of No. 4 of the Courtis Test 8.

This attaches no blame to Dr. Stone or to Mr. Courtis. Until very
recently we were all so used to the artificial problems of the
traditional sort that we did not expect anything better; and so blind to
the language demands of described problems that we did not see their
very great influence. Courtis himself has been active in reform and has
pointed out ('13, p. 4 f.) the defects in his Tests 6 and 8.

"Tests Nos. 6 and 8, the so-called reasoning tests, have proved the
least satisfactory of the series. The judgments of various teachers and
superintendents as to the inequalities of the units in any one test, and
of the differences between the different editions of the same test, have
proved the need of investigating these questions. Tests of adults in
many lines of commercial work have yielded in many cases lower scores
than those of the average eighth grade children. At the same time the
scores of certain individuals of marked ability have been high, and
there appears to be a general relation between ability in these tests
and accuracy in the abstract work. The most significant facts, however,
have been the difficulties experienced by teachers in attempting to
remedy the defects in reasoning. It is certain that the tests measure
abilities of value but the abilities are probably not what they seem to
be. In an attempt to measure the value of different units, for instance,
as many problems as possible were constructed based upon a single
situation. Twenty-one varieties were secured by varying the relative
form of the question and the relative position of the different phrases.
One of these proved nineteen times as hard as another as measured by the
number of mistakes made by the children; yet the cause of the difference
was merely the changes in the phrasing. This and other facts of the same
kind seem to show that Tests 6 and 8 measure mainly the ability to
read."

The scientific measurement of the abilities and achievements concerned
with applied arithmetic or problem-solving is thus a matter for the
future. In the case of described problems a beginning has been made in
the series which form a part of the National Intelligence Tests ['20],
one of which is shown on page 49 f. In the case of problems with real
situations, nothing in systematic form is yet available.

Systematic tests and scales, besides defining the abilities we are to
establish and improve, are of very great service in measuring the status
and improvement of individuals and of classes, and the effects of
various methods of instruction and of study. They are thus helpful to
pupils, teachers, supervisors, and scientific investigators; and are
being more and more widely used every year. Information concerning the
merits of the different tests, the procedure to follow in giving and
scoring them, the age and grade standards to be used in interpreting
results, and the like, is available in the manuals of Educational
Measurement, such as Courtis, _Manual of Instructions for Giving and
Scoring the Courtis Standard Tests in the Three R's_ ['14]; Starch,
_Educational Measurements_ ['16]; Chapman and Rush, _Scientific
Measurement of Classroom Products_ ['17]; Monroe, DeVoss, and Kelly,
_Educational Tests and Measurements_ ['17]; Wilson and Hoke, _How to
Measure_ ['20]; and McCall, _How to Measure in Education_ ['21].

TEST 1

  National Intelligence Tests.
  Scale A. Form 1, Edition 1

  Find the answers as quickly as you can.
  Write the answers on the dotted lines.
  Use the side of the page to figure on.

  #Begin here#

  1  Five cents make 1 nickel. How many nickels make a
     dime?                                                _Answer_ .....

  2  John paid 5 dollars for a watch and 3 dollars for a chain.
     How many dollars did he pay for the watch and chain? _Answer_ .....

  3  Nell is 13 years old. Mary is 9 years old. How much
     younger is Mary than Nell?                           _Answer_ .....

  4  One quart of ice cream is enough for 5 persons. How
     many quarts of ice cream are needed for 25 persons?  _Answer_ .....

  5  John's grandmother is 86 years old. If she lives, in
     how many years will she be 100 years old?            _Answer_ .....

  6  If a man gets $2.50 a day, what will he be paid for six
     days' work?                                          _Answer_ .....

  7  How many inches are there in a foot and a half?      _Answer_ .....

  8  What is the cost of 12 cakes at 6 for 5 cents?       _Answer_ .....

  9  The uniforms for a baseball team of nine boys cost $2.50
     each. The shoes cost $2 a pair. What was the total
     cost of uniforms and shoes for the nine?             _Answer_ .....

  10 A train that usually arrives at half-past ten was 17
     minutes late. When did it arrive?                    _Answer_ .....

  11 At 10¢ a yard, what is the cost of a piece 10-1/2 ft. long?
                                                          _Answer_ .....

  12 A man earns $6 a day half the time, $4.50 a day one
     fourth of the time, and nothing on the remaining days
     for a total period of 40 days. What did he earn in all
     in the 40 days?                                      _Answer_ .....

  13 What per cent of $800 is 4% of $1000?                _Answer_ .....

  14 If 60 men need 1500 lb. flour per month, what is the
     requirement per man per day counting a month as 30
     days?                                                _Answer_ .....

  15 A car goes at the rate of a mile a minute. A truck goes
     20 miles an hour. How many times as far will the car
     go as the truck in 10 seconds?                       _Answer_ .....

  16 The area of the base (inside measure) of a cylindrical
     tank is 90 square feet. How tall must it be to hold
     100 cubic yards?                                     _Answer_ .....

  From National Intelligence Tests by National Research Council.

  Copyright, 1920, by The World Book Company, Yonkers-on-Hudson,
  New York.

  Used by permission of the publishers.




CHAPTER III

THE CONSTITUTION OF ARITHMETICAL ABILITIES


THE ELEMENTARY FUNCTIONS OF ARITHMETICAL LEARNING

It would be a useful work for some one to try to analyze arithmetical
learning into the unitary abilities which compose it, showing just what,
in detail, the mind has to do in order to be prepared to pass a thorough
test on the whole of arithmetic. These unitary abilities would make a
very long list. Examination of a well-planned textbook will show that
such an ability as multiplication is treated as a composite of the
following: knowledge of the multiplications up to 9 × 9; ability to
multiply two (or more)-place numbers by 2, 3, and 4 when 'carrying' is
not required and no zeros occur in the multiplicand; ability to multiply
by 2, 3, ... 9, with carrying; the ability to handle zeros in the
multiplicand; the ability to multiply with two-place numbers not ending
in zero; the ability to handle zero in the multiplier as last number;
the ability to multiply with three (or more)-place numbers not including
a zero; the ability to multiply with three- and four-place numbers with
zero in second or third, or second and third, as well as in last place;
the ability to save time by annexing zeros; and so on and on through a
long list of further abilities required to multiply with United States
money, decimal fractions, common fractions, mixed numbers, and
denominate numbers.

The units or 'steps' thus recognized by careful teaching would make a
long list, but it is probable that a still more careful study of
arithmetical ability as a hierarchy of mental habits or connections
would greatly increase the list. Consider, for example, ordinary column
addition. The majority of teachers probably treat this as a simple
application of the knowledge of the additions to 9 + 9, plus
understanding of 'carrying.' On the contrary there are at least seven
processes or minor functions involved in two-place column addition, each
of which is psychologically distinct and requires distinct educational
treatment.

These are:--

  A. Learning to keep one's place in the column as one adds.

  B. Learning to keep in mind the result of each addition until the
        next number is added to it.

  C. Learning to add a seen to a thought-of number.

  D. Learning to neglect an empty space in the columns.

  E. Learning to neglect 0s in the columns.

  F. Learning the application of the combinations to higher decades
        may for the less gifted pupils involve as much time and labor
        as learning all the original addition tables. And even for
        the most gifted child the formation of the connection
        '8 and 7 = 15' probably never quite insures the presence
        of the connections '38 and 7 = 45' and '18 + 7 = 25.'

  G. Learning to write the figure signifying units rather than the
        total sum of a column. In particular, learning to write 0 in
        the cases where the sum of the column is 10, 20, etc. Learning
        to 'carry' also involves in itself at least two distinct
        processes, by whatever way it is taught.

We find evidence of such specialization of functions in the results with
such tests as Woody's. For example, 2 + 5 + 1 = .... surely involves
abilities in part different from

    2
    4
    3
    -

because only 77 percent of children in grade 3 do the former correctly,
whereas 95 percent of children in that grade do the latter correctly. In
grade 2 the difference is even more marked. In the case of subtraction

    4
    4
    -

involves abilities different from those involved in

    9
    3,
    -

being much less often solved correctly in grades 2 and 4.

    6
    0
    -

is much harder than either of the above.

   43
    1                       21
    2                       33
   13  is much harder than  35.
   --                       --

It may be said that these differences in difficulty are due to different
amounts of practice. This is probably not true, but if it were, it would
not change the argument; if the two abilities were identical, the
practice of one would improve the other equally.

I shall not undertake here this task of listing and describing the
elementary functions which constitute arithmetical learning, partly
because what they are is not fully known, partly because in many cases a
final ability may be constituted in several different ways whose
descriptions become necessarily tedious, and partly because an adequate
statement of what is known would far outrun the space limits of this
chapter. Instead, I shall illustrate the results by some samples.


KNOWLEDGE OF THE MEANING OF A FRACTION

As a first sample, consider knowledge of the meaning of a fraction. Is
the ability in question simply to understand that a fraction is a
statement of the number of parts, each of a certain size, the upper
number or numerator telling how many parts are taken and the lower
number or denominator telling what fraction of unity each part is? And
is the educational treatment required simply to describe and illustrate
such a statement and have the pupils apply it to the recognition of
fractions and the interpretation of each of them? And is the learning
process (1) the formation of the notions of part, size of part, number
of part, (2) relating the last two to the numbers in a fraction, and, as
a necessary consequence, (3) applying these notions adequately whenever
one encounters a fraction in operation?

Precisely this was the notion a few generations ago. The nature of
fractions was taught as one principle, in one step, and the habits of
dealing with fractions were supposed to be deduced from the general law
of a fraction's nature. As a result the subject of fractions had to be
long delayed, was studied at great cost of time and effort, and, even
so, remained a mystery to all save gifted pupils. These gifted pupils
probably of their own accord built up the ability piecemeal out of
constituent insights and habits.

At all events, scientific teaching now does build up the total ability
as a fusion or organization of lesser abilities. What these are will be
seen best by examining the means taken to get them. (1) First comes the
association of 1/2 of a pie, 1/2 of a cake, 1/2 of an apple, and such
like with their concrete meanings so that a pupil can properly name a
clearly designated half of an obvious unit like an orange, pear, or
piece of chalk. The same degree of understanding of 1/4, 1/8, 1/3, 1/6,
and 1/5 is secured. The pupil is taught that 1 pie = 2 1/2s, 3 1/3s, 4
1/4s, 5 1/5s, 6 1/6s, and 8 1/8s; similarly for 1 cake, 1 apple, and the
like.

So far he understands 1/_x_ of _y_ in the sense of certain simple parts
of obviously unitary _y_s.

(2) Next comes the association with 1/2 of an inch, 1/2 of a foot, 1/2
of a glassful and other cases where _y_ is not so obviously a unitary
object whose pieces still show their derivation from it. Similarly for
1/4, 1/3, etc.

(3) Next comes the association with 1/2 of a collection of eight pieces
of candy, 1/3 of a dozen eggs, 1/5 of a squad of ten soldiers, etc.,
until 1/2, 1/3, 1/4, 1/5, 1/6, and 1/8 are understood as names of
certain parts of a collection of objects.

(4) Next comes the similar association when the nature of the collection
is left undefined, the pupil responding to

  1/2 of 6 is ..., 1/4 of 8 is ..., 2 is 1/5 of ...,
  1/3 of 6 is ..., 1/3 of 9 is ..., 2 is 1/3 of ..., and the like.

Each of these abilities is justified in teaching by its intrinsic
merits, irrespective of its later service in helping to constitute the
general understanding of the meaning of a fraction. The habits thus
formed in grades 3 or 4 are of constant service then and thereafter in
and out of school.

(5) With these comes the use of 1/5 of 10, 15, 20, etc., 1/6 of 12, 18,
42, etc., as a useful variety of drill on the division tables, valuable
in itself, and a means of making the notion of a unit fraction more
general by adding 1/7 and 1/9 to the scheme.

(6) Next comes the connection of 3/4, 2/5, 3/5, 4/5, 2/3, 1/6, 5/6, 3/8,
5/8, 7/8, 3/10, 7/10, and 9/10, each with its meaning as a certain part
of some conveniently divisible unit, and, (7) and (8), connections
between these fractions and their meanings as parts of certain
magnitudes (7) and collections (8) of convenient size, and (9)
connections between these fractions and their meanings when the nature
of the magnitude or collection is unstated, as in 4/5 of 15 = ...,
5/8 of 32 = ....

(10) That the relation is general is shown by using it with
numbers requiring written division and multiplication, such as
7/8 of 1736 = ..., and with United States money.

Elements (6) to (10) again are useful even if the pupil never goes
farther in arithmetic. One of the commonest uses of fractions is in
calculating the cost of fractions of yards of cloth, and fractions of
pounds of meat, cheese, etc.

The next step (11) is to understand to some extent the principle that
the value of any of these fractions is unaltered by multiplying or
dividing the numerator and denominator by the same number. The drills in
expressing fractions in lower and higher terms which accomplish this are
paralleled by (12) and (13) simple exercises in adding and subtracting
fractions to show that fractions are quantities that can be operated on
like any quantities, and by (14) simple work with mixed numbers
(addition and subtraction and reductions), and (15) improper fractions.
All that is done with improper fractions is (_a_) to have the pupil use
a few of them as he would any fractions and (_b_) to note their
equivalent mixed numbers. In (12), (13), and (14) only fractions of the
same denominators are added or subtracted, and in (12) (13), (14), and
(15) only fractions with 2, 3, 4, 5, 6, 8, or 10 in the denominator are
used. As hitherto, the work of (11) to (15) is useful in and of itself.
(16) Definitions are given of the following type:--

Numbers like 2, 3, 4, 7, 11, 20, 36, 140, 921 are called whole numbers.

Numbers like 7/8, 1/5, 2/3, 3/4, 11/8, 7/6, 1/3, 4/3, 1/8, 1/6 are
called fractions.

Numbers like 5-1/4, 7-3/8, 9-1/2, 16-4/5, 315-7/8, 1-1/3, 1-2/3 are
called mixed numbers.

(17) The terms numerator and denominator are connected with the upper
and lower numbers composing a fraction.

Building this somewhat elaborate series of minor abilities seems to be a
very roundabout way of getting knowledge of the meaning of a fraction,
and is, if we take no account of what is got along with this knowledge.
Taking account of the intrinsically useful habits that are built up, one
might retort that the pupil gets his knowledge of the meaning of a
fraction at zero cost.


KNOWLEDGE OF THE SUBTRACTION AND DIVISION TABLES

Consider next the knowledge of the subtraction and division 'Tables.'
The usual treatment presupposes that learning them consists of forming
independently the bonds:--

   3 - 1 = 2                4 ÷ 2 = 2
   3 - 2 = 1                6 ÷ 2 = 3
   4 - 1 = 3                6 ÷ 3 = 2
     .                        .
     .                        .
     .                        .
     .                        .
     .                        .
     .                        .
  18 - 9 = 9               81 ÷ 9 = 9

In fact, however, these 126 bonds are not formed independently. Except
perhaps in the case of the dullest twentieth of pupils, they are
somewhat facilitated by the already learned additions and
multiplications. And by proper arrangement of the learning they may be
enormously facilitated thereby. Indeed, we may replace the independent
memorizing of these facts by a set of instructive exercises wherein the
pupil derives the subtractions from the corresponding additions by
simple acts of reasoning or selective thinking. As soon as the additions
giving sums of 9 or less are learned, let the pupil attack an exercise
like the following:--

Write the missing numbers:--

         A                 B                 C                 D
  3 and ... are 5.  5 and ... are 8.  4 and ... are 5.  4 and ... are 8.
  3 and ... are 9.  3 and ... are 6.  5 and ... are 6.  1 and ... are 7.
  4 and ... are 7.  4 and ... are 9.  6 and ... are 9.  6 and ... are 7.
  5 and ... are 7.  2 and ...  =  6.  1 and ... are 8.  8 and ... are 9.
  6 and ... are 8.  5 and ...  =  9.  3 and ... are 7.  3  +  ... are 4.
  4 and ... are 6.  2 and ...  =  7.  1  +  ... are 3.  7  +  ... are 8.
  2 and ... are 5.  3 and ...  =  8.  1  +  ... are 5.  4  +  ... are 9.
  2 and ...  =  8.  1 and ...  =  4.  4  +  ... are 8.  2  +  ... are 3.
  3 and ...  =  6.  2 and ...  =  4.  7  +  ... are 9.  1  +  ... are 9.
  6 and ...  =  9.  3 and ...  =  8.  2  +  ...  =  4.  3  +  ...  =  6.
  4 and ...  =  6.  6 and ...  =  7.  3  +  ...  =  8.  5  +  ...  =  9.
  4 and ...  =  7.  2 and ...  =  5.  4  +  ...  =  5.  1  +  ...  =  3.

The task for reasoning is only to try, one after another, numbers that
seem promising and to select the right one when found. With a little
stimulus and direction children can thus derive the subtractions up to
those with 9 as the larger number. Let them then be taught to do the
same with the printed forms:--

Subtract

  9   7   8   5   8   6
  3   5   6   2   2   4   etc.
  -   -   -   -   -   -

and 9 - 7 = ..., 9 - 5 = ..., 7 - 5 = ..., etc.

In the case of the divisions, suppose that the pupil has learned his
first table and gained surety in such exercises as:--

  4 5s = ....    6 × 5 = ....    9 nickels = .... cents.
  8 5s = ....    4 × 5 = ....    6    "    = ....   "
  3 5s = ....    2 × 5 = ....    5    "    = ....   "
  7 5s = ....    9 × 5 = ....    7    "    = ....   "

  If one ball costs 5 cents,
                two balls cost .... cents,
                three balls cost .... cents, etc.

He may then be set at once to work at the answers to exercises like the
following:--

Write the answers and the missing numbers:--

       A            B                 C                 D
  .... 5s = 15   40 = .... 5s   .... × 5 = 25   20 cents = .... nickels.
  .... 5s = 20   20 = .... 5s   .... × 5 = 50   30 cents = .... nickels.
  .... 5s = 40   15 = .... 5s   .... × 5 = 35   15 cents = .... nickels.
  .... 5s = 25   45 = .... 5s   .... × 5 = 10   40 cents = .... nickels.
  .... 5s = 30   50 = .... 5s   .... × 5 = 40
  .... 5s = 35   25 = .... 5s   .... × 5 = 45

                          E
  For  5 cents you can buy 1 small loaf of bread.
  For 10 cents you can buy 2 small loaves of bread.
  For 25 cents you can buy .... small loaves of bread.
  For 45 cents you can buy .... small loaves of bread.
  For 35 cents you can buy .... small loaves of bread.

               F
   5 cents pays 1 car fare.
  15 cents pays .... car fares.
  10 cents pays .... car fares.
  20 cents pays .... car fares.

                           G
  How many 5 cent balls can you buy with 30 cents? ....
  How many 5 cent balls can you buy with 35 cents? ....
  How many 5 cent balls can you buy with 25 cents? ....
  How many 5 cent balls can you buy with 15 cents? ....

In the case of the meaning of a fraction, the ability, and so the
learning, is much more elaborate than common practice has assumed; in
the case of the subtraction and division tables the learning is much
less so. In neither case is the learning either mere memorizing of facts
or the mere understanding of a principle _in abstracto_ followed by its
application to concrete cases. It is (and this we shall find true of
almost all efficient learning in arithmetic) the formation of
connections and their use in such an order that each helps the others to
the maximum degree, and so that each will do the maximum amount for
arithmetical abilities other than the one specially concerned, and for
the general competence of the learner.


LEARNING THE PROCESSES OF COMPUTATION

As another instructive topic in the constitution of arithmetical
abilities, we may take the case of the reasoning involved in
understanding the manipulations of figures in two (or more)-place
addition and subtraction, multiplication and division involving a two
(or more)-place number, and the manipulations of decimals in all four
operations. The psychology of these is of special interest and
importance. For there are two opposite explanations possible here,
leading to two opposite theories of teaching.

The common explanation is that these methods of manipulation, if
understood at all, are understood as deductions from the properties of
our system of decimal notation. The other is that they are understood
partly as inductions from the experience that they always give the right
answer. The first explanation leads to the common preliminary deductive
explanations of the textbooks. The other leads to explanations by
verification; _e.g._, of addition by counting, of subtraction by
addition, of multiplication by addition, of division by multiplication.
Samples of these two sorts of explanation are given below.


SHORT MULTIPLICATION WITHOUT CARRYING: DEDUCTIVE EXPLANATION

MULTIPLICATION is the process of taking one number as many times as
there are units in another number.

The PRODUCT is the result of the multiplication.

The MULTIPLICAND is the number to be taken.

The MULTIPLIER is the number denoting how many times the multiplicand is
to be taken.

The multiplier and multiplicand are the FACTORS.

    Multiply 623 by 3

    OPERATION

      _Multiplicand_    623
      _Multiplier_        3
                       ----
      _Product_        1869

    EXPLANATION.--For convenience we write the multiplier under the
    multiplicand, and begin with units to multiply. 3 times 3 units are
    9 units. We write the nine units in units' place in the product. 3
    times 2 tens are 6 tens. We write the 6 tens in tens' place in the
    product. 3 times 6 hundreds are 18 hundreds, or 1 thousand and 8
    hundreds. The 1 thousand we write in thousands' place and the 8
    hundreds in hundreds' place in the product. Therefore, the product
    is 1 thousand 8 hundreds, 6 tens and 9 units, or 1869.


SHORT MULTIPLICATION WITHOUT CARRYING: INDUCTIVE EXPLANATION

  1. The children of the third grade are to have a picnic. 32 are going.
  How many sandwiches will they need if each of the 32 children has four
  sandwiches?

      _Here is a quick way to find out_:--

       32   _Think "4 × 2," write 8 under the 2 in the ones column._
        4   _Think "4 × 3," write 12 under the 3 in the tens column._
       --

  2. How many bananas will they need if each of the 32 children has
  two bananas? 32 × 2 or 2 × 32 will give the answer.

  3. How many little cakes will they need if each child has three
  cakes? 32 × 3 or 3 × 32 will give the answer.

    32           3 × 2 = ....  _Where do you write the 6?_
     3           3 × 3 = ....  _Where do you write the 9?_
    --

  4.  Prove that 128, 64, and 96 are right by adding four 32s, two 32s,
  and three 32s.

      32
      32  32
      32  32  32
      32  32  32
      --  --  --


Multiplication

  You #multiply# when you find the answers to questions like

    How many are 9 × 3?
    How many are 3 × 32?
    How many are 8 × 5?
    How many are 4 × 42?

  1. Read these lines. Say the right numbers where the dots are:

     If you #add# 3 to 32, you have .... 35 is the #sum#.
     If you #subtract# 3 from 32, the result is .... 29 is the
          #difference# or #remainder#.
     If you #multiply# 3 by 32 or 32 by 3, you have .... 96 is the
          #product#.

  Find the products. Check your answers to the first line by adding.

   2.    3.    4.    5.    6.    7.    8.    9.

   41    33    42    44    53    43    34    24
    3     2     4     2     3     2     2     2
   --    --    --    --    --    --    --    --

      10.   11.   12.   13.   14.   15.   16.

      43    52    32    23    41    51    14
       3     3     3     3     2     4     2
      --    --    --    --    --    --    --

  17.     213  _Write the 9 in the ones column._
            3  _Write the 3 in the tens column._
          ---  _Write the 6 in the hundreds column._

  _Check your answer by adding._    Add
                                    213
                                    213
                                    213
                                    ---

   18.    19.    20.    21.    22.    23.    24.

   214    312    432    231    132    314    243
     2      3      2      3      3      2      2
   ---    ---    ---    ---    ---    ---    ---


SHORT DIVISION: DEDUCTIVE EXPLANATION

Divide 1825 by 4

  Divisor  4 |1825    Dividend
             --------
             456-1/4
             Quotient

    EXPLANATION.--For convenience we write the divisor at the left of
    the dividend, and the quotient below it, and begin at the left to
    divide. 4 is not contained in 1 thousand any thousand times,
    therefore the quotient contains no unit of any order higher than
    hundreds. Consequently we find how many times 4 is contained in the
    hundreds of the dividend. 1 thousand and 8 hundreds are 18
    hundreds. 4 is contained in 18 hundreds 4 hundred times and 2
    hundreds remaining. We write the 4 hundreds in the quotient. The 2
    hundreds we consider as united with the 2 tens, making 22 tens. 4
    is contained in 22 tens 5 tens times, and 2 tens remaining. We
    write the 5 tens in the quotient, and the remaining 2 tens we
    consider as united with the 5 units, making 25 units. 4 is
    contained in 25 units 6 units times and 1 unit remaining. We write
    the 6 units in the quotient and indicate the division of the
    remainder, 1 unit, by the divisor 4.

    Therefore the quotient of 1825 divided by 4 is 456-1/4, or 456 and
    1 remainder.


SHORT DIVISION: INDUCTIVE EXPLANATION

Dividing Large Numbers

  1. Tom, Dick, Will, and Fred put in 2 cents each to buy an eight-cent
  bag of marbles. There are 128 marbles in it. How many should each boy
  have, if they divide the marbles equally among the four boys?

      -----
    4 |128

    _Think "12 = three 4s." Write the 3 over the 2 in the tens column._
    _Think "8 = two 4s." Write the 2 over the 8 in the ones column._
    _32 is right, because 4 × 32 = 128._

  2. Mary, Nell, and Alice are going to buy a book as a present for
  their Sunday-school teacher. The present costs 69 cents. How much
  should each girl pay, if they divide the cost equally among the
  three girls?
     ----
    3|69

    _Think "6 = .... 3s." Write the 2 over the 6 in the tens column._
    _Think "9 = .... 3s." Write the 3 over the 9 in the ones column._
    _23 is right, for 3 × 23 = 69._

  3. Divide the cost of a 96-cent present equally among three girls. How
  much should each girl pay?
     ------
    3|96

  4. Divide the cost of an 84-cent present equally among 4 girls. How
  much should each girl pay?

  5. Learn this: (Read ÷ as "_divided by_.")

      12 + 4 = 16.   16 is the sum.
      12 - 4 =  8.    8 is the difference or remainder.
      12 × 4 = 48.   48 is the product.
      12 ÷ 4 =  3.    3 is the quotient.

  6. Find the quotients. Check your answers by multiplying.
     ----     ----     ----     -----     -----     -----
    3|99     2|86     5|155    6|246     4|168     3|219

[Uneven division is taught by the same general plan, extended.]


LONG DIVISION: DEDUCTIVE EXPLANATION

To Divide by Long Division

1. Let it be required to divide 34531 by 15.

              _Operation_

               Divided
  Divisor  15)34531(2302-1/15  Quotient
              30
              --
               45
               45
               --
                 31
                 30
                 --
                  1 Remainder

For convenience we write the divisor at the left and the quotient at the
right of the dividend, and begin to divide as in Short Division.

15 is contained in 3 ten-thousands 0 ten-thousands times; therefore,
there will be 0 ten-thousands in the quotient. Take 34 thousands; 15 is
contained in 34 thousands 2 thousands times; we write the 2 thousands in
the quotient. 15 × 2 thousands = 30 thousands, which, subtracted from 34
thousands, leaves 4 thousands = 40 hundreds. Adding the 5 hundreds, we
have 45 hundreds.

15 in 45 hundreds 3 hundreds times; we write the 3 hundreds in the
quotient. 15 × 3 hundreds = 45 hundreds, which subtracted from 45
hundreds, leaves nothing. Adding the 3 tens, we have 3 tens.

15 in 3 tens 0 tens times; we write 0 tens in the quotient. Adding to
the three tens, which equal 30 units, the 1 unit, we have 31 units.

15 in 31 units 2 units times; we write the 2 units in the quotient. 15 ×
2 units = 30 units, which, subtracted from 31 units, leaves 1 unit as a
remainder. Indicating the division of the 1 unit, we annex the
fractional expression, 1/15 unit, to the integral part of the quotient.

Therefore, 34531 divided by 15 is equal to 2302-1/15.

[B. Greenleaf, _Practical Arithmetic_, '73, p. 49.]


LONG DIVISION: INDUCTIVE EXPLANATION

Dividing by Large Numbers

  1. Just before Christmas Frank's father sent 360 oranges to be divided
     among the children in Frank's class. There are 29 children. How
     many oranges should each child receive? How many oranges will be
     left over?

  _Here is the best way to find out:_

       12    and 12    _Think how many 29s there are in 36. 1 is right._
     ______ remainder  _Write 1 over the 6 of 36. Multiply 29 by 1._
  29 )360              _Write the 29 under the 36. Subtract 29 from 36._
       29              _Write the 0 of 360 after the 7._
      ---              _Think how many 29s there are in 70. 2 is right._
       70              _Write 2 over the 0 of 360. Multiply 29 by 2._
       58              _Write the 58 under 70. Subtract 58 from 70._
       --              _There is 12 remainder._
       12              _Each child gets 12 oranges, and there are 12
                         left over. This is right, for 12 multiplied
                         by 29 = 348, and 348 + 12 = 360._

       *       *       *       *       *

  8.                  _In No. 8, keep on dividing by 31 until you have
    ________            used the 5, the 8, and the 7, and have four
  31)99,587             figures in the quotient._

  9.         10.         11.         12.        13.
    _____      _____       _____       ____       _______
  22)253     22)2895     21)8891     22)290     32)16,368

Check your results for 9, 10, 11, 12, and 13.

  1. The boys and girls of the Welfare Club plan to earn money to buy
     a victrola. There are 23 boys and girls. They can get a good
     second-hand victrola for $5.75. How much must each earn if they
     divide the cost equally?

     _Here is the best way to find out_:

      $.25     _Think how many 23s there are in 57. 2 is right._
     -----     _Write 2 over the 7 of 57. Multiply 23 by 2._
  23|$5.75     _Write 46 under 57 and subtract. Write the 5 of 575
        46       after the 11._
      ----     _Think how many 23s there are in 115. 5 is right._
       115     _Write 5 over the 5 of 575. Multiply 23 by 5._
       115     _Write the 115 under the 115 that is there and subtract._
      ----     _There is no remainder._
               _Put $ and the decimal point where they belong._
               _Each child must earn 25 cents. This is right, for $.25
                 multiplied by 23 = $5.75._

  2. Divide $71.76 equally among 23 persons. How much is each person's
     share?

  3. Check your result for No. 2 by multiplying the quotient by the
     divisor.

  Find the quotients. Check each quotient by multiplying it by the
  divisor.

  4.           5.           6.            7.           8.
    _______      _______      ________      _______      _______
  23)$99.13    25)$18.50    21)$129.15    13)$29.25    32)$73.92

    1 bushel = 32 qt.

  9. How many bushels are there in 288 qt.?

  10. In 192 qt.?

  11. In 416 qt.?

Crucial experiments are lacking, but there are several lines of
well-attested evidence. First of all, there can be no doubt that the
great majority of pupils learn these manipulations at the start from the
placing of units under units, tens under tens, etc., in adding, to the
placing of the decimal point in division with decimals, by imitation and
blind following of specific instructions, and that a very large
proportion of the pupils do not to the end, that is to the fifth
school-year, understand them as necessary deductions from decimal
notation. It also seems probable that this proportion would not be much
reduced no matter how ingeniously and carefully the deductions were
explained by textbooks and teachers. Evidence of this fact will appear
abundantly to any one who will observe schoolroom life. It also appears
in the fact that after the properties of the decimal notation have been
thus used again and again; _e.g._, for deducing 'carrying' in addition,
'borrowing' in subtraction, 'carrying' in multiplication, the value of
the digits in the partial product, the value of each remainder in short
division, the value of the quotient figures in division, the addition,
subtraction, multiplication, and division of United States money, and
the placing of the decimal point in multiplication, no competent teacher
dares to rely upon the pupil, even though he now has four or more years'
experience with decimal notation, to deduce the placing of the decimal
point in division with decimals. It may be an illusion, but one seems to
sense in the better textbooks a recognition of the futility of the
attempt to secure deductive derivations of those manipulations. I refer
to the brevity of the explanations and their insertion in such a form
that they will influence the pupils' thinking as little as possible. At
any rate the fact is sure that most pupils do not learn the
manipulations by deductive reasoning, or understand them as necessary
consequences of abstract principles.

It is a common opinion that the only alternative is knowing them by
rote. This, of course, is one common alternative, but the other
explanation suggests that understanding the manipulations by inductive
reasoning from their results is another and an important alternative.
The manipulations of 'long' multiplication, for instance, learned by
imitation or mechanical drill, are found to give for 25 × _A_ a result
about twice as large as for 13 × _A_, for 38 or 39 × _A_ a result about
three times as large; for 115 × _A_ a result about ten times as large as
for 11 × _A_. With even the very dull pupils the procedure is verified
at least to the extent that it gives a result which the scientific
expert in the case--the teacher--calls right. With even the very bright
pupils, who can appreciate the relation of the procedure to decimal
notation, this relation may be used not as the sole deduction of the
procedure beforehand, but as one partial means of verifying it
afterward. Or there may be the condition of half-appreciation of the
relation in which the pupil uses knowledge of the decimal notation to
convince himself that the procedure _does_, but not that it _must_ give
the right answer, the answer being 'right' because the teacher, the
answer-list, and collateral evidence assure him of it.

I have taken the manipulation of the partial products as an illustration
because it is one of the least favored cases for the explanation I am
presenting. If we take the first case where a manipulation may be
deduced from decimal notation, known merely by rote, or verified
inductively, namely, the addition of two-place numbers, it seems sure
that the mental processes just described are almost the universal rule.

Surely in our schools at present children add the 3 of 23 to the 3 of 53
and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten
because they see the teacher do so and are told to do so. They are
protected from adding 3 + 3 + 2 + 5 not by any deduction of any sort but
because they do not know how to add 8 and 5, because they have been
taught the habit of adding figures that stand one above the other, or
with a + between them; and because they are shown or told what they are
to do. They are protected from adding 3 + 5 and 2 + 3, again, by no
deductive reasoning but for the second and third reasons just given.
In nine cases out of ten they do not even think of the possibility
of adding in any other way than the '3 + 3, 2 + 5' way, much less
do they select that way on account of the facts that 53 = 50 + 3
and 23 = 20 + 3, that 50 + 20 = 70, that 3 + 3 = 6, and that
(_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_)!

Just as surely all but the very dullest twentieth or so of children come
in the end to something more than rote knowledge,--to _understand_, to
_know_ that the procedure in question is right.

Whether they know _why_ 76 is right depends upon what is meant by
_why_. If it means that 76 is the result which competent people
agree upon, they do. If it means that 76 is the result which would
come from accurate counting they perhaps know why as well as they
would have, had they been given full explanations of the relation
of the procedure in two-place addition to decimal notation.
If _why_ means because 53 = 50 + 3, 23 = 20 + 3, 50 + 20 = 70, and
(_a_ + _b_) + (_c_ + _d_) = (_a_ + _c_) + (_b_ + _d_), they do not.
Nor, I am tempted to add, would most of them by any sort of teaching
whatever.

I conclude, therefore, that school children may and do reason about and
understand the manipulations of numbers in this inductive, verifying way
without being able to, or at least without, under present conditions,
finding it profitable to derive them deductively. I believe, in fact,
that pure arithmetic _as it is learned and known_ is largely an
_inductive science_. At one extreme is a minority to whom it is a series
of deductions from principles; at the other extreme is a minority to
whom it is a series of blind habits; between the two is the great
majority, representing every gradation but centering about the type of
the inductive thinker.




CHAPTER IV

THE CONSTITUTION OF ARITHMETICAL ABILITIES (CONTINUED): THE SELECTION OF
THE BONDS TO BE FORMED


When the analysis of the mental functions involved in arithmetical
learning is made thorough it turns into the question, 'What are the
elementary bonds or connections that constitute these functions?' and
when the problem of teaching arithmetic is regarded, as it should be in
the light of present psychology, as a problem in the development of a
hierarchy of intellectual habits, it becomes in large measure a problem
of the choice of the bonds to be formed and of the discovery of the best
order in which to form them and the best means of forming each in that
order.


THE IMPORTANCE OF HABIT-FORMATION

The importance of habit-formation or connection-making has been grossly
underestimated by the majority of teachers and writers of textbooks.
For, in the first place, mastery by deductive reasoning of such matters
as 'carrying' in addition, 'borrowing' in subtraction, the value of the
digits in the partial products in multiplication, the manipulation of
the figures in division, the placing of the decimal point after
multiplication or division with decimals, or the manipulation of the
figures in the multiplication and division of fractions, is impossible
or extremely unlikely in the case of children of the ages and experience
in question. They do not as a rule deduce the method of manipulation
from their knowledge of decimal notation. Rather they learn about
decimal notation by carrying, borrowing, writing the last figure of each
partial product under the multiplier which gives that product, etc. They
learn the method of manipulating numbers by seeing them employed, and by
more or less blindly acquiring them as associative habits.

In the second place, we, who have already formed and long used the right
habits and are thereby protected against the casual misleadings of
unfortunate mental connections, can hardly realize the force of mere
association. When a child writes sixteen as 61, or finds 428 as the sum
of

  15
  19
  16
  18
  --

or gives 642 as an answer to 27 × 36, or says that 4 divided by 1/4 = 1,
we are tempted to consider him mentally perverse, forgetting or perhaps
never having understood that he goes wrong for exactly the same general
reason that we go right; namely, the general law of habit-formation. If
we study the cases of 61 for 16, we shall find them occurring in the
work of pupils who after having been drilled in writing 26, 36, 46, 62,
63, and so on, in which the order of the six in writing is the same as
it is in speech, return to writing the 'teen numbers. If our language
said onety-one for eleven and onety-six for sixteen, we should probably
never find such errors except as 'lapses' or as the results of
misperception or lack of memory. They would then be more frequent
_before_ the 20s, 30s, etc., were learned.

If pupils are given much drill on written single column addition
involving the higher decades (each time writing the two-figure sum),
they are forming a habit of writing 28 after the sum of 8, 6, 9, and 5
is reached; and it should not surprise us if the pupil still
occasionally writes the two-figure sum for the first column though a
second column is to be added also. On the contrary, unless some counter
force influences him, he is absolutely sure to make this mistake.

The last mistake quoted (4 ÷ 1/4 = 1) is interesting because here we
have possibly one of the cases where deduction from psychology alone can
give constructive aid to teaching. Multiplication and division by
fractions have been notorious for their difficulty. The former is now
alleviated by using _of_ instead of × until the new habit is fixed. The
latter is still approached with elaborate caution and with various means
of showing why one must 'invert and multiply' or 'multiply by the
reciprocal.'

But in the author's opinion it seems clear that the difficulty in
multiplying and dividing by a fraction was not that children felt any
logical objections to canceling or inverting. I fancy that the majority
of them would cheerfully invert any fraction three times over or cancel
numbers at random in a column if they were shown how to do so. But if
you are a youngster inexperienced in numerical abstractions and if you
have had _divide_ connected with 'make smaller' three thousand times and
never once connected with 'make bigger,' you are sure to be somewhat
impelled to make the number smaller the three thousand and first time
you are asked to divide it. Some of my readers will probably confess
that even now they feel a slight irritation or doubt in saying or
writing that 16/1 ÷ 1/8 = 128.

The habits that have been confirmed by every multiplication and division
by integers are, in this particular of '_the ratio of result to number
operated upon_,' directly opposed to the formation of the habits
required with fractions. And that is, I believe, the main cause of the
difficulty. Its treatment then becomes easy, as will be shown later.

These illustrations could be added to almost indefinitely, especially in
the case of the responses made to the so-called 'catch' problems. The
fact is that the learner rarely can, and almost never does, survey and
analyze an arithmetical situation and justify what he is going to do by
articulate deductions from principles. He usually feels the situation
more or less vaguely and responds to it as he has responded to it or
some situation like it in the past. Arithmetic is to him not a logical
doctrine which he applies to various special instances, but a set of
rather specialized habits of behavior toward certain sorts of quantities
and relations. And in so far as he does come to know the doctrine it is
chiefly by doing the will of the master. This is true even with the
clearest expositions, the wisest use of objective aids, and full
encouragement of originality on the pupil's part.

Lest the last few paragraphs be misunderstood, I hasten to add that the
psychologists of to-day do not wish to make the learning of arithmetic a
mere matter of acquiring thousands of disconnected habits, nor to
decrease by one jot the pupil's genuine comprehension of its general
truths. They wish him to reason not less than he has in the past, but
more. They find, however, that you do not secure reasoning in a pupil by
demanding it, and that his learning of a general truth without the
proper development of organized habits back of it is likely to be, not a
rational learning of that general truth, but only a mechanical
memorizing of a verbal statement of it. They have come to know that
reasoning is not a magic force working in independence of ordinary
habits of thought, but an organization and coöperation of those very
habits on a higher level.

The older pedagogy of arithmetic stated a general law or truth or
principle, ordered the pupil to learn it, and gave him tasks to do which
he could not do profitably unless he understood the principle. It left
him to build up himself the particular habits needed to give him
understanding and mastery of the principle. The newer pedagogy is
careful to help him build up these connections or bonds ahead of and
along with the general truth or principle, so that he can understand it
better. The older pedagogy commanded the pupil to reason and let him
suffer the penalty of small profit from the work if he did not. The
newer provides instructive experiences with numbers which will stimulate
the pupil to reason so far as he has the capacity, but will still be
profitable to him in concrete knowledge and skill, even if he lacks the
ability to develop the experiences into a general understanding of the
principles of numbers. The newer pedagogy secures more reasoning in
reality by not pretending to secure so much.

The newer pedagogy of arithmetic, then, scrutinizes every element of
knowledge, every connection made in the mind of the learner, so as to
choose those which provide the most instructive experiences, those which
will grow together into an orderly, rational system of thinking about
numbers and quantitative facts. It is not enough for a problem to be a
test of understanding of a principle; it must also be helpful in and of
itself. It is not enough for an example to be a case of some rule; it
must help review and consolidate habits already acquired or lead up to
and facilitate habits to be acquired. Every detail of the pupil's work
must do the maximum service in arithmetical learning.


DESIRABLE BONDS NOW OFTEN NEGLECTED

As hitherto, I shall not try to list completely the elementary bonds
that the course of study in arithmetic should provide for. The best
means of preparing the student of this topic for sound criticism and
helpful invention is to let him examine representative cases of bonds
now often neglected which should be formed and representative cases of
useless, or even harmful, bonds now often formed at considerable waste
of time and effort.

(1) _Numbers as measures of continuous quantities._--The numbers one,
two, three, 1, 2, 3, etc., should be connected soon after the beginning
of arithmetic each with the appropriate amount of some continuous
quantity like length or volume or weight, as well as with the
appropriate sized collection of apples, counters, blocks, and the like.
Lines should be labeled 1 foot, 2 feet, 3 feet, etc.; one inch, two
inches, three inches, etc.; weights should be lifted and called one
pound, two pounds, etc.; things should be measured in glassfuls,
handfuls, pints, and quarts. Otherwise the pupil is likely to limit the
meaning of, say, _four_ to four sensibly discrete things and to have
difficulty in multiplication and division. Measuring, or counting by
insensibly marked off repetitions of a unit, binds each number name to
its meaning as ---- _times whatever 1 is_, more surely than mere
counting of the units in a collection can, and should reënforce the
latter.

(2) _Additions in the higher decades._--In the case of all save the very
gifted children, the additions with higher decades--that is, the bonds,
16 + 7 = 23, 26 + 7 = 33, 36 + 7 = 43, 14 + 8 = 22, 24 + 8 = 32, and the
like--need to be specifically practiced until the tendency becomes
generalized. 'Counting' by 2s beginning with 1, and with 2, counting by
3s beginning with 1, with 2, and with 3, counting by 4s beginning with
1, with 2, with 3, and with 4, and so on, make easy beginnings in the
formation of the decade connections. Practice with isolated bonds should
soon be added to get freer use of the bonds. The work of column addition
should be checked for accuracy so that a pupil will continually get
beneficial practice rather than 'practice in error.'

(3) _The uneven divisions._--The quotients with remainders for the
divisions of every number to 19 by 2, every number to 29 by 3, every
number to 39 by 4, and so on should be taught as well as the even
divisions. A table like the following will be found a convenient means
of making these connections:--

  10 = .... 2s
  10 = .... 3s and .... rem.
  10 = .... 4s and .... rem.
  10 = .... 5s
  11 = .... 2s and .... rem.
  11 = .... 3s and .... rem.
  .
  .
  .
  89 = .... 9s and .... rem.

These bonds must be formed before short division can be efficient, are
useful as a partial help toward selection of the proper quotient figures
in long division, and are the chief instruments for one of the important
problem series in applied arithmetic,--"How many _x_s can I buy for _y_
cents at _z_ cents per _x_ and how much will I have left?" That these
bonds are at present sadly neglected is shown by Kirby ['13], who found
that pupils in the last half of grade 3 and the first half of grade 4
could do only about four such examples per minute (in a ten-minute
test), and even at that rate made far from perfect records, though they
had been taught the regular division tables. Sixty minutes of practice
resulted in a gain of nearly 75 percent in number done per minute, with
an increase in accuracy as well.

(4) _The equation form._--The equation form with an unknown quantity to
be determined, or a missing number to be found, should be connected with
its meaning and with the problem attitude long before a pupil begins
algebra, and in the minds of pupils who never will study algebra.

Children who have just barely learned to add and subtract learn easily
to do such work as the following:--

Write the missing numbers:--

  4 + 8 = ....
  5 + .... = 14
  .... + 3 = 11
  .... = 5 + 2
  16 = 7 + ....
  12 = .... + 5

The equation form is the simplest uniform way yet devised to state a
quantitative issue. It is capable of indefinite extension if certain
easily understood conventions about parentheses and fraction signs are
learned. It should be employed widely in accounting and the treatment of
commercial problems, and would be except for outworn conventions. It is
a leading contribution of algebra to business and industrial life.
Arithmetic can make it nearly as well. It saves more time in the case of
drills on reducing fractions to higher and lower terms alone than is
required to learn its meaning and use. To rewrite a quantitative
problem as an equation and then make the easy selection of the
necessary technique to solve the equation is one of the most universally
useful intellectual devices known to man. The words 'equals,' 'equal,'
'is,' 'are,' 'makes,' 'make,' 'gives,' 'give,' and their rarer
equivalents should therefore early give way on many occasions to the '='
which so far surpasses them in ultimate convenience and simplicity.

(5) _Addition and subtraction facts in the case of fractions._--In the
case of adding and subtracting fractions, certain specific
bonds--between the situation of halves and thirds to be added and the
responses of thinking of the numbers as equal to so many sixths, between
the situation thirds and fourths to be added and thinking of them as so
many twelfths, between fourths and eighths to be added and thinking of
them as eighths, and the like--should be formed separately. The general
rule of thinking of fractions as their equivalents with some convenient
denominator should come as an organization and extension of such special
habits, not as an edict from the textbook or teacher.

(6) _Fractional equivalents._--Efficiency requires that in the end the
much used reductions should be firmly connected with the situations
where they are needed. They may as well, therefore, be so connected from
the beginning, with the gain of making the general process far easier
for the dull pupils to master. We shall see later that, for all save the
very gifted pupils, the economical way to get an understanding of
arithmetical principles is not, usually, to learn a rule and then apply
it, but to perform instructive operations and, in the course of
performing them, to get insight into the principles.

(7) _Protective habits in multiplying and dividing with fractions._--In
multiplying and dividing with fractions special bonds should be formed
to counteract the now harmful influence of the 'multiply = get a larger
number' and 'divide = get a smaller number' bonds which all work with
integers has been reënforcing.

For example, at the beginning of the systematic work with multiplication
by a fraction, let the following be printed clearly at the top of every
relevant page of the textbook and displayed on the blackboard:--

_When you multiply a number by anything more than 1 the result is larger
than the number._

_When you multiply a number by 1 the result is the same as the number._

_When you multiply a number by anything less than 1 the result is
smaller than the number._

Let the pupils establish the new habit by many such exercises as:--

   18 × 4 = ....          9 × 2 = ....
    4 × 4 = ....          6 × 2 = ....
    2 × 4 = ....          3 × 2 = ....
    1 × 4 = ....          1 × 2 = ....
  1/2 × 4 = ....        1/3 × 2 = ....
  1/4 × 4 = ....        1/6 × 2 = ....
  1/8 × 4 = ....        1/9 × 2 = ....

In the case of division by a fraction the old harmful habit should be
counteracted and refined by similar rules and exercises as follows:--

_When you divide a number by anything more than 1 the result is smaller
than the number._

_When you divide a number by 1 the result is the same as the number._

_When you divide a number by anything less than 1 the result is larger
than the number._

State the missing numbers:--

    8 = .... 4s      12 = .... 6s      9 = .... 9s
    8 = .... 2s      12 = .... 4s      9 = .... 3s
    8 = .... 1s      12 = .... 3s      9 = .... 1s
    8 = .... 1/2s    12 = .... 2s      9 = .... 1/3s
    8 = .... 1/4s    12 = .... 1s      9 = .... 1/9s
    8 = .... 1/8s    12 = .... 1/2s
                     12 = .... 1/3s
                     12 = .... 1/4s

  16 ÷ 16  =         9 ÷  9  =      10 ÷ 10   =    12 ÷  6  =
  16 ÷  8  =         9 ÷  3  =      10 ÷  5   =    12 ÷  4  =
  16 ÷  4  =         9 ÷  1  =      10 ÷  1   =    12 ÷  3  =
  16 ÷  2  =         9 ÷ 1/3 =      10 ÷ 1/5  =    12 ÷  2  =
  16 ÷  1  =         9 ÷ 1/9 =      10 ÷ 1/10 =    12 ÷  1  =
  16 ÷ 1/2 =                                       12 ÷ 1/2 =
  16 ÷ 1/4 =                                       12 ÷ 1/3 =
  16 ÷ 1/8 =                                       12 ÷ 1/4 =
                                                   12 ÷ 1/6 =

(8) _'% of' means 'hundredths times'._--In the case of percentage a
series of bonds like the following should be formed:--

   5 percent of = .05 times
  20  "   "   " = .20   "
   6  "   "   " = .06   "
  25%         " = .25 ×
  12%         " = .12 ×
   3%         " = .03 ×

Four five-minute drills on such connections between '_x_ percent of' and
'its decimal equivalent times' are worth an hour's study of verbal
definitions of the meaning of percent as per hundred or the like. The
only use of the study of such definitions is to facilitate the later
formation of the bonds, and, with all save the brighter pupils, the
bonds are more needed for an understanding of the definitions than the
definitions are needed for the formation of the bonds.

(9) _Habits of verifying results._--Bonds should early be formed between
certain manipulations of numbers and certain means of checking, or
verifying the correctness of, the manipulation in question. The
additions to 9 + 9 and the subtractions to 18 - 9 should be verified by
objective addition and subtraction and counting until the pupil has sure
command; the multiplications to 9 × 9 should be verified by objective
multiplication and counting of the result (in piles of tens and a pile
of ones) eight or ten times,[4] and by addition eight or ten times;[4]
the divisions to 81 ÷ 9 should be verified by multiplication and
occasionally objectively until the pupil has sure command; column
addition should be checked by adding the columns separately and adding
the sums so obtained, and by making two shorter tasks of the given task
and adding the two sums; 'short' multiplication should be verified eight
or ten times by addition; 'long' multiplication should be checked by
reversing multiplier and multiplicand and in other ways; 'short' and
'long' division should be verified by multiplication.

  [4] Eight or ten times _in all_, not eight or ten times for each fact
      of the tables.

These habits of testing an obtained result are of threefold value. They
enable the pupil to find his own errors, and to maintain a standard of
accuracy by himself. They give him a sense of the relations of the
processes and the reasons why the right ways of adding, subtracting,
multiplying, and dividing are right, such as only the very bright pupils
can get from verbal explanations. They put his acquisition of a certain
power, say multiplication, to a real and intelligible use, in checking
the results of his practice of a new power, and so instill a respect for
arithmetical power and skill in general. The time spent in such
verification produces these results at little cost; for the practice in
adding to verify multiplications, in multiplying to verify divisions,
and the like is nearly as good for general drill and review of the
addition and multiplication themselves as practice devised for that
special purpose.

Early work in adding, subtracting, and reducing fractions should be
verified by objective aids in the shape of lines and areas divided
in suitable fractional parts. Early work with decimal fractions
should be verified by the use of the equivalent common fractions
for .25, .75, .125, .375, and the like. Multiplication and division
with fractions, both common and decimal, should in the early stages
be verified by objective aids. The placing of the decimal point in
multiplication and division with decimal fractions should be verified
by such exercises as:--

           20  It cannot be 200; for 200 × 1.23 is much more than 24.6.
       ______  It cannot be 2; for 2 × 1.23 is much less than 24.6.
  1.23 )24.60
        246
       ----

The establishment of habits of verifying results and their use is very
greatly needed. The percentage of wrong answers in arithmetical work in
schools is now so high that the pupils are often being practiced in
error. In many cases they can feel no genuine and effective confidence
in the processes, since their own use of the processes brings wrong
answers as often as right. In solving problems they often cannot decide
whether they have done the right thing or the wrong, since even if they
have done the right thing, they may have done it inaccurately. A wrong
answer to a problem is therefore too often ambiguous and uninstructive
to them.[5]

  [5] The facts concerning the present inaccuracy of school work in
      arithmetic will be found on pages 102 to 105.

These illustrations of the last few pages are samples of the procedures
recommended by a consideration of all the bonds that one might form and
of the contribution that each would make toward the abilities that the
study of arithmetic should develop and improve. It is by doing more or
less at haphazard what psychology teaches us to do deliberately and
systematically in this respect that many of the past advances in the
teaching of arithmetic have been made.


WASTEFUL AND HARMFUL BONDS

A scrutiny of the bonds now formed in the teaching of arithmetic with
questions concerning the exact service of each, results in a list of
bonds of small value or even no value, so far as a psychologist can
determine. I present here samples of such psychologically unjustifiable
bonds with some of the reasons for their deficiencies.

(1) _Arbitrary units._--In drills intended to improve the ability to see
and use the meanings of numbers as names for ratios or relative
magnitudes, it is unwise to employ entirely arbitrary units. The
procedure in II (on page 84) is better than that in I. Inches,
half-inches, feet, and centimeters are better as units of length than
arbitrary As. Square inches, square centimeters, and square feet are
better for areas. Ounces and pounds should be lifted rather than
arbitrary weights. Pints, quarts, glassfuls, cupfuls, handfuls, and
cubic inches are better for volume.

All the real merit in the drills on relative magnitude advocated by
Speer, McLellan and Dewey, and others can be secured without spending
time in relating magnitudes for the sake of relative magnitude alone.
The use of units of measure in drills which will never be used in _bona
fide_ measuring is like the use of fractions like sevenths, elevenths,
and thirteenths. A very little of it is perhaps desirable to test the
appreciation of certain general principles, but for regular training it
should give place to the use of units of practical significance.

    [Illustration: FIG. 3.

      A ----------
      B ------------------------------
      C --------------------
      D ----------------------------------------

    I. If _A_ is 1 which line is 2? Which line is 4? Which line is 3?
    _A_ and _C_ together equal what line? _A_ and _B_ together equal
    what line? How much longer is _B_ than _A_? How much longer is _B_
    than _C_? How much longer is _D_ than _A_?]

    [Illustration: FIG. 4.

      A ----------
      B ------------------------------
      C --------------------
      D ----------------------------------------

    II. _A_ is 1 inch long. Which line is 2 inches long? Which line is
    4 inches long? Which line is 3 inches long? _A_ and _C_ together
    make ... inches? _A_ and _B_ together make ... inches? _B_ is ...
    ... longer than _A_? _B_ is ... ... longer than _C_? _D_ is ...
    ... longer than _A_?]

(2) _Multiples of 11._--The multiplications of 2 to 12 by 11 and 12 as
single connections should be left for the pupil to acquire by himself as
he needs them. These connections interfere with the process of learning
two-place multiplication. The manipulations of numbers there required
can be learned much more easily if 11 and 12 are used as multipliers in
just the same way that 78 or 96 would be. Later the 12 × 2, 12 × 3,
etc., may be taught. There is less reason for knowing the multiples
of 11 than for knowing the multiples of 15, 16, or 25.

(3) _Abstract and concrete numbers._--The elaborate emphasis of the
supposed fact that we cannot multiply 726 by 8 dollars and the still
more elaborate explanations of why nevertheless we find the cost of 726
articles at $8 each by multiplying 726 by 8 and calling the answer
dollars are wasteful. The same holds of the corresponding pedantry about
division. These imaginary difficulties should not be raised at all. The
pupil should not think of multiplying or dividing men or dollars, but
simply of the necessary equation and of the sort of thing that the
missing number represents. "8 × 726 = .... Answer is dollars," or
"8, 726, multiply. Answer is dollars," is all that he needs to think,
and is in the best form for his thought. Concerning the distinction
between abstract and concrete numbers, both logic and common sense as
well as psychology support the contention of McDougle ['14, p. 206f.],
who writes:--

"The most elementary counting, even that stage when the counts were not
carried in the mind, but merely in notches on a stick or by DeMorgan's
stones in a pot, requires some thought; and the most advanced counting
implies memory of things. The terms, therefore, abstract and concrete
number, have long since ceased to be used by thinking people.

"Recently the writer visited an arithmetic class in a State Normal
School and saw a group of practically adult students confused about this
very question concerning abstract and concrete numbers, according to
their previous training in the conventionalities of the textbook. Their
teacher diverted the work of the hour and she and the class spent almost
the whole period in reëstablishing the requirements 'that the product
must always be the same kind of unit as the multiplicand,' and 'addends
must all be alike to be added.' This is not an exceptional case.
Throughout the whole range of teaching arithmetic in the public schools
pupils are obfuscated by the philosophical encumbrances which have been
imposed upon the simplest processes of numerical work. The time is
surely ripe, now that we are readjusting our ideas of the subject of
arithmetic, to revise some of these wasteful and disheartening
practices. Algebra historically grew out of arithmetic, yet it has not
been laden with this distinction. No pupil in algebra lets _x_ equal the
horses; he lets _x_ equal the _number_ of horses, and proceeds to drop
the idea of horses out of his consideration. He multiplies, divides, and
extracts the root of the _number_, sometimes handling fractions in the
process, and finally interprets the result according to the conditions
of his problem. Of course, in the early number work there have been the
sense-objects from which number has been perceived, but the mind
retreats naturally from objectivity to the pure conception of number,
and then to the number symbol. The following is taken from the appendix
to Horn's thesis, where a seventh grade girl gets the population of the
United States in 1820:--

  7,862,166 whites
    233,634 free negroes
  1,538,022 slaves
  ---------
  9,633,822

In this problem three different kinds of addends are combined, if we
accept the usual distinction. Some may say that this is a mistake,--that
the pupil transformed the 'whites,' 'free negroes,' and 'slaves' into a
common unit, such as 'people' of 'population' and then added these
common units. But this 'explanation' is entirely gratuitous, as one will
find if he questions the pupil about the process. It will be found that
the child simply added the figures as numbers only and then interpreted
the result, according to the statement of the problem, without so much
mental gymnastics. The writer has questioned hundreds of students in
Normal School work on this point, and he believes that the ordinary
mind-movement is correctly set forth here, no matter how well one may
maintain as an academic proposition that this is not logical. Many
classes in the Eastern Kentucky State Normal have been given this
problem to solve, and they invariably get the same result:--

'In a garden on the Summit are as many cabbage-heads as the total number
of ladies and gentlemen in this class. How many cabbage-heads in the
garden?'

And the blackboard solution looks like this each time:--

  29 ladies
  15 gentlemen
  --
  44 cabbage-heads

So, also, one may say: I have 6 times as many sheep as you have cows. If
you have 5 cows, how many sheep have I? Here we would multiply the
number of cows, which is 5, by 6 and call the result 30, which must be
linked with the idea of sheep because the conditions imposed by the
problem demand it. The mind naturally in this work separates the pure
number from its situation, as in algebra, handles it according to the
laws governing arithmetical combinations, and labels the result as the
statement of the problem demands. This is expressed in the following,
which is tacitly accepted in algebra, and should be accepted equally in
arithmetic:

'In all computations and operations in arithmetic, all numbers are
essentially abstract and should be so treated. They are concrete only in
the thought process that attends the operation and interprets the
result.'"

(4) _Least common multiple._--The whole set of bonds involved in
learning 'least common multiple' should be left out. In adding and
subtracting fractions the pupil should _not_ find the least common
multiple of their denominators but should find any common multiple that
he can find quickly and correctly. No intelligent person would ever
waste time in searching for the least common multiple of sixths, thirds,
and halves except for the unfortunate traditions of an oversystematized
arithmetic, but would think of their equivalents in sixths or twelfths
or twenty-fourths or _any other convenient common multiple_. The process
of finding the least common multiple is of such exceedingly rare
application in science or business or life generally that the textbooks
have to resort to purely fantastic problems to give drill in its use.

(5) _Greatest common divisor._--The whole set of bonds involved in
learning 'greatest common divisor' should also be left out. In reducing
fractions to lowest terms the pupil should divide by anything that he
sees that he can divide by, favoring large divisors, and continue doing
so until he gets the fraction in terms suitable for the purpose in hand.
The reader probably never has had occasion to compute a greatest common
divisor since he left school. If he has computed any, the chances are
that he would have saved time by solving the problem in some other way!

The following problems are taken at random from those given by one of
the best of the textbooks that make the attempt to apply the facts of
Greatest Common Divisor and Least Common Multiple to problems.[6] Most
of these problems are fantastic. The others are trivial, or are better
solved by trial and adaptation.

  1. A certain school consists of 132 pupils in the high school, 154
     in the grammar, and 198 in the primary grades. If each group is
     divided into sections of the same number containing as many pupils
     as possible, how many pupils will there be in each section?

  2. A farmer has 240 bu. of wheat and 920 bu. of oats, which he
     desires to put into the least number of boxes of the same capacity,
     without mixing the two kinds of grain. Find how many bushels each
     box must hold.

  3. Four bells toll at intervals of 3, 7, 12, and 14 seconds
     respectively, and begin to toll at the same instant. When will
     they next toll together?

  4. A, B, C, and D start together, and travel the same way around an
     island which is 600 mi. in circuit. A goes 20 mi. per day, B 30,
     C 25, and D 40. How long must their journeying continue, in order
     that they may all come together again?

  5. The periods of three planets which move uniformly in circular
     orbits round the sun, are respectively 200, 250, and 300 da.
     Supposing their positions relatively to each other and the sun
     to be given at any moment, determine how many da. must elapse
     before they again have exactly the same relative positions.

  [6] McLellan and Ames, _Public School Arithmetic_ [1900].

(6) _Rare and unimportant words._--The bonds between rare or unimportant
words and their meanings should not be formed for the mere sake of
verbal variety in the problems of the textbook. A pupil should not be
expected to solve a problem that he cannot read. He should not be
expected in grades 2 and 3, or even in grade 4, to read words that he
has rarely or never seen before. He should not be given elaborate drill
in reading during the time devoted to the treatment of quantitative
facts and relations.

All this is so obvious that it may seem needless to relate. It is not.
With many textbooks it is now necessary to give definite drill in
reading the words in the printed problems intended for grades 2, 3, and
4, or to replace them by oral statements, or to leave the pupils in
confusion concerning what the problems are that they are to solve. Many
good teachers make a regular reading-lesson out of every page of
problems before having them solved. There should be no such necessity.

To define _rare_ and _unimportant_ concretely, I will say that for
pupils up to the middle of grade 3, such words as the following are rare
and unimportant (though each of them occurs in the very first fifty
pages of some well-known beginner's book in arithmetic).

  absentees
  account
  Adele
  admitted
  Agnes
  agreed
  Albany
  Allen
  allowed
  alternate
  Andrew
  Arkansas
  arrived
  assembly
  automobile
  baking powder
  balance
  barley
  beggar
  Bertie
  Bessie
  bin
  Boston
  bouquet
  bronze
  buckwheat
  Byron
  camphor
  Carl
  Carrie
  Cecil
  Charlotte
  charity
  Chicago
  cinnamon
  Clara
  clothespins
  collect
  comma
  committee
  concert
  confectioner
  cranberries
  crane
  currants
  dairyman
  Daniel
  David
  dealer
  debt
  delivered
  Denver
  department
  deposited
  dictation
  discharged
  discover
  discovery
  dish-water
  drug
  due
  Edgar
  Eddie
  Edwin
  election
  electric
  Ella
  Emily
  enrolled
  entertainment
  envelope
  Esther
  Ethel
  exceeds
  explanation
  expression
  generally
  gentlemen
  Gilbert
  Grace
  grading
  Graham
  grammar
  Harold
  hatchet
  Heralds
  hesitation
  Horace Mann
  impossible
  income
  indicated
  inmost
  inserts
  installments
  instantly
  insurance
  Iowa
  Jack
  Jennie
  Johnny
  Joseph
  journey
  Julia
  Katherine
  lettuce-plant
  library
  Lottie
  Lula
  margin
  Martha
  Matthew
  Maud
  meadow
  mentally
  mercury
  mineral
  Missouri
  molasses
  Morton
  movements
  muslin
  Nellie
  nieces
  Oakland
  observing
  obtained
  offered
  office
  onions
  opposite
  original
  package
  packet
  palm
  Patrick
  Paul
  payments
  peep
  Peter
  perch
  phaeton
  photograph
  piano
  pigeons
  Pilgrims
  preserving
  proprietor
  purchased
  Rachel
  Ralph
  rapidity
  rather
  readily
  receipts
  register
  remanded
  respectively
  Robert
  Roger
  Ruth
  rye
  Samuel
  San Francisco
  seldom
  sheared
  shingles
  skyrockets
  sloop
  solve
  speckled
  sponges
  sprout
  stack
  Stephen
  strap
  successfully
  suggested
  sunny
  supply
  Susan
  Susie's
  syllable
  talcum
  term
  test
  thermometer
  Thomas
  torpedoes
  trader
  transaction
  treasury
  tricycle
  tube
  two-seated
  united
  usually
  vacant
  various
  vase
  velocipede
  votes
  walnuts
  Walter
  Washington
  watched
  whistle
  woodland
  worsted

(7) _Misleading facts and procedures._--Bonds should not be formed
between articles of commerce and grossly inaccurate prices therefor,
between events and grossly improbable consequences, or causes or
accompaniments thereof, nor between things, qualities, and events which
have no important connections one with another in the real world. In
general, things should not be put together in the pupil's mind that do
not belong together.

If the reader doubts the need of this warning let him examine problems 1
to 5, all from reputable books that are in common use, or have been
within a few years, and consider how addition, subtraction, and the
habits belonging with each are confused by exercise 6.

  1. If a duck flying 3/5 as fast as a hawk flies 90 miles in an hour,
     how fast does the hawk fly?

  2. At 5/8 of a cent apiece how many eggs can I buy for $60?

  3. At $.68 a pair how many pairs of overshoes can you buy for $816?

  4. At $.13 a dozen how many dozen bananas can you buy for $3.12?

  5. How many pecks of beans can be put into a box that will hold just
     21 bushels?

  6. Write answers:

     537    Beginning at the bottom say 11, 18, and 2 (writing it in
     365    its place) are 20. 5, 11, 14, and 6 (writing it) are 20,
       ?    5, 10. The number, omitted, is 62.
      36
    ----
    1000

  _a._  581     _b._  625     _c._  752     _d._  314     _e._    ?
         97             ?           414           429           845
        364            90           130             ?           223
          ?           417             ?            76            95
       ----          ----          ----          ----          ----
       1758          2050          2460          1000          2367

(8) _Trivialities and absurdities._--Bonds should not be formed between
insignificant or foolish questions and the labor of answering them,
nor between the general arithmetical work of the school and such
insignificant or foolish questions. The following are samples from
recent textbooks of excellent standing:--

  On one side of George's slate there are 32 words, and on the other
  side 26 words. If he erases 6 words from one side, and 8 from the
  other, how many words remain on his slate?

  A certain school has 14 rooms, and an average of 40 children in a
  room. If every one in the school should make 500 straight marks on
  each side of his slate, how many would be made in all?

  8 times the number of stripes in our flag is the number of years
  from 1800 until Roosevelt was elected President. In what year was
  he elected President?

  From the Declaration of Independence to the World's Fair in Chicago
  was 9 times as many years as there are stripes in the flag. How
  many years was it?

(9) _Useless methods._--Bonds should not be formed between a described
situation and a method of treating the situation which would not be a
useful one to follow in the case of the real situation. For example, "If
I set 96 trees in rows, sixteen trees in a row, how many rows will I
have?" forms the habit of treating by division a problem that in reality
would be solved by counting the rows. So also "I wish to give 25 cents
to each of a group of boys and find that it will require $2.75. How many
boys are in the group?" forms the habit of answering a question by
division whose answer must already have been present to give the data of
the problem.

(10) _Problems whose answers would, in real life, be already
known._--The custom of giving problems in textbooks which could not
occur in reality because the answer has to be known to frame the problem
is a natural result of the lazy author's tendency to work out a problem
to fit a certain process and a certain answer. Such bogus problems are
very, very common. In a random sampling of a dozen pages of "General
Review" problems in one of the most widely used of recent textbooks, I
find that about 6 percent of the problems are of this sort. Among the
problems extemporized by teachers these bogus problems are probably
still more frequent. Such are:--

  A clerk in an office addressed letters according to a given list.
  After she had addressed 2500, 4/9 of the names on the list had not
  been used; how many names were in the entire list?

  The Canadian power canal at Sault Ste. Marie furnished 20,000
  horse power. The canal on the Michigan side furnished 2-1/2
  times as much. How many horse power does the latter furnish?

It may be asserted that the ideal of giving as described problems only
problems that might occur and demand the same sort of process for
solution with a real situation, is too exacting. If a problem is
comprehensible and serves to illustrate a principle or give useful
drill, that is enough, teachers may say. For really scientific teaching
it is not enough. Moreover, if problems are given merely as tests of
knowledge of a principle or as means to make some fact or principle
clear or emphatic, and are not expected to be of direct service in the
quantitative work of life, it is better to let the fact be known. For
example, "I am thinking of a number. Half of this number is twice six.
What is the number?" is better than "A man left his wife a certain sum
of money. Half of what he left her was twice as much as he left to his
son, who receives $6000. How much did he leave his wife?" The former is
better because it makes no false pretenses.

(11) _Needless linguistic difficulties._--It should be unnecessary to
add that bonds should not be formed between the pupil's general attitude
toward arithmetic and needless, useless difficulty in language or
needless, useless, wrong reasoning. Our teaching is, however, still
tainted by both of these unfortunate connections, which dispose the
pupil to think of arithmetic as a mystery and folly.

Consider, for example, the profitless linguistic difficulty of problems
1-6, whose quantitative difficulties are simply those of:--

  1.  5 + 8 + 3 + 7
  2.  64 ÷ 8, and knowledge that 1 peck = 8 quarts
  3.  12 ÷ 4
  4.  6 ÷ 2
  5.  3 × 2
  6.  4 × 4

  1. What amount should you obtain by putting together 5 cents, 8
     cents, 3 cents, and 7 cents? Did you find this result by adding or
     multiplying?

  2. How many times must you empty a peck measure to fill a basket
     holding 64 quarts of beans?

  3. If a girl commits to memory 4 pages of history in one day, in
     how many days will she commit to memory 12 pages?

  4. If Fred had 6 chickens how many times could he give away 2
     chickens to his companions?

  5. If a croquet-player drove a ball through 2 arches at each
     stroke, through how many arches will he drive it by 3 strokes?

  6. If mamma cut the pie into 4 pieces and gave each person a piece,
     how many persons did she have for dinner if she used 4 whole pies
     for dessert?

Arithmetically this work belongs in the first or second years of
learning. But children of grades 2 and 3, save a few, would be utterly
at a loss to understand the language.

We are not yet free from the follies illustrated in the lessons of pages
96 to 99, which mystified our parents.

LESSON I

    [Illustration: FIG. 5.]

  1. In this picture, how many girls are in the swing?

  2. How many girls are pulling the swing?

  3. If you count both girls together, how many are they?
     _One_ girl and _one_ other girl are how many?

  4. How many kittens do you see on the stump?

  5. How many on the ground?

  6. How many kittens are in the picture? One kitten and one other
     kitten are how many?

  7. If you should ask me how many girls are in the swing, or how
     many kittens are on the stump, I could answer aloud, _One_; or I
     could write _One_; or thus, _1_.

  8. If I write _One_, this is called the _word One_.

  9. This, _1_, is named a _figure One_, because it means the same as
     the word _One_, and stands for _One_.

 10. Write 1. What is this named? Why?

 11. A figure 1 may stand for _one_ girl, _one_ kitten, or _one_
     anything.

 12. When children first attend school, what do they begin to learn?
     _Ans._ Letters and words.

 13. Could you read or write before you had learned either letters
     or words?

 14. If we have all the _letters_ together, they are named the
     Alphabet.

 15. If we write or speak _words_, they are named Language.

 16. You are commencing to study Arithmetic; and you can read and
     write in Arithmetic only as you learn the Alphabet and Language
     of Arithmetic. But little time will be required for this purpose.

LESSON II

    [Illustration: FIG. 6.]

  1. If we speak or write words, what do we name them, when taken
     together?

  2. What are you commencing to study? _Ans._ Arithmetic.

  3. What Language must you now learn?

  4. What do we name this, 1? Why?

  5. This figure, 1, is part of the Language of Arithmetic.

  6. If I should write something to stand for _Two_--_two_ girls,
     _two_ kittens, or _two_ things of any kind--what do you think we
     would name it?

  7. A _figure Two_ is written thus: _2._ Make a _figure two_.

  8. Why do we name this a _figure two_?

  9. This figure two (2) is part of the Language of Arithmetic.

 10. In this picture one boy is sitting, playing a flageolet. What
     is the other boy doing? If the boy standing should sit down by the
     other, how many boys would be sitting together? One boy and one
     other boy are how many boys?

 11. You see a flageolet and a violin. They are musical instruments.
     One musical instrument and one other musical instrument are how
     many?

 12. I will write thus: 1 1 2. We say that 1 boy and 1 other boy,
     counted together, are 2 boys; or are equal to 2 boys. We will now
     write something to show that the first 1 and the other 1 are to be
     counted together.

 13. We name a line drawn thus, -, a _horizontal line_. Draw such a
     line. Name it.

 14. A line drawn thus, |, we name a _vertical line_. Draw such a
     line. Name it.

 15. Now I will put two such lines together; thus, +. What kind of a
     line do we name the first (-)? And what do we name the last? (|)?
     Are these lines long or short? Where do they cross each other?

 16. Each of you write thus: -, |, +.

 17. This, +, is named _Plus_. _Plus_ means _more_; and + also means
     _more_.

 18. I will write.

        _One and One More Equal Two._

 19. Now I will write part of this in the Language of Arithmetic.
     I write the first _One_ thus, 1; then the other _One_ thus, 1.
     Afterward I write, for the word _More_, thus, +, placing
     the + between 1 and 1, so that the whole stands thus: 1 + 1.
     As I write, I say, _One and One more_.

 20. Each of you write 1 + 1. Read what you have written.

 21. This +, when written between the 1s, shows that they are to be
     put together, or counted together, so as to make 2.

 22. Because + shows what is to be done, it is called a _Sign_. If
     we take its name, _Plus_, and the word _Sign_, and put both words
     together, we have _Sign Plus_, or _Plus Sign_. In speaking of this
     we may call it _Sign Plus_, or _Plus Sign_, or _Plus_.

 23. 1, 2, +, are part of the Language of Arithmetic.

        _Write the following in the Language of Arithmetic_:

 24. One and one more.

 25. One and two more.

 26. Two and one more.

(12) _Ambiguities and falsities._--Consider the ambiguities and false
reasoning of these problems.

  1. If you can earn 4 cents a day, how much can you earn in 6 weeks?
     (Are Sundays counted? Should a child who earns 4 cents some day
     expect to repeat the feat daily?)

  2. How many lines must you make to draw ten triangles and five
     squares? (I can do this with 8 lines, though the answer the book
     requires is 50.)

  3. A runner ran twice around an 1/8 mile track in two minutes. What
     distance did he run in 2/3 of a minute? (I do not know, but I do
     know that, save by chance, he did not run exactly 2/3 of 1/8 mile.)

  4. John earned $4.35 in a week, and Henry earned $1.93. They put
     their money together and bought a gun. What did it cost? (Maybe $5,
     maybe $10. Did they pay for the whole of it? Did they use all their
     earnings, or less, or more?)

  5. Richard has 12 nickels in his purse. How much more than 50 cents
     would you give him for them? (Would a wise child give 60 cents to a
     boy who wanted to swap 12 nickels therefor, or would he suspect a
     trick and hold on to his own coins?)

  6. If a horse trots 10 miles in one hour how far will he travel in
     9 hours?

  7. If a girl can pick 3 quarts of berries in 1 hour how many quarts
     can she pick in 3 hours?

       (These last two, with a teacher insisting on the 90 and 9, might
       well deprive a matter-of-fact boy of respect for arithmetic for
       weeks thereafter.)

     The economics and physics of the next four problems speak for
     themselves.

  8. I lost $15 by selling a horse for $85. What was the value of the
     horse?

  9. If floating ice has 7 times as much of it under the surface of
     the water as above it, what part is above water? If an iceberg is
     50 ft. above water, what is the entire height of the iceberg? How
     high above water would an iceberg 300 ft. high have to be?

 10. A man's salary is $1000 a year and his expenses $625. How many
     years will elapse before he is worth $10,000 if he is worth $2500
     at the present time?

 11. Sound travels 1120 ft. a second. How long after a cannon is
     fired in New York will the report be heard in Philadelphia, a
     distance of 90 miles?


GUIDING PRINCIPLES

The reader may be wearied of these special details concerning bonds now
neglected that should be formed and useless or harmful bonds formed for
no valid reason. Any one of them by itself is perhaps a minor matter,
but when we have cured all our faults in this respect and found all the
possibilities for wiser selection of bonds, we shall have enormously
improved the teaching of arithmetic. The ideal is such choice of bonds
(and, as will be shown later, such arrangement of them) as will most
improve the functions in question at the least cost of time and effort.
The guiding principles may be kept in mind in the form of seven simple
but golden rules:--

1. Consider the situation the pupil faces.

2. Consider the response you wish to connect with it.

3. Form the bond; do not expect it to come by a miracle.

4. Other things being equal, form no bond that will have to be broken.

5. Other things being equal, do not form two or three bonds when one
will serve.

6. Other things being equal, form bonds in the way that they are
required later to act.

7. Favor, therefore, the situations which life itself will offer, and
the responses which life itself will demand.




CHAPTER V

THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE STRENGTH OF BONDS


An inventory of the bonds to be formed in learning arithmetic should be
accompanied by a statement of how strong each bond is to be made and
kept year by year. Since, however, the inventory itself has been
presented here only in samples, the detailed statement of desired
strength for each bond cannot be made. Only certain general facts will
be noted here.


THE NEED OF STRONGER ELEMENTARY BONDS

The constituent bonds involved in the fundamental operations with
numbers need to be much stronger than they now are. Inaccuracy in these
operations means weakness of the constituent bonds. Inaccuracy exists,
and to a degree that deprives the subject of much of its possible
disciplinary value, makes the pupil's achievements of slight value for
use in business or industry, and prevents the pupil from verifying his
work with new processes by some previously acquired process.

The inaccuracy that exists may be seen in the measurements made by the
many investigators who have used arithmetical tasks as tests of fatigue,
practice, individual differences and the like, and in the special
studies of arithmetical achievements for their own sake made by Courtis
and others.

Burgerstein ['91], using such examples as

    28704516938276546397
  + 35869427359163827263
  ----------------------

and similar long numbers to be multiplied by 2 or by 3 or by 4 or by 5
or by 6, found 851 errors in 28,267 answer-figures, or 3 per hundred
answer-figures, or 3/5 of an error per example. The children were 9-1/2
to 15 years old. Laser ['94], using the same sort of addition and
multiplication, found somewhat over 3 errors per hundred answer-figures
in the case of boys and girls averaging 11-1/2 years, during the period
of their most accurate work. Holmes ['95], using addition of the sort
just described, found 346 errors in 23,713 answer-figures or about 1-1/2
per hundred. The children were from all grades from the third to the
eighth. In Laser's work, 21, 19, 13, and 10 answer-figures were obtained
per minute. Friedrich ['97] with similar examples, giving the very long
time of 20 minutes for obtaining about 200 answer-figures, found from 1
to 2 per hundred wrong. King ['07] had children in grade 5 do sums, each
consisting of 5 two-place numbers. In the most accurate work-period,
they made 1 error per 20 columns. In multiplying a four-place by a
four-place number they had less than one total answer right out of
three. In New York City Courtis found ['11-'12] with his Test 7 that in
12 minutes the average achievement of fourth-grade children is 8.8 units
attempted with 4.2 right. In grade 5 the facts are 10.9 attempts with
5.8 right; in grade 6, 12.5 attempts with 7.0 right; in grade 7, 15
attempts with 8.5 right; in grade 8, 15.7 attempts with 10.1 right.
These results are near enough to those obtained from the country at
large to serve as a text here.

The following were set as official standards, in an excellent school
system, Courtis Series B being used:--


                            SPEED        PERCENT OF
                  GRADE.    ATTEMPTS.    CORRECT ANSWERS.
  Addition          8         12           80
                    7         11           80
                    6         10           70
                    5          9           70
                    4          8           70

  Subtraction       8         12           90
                    7         11           90
                    6         10           90
                    5          9           80
                    4          7           80

  Multiplication    8         11           80
                    7         10           80
                    6          9           80
                    5          7           70
                    4          6           60

  Division          8         11           90
                    7         10           90
                    6          8           80
                    5          6           70
                    4          4           60

Kirby ['13, pp. 16 ff. and 55 ff.] found that, in adding columns like
those printed below, children in grade 4 got on the average less than 80
percent of correct answers. Their average speed was about 2 columns per
minute. In doing division of the sort printed below children of grades 3
_B_ and 4 _A_ got less than 95 percent of correct answers, the average
speed being 4 divisions per minute. In both cases the slower computers
were no more accurate than the faster ones. Practice improved the speed
very rapidly, but the accuracy remained substantially unchanged. Brown
['11 and '12] found a similar low status of ability and notable
improvement from a moderate amount of special practice.

   3    5    6    2    3    8    9    7    4    9
   7    9    6    5    5    6    4    5    8    2
   3    4    7    8    7    3    7    9    3    7
   8    8    4    8    2    6    8    2    9    8
   2    2    4    7    6    9    8    5    6    2
   6    9    5    7    8    5    2    3    2    4
   9    6    4    2    7    2    9    4    4    5
   3    3    7    9    9    9    2    8    9    7
   6    8    9    6    4    7    7    9    2    4
   8    4    6    9    9    2    6    9    8    9
  --   --   --   --   --   --   --   --   --   --

  20 = .... 5s
  56 = .... 9s and .... _r_.
  30 = .... 7s and .... _r_.
  89 = .... 9s and .... _r_.
  20 = .... 8s and .... _r_.
  56 = .... 6s and .... _r_.
  31 = .... 4s and .... _r_.
  86 = .... 9s and .... _r_.

It is clear that numerical work as inaccurate as this has little or no
commercial or industrial value. If clerks got only six answers out of
ten right as in the Courtis tests, one would need to have at least four
clerks make each computation and would even then have to check many of
their discrepancies by the work of still other clerks, if he wanted his
accounts to show less than one error per hundred accounting units of the
Courtis size.

It is also clear that the "habits of ... absolute accuracy, and
satisfaction in truth as a result" which arithmetic is supposed to
further must be largely mythical in pupils who get right answers only
from three to nine times out of ten!


EARLY MASTERY

The bonds in question clearly must be made far stronger than they now
are. They should in fact be strong enough to abolish errors in
computation, except for those due to temporary lapses. It is much
better for a child to know half of the multiplication tables, and to
know that he does not know the rest, than to half-know them all; and
this holds good of all the elementary bonds required for computation.
Any bond should be made to work perfectly, though slowly, very soon
after its formation is begun. Speed can easily be added by proper
practice.

The chief reasons why this is not done now seem to be the following:
(1) Certain important bonds (like the additions with higher decades)
are not given enough attention when they are first used. (2) The special
training necessary when a bond is used in a different connection (as
when the multiplications to 9 × 9 are used in examples like

  729
    8
  ---

where the pupil has also to choose the right number to multiply, keep in
mind what is carried, use it properly, and write the right figure in the
right place, and carry a figure, or remember that he carries none) is
neglected. (3) The pupil is not taught to check his work. (4) He is not
made responsible for substantially accurate results. Furthermore, the
requirement of (4) without the training of (1), (2), and (3) will
involve either a fruitless failure on the part of many pupils, or an
utterly unjust requirement of time. The common error of supposing that
the task of computation with integers consists merely in learning the
additions to 9 + 9, the subtractions to 18 - 9, the multiplications to
8 × 9, and the divisions to 81 ÷ 9, and in applying this knowledge in
connection with the principles of decimal notation, has had a large
share in permitting the gross inaccuracy of arithmetical work. The bonds
involved in 'knowing the tables' do not make up one fourth of the bonds
involved in real adding, subtracting, multiplying, and dividing (with
integers alone).

It should be noted that if the training mentioned in (1) and (2) is
well cared for, the checking of results as recommended in (3) becomes
enormously more valuable than it is under present conditions, though
even now it is one of our soundest practices. If a child knows the
additions to higher decades so that he can add a seen one-place number
to a thought-of two-place number in three seconds or less with a correct
answer 199 times out of 200, there is only an infinitesimal chance that
a ten-figure column twice added (once up, once down) a few minutes apart
with identical answers will be wrong. Suppose that, in long
multiplication, a pupil can multiply to 9 × 9 while keeping his place
and keeping track of what he is 'carrying' and of where to write the
figure he writes, and can add what he carries without losing track of
what he is to add it to, where he is to write the unit figure, what he
is to multiply next and by what, and what he will then have to carry, in
each case to a surety of 99 percent of correct responses. Then two
identical answers got by multiplying one three-place number by another a
few minutes apart, and with reversal of the numbers, will not be wrong
more than twice in his entire school career. Checks approach proofs when
the constituent bonds are strong.

If, on the contrary, the fundamental bonds are so weak that they do not
work accurately, checking becomes much less trustworthy and also very
much more laborious. In fact, it is possible to show that below a
certain point of strength of the fundamental bonds, the time required
for checking is so great that part of it might better be spent in
improving the fundamental bonds.

For example, suppose that a pupil has to find the sum of five numbers
like $2.49, $5.25, $6.50, $7.89, and $3.75. Counting each act of
holding in mind the number to be carried and each writing of a column's
result as equivalent in difficulty to one addition, such a sum equals
nineteen single additions. On this basis and with certain additional
estimates[7] we can compute the practical consequences for a pupil's use
of addition in life according to the mastery of it that he has gained in
school.

  [7] These concern allowances for two errors occurring in the same
      example and for the same wrong answer being obtained in both
      original work and check work.

I have so computed the amount of checking a pupil will have to do to
reach two agreeing numbers (out of two, or three, or four, or five, or
whatever the number before he gets two that are alike), according to his
mastery of the elementary processes. The facts appear in Table 1.

It is obvious that a pupil whose mastery of the elements is that denoted
by getting them right 96 times out of 100 will require so much time for
checking that, even if he were never to use this ability for anything
save a few thousand sums in addition, he would do well to improve this
ability before he tried to do the sums. An ability of 199 out of 200, or
995 out of 1000, seems likely to save much more time than would be taken
to acquire it, and a reasonable defense could be made for requiring 996
or 997 out of 1000.

A precision of from 995 to 997 out of 1000 being required, and ordinary
sagacity being used in the teaching, speed will substantially take care
of itself. Counting on the fingers or in words will not give that
precision. Slow recourse to memory of serial addition tables will not
give that precision. Nothing save sure memory of the facts operating
under the conditions of actual examples will give it. And such memories
will operate with sufficient speed.

TABLE 1

THE EFFECT OF MASTERY OF THE ELEMENTARY FACTS OF ADDITION UPON THE LABOR
REQUIRED TO SECURE TWO AGREEING ANSWERS WHEN ADDING FIVE THREE-FIGURE
NUMBERS

  ======================================================================
  MASTERY OF   |APPROXIMATE  |APPROXIMATE  |APPROXIMATE  |APPROXIMATE
  THE          |NUMBER OF    |NUMBER OF    |NUMBER OF    |NUMBER OF
  ELEMENTARY   |WRONG ANSWERS|AGREEING     |AGREEING     |CHECKINGS
  ADDITIONS    |IN SUMS OF 5 |ANSWERS,     |ANSWERS,     |REQUIRED (OVER
  TIMES RIGHT  |THREE-PLACE  |AFTER ONE    |AFTER A      |AND ABOVE THE
  IN 1000      |NUMBERS PER  |CHECKING,    |CHECKING OF  |FIRST GENERAL
               |1000         |PER 1000     |THE FIRST    |CHECKING OF
               |             |             |DISCREPANCIES|THE 1000 SUMS)
               |             |             |             |TO SECURE TWO
               |             |             |             |AGREEING
               |             |             |             |RESULTS
  -------------+-------------+-------------+-------------+--------------
       960     |     700     |      90     |     216     |     4500
       980     |     380     |     384     |     676     |     1200
       990     |     190     |     656     |     906     |      470
       995     |      95     |     819     |     975     |      210
       996     |      76     |     854     |     984     |      165
       997     |      54     |     895     |     992     |      115
       998     |      38     |     925     |     996     |       80
       999     |      19     |     962     |     999     |       40
  -------------+-------------+-------------+-------------+--------------

There is one intelligent objection to the special practice necessary to
establish arithmetical connections so fully as to give the accuracy
which both utilitarian and disciplinary aims require. It may be said
that the pupils in grades 3, 4, and 5 cannot appreciate the need and
that consequently the work will be dull, barren, and alien, without
close personal appropriation by the pupil's nature. It is true that no
vehement life-purpose is directly involved by the problem of perfecting
one's power to add 7 to 28 in grade 2, or by the problem of multiplying
253 by 8 accurately in grade 3 or by precise subtraction in long
division in grade 4. It is also true, however, that the most humanly
interesting of problems--one that the pupil attacks most
whole-heartedly--will not be solved correctly unless the pupil has the
necessary associative mechanisms in order; and the surer he is of them,
the freer he is to think out the problem as such. Further, computation
is not dull if the pupil can compute. He does not himself object to its
barrenness of vital meaning, so long as the barrenness of failure is
prevented. We must not forget that pupils like to learn. In teaching
excessively dull individuals, who has not often observed the great
interest which they display in anything that they are enabled to master?
There is pathos in their joy in learning to recognize parts of speech,
perform algebraic simplifications, or translate Latin sentences, and in
other accomplishments equally meaningless to all their interests save
the universal human interest in success and recognition. Still further,
it is not very hard to show to pupils the imperative need of accuracy in
scoring games, in the shop, in the store, and in the office. Finally,
the argument that accurate work of this sort is alien to the pupil in
these grades is still stronger against _inaccurate_ work of the same
sort. If we are to teach computation with two- and three- and four-place
numbers at all, it should be taught as a reliable instrument, not as a
combination of vague memories and faith. The author is ready to cut
computation with numbers above 10 out of the curriculum of grades 1-6 as
soon as more valuable educational instruments are offered in its place,
but he is convinced that nothing in child-nature makes a large variety
of inaccurate computing more interesting or educative or germane to felt
needs, than a smaller variety of accurate computing!


THE STRENGTH OF BONDS FOR TEMPORARY SERVICE

The second general fact is that certain bonds are of service for only a
limited time and so need to be formed only to a limited and slight
degree of strength. The data of problems set to illustrate a principle
or improve some habit of computation are, of course, the clearest cases.
The pupil needs to remember that John bought 3 loaves of bread and that
they were 5-cent loaves and that he gave 25 cents to the baker only long
enough to use the data to decide what change John should receive. The
connections between the total described situation and the answer
obtained, supposing some considerable computation to intervene, is a
bond that we let expire almost as soon as it is born.

It is sometimes assumed that the bond between a certain group of
features which make a problem a 'Buy _a_ things at _b_ per thing, find
total cost' problem or a 'Buy _a_ things at _b_ per thing, what change
from _c_' problem or a 'What gain on buying for _a_ and selling for _b_'
problem or a 'How many things at _a_ each can I buy for _b_ cents'
problem--it is assumed that the bond between these essential defining
features and the operation or operations required for solution is as
temporary as the bonds with the name of the buyer or the price of the
thing. It is assumed that all problems are and should be solved by some
pure act of reasoning without help or hindrance from bonds with the
particular verbal structure and vocabulary of the problems. Whether or
not they _should_ be, they _are not_. Every time that a pupil solves a
'bought-sold' problem by subtraction he strengthens the tendency to
respond to any problem whatsoever that contains the words 'bought for'
and 'sold for' by subtraction; and he will by no means surely stop and
survey every such problem in all its elements to make sure that no
other feature makes inapplicable the tendency to subtract which the
'bought sold' evokes.

To prevent pupils from responding to the form of statement rather than
the essential facts, we should then not teach them to forget the form of
statement, but rather give them all the common forms of statement to
which the response in question is an appropriate response, and only
such. If a certain form of statement does in life always signify a
certain arithmetical procedure, the bond between it and that procedure
may properly be made very strong.

Another case of the formation of bonds to only a slight degree
of strength concerns the use of so-called 'crutches' such as
writing +, -, and × in copying problems like those below:--

  Add       Subtract       Multiply
  23           79             32
  61           24              3
  --           --             --

or altering the figures when 'borrowing' in subtraction, and the like.
Since it is undesirable that the pupil should regard the 'crutch'
response as essential to the total procedure, or become so used to
having it that he will be disturbed by its absence later, it is supposed
that the bond between the situation and the crutch should not be fully
formed. There is a better way out of the difficulty, in case crutches
are used at all. This is to associate the crutch with a special 'set,'
and its non-use with the general set which is to be the permanent one.
For example, children may be taught from the start never to write
the crutch sign or crutch figure unless the work is accompanied by
"Write ... to help you to...."

  Write - to help you to        Find the differences:--
  remember that you must        39     67     78     56     45
  subtract in this row.         23     44     36     26     24
                                --     --     --     --     --

  Remember that you must        Find the differences:--
  subtract in this row.         85     27     96     38     78
                                63     14     51     45     32
                                --      --    --     --     --

The bond evoking the use of the crutch may then be formed thoroughly
enough so that there is no hesitation, insecurity, or error, without
interfering to any harmful extent with the more general bond from the
situation to work without the crutch.


THE STRENGTH OF BONDS WITH TECHNICAL FACTS AND TERMS

Another instructive case concerns the bonds between certain words and
their meanings, and between certain situations of commerce, industry, or
agriculture and useful facts about these situations. Illustrations of
the former are the bonds between _cube root_, _hectare_, _brokerage_,
_commission_, _indorsement_, _vertex_, _adjacent_, _nonagon_, _sector_,
_draft_, _bill of exchange_, and their meanings. Illustrations of the
latter are the bonds from "Money being lent 'with interest' at no
specified rate, what rate is charged?" to "The legal rate of the state,"
from "$_X_ per M as a rate for lumber" to "Means $_X_ per thousand board
feet, a board foot being 1 ft. by 1 ft. by 1 in."

It is argued by many that such bonds are valuable for a short time;
namely, while arithmetical procedures in connection with which they
serve are learned, but that their value is only to serve as a means for
learning these procedures and that thereafter they may be forgotten.
"They are formed only as accessory means to certain more purely
arithmetical knowledge or discipline; after this is acquired they may
be forgotten. Everybody does in fact forget them, relearning them later
if life requires." So runs the argument.

In some cases learning such words and facts only to use them in solving
a certain sort of problems and then forget them may be profitable. The
practice is, however, exceedingly risky. It is true that everybody does
in fact forget many such meanings and facts, but this commonly means
either that they should not have been learned at all at the time that
they were learned, or that they should have been learned more
permanently, or that details should have been learned with the
expectation that they themselves would be forgotten but that a general
fact or attitude would remain. For example, duodecagon should not be
learned at all in the elementary school; indorsement should either not
be learned at all there, or be learned for permanence of a year or more;
the details of the metric system should be so taught as to leave for
several years at least knowledge of the facts that there is a system so
named that is important, whose tables go by tens, hundreds, or
thousands, and a tendency (not necessarily strong) to connect meter,
kilogram, and liter with measurement by the metric system and with
approximate estimates of their several magnitudes.

If an arithmetical procedure seems to require accessory bonds which are
to be forgotten, once the procedure is mastered, we should be suspicious
of the value of the procedure itself. If pupils forget what compound
interest is, we may be sure that they will usually also have forgotten
how to compute it. Surely there is waste if they have learned what it is
only to learn how to compute it only to forget how to compute it!


THE STRENGTH OF BONDS CONCERNING THE REASONS FOR ARITHMETICAL PROCESSES

The next case of the formation of bonds to slight strength is the
problematic one of forming the bonds involved in understanding the
reasons for certain processes only to forget them after the process has
become a habit. Should a pupil, that is, learn why he inverts and
multiplies, only to forget it as soon as he can be trusted to divide by
a fraction? Should he learn why he puts the units figure of each partial
product in multiplication under the figure that he multiplies by, only
to forget the reason as soon as he has command of the process? Should he
learn why he gets the number of square inches in a rectangle by
multiplying the length by the width, both being expressed in linear
inches, and forget why as soon as he is competent to make computations
of the areas of rectangles?

On general psychological grounds we should be suspicious of forming
bonds only to let them die of starvation later, and tend to expect that
elaborate explanations learned only to be forgotten either should not be
learned at all, or should be learned at such a time and in such a way
that they would not be forgotten. Especially we should expect that the
general principles of arithmetic, the whys and wherefores of its
fundamental ways of manipulating numbers, ought to be the last bonds of
all to be forgotten. Details of _how_ you arranged numbers to multiply
might vanish, but the general reasons for the placing would be expected
to persist and enable one to invent the detailed manipulations that had
been forgotten.

This suspicion is, I think, justified by facts. The doctrine that the
customary deductive explanations of why we invert and multiply, or place
the partial products as we do before adding, may be allowed to be
forgotten once the actual habits are in working order, has a suspicious
source. It arose to meet the criticism that so much time and effort were
required to keep these deductive explanations in memory. The fact was
that the pupil learned to compute correctly _irrespective of_ the
deductive explanations. They were only an added burden. His inductive
learning that the procedure gave the right answer really taught him. So
he wisely shuffled off the extra burden of facts about the consequences
of the nature of a fraction or the place values of our decimal notation.
The bonds weakened because they were not used. They were not used
because they were not useful in the shape and at the time that they were
formed, or because the pupil was unable to understand the explanations
so as to form them at all.

The criticism was valid and should have been met in part by replacing
the deductive explanations by inductive verifications, and in part by
using the deductive reasoning as a check after the process itself is
mastered. The very same discussions of place-value which are futile as
proof that you must do a certain thing before you have done it, often
become instructive as an explanation of why the thing that you have
learned to do and are familiar with and have verified by other tests
works as well as it does. The general deductive theory of arithmetic
should not be learned only to be forgotten. Much of it should, by most
pupils, not be learned at all. What is learned should be learned much
later than now, as a synthesis and rationale of habits, not as their
creator. What is learned of such deductive theory should rank among the
most rather than least permanent of a pupil's stock of arithmetical
knowledge and power. There are bonds which are formed only to be lost,
and bonds formed only to be lost _in their first form_, being used in a
new organization as material for bonds of a higher order; but the bonds
involved in deductive explanations of why certain processes are right
are not such: they are not to be formed just to be forgotten, nor as
mere propædeutics to routine manipulations.


PROPÆDEUTIC BONDS

The formation of bonds to a limited strength because they are to be lost
in their first form, being worked over in different ways in other bonds
to which they are propædeutic or contributing is the most important case
of low strength, or rather low permanence, in bonds.

The bond between four 5s in a column to be added and the response of
thinking '10, 15, 20' is worth forming, but it is displaced later by the
multiplication bond or direct connection of 'four 5s to be added' with
'20.' Counting by 2s from 2, 3s from 3, 4s from 4, 5s from 5, etc.,
forms serial bonds which as series might well be left to disappear.
Their separate steps are kept as permanent bonds for use in column
addition, but their serial nature is changed from 2 (and 2) 4, (and 2)
6, (and 2) 8, etc., to two 2s = 4, three 2s = 6, four 2s = 8, etc.;
after playing their part in producing the bonds whereby any multiple of
2 by 2 to 9, can be got, the original serial bonds are, as series,
needed no longer. The verbal response of saying 'and' in adding, after
helping to establish the bonds whereby the general set of the mind
toward adding coöperates with the numbers seen or thought of to produce
their sum, should disappear; or remain so slurred in inner speech as to
offer no bar to speed.

The rule for such bonds is, of course, to form them strongly enough so
that they work quickly and accurately for the time being and facilitate
the bonds that are to replace them, but not to overlearn them. There is
a difference between learning something to be held for a short time, and
the same amount of energy spent in learning for long retention. The
former sort of learning is, of course, appropriate with many of these
propædeutic bonds.

The bonds mentioned as illustrations are not _purely_ propædeutic, nor
formed _only_ to be transmuted into something else. Even the saying of
'and' in addition has some genuine, intrinsic value in distinguishing
the process of addition, and may perhaps be usefully reviewed for a
brief space during the first steps in adding common fractions. Some such
propædeutic bonds may be worth while apart from their value in preparing
for other bonds. Consider, for example, exercises like those shown below
which are propædeutic to long division, giving the pupil some basis in
experience for his selection of the quotient figures. These
multiplications are intrinsically worth doing, especially the 12s and
25s. Whatever the pupil remembers of them will be to his advantage.

  1. Count by 11s to 132, beginning 11, 22, 33.

  2. Count by 12s to 144, beginning 12, 24, 36.

  3. Count by 25s to 300, beginning 25, 50, 75.

  4. State the missing numbers:--

          A.          B.             C.                D.
       3 11s =     5 11s =     8 ft. = .... in.     2 dozen =
       4 12s =     3 12s =    10 ft. = .... in.     4 dozen =
       5 12s =     6 12s =     7 ft. = .... in.    10 dozen =
       6 11s =    12 11s =     4 ft. = .... in.     5 dozen =
       9 11s =     2 12s =     6 ft. = .... in.     7 dozen =
       7 12s =     9 12s =     9 ft. = .... in.    12 dozen =
       8 12s =     7 11s =    11 ft. = .... in.     9 dozen =
      11 11s =    12 12s =     5 ft. = .... in.     6 dozen =

  5. Count by 25s to $2.50, saying, "25 cents, 50 cents, 75 cents,
     one dollar," and so on.

  6. Count by 15s to $1.50.

  7. Find the products. Do not use pencil. Think what they are.

         A.        B.         C.          D.          E.
       2 × 25    3 × 15     2 × 12      4 × 11      6 × 25
       3 × 25   10 × 15     2 × 15      4 × 15      6 × 15
       5 × 25    4 × 15     2 × 25      4 × 12      6 × 12
      10 × 25    2 × 15     2 × 11      4 × 25      6 × 11
       4 × 25    7 × 15     3 × 25      5 × 11      7 × 12
       6 × 25    9 × 15     3 × 15      5 × 12      7 × 15
       8 × 25    5 × 15     3 × 11      5 × 15      7 × 25
       7 × 25    8 × 15     3 × 12      5 × 25      7 × 11
       9 × 25    6 × 15     8 × 12      9 × 12      8 × 25

  State the missing numbers:--

    A.  36 = .... 12s    B.  44 = .... 11s    C.  50 = .... 25s
        60 = .... 12s        88 = .... 11s       125 = .... 25s
        24 = .... 12s        77 = .... 11s        75 = .... 25s
        48 = .... 12s        55 = .... 11s       200 = .... 25s
       144 = .... 12s        99 = .... 11s       250 = .... 25s
       108 = .... 12s       110 = .... 11s       175 = .... 25s
        72 = .... 12s        33 = .... 11s       225 = .... 25s
        96 = .... 12s        66 = .... 11s       150 = .... 25s
        84 = .... 12s        22 = .... 11s       100 = .... 25s

  Find the quotients and remainders. If you need to use paper and pencil
  to find them, you may. But find as many as you can without pencil and
  paper. Do Row A first. Then do Row B. Then Row C, etc.
             __      __      __      __      __      __
  Row A.  11|45   12|45   25|45   15|45   21|45   22|45
             __      __      __      __      __      __
  Row B.  25|55   11|55   12|55   15|55   22|55   30|55
             __      __      __      __      __      __
  Row C.  12|60   25|60   15|60   11|60   30|60   21|60
             __      __      __      __      __      __
  Row D.  12|75   11|75   15|75   25|75   30|75   35|75
             ___     ___     ___     ___     ___     ___
  Row E.  11|100  12|100  25|100  15|100  30|100  22|100
             __      __      __      __      __      __
  Row F.  11|96   12|96   25|96   15|96   30|96   22|96
             ___     ___     ___     ___     ___     ___
  Row G.  25|105  11|105  15|105  12|105  22|105  35|105
             __      __      __      __      __      __
  Row H.  12|64   15|64   25|64   11|64   22|64   21|64
             __      __      __      __      __      __
  Row I.  11|80   12|80   15|80   25|80   35|80   21|80
             ___     ___     ___     ___     ___     ___
  Row J.  25|200  30|200  75|200  63|200  65|200  66|200

  Do this section again. Do all the first column first. Then do
  the second column, then the third, and so on.

Consider, from the same point of view, exercises like (3 × 4) + 2,
(7 × 6) + 5, (9 × 4) + 6, given as a preparation for written
multiplication. The work of

  48   68   47
   3    7    9
  --   --   --

and the like is facilitated if the pupil has easy control of the process
of getting a product, and keeping it in mind while he adds a one-place
number to it. The practice with (3 × 4) + 2 and the like is also good
practice intrinsically. So some teachers provide systematic preparatory
drills of this type just before or along with the beginning of short
multiplication.

In some cases the bonds are purely propædeutic or are formed _only_ for
later reconstruction. They then differ little from 'crutches.' The
typical crutch forms a habit which has actually to be broken, whereas
the purely propædeutic bond forms a habit which is left to rust out from
disuse.

For example, as an introduction to long division, a pupil may be given
exercises using one-figure divisors in the long form, as:--

     773 and 5 remainder
   ______
  7)5416
    49
    --
     51
     49
     --
      26
      21
      --
       5

The important recommendation concerning these purely propædeutic bonds,
and bonds formed only for later reconstruction, is to be very critical
of them, and not indulge in them when, by the exercise of enough
ingenuity, some bond worthy of a permanent place in the individual's
equipment can be devised which will do the work as well. Arithmetical
teaching has done very well in this respect, tending to err by leaving
out really valuable preparatory drills rather than by inserting
uneconomical ones. It is in the teaching of reading that we find the
formation of propædeutic bonds of dubious value (with letters,
phonograms, diacritical marks, and the like) often carried to
demonstrably wasteful extremes.




CHAPTER VI

THE PSYCHOLOGY OF DRILL IN ARITHMETIC: THE AMOUNT OF PRACTICE AND THE
ORGANIZATION OF ABILITIES


THE AMOUNT OF PRACTICE

It will be instructive if the reader will perform the following
experiment as an introduction to the discussion of this chapter, before
reading any of the discussion.

Suppose that a pupil does all the work, oral and written, computation
and problem-solving, presented for grades 1 to 6 inclusive (that is, in
the first two books of a three-book series) in the average textbook now
used in the elementary school. How many times will he have exercised
each of the various bonds involved in the four operations with integers
shown below? That is, how many times will he have thought, "1 and 1
are 2," "1 and 2 are 3," etc.? Every case of the action of each bond
is to be counted.


THE FUNDAMENTAL BONDS

  1 + 1                             2 - 1        1 × 1      2 ÷ 1
  1 + 2                             2 - 2        2 × 1      2 ÷ 2
  1 + 3                                          3 × 1
  1 + 4                                          4 × 1
  1 + 5                             3 - 1        5 × 1      3 ÷ 1
  1 + 6                             3 - 2        6 × 1      3 ÷ 2
  1 + 7                             3 - 3        7 × 1      3 ÷ 3
  1 + 8                                          8 × 1
  1 + 9                                          9 × 1

                                    4 - 1                   4 ÷ 1
                                    4 - 2                   4 ÷ 2
  11 (or 21 or 31, etc.) + 1        4 - 3        1 × 2      4 ÷ 3
  11        "            + 2        4 - 4        2 × 2      4 ÷ 4
  11        "            + 3                     3 × 2
  11        "            + 4                     4 × 2
  11        "            + 5        5 - 1        5 × 2      5 ÷ 1
  11        "            + 6        5 - 2        6 × 2      5 ÷ 2
  11        "            + 7        5 - 3        7 × 2      5 ÷ 3
  11        "            + 8        5 - 4        8 × 2      5 ÷ 4
  11        "            + 9        5 - 5        9 × 2      5 ÷ 5

                                    6 - 1        1 × 3      6 ÷ 1
  2 + 1                             6 - 2        2 × 3      6 ÷ 2
  2 + 2                             6 - 3        3 × 3      6 ÷ 3
  2 + 3                             6 - 4        4 × 3      6 ÷ 4
  2 + 4                             6 - 5        5 × 3      6 ÷ 5
  2 + 5                             6 - 6        6 × 3      6 ÷ 6
  2 + 6                                          7 × 3
  2 + 7                                          8 × 3
  2 + 8                             7 - 1        9 × 3      7 ÷ 1
  2 + 9                             7 - 2                   7 ÷ 2
                                    7 - 3                   7 ÷ 3
                                    7 - 4        1 × 4      7 ÷ 4
  12 (or 22 or 32, etc.) + 1        7 - 5        2 × 4      7 ÷ 5
  12        "            + 2        7 - 6        and so on  7 ÷ 6
                                    7 - 7        to 9 × 9   7 ÷ 7
  and so on to                      and so on               and so on to
  9 + 9                             to 18 - 9               82 ÷ 9
  19 (or 29 or 39, etc.) + 9                                83 ÷ 9, etc.

If estimating for the entire series is too long a task, it will be
sufficient to use eight or ten from each, say:--

   3 + 2    13, 23, etc. + 2     7 + 2    17, 27, etc. + 2
     " 3           "       3       " 3            "      3
     " 4           "       4       " 4            "      4
     " 5           "       5       " 5            "      5
     " 6           "       6       " 6            "      6
     " 7           "       7       " 7            "      7
     " 8           "       8       " 8            "      8
     " 9           "       9       " 9            "      9

   3 - 3     7 - 7     9 × 7    63 ÷ 9
   4   "     8   "     7 × 9    64   "
   5   "     9   "     8 × 6    65   "
   6   "    10   "     6 × 8    66   "
   7   "    11   "              67   "
   8   "    12   "              68   "
   9   "    13   "              69   "
  10   "    14   "              70   "
  11   "    15   "              71   "
  12   "    16   "


TABLE 2

ESTIMATES OF THE AMOUNT OF PRACTICE PROVIDED IN BOOKS I AND II OF THE
AVERAGE THREE-BOOK TEXT IN ARITHMETIC; BY 50 EXPERIENCED TEACHERS

  ======================================================================
                         | LOWEST | MEDIAN | HIGHEST |RANGE REQUIRED TO
    ARITHMETICAL FACT    |ESTIMATE|ESTIMATE|ESTIMATE | INCLUDE HALF OF
                         |        |        |         |  THE ESTIMATES
  -----------------------+--------+--------+---------+------------------
  3 or 13 or 23, etc. + 2|   25   |  1500  |1,000,000|    800-5000
       "         "      3|   24   |  1450  |   80,000|    475-5000
       "         "      4|   23   |  1150  |   50,000|    750-5000
       "         "      5|   22   |  1400  |   44,000|    700-5000
       "         "      6|   21   |  1350  |   41,000|    700-4500
       "         "      7|   21   |  1500  |   37,000|    600-4000
       "         "      8|   20   |  1400  |   33,000|    550-4100
       "         "      9|   20   |  1150  |   28,000|    650-4500
                         |        |        |         |
  7 or 17 or 27, etc. + 2|   20   |  1250  |2,000,000|    600-5000
       "         "      3|   19   |  1100  |1,000,000|    650-4900
       "         "      4|   18   |  1000  |   80,000|    650-4900
       "         "      5|   17   |  1300  |   80,000|    650-4400
       "         "      6|   16   |  1100  |   29,000|    650-4500
       "         "      7|   15   |  1100  |   25,000|    500-4500
       "         "      8|   13   |  1100  |   21,000|    650-3800
       "         "      9|   10   |  1275  |   17,000|    500-4000
                         |        |        |         |
          3 - 3          |   25   |  1000  |  100,000|    500-4000
          4 - 3          |   20   |  1050  |  500,000|    525-3000
          5 - 3          |   20   |  1100  |2,500,000|    650-4200
          6 - 3          |   10   |  1050  |   21,000|    650-3250
          7 - 3          |   22   |  1100  |   15,000|    550-3050
          8 - 3          |   21   |  1075  |   15,000|    650-3000
          9 - 3          |   21   |  1000  |   15,000|    700-2600
         10 - 3          |   20   |  1000  |   20,000|    600-2500
         11 - 3          |   20   |  1000  |   15,000|    465-2550
         12 - 3          |   18   |  1000  |   15,000|    650-2100
                         |        |        |         |
          7 - 7          |   10   |  1000  |   18,000|    425-3000
          8 - 7          |   15   |  1000  |   18,000|    413-3100
          9 - 7          |   15   |   950  |   18,000|    550-3000
         10 - 7          |   15   |   950  |   18,000|    600-3950
         11 - 7          |   10   |   900  |   18,000|    550-3000
         12 - 7          |   10   |   925  |   18,000|    525-3100
         13 - 7          |   10   |   900  |   18,000|    500-2600
         14 - 7          |   10   |   900  |   18,000|    500-3100
         15 - 7          |   10   |   925  |   18,000|    500-3000
         16 - 7          |   10   |   875  |   18,000|    500-2500
                         |        |        |         |
          9 × 7          |   10   |   700  |   20,000|    500-2000
          7 × 9          |   10   |   700  |   20,000|    500-1750
          8 × 6          |   10   |   750  |   20,000|    500-2500
          6 × 8          |    9   |   700  |   20,000|    500-2500
                         |        |        |         |
         63 ÷ 9          |    9   |   500  |    4,500|    300-2500
         64 ÷ 9          |    9   |   200  |    4,000|    100- 700
         65 ÷ 9          |    8   |   200  |    4,000|    100- 600
         66 ÷ 9          |    7   |   200  |    4,000|    100- 550
         67 ÷ 9          |    7   |   200  |    4,000|     75- 450
         68 ÷ 9          |    6   |   200  |    4,000|     87- 575
         69 ÷ 9          |    6   |   200  |    4,000|     87- 450
         70 ÷ 9          |    5   |   200  |    4,000|     75- 575
         71 ÷ 9          |    5   |   200  |    4,000|     75- 700
                         |        |        |         |
           _XX_          |   40   |   550  |1,000,000|    300-2000
           _XO_          |   20   |   500  |   11,500|    150-2000
          _XXX_          |   15   |   450  |   12,000|    100-1000
          _XXO_          |   25   |   400  |   15,000|    150-1000
          _XOO_          |   15   |   400  |    5,000|    100-1000
          _XOX_          |   10   |   400  |   10,000|    100- 975
  ======================================================================

Having made his estimates the reader should compare them first with
similar estimates made by experienced teachers (shown on page 124 f.),
and then with the results of actual counts for representative textbooks
in arithmetic (shown on pages 126 to 132).

It will be observed in Table 2 that even experienced teachers vary
enormously in their estimates of the amount of practice given by an
average textbook in arithmetic, and that most of them are in serious
error by overestimating the amount of practice. In general it is the
fact that we use textbooks in arithmetic with very vague and erroneous
ideas of what is in them, and think they give much more practice than
they do.

The authors of the textbooks as a rule also probably had only very vague
and erroneous ideas of what was in them. If they had known, they would
almost certainly have revised their books. Surely no author would
intentionally provide nearly four times as much practice on 2 + 2 as on
8 + 8, or eight times as much practice on 2 × 2 as on 9 × 8, or eleven
times as much practice on 2 - 2 as on 17 - 8, or over forty times as
much practice on 2 ÷ 2 as on 75 ÷ 8 and 75 ÷ 9, both together. Surely
no author would have provided intentionally only twenty to thirty
occurrences each of 16 - 7, 16 - 8, 16 - 9, 17 - 8, 17 - 9, and 18 - 9
for the entire course through grade 6; or have left the practice on
60 ÷ 7, 60 ÷ 8, 60 ÷ 9, 61 ÷ 7, 61 ÷ 8, 61 ÷ 9, and the like to occur
only about once a year!


TABLE 3

AMOUNT OF PRACTICE: ADDITION BONDS IN A RECENT TEXTBOOK (A) OF EXCELLENT
REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF SUPPLEMENTARY
MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION

The Table reads: 2 + 2 was used 226 times, 12 + 2 was used 74 times,
22 + 2, 32 + 2, 42 + 2, and so on were used 50 times.

  ======================================================================
                  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9 | TOTAL
  ----------------+-----+-----+-----+-----+-----+-----+-----+----+------
    2             | 226 | 154 | 162 | 150 |  97 |  87 |  66 |  45|
   12             |  74 |  53 |  76 |  46 |  51 |  37 |  36 |  33|
   22, etc.       |  50 |  60 |  68 |  63 |  42 |  50 |  38 |  26|
                  |     |     |     |     |     |     |     |    |
    3             | 216 | 141 | 127 |  89 |  82 |  54 |  58 |  40|
   13             |  43 |  43 |  60 |  70 |  52 |  30 |  22 |  18|
   23, etc.       |  15 |  30 |  51 |  50 |  42 |  32 |  29 |  30|
                  |     |     |     |     |     |     |     |    |
    7             |  85 |  90 | 103 | 103 |  84 |  81 |  61 |  47|
   17             |  35 |  25 |  42 |  32 |  35 |  21 |  29 |  16|
   27, etc.       |  30 |  23 |  32 |  29 |  24 |  23 |  25 |  28|
                  |     |     |     |     |     |     |     |    |
    8             | 185 | 112 | 146 |  99 |  75 |  71 |  73 |  61|
   18             |  28 |  35 |  52 |  46 |  28 |  29 |  24 |  14|
   28, etc.       |  53 |  36 |  34 |  38 |  23 |  36 |  27 |  27|
                  |     |     |     |     |     |     |     |    |
    9             | 104 |  81 | 112 |  96 |  63 |  74 |  58 |  57|
   19             |  13 |  11 |  31 |  38 |  25 |  14 |  22 |  11|
   29, etc.       |  19 |  17 |  27 |  20 |  32 |  32 |  19 |  18|
                  |     |     |     |     |     |     |     |    |
  2, 12, 22, etc. | 350 | 277 | 306 | 260 | 190 | 174 | 140 | 104| 1801
  3, 13, 23, etc. | 274 | 214 | 230 | 209 | 176 | 116 | 109 |  88| 1406
                  |     |     |     |     |     |     |     |    |
  7, 17, 27, etc. | 148 | 138 | 187 | 164 | 141 | 125 | 115 |  91| 1109
  8, 18, 28, etc. | 266 | 183 | 232 | 185 | 126 | 136 | 124 | 102| 1354
  9, 19, 29, etc. | 136 | 109 | 170 | 154 | 120 | 120 |  99 |  86|  994
                  |     |     |     |     |     |     |     |    |
  Totals          |1164 | 921 |1125 | 972 | 753 | 671 | 687 | 471|
  ======================================================================


TABLE  4

AMOUNT OF PRACTICE: SUBTRACTION BONDS IN A RECENT TEXTBOOK (A)
OF EXCELLENT REPUTE. BOOKS I AND II, ALL SAVE FOUR SECTIONS OF
SUPPLEMENTARY MATERIAL, TO BE USED AT THE TEACHER'S DISCRETION

  ================================================================
            |                     SUBTRAHENDS
   MINUENDS |-----------------------------------------------------
            |  1  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9
  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----
    1       | 372 |     |     |     |     |     |     |     |
    2       | 214 | 311 |     |     |     |     |     |     |
    3       | 136 | 149 | 189 |     |     |     |     |     |
    4       | 146 | 142 | 103 | 205 |     |     |     |     |
    5       | 171 |  91 |  92 | 164 | 136 |     |     |     |
    6       |  80 |  59 |  69 |  71 |  81 | 192 |     |     |
    7       | 106 |  57 |  55 |  67 |  59 | 156 |  80 |     |
    8       |  73 |  50 |  50 |  75 |  50 |  62 |  48 | 152 |
    9       |  71 |  75 |  54 |  74 |  48 |  55 |  55 | 124 | 133
   10       | 261 |  84 |  63 | 100 | 193 |  83 |  57 | 124 |  91
            |     |     |     |     |     |     |     |     |
   11       |     |  48 |  31 |  50 |  36 |  41 |  32 |  46 |  35
   12       |     |     |  48 |  77 |  57 |  51 |  35 |  80 |  30
   13       |     |     |     |  35 |  22 |  40 |  29 |  35 |  28
   14       |     |     |     |     |  25 |  37 |  36 |  49 |  32
   15       |     |     |     |     |     |  33 |  19 |  48 |  20
            |     |     |     |     |     |     |     |     |
   16       |     |     |     |     |     |     |  16 |  36 |  26
   17       |     |     |     |     |     |     |     |  27 |  20
   18       |     |     |     |     |     |     |     |     |  19
            |     |     |     |     |     |     |     |     |
  Total     |     |     |     |     |     |     |     |     |
  excluding |     |     |     |     |     |     |     |     |
  1-1, 2-2, |     |     |     |     |     |     |     |     |
  etc.      |1258 | 755 | 565 | 713 | 571 | 558 | 327 | 569 | 301
  ================================================================


TABLE 5

FREQUENCIES OF SUBTRACTIONS NOT INCLUDED IN TABLE 4

These are cases where the pupil would, by reason of his stage of
advancement, probably operate 35-30, 46-46, etc., as one bond.

  ======================================================================
                      |                    SUBTRAHENDS
                      |----+----+----+----+----+----+----+----+----+----
                      |   1|  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |
        MINUENDS      |  11| 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |  10
                      |  21| 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 |  20
                      |etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.|etc.
  --------------------+----+----+----+----+----+----+----+----+----+----
  10, 20, 30, 40, etc.| 11 | 29 | 16 | 52 | 32 | 51 |  7 | 30 | 22 | 60
  11, 21, 31, 41, etc.| 42 | 14 | 22 | 32 | 12 | 26 | 19 | 52 | 17 | 10
  12, 22, 32, 42, etc.| 47 | 97 |  5 | 13 |  9 | 21 | 11 | 24 | 19 | 17
  13, 23, 33, 43, etc.|  7 | 40 |  7 | 14 | 15 | 13 | 19 | 19 | 22 |  3
  14, 24, 34, 44, etc.|  8 | 28 | 14 | 58 | 13 | 16 | 14 | 26 | 19 |  7
  15, 25, 35, 45, etc.| 21 | 28 | 29 | 54 | 51 | 15 | 21 | 12 | 24 |  8
  16, 26, 36, 46, etc.|  5 | 18 | 12 | 27 | 35 | 69 | 13 | 17 | 19 |  2
  17, 27, 37, 47, etc.|  5 |  9 | 12 | 40 | 32 | 54 | 24 | 12 | 12 |  1
  18, 28, 38, 48, etc.|  2 | 16 | 10 | 23 | 22 | 36 | 18 | 47 | 16 |  0
  19, 29, 39, etc.    |  5 |  7 |  7 | 10 | 13 | 28 | 14 | 23 | 16 |  0
                      |    |    |    |    |    |    |    |    |    |
  Totals              |153 |286 |134 |323 |234 |329 |160 |262 |186 |108
  =====================================================================


TABLE 6

AMOUNT OF PRACTICE: MULTIPLICATION BONDS IN ANOTHER RECENT TEXTBOOK (B)
OF EXCELLENT REPUTE. BOOKS I AND II

  ======================================================================
              |                     MULTIPLICANDS
  MULTIPLIERS |---------------------------------------------------------
              |  0 |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |Totals
  ------------+----+----+----+----+----+----+----+----+----+----+-------
       1      | 299| 534| 472| 271| 310| 293| 261| 178| 195|  99| 2912
       2      | 350| 644| 668| 480| 458| 377| 332| 238| 239| 155| 3941
       3      | 280| 487| 509| 388| 318| 302| 247| 199| 227| 152| 3109
       4      | 186| 375| 398| 242| 203| 265| 197| 163| 159|  93| 2281
       5      | 268| 359| 393| 234| 263| 243| 217| 192| 197| 114| 2480
       6      | 180| 284| 265| 199| 196| 191| 168| 169| 165| 106| 1923
       7      | 135| 283| 277| 176| 187| 158| 155| 121| 145| 118| 1755
       8      | 137| 272| 292| 175| 192| 164| 158| 157| 126| 126| 1799
       9      |  71| 173| 140| 122|  97| 102| 101| 100|  82| 110| 1098
              |    |    |    |    |    |    |    |    |    |    |
  Totals      |1906|3411|3414|2287|2224|2095|1836|1517|1535|1073|
  ======================================================================


TABLE  7

AMOUNT OF PRACTICE: DIVISIONS WITHOUT REMAINDER IN TEXTBOOK B,
PARTS I AND II

  ======================================================================
                         |                   DIVISORS
        DIVIDENDS        |----------------------------------------------
                         |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 |Totals
  -----------------------+----+----+----+----+----+----+----+----+------
  Integral               | 397| 224| 250| 130|  93|  44|  98|  23| 1259
  multiples              |    |    |    |    |    |    |    |    |
  of 2 to 9              | 256| 124| 152|  79|  28|  43|  61|  25|  768
  in sequence;           |    |    |    |    |    |    |    |    |
  _i.e._, 4 ÷ 2          | 318| 123| 130|  65|  50|  19|  39|  19|  763
  occurred               |    |    |    |    |    |    |    |    |
  397 times,             | 258|  98|  86| 105|  25|  24|  34|  20|  650
  6 ÷ 2 occurred         |    |    |    |    |    |    |    |    |
  256 times,             | 198|  49|  76|  27|  22|  30|  33|  16|  451
  6 ÷ 3, 224 times,      |    |    |    |    |    |    |    |    |
  9 ÷ 3, 124 times.      |  77|  54|  36|  31|  28|  27|  16|   9|  278
                         | 180|  91|  50|  38|  17|  13|  22|  16|  427
                         |  69|  46|  37|  24|  12|  17|  16|  15|  236
                         |    |    |    |    |    |    |    |    |
  Totals                 |1753| 809| 817| 499| 275| 217| 319| 142|
  ======================================================================


TABLE 8

DIVISION BONDS, WITH AND WITHOUT REMAINDERS. BOOK B

All work through grade 6, except estimates of quotient figures in long
division.

  Dividend       2       3           4               5
  Divisor        1   2   1   2   3   1   2   3   4   1   2   3   4   5
  Number of
  Occurrences   41 386  27 189 240  26 397  66 185  23 136  43  53 135

  Dividend       6                       7
  Divisor        1   2   3   4   5   6   1   2   3   4   5   6   7
  Number of
  Occurrences   21 256 224  68  43  83  23  72  55  38  46  32  54

  Dividend       8                            9
  Divisor        1   2   3   4   5  6  7  8   1  2   3  4  5  6  7  8  9
  Number of
  Occurrences   17 318  30 250  22 28 39 91  19 50 124 49 25 15 18 30 38

  Dividend      10                         11
  Divisor        2  3  4   5  6 7  8  9     2  3  4  5  6  7  8  9
  Number of
  Occurrences  258 38 46 120 19 9 24 24    32 21 16  3  7 11 14  3

  Dividend      12                         13
  Divisor        2   3   4  5  6 7  8 9     2  3  4  5  6  7  8  9
  Number of
  Occurrences  198 123 152 29 93 9 16 7    45 16 15 11  7  4  5  3

  Dividend      14                         15
  Divisor        2  3  4  5  6  7  8  9     2  3  4  5  6  7  8  9
  Number of
  Occurrences   77 20 13  5  8 44  8  6    69 98 16 79  8  8  4  6

  Dividend      16                         17
  Divisor        2  3   4  5 6  7  8  9     2  3  4  5  6  7  8  9
  Number of
  Occurrences  180 19 130 14 6  9 98  3    61  9 15 14  6  6 12  3

  Dividend      18                         19
  Divisor        2  3  4  5  6  7  8  9     2  3  4  5  6  7  8  9
  Number of
  Occurrences   69 49 13  6 28  7  7 23    21  6 10  5  3  4 10  4

  Dividend      20                      21
  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
  Number of
  Occurrences   24 86 65 11  3 23  5    54 12  8  5 43 10  5

  Dividend      22                      23
  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
  Number of
  Occurrences   17 16 15  8 13  6 15     7  8 11  8  6  3  2

  Dividend      24                      25
  Divisor        3  4  5  6  7  8  9     3  4   5 6  7  8  9
  Number of
  Occurrences   91 76 18 50  5 61  1    11 13 105 5  6  5  3

  Dividend      26                      27
  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
  Number of
  Occurrences    5  6  3  3  4  6  3    46  8 10  4  2  6 25

  Dividend      28                       29
  Divisor        3  4  5  6  7  8  9     3  4  5  6  7  8  9
  Number of
  Occurrences    4 36  8  3 19  3  7     6  8  0  5 11  2  3

  Dividend      30                 31                 32
  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
  Number of
  Occurrences   21 27 25  6  7 13   4  3  1  1  4  2  50 11  3  6 39  5

  Dividend      33                 34                 35
  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
  Number of
  Occurrences    8  7  7  2  6  1   8  3  5  2  1  1  10 31  5 24  5  3

  Dividend      36                 37                 38
  Divisor        4  5  6  7  8  9   4  5  6  7  8  9   4  5  6  7  8  9
  Number of
  Occurrences   37 16 22  2  6 19   12 8  7  5  3  9   7  8  7  1  1  5

  Dividend      39             40            41            42
  Divisor        4 5 6 7 8 9    5 6 7  8 9    5 6 7 8 9     5  6  7  8 9
  Number of
  Occurrences    4 3 7 4 3 1   38 9 2 34 2    6 6 3 7 5     7 28 30 10 3

  Dividend      43             44            45            46
  Divisor        5 6  7 8 9     5 6 7 8 9     5 6 7  8  9   5 6 7 8 9
  Number of
  Occurrences    7 5 10 3 2     7 6 4 5 0    24 6 7 10 20   3 3 2 2 2

  Dividend      47             48            49            50
  Divisor        5 6 7 8 9      5  6 7  8 9   5 6  7 8 9    6 7 8 9
  Number of
  Occurrences    6 2 2 0 3      7 17 4 33 2   4 7 27 9 2    4 6 3 8

  Dividend      51         52           53         54
  Divisor        6 7 8 9    6 7 8 9      6 7 8 9    6 7 8  9
  Number of
  Occurrences    2 3 1 2    5 5 5 3      4 3 2 2    12 5 1 16

  Dividend      55         56           57         58         59
  Divisor        6 7 8 9    6  7  8 9    6 7 8 9    6 7 8 9    6 7 8 9
  Number of
  Occurrences    5 3 4 2    0 13 16 8    0 3 1 3    2 2 3 1    2 3 0 3

  Dividend      60       61       62       63       64         65
  Divisor        7 8 9    7 8 9    7 8 9    7 8 9    7  8  9    7  8 9
  Number of
  Occurrences    3 9 1    1 2 5    4 6 1   17 5 9    5 22  0    1 10 1

  Dividend      66       67       68       69       70     71
  Divisor        7 8 9    7 8 9    7 8 9    7 8 9    8 9    8 9
  Number of
  Occurrences    2 1 4    0 1 1    1 3 2    0 6 1    6 2    1 0

  Dividend      72      73     74     75     76     77     78     79
  Divisor        8  9    8 9    8 9    8 9    8 9    8 9    8 9    8 9
  Number of
  Occurrences   16 10    7 5    3 3    5 3    3 2    3 0    4 1    0 2

  Dividend      80   81   82   83   84   85   86   87   88   89
  Divisor        9    9    9    9    9    9    9    9    9    9
  Number of
  Occurrences    4   15    2    4    1    2    0    3    2    7

Tables 3 to 8 show that even gifted authors make instruments for
instruction in arithmetic which contain much less practice on certain
elementary facts than teachers suppose; and which contain relatively
much more practice on the more easily learned facts than on those which
are harder to learn.

How much practice should be given in arithmetic? How should it be
divided among the different bonds to be formed? Below a certain amount
there is waste because, as has been shown in Chapter VI, the pupil will
need more time to detect and correct his errors than would have been
required to give him mastery. Above a certain amount there is waste
because of unproductive overlearning. If 668 is just enough for 2 × 2,
82 is not enough for 9 × 8. If 82 is just enough for 9 × 8, 668 is too
much for 2 × 2.

It is possible to find the answers to these questions for the pupil of
median ability (or any stated ability) by suitable experiments. The
amount of practice will, of course, vary according to the ability of
the pupil. It will also vary according to the interest aroused in him
and the satisfaction he feels in progress and mastery. It will also vary
according to the amount of practice of other related bonds; 7 + 7 = 14
and 60 ÷ 7 = 8 and 4 remainder will help the formation of 7 + 8 = 15
and 61 ÷ 7 = 8 and 5 remainder. It will also, of course, vary with the
general difficulty of the bond, 17 - 8 = 9 being under ordinary
conditions of teaching harder to form than 7 - 2 = 5.

Until suitable experiments are at hand we may estimate for the
fundamental bonds as follows, assuming that by the end of grade 6 a
strength of 199 correct out of 200 is to be had, and that the teaching
is by an intelligent person working in accord with psychological
principles as to both ability and interest.

For one of the easier bonds, most facilitated by other bonds (such
as 2 × 5 = 10, or 10 - 2 = 8, or the double bond 7 = two 3s and 1
remainder) in the case of the median or average pupil, twelve practices
in the week of first learning, supported by twenty-five practices during
the two months following, and maintained by thirty practices well spread
over the later periods should be enough. For the more gifted pupils
lesser amounts down to six, twelve, and fifteen may suffice. For the
less gifted pupils more may be required up to thirty, fifty, and a
hundred. It is to be doubted, however, whether pupils requiring nearly
two hundred repetitions of each of these easy bonds should be taught
arithmetic beyond a few matters of practical necessity.

For bonds of ordinary difficulty, with average facilitation from other
bonds (such as 11 - 3, 4 × 7, or 48 ÷ 8 = 6) in the case of the median
or average pupil, we may estimate twenty practices in the week of first
learning, supported by thirty, and maintained by fifty practices well
spread over the later periods. Gifted pupils may gain and keep mastery
with twelve, fifteen, and twenty practices respectively. Pupils dull at
arithmetic may need up to twenty, sixty, and two hundred. Here, again,
it is to be doubted whether a pupil for whom arithmetical facts, well
taught and made interesting, are so hard to acquire as this, should
learn many of them.

For bonds of greater difficulty, less facilitated by other bonds (such
as 17 - 9, 8 × 7, or 12-1/2% of = 1/8 of), the practice may be from ten
to a hundred percent more than the above.


UNDERLEARNING AND OVERLEARNING

If we accept the above provisional estimates as reasonable, we may
consider the harm done by giving less and by giving more than these
reasonable amounts. Giving less is indefensible. The pupil's time is
wasted in excessive checking to find his errors. He is in danger of
being practiced in error. His attention is diverted from the learning of
new facts and processes by the necessity of thinking out these
supposedly mastered facts. All new bonds are harder to learn than they
should be because the bonds which should facilitate them are not strong
enough to do so. Giving more does harm to some extent by using up time
that could be spent better for other purposes, and (though not
necessarily) by detracting from the pupil's interest in arithmetic. In
certain cases, however, such excess practice and overlearning are
actually desirable. Three cases are of special importance.

The first is the case of a bond operating under a changed mental set or
adjustment. A pupil may know 7 × 8 adequately as a thing by itself, but
need more practice to operate it in

  285
    7
  ---

where he has to remember that 3 is to be added to the 56 when he
obtains it, and that only the 9 is to be written down, the 5 to be held
in mind for later use. The practice required to operate the bond
efficiently in this new set is desirable, even though it is excess from
a narrower point of view, and causes the straightforward 'seven eights
are fifty-six' to be overlearned. So also a pupil's work with 24, 34,
44, etc., +9 may react to give what would be excess practice from the
point of view of 4 + 9 alone; his work in estimating approximate
quotient figures in long division may give excess practice on the
division tables. There are many such cases. Even adding the 5 and 7 in
5/12 + 7/12 is not quite the same task as adding 5 and 7 undisturbed by
the fact that they are twelfths. We know far too little about the amount
of practice needed to adapt arithmetical bonds to efficient operation in
these more complicated conditions to estimate even approximately the
allowances to be made. But some allowance, and often a rather large
allowance, must be made.

The second is the case where the computation in general should be made
very easy and sure for the pupil except for some one new element that
is being learned. For example, in teaching the meaning and uses of
'Averages' and of uneven division, we may deliberately use 2, 3, and 4
as divisors rather than 7 and 9, so as to let all the pupil's energy be
spent in learning the new facts, and so that the fraction in the
quotient may be something easily understood, real, and significant. In
teaching the addition of mixed numbers, we may use, in the early steps,

  11-1/2
  13-1/2
  24
  ------

rather than

  79-1/2
  98-1/2
  67
  ------

so as to save attention for the new process itself. In cancellation, we
may give excess practice to divisions by 2, 3, 4, and 5 in order to make
the transfer to the new habits of considering two numbers together from
the point of view of their divisibility by some number. In introducing
trade discount, we may give excess practice on '5% of' and '10% of'
deliberately, so that the meaning of discount may not be obscured by
difficulties in the computation itself. Excess practice on, and
overlearning of, certain bonds is thus very often justifiable.

The third case concerns bonds whose importance for practical uses in
life or as notable facilitators of other bonds is so great that they may
profitably be brought to a greater strength than 199 correct out of 200
at a speed of 2 sec. or less, or be brought to that degree of strength
very early. Examples of bonds of such special practical use are the
subtractions from 10, 1/2 + 1/2, 1/2 + 1/4, 1/2 of 60, 1/4 of 60,
and the fractional parts of 12 and of $1.00. Examples of notable
facilitating bonds are ten 10s = 100, ten 100s = 1000, additions like
2 + 2, 3 + 3, and 4 + 4, and all the multiplication tables to 9 × 9.

In consideration of these three modifying cases or principles, a volume
could well be written concerning just how much practice to give to each
bond, in each of the types of complex situations where it has to
operate. There is evidently need for much experimentation to expose the
facts, and for much sagacity and inventiveness in making sure of
effective learning without wasteful overlearning.

The facts of primary importance are:--

  (1) The textbook or other instrument of instruction which is a
  teacher's general guide may give far too little practice on
  certain bonds.

  (2) It may divide the practice given in ways that are apparently
  unjustifiable.

  (3) The teacher needs therefore to know how much practice it does
  give, where to supplement it, and what to omit.

  (4) The omissions, on grounds of apparent excess practice, should
  be made only after careful consideration of the third principle
  described above.

  (5) The amount of practice should always be considered in the
  light of its interest and appeal to the pupil's tendency to work
  with full power and zeal. Mere repetition of bonds when the
  learner does not care whether he is improving is rarely
  justifiable on any grounds.

  (6) Practice that is actually in excess is not a very grave defect
  if it is enjoyed and improves the pupil's attitude toward
  arithmetic. Not much time is lost; a hundred practices for each of
  a thousand bonds after mastery to 199 in 200 at 2 seconds will use
  up less than 60 hours, or 15 hours per year in grades 3 to 6.

  (7) By the proper division of practice among bonds, the
  arrangement of learning so that each bond helps the others, the
  adroit shifting of practice of a bond to each new type of
  situation requiring it to operate under changed conditions, and
  the elimination of excess practice where nothing substantial is
  gained, notable improvements over the past hit-and-miss customs
  may be expected.

  (8) Unless the material for practice is adequate, well balanced
  and sufficiently motivated, the teacher must keep close account of
  the learning of pupils. Otherwise disastrous underlearning of many
  bonds is almost sure to occur and retard the pupil's development.


THE ORGANIZATION OF ABILITIES

There is danger that the need of brevity and simplicity which has made
us speak so often of a bond or an ability, and of the amount of
practice it requires, may mislead the reader into thinking that these
bonds and abilities are to be formed each by itself alone and kept so.
They should rarely be formed so and never kept so. This we have
indicated from time to time by references to the importance of forming a
bond in the way in which it is to be used, to the action of bonds in
changed situations, to facilitation of one bond by others, to the
coöperation of abilities, and to their integration into a total
arithmetical ability.

As a matter of fact, only a small part of drill work in arithmetic
should be the formation of isolated bonds. Even the very young
pupil learning 5 and 3 are 8 should learn it with '5 and 5 = 10,'
'5 and 2 = 7,' at the back of his mind, so to speak. Even so early,
5 + 3 = 8 should be part of an organized, coöperating system of bonds.
Later 50 + 30 = 80 should become allied to it. Each bond should be
considered, not simply as a separate tool to be put in a compartment
until needed, but also as an improvement of one total tool or machine,
arithmetical ability.

There are differences of course. Knowledge of square root can be
regarded somewhat as a separate tool to be sharpened, polished, and used
by itself, whereas knowledge of the multiplication tables cannot. Yet
even square root is probably best made more closely a part of the total
ability, being taught as a special case of dividing where divisor is to
be the same as quotient, the process being one of estimating and
correcting.

In general we do not wish the pupil to be a repository of separated
abilities, each of which may operate only if you ask him the sort of
questions which the teacher used to ask him, or otherwise indicate to
him which particular arithmetical tool he is to use. Rather he is to be
an effective organization of abilities, coöperating in useful ways to
meet the quantitative problems life offers. He should not as a rule
have to think in such fashion as: "Is this interest or discount? Is it
simple interest or compound interest? What did I do in compound
interest? How do I multiply by 2 percent?" The situation that calls up
interest should also call up the kind of interest that is appropriate,
and the technique of operating with percents should be so welded
together with interest in his mind that the right coöperation will occur
almost without supervision by him.

As each new ability is acquired, then, we seek to have it take its place
as an improvement of a thinking being, as a coöperative member of a
total organization, as a soldier fighting together with others, as an
element in an educated personality. Such an organization of bonds will
not form itself any more than any one bond will create itself. If the
elements of arithmetical ability are to act together as a total
organized unified force they must be made to act together in the course
of learning. What we wish to have work together we must put together and
give practice in teamwork.

We can do much to secure such coöperative action when and where and as
it is needed by a very simple expedient; namely, to give practice with
computation and problems such as life provides, instead of making up
drills and problems merely to apply each fact or principle by itself.
Though a pupil has solved scores of problems reading, "A triangle has a
base of _a_ feet and an altitude of _b_ feet, what is its area?" he may
still be practically helpless in finding the area of a triangular plot
of ground; still more helpless in using the formula for a triangle which
is one of two into which a trapezoid is divided. Though a pupil has
learned to solve problems in trade discount, simple interest, compound
interest, and bank discount one at a time, stated in a few set forms,
he may be practically helpless before the actual series of problems
confronting him in starting in business, and may take money out of the
savings bank when he ought to borrow on a time loan, or delay payment on
his bills when by paying cash he could save money as well as improve his
standing with the wholesaler.

Instead of making up problems to fit the abilities given by school
instruction, we should preferably modify school instruction so that
arithmetical abilities will be organized into an effective total ability
to meet the problems that life will offer. Still more generally, _every
bond formed should be formed with due consideration of every other bond
that has been or will be formed; every ability should be practiced in
the most effective possible relations with other abilities_.




CHAPTER VII

THE SEQUENCE OF TOPICS: THE ORDER OF FORMATION OF BONDS


The bonds to be formed having been chosen, the next step is to arrange
for their most economical order of formation--to arrange to have each
help the others as much as possible--to arrange for the maximum of
facilitation and the minimum of inhibition.

The principle is obvious enough and would probably be admitted in theory
by any intelligent teacher, but in practice we are still wedded to
conventional usages which arose long before the psychology of arithmetic
was studied. For example, we inherit the convention of studying addition
of integers thoroughly, and then subtraction, and then multiplication,
and then division, and many of us follow it though nobody has ever given
a proof that this is the best order for arithmetical learning. We
inherit also the opposite convention of studying in a so-called "spiral"
plan, a little addition, subtraction, multiplication, and division, and
then some more of each, and then some more, and many of us follow this
custom, with an unreasoned faith that changing about from one process to
another is _per se_ helpful.

Such conventions are very strong, illustrating our common tendency to
cherish most those customs which we cannot justify! The reductions of
denominate numbers ascending and descending were, until recently, in
most courses of study, kept until grade 4 or grade 5 was reached,
although this material is of far greater value for drills on the
multiplication and division tables than the customary problems about
apples, eggs, oranges, tablets, and penholders. By some historical
accident or for good reasons the general treatment of denominate numbers
was put late; by our naïve notions of order and system we felt that any
use of denominate numbers before this time was heretical; we thus became
blind to the advantages of quarts and pints for the tables of 2s; yards
and feet for the tables of 3s; gallons and quarts for the tables of 4s;
nickels and cents for the 5s; weeks and days for the 7s; pecks and
quarts for the 8s; and square yards and square feet for the 9s.
Problems like 5 yards = __ feet or 15 feet = __ yards have not only the
advantages of brevity, clearness, practical use, real reference, and
ready variation, but also the very great advantage that part of the data
have to be _thought of_ in a useful way instead of _read off_ from the
page. In life, when a person has twenty cents with which to buy tablets
of a certain sort, he _thinks of_ the price in making his purchase,
asking it of the clerk only in case he does not know it, and in planning
his purchases beforehand he _thinks of_ prices as a rule. In spite of
these and other advantages, not one textbook in ten up to 1900 made
early use of these exercises with denominate numbers. So strong is mere
use and wont.

Besides these conventional customs, there has been, in those responsible
for arithmetical instruction, an admiration for an arrangement of topics
that is easy for a person, after he knows the subject, to use in
thinking of its constituent parts and their relations. Such arrangements
are often called 'logical' arrangements of subject matter, though they
are often far from logical in any useful sense. Now the easiest order in
which to think of a hierarchy of habits after you have formed them all
may be an extremely difficult order in which to form them. The criticism
of other orders as 'scrappy,' or 'unsystematic,' valid enough if the
course of study is thought of as an object of contemplation, may be
foolish if the course of study is regarded as a working instrument for
furthering arithmetical learning.

We must remember that all our systematizing and labeling is largely
without meaning to the pupils. They cannot at any point appreciate the
system as a progression from that point toward this and that, since they
have no knowledge of the 'this or that.' They do not as a rule think of
their work in grade 4 as an outcome of their work in grade 3 with
extensions of a to a_1, and additions of b_2 and b_3 to b and b_1, and
refinements of c and d by c_4 and d_5. They could give only the vaguest
account of what they did in grade 3, much less of why it should have
been done then. They are not much disturbed by a lack of so-called
'system' and 'logical' progression for the same reason that they are not
much helped by their presence. What they need and can use is a
_dynamically_ effective system or order, one that they can learn easily
and retain long by, regardless of how it would look in a museum of
arithmetical systems. Unless their actual arithmetical habits are
usefully related it does no good to see the so-called logical relations;
and if their habits are usefully related, it does not very much matter
whether or not they do see these; finally, they can be brought to see
them best by first acquiring the right habits in a dynamically effective
order.


DECREASING INTERFERENCE AND INCREASING FACILITATION

Psychology offers no single, easy, royal road to discovering this
dynamically best order. It can only survey the bonds, think what each
demands as prerequisite and offers as future help, recommend certain
orders for trial, and measure the efficiency of each order as a means of
attaining the ends desired. The ingenious thought and careful
experimentation of many able workers will be required for many years to
come.

Psychology can, however, even now, give solid constructive help in many
instances, either by recommending orders that seem almost certainly
better than those in vogue, or by proposing orders for trial which can
be justified or rejected by crucial tests.

Consider, for example, the situation, 'a column of one-place numbers to
be added, whose sum is over 9,' and the response 'writing down the sum.'
This bond is commonly firmly fixed before addition with two-place
numbers is undertaken. As a result the pupil has fixed a habit that he
has to break when he learns two-place addition. If _oral_ answers only
are given with such single columns until two-place addition is well
under way, the interference is avoided.

In many courses of study the order of systematic formation of the
multiplication table bonds is: 1 × 1, 2 × 1, etc., 1 × 2, 2 × 2, etc.,
1 × 3, 2 × 3, etc., 1 × 9, 2 × 9, etc. This is probably wrong in two
respects. There is abundant reason to believe that the × 5s should be
learned first, since they are easier to learn than the 1s or the 2s, and
give the idea of multiplying more emphatically and clearly. There is
also abundant reason to believe that the 1 × 5, 1 × 2, 1 × 3, etc.,
should be put very late--after at least three or four tables are
learned, since the question "What is 1 times 2?" (or 3 or 5) is
unnecessary until we come to multiplication of two- and three-place
numbers, seems a foolish question until then, and obscures the notion of
multiplication if put early. Also the facts are best learned once for
all as the habits "1 times _k_ is the same as _k_," and "_k_ times 1 is
the same as _k_."[8]

  [8] The very early learning of 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2,
      3 × 3, and perhaps a few more multiplications is not considered
      here. It is advisable. The treatment of 0 × 0, 0 × 1, 1 × 0, etc.,
      is not considered here. It is probably best to defer the '× 0'
      bonds until after all the others are formed and are being used
      in short multiplication, and to form them in close connection
      with their use in short multiplication. The '0 ×' bonds may well
      be deferred until they are needed in 'long' multiplication,
      0 × 0 coming last of all.

In another connection it was recommended that the divisions to 81 ÷ 9
be learned by selective thinking or reasoning from the multiplications.
This determines the order of bonds so far as to place the formation of
the division bonds soon after the learning of the multiplications. For
other reasons it is well to make the proximity close.

One of the arbitrary systematizations of the order of formation of bonds
restricts operations at first to the numbers 1 to 10, then to numbers
under 100, then to numbers under 1000, then to numbers under 10,000.
Apart from the avoidance of unreal and pedantic problems in applied
arithmetic to which work with large numbers in low grades does somewhat
predispose a teacher, there is little merit in this restriction of the
order of formation of bonds. Its demerits are many. For example, when
the pupil is learning to 'carry' in addition he can be given better
practice by soon including tasks with sums above 100, and can get a
valuable sense of the general use of the process by being given a few
examples with three- and four-place numbers to be added. The same holds
for subtraction. Indeed, there is something to be said in favor of using
six- or seven-place numbers in subtraction, enforcing the 'borrowing'
process by having it done again and again in the same example, and
putting it under control by having the decision between 'borrowing' and
'not borrowing' made again and again in the same example. When the
multiplication tables are learned the most important use for them is not
in tedious reviews or trivial problems with answers under 100, but in
regular 'short' multiplication of two- and three- and even four-place
numbers. Just as the addition combinations function mainly in the
higher-decade modifications of them, so the multiplication combinations
function chiefly in the cases where the bond has to operate while the
added tasks of keeping one's place, adding what has been carried,
writing down the right figure in the right place, and holding the right
number for later addition, are also taken care of. It seems best to
introduce such short multiplication as soon as the × 5s, × 2s, × 3s,
and × 4s are learned and to put the × 6s, × 7s, and the rest to work
in such short multiplication as soon as each is learned.

Still surer is the need for four-, five-, and six-place numbers when
two-place numbers are used in multiplying. When the process with a
two-place multiplier is learned, multiplications by three-place numbers
should soon follow. They are not more difficult then than later. On the
contrary, if the pupil gets used to multiplying only as one does with
two-place multipliers, he will suffer more by the resulting interference
than he does from getting six- or seven-place answers whose meaning he
cannot exactly realize. They teach the rationale and the manipulations
of long multiplication with especial economy because the principles and
the procedures are used two or three times over and the contrasts
between the values which the partial products have in adding become
three instead of one.

The entire matter of long multiplication with integers and United States
money should be treated as a teaching unit and the bonds formed in close
organization, even though numbers as large as 900,000 are occasionally
involved. The reason is not that it is more logical, or less scrappy,
but that each of the bonds in question thus gets much help from, and
gives much help to, the others.

In sharp contrast to a topic like 'long multiplication' stands a topic
like denominate numbers. It most certainly should not be treated as a
large teaching unit, and all the bonds involved in adding, subtracting,
multiplying, and dividing with all the ordinary sorts of measures should
certainly not be formed in close sequence. The reductions ascending and
descending for many of the measures should be taught as drills on the
appropriate multiplication and division tables. The reduction of feet
and inches to inches, yards and feet to yards, gallons and quarts to
quarts, and the like are admirable exercises in connection with the
(_a_ × _b_) + _c_ = .... problems,--the 'Bought 3 lbs. of sugar at
7 cents and 5 cents worth of matches' problems. The reductions of
inches to feet and inches and the like are admirable exercises in
the _d_ = (.... × _b_) + _c_ or 'making change' problem, which in its
small-number forms is an excellent preparatory step for short division.
They are also of great service in early work with fractions. The
feet-mile, square-foot-square-inch, and other simple relations give a
genuine and intelligible demand for multiplication with large numbers.

Knowledge of the metric system for linear and square measure would
perhaps, as an introduction to decimal fractions, more than save the
time spent to learn it. It would even perhaps be worth while to invent a
measure (call it the _twoqua_) midway between the quart and gallon and
teach carrying in addition and borrowing in subtraction by teaching
first the addition and subtraction of 'gallon, twoqua, quart, and pint'
series! Many of the bonds which a system-made tradition huddled together
uselessly in a chapter on denominate numbers should thus be formed as
helpful preparations for and applications of other bonds all the way
from the first to the eighth half-year of instruction in arithmetic.

The bonds involved in the ability to respond correctly to the series:--

   5 = .... 2s and .... remainder
   5 = .... 3s and .... remainder
  88 = .... 9s and .... remainder

should be formed before, not during, the training in short division.
They are admirable at that point as practice on the division tables; are
of practical service in the making-change problems of the small purchase
and the like; and simplify the otherwise intricate task of keeping one's
place, choosing the quotient figure, multiplying by it, subtracting and
holding in mind the new number to be divided, which is composed half of
the remainder and half of a figure in the written dividend. This change
of order is a good illustration of the nearly general rule that "_When
the practice or review required to perfect or hold certain bonds can, by
an inexpensive modification, be turned into a useful preparation for new
bonds, that modification should be made._"

The bonds involved in the four operations with United States money
should be formed in grades 3 and 4 along with or very soon after the
corresponding bonds with three-place and four-place integers. This
statement would have seemed preposterous to the pedagogues of fifty
years ago. "United States money," they would have said, "is an
application of decimals. How can it be learned until the essentials of
decimal fractions are known? How will the child understand when
multiplying $.75 by 3 that 3 times 5 cents is 1 dime and 5 cents, or
that 3 times 70 cents is 2 dollars and 1 dime? Why perplex the young
pupils with the difficulties of placing the decimal point? Why disturb
the learning of the four operations with integers by adding at each step
a second 'procedure with United States money'?"

The case illustrates very well the error of the older oversystematic
treatment of the order of topics and the still more important error of
confusing the logic of proof with the psychology of learning. To prove
that 3 × $.75 = $2.25 to the satisfaction of certain arithmeticians, you
may need to know the theory of decimal fractions; but to do such
multiplication all a child needs is to do just what he has been doing
with integers and then "Put a $ before the answer to show that it means
dollars and cents, and put a decimal point in the answer to show which
figures mean dollars and which figures mean cents." And this is general.
The ability to operate with integers plus the two habits of prefixing $
and separating dollars from cents in the result will enable him to
operate with United States money.

Consequently good practice came to use United States money not as a
consequence of decimal fractions, learned by their aid, but as an
introduction to decimal fractions which aids the pupil to learn them. So
it has gradually pushed work with United States money further and
further back, though somewhat timidly.

We need not be timid. The pupil will have no difficulty in adding,
subtracting, multiplying, and dividing with United States money--unless
we create it by our explanations! If we simply form the two bonds
described above and show by proper verification that the procedure
always gives the right answer, the early teaching of the four operations
with United States money will in fact actually show a learning profit!
It will save more time in the work with integers than was spent in
teaching it! For, in the first place, it will help to make work with
four-place and five-place numbers more intelligible and vital. A pupil
can understand $16.75 or $28.79 more easily than 1675 or 2879. The
former may be the prices of a suit or sewing machine or bicycle. In the
second place, it permits the use of a large stock of genuine problems
about spending, saving, sharing, and the like with advertisements and
catalogues and school enterprises. In the third place, it permits the
use of common-sense checks. A boy may find one fourth of 3000 as 7050 or
75 and not be disturbed, but he will much more easily realize that one
fourth of $30.00 is not over $70 or less than $1. Even the decimal point
of which we used to be so afraid may actually help the eye to keep its
place in adding.


INTEREST

So far, the illustrations of improvements in the order of bonds so as to
get less interference and more facilitation than the customary orders
secure have sought chiefly to improve the mechanical organization of the
bonds. Any gain in interest which the changes described effected would
be largely due to the greater achievement itself. Dewey and others have
emphasized a very different principle of improving the order of
formation of bonds--the principle of determination of the bonds to be
formed by some vital, engaging problem which arouses interest enough to
lighten the labor and which goes beyond or even against cut-and-dried
plans for sequences in order to get effective problems. For example, the
work of the first month in grade 2B might sacrifice facilitations of the
mechanical sort in order to put arithmetic to use in deciding what
dimensions a rabbit's cage should have to give him 12 square feet of
floor space, how much bread he should have per meal to get 6 ounces a
day, how long a ten-cent loaf would last, how many loaves should be
bought per week, how much it costs to feed the rabbit, how much he has
gained in weight since he was brought to the school, and so on.

Such sacrifices of the optimal order if interest were equal, in order to
get greater interest or a healthier interest, are justifiable. Vital
problems as nuclei around which to organize arithmetical learning are of
prime importance. It is even safe probably to insist that some genuine
problem-situation requiring a new process, such as addition with
carrying, multiplication by two-place numbers, or division with
decimals, be provided in every case as a part of the introduction to
that process. The sacrifice should not be too great, however; the search
for vital problems that fit an economical order of subject matter is as
much needed as the amendment of that order to fit known interests; and
the assurance that a problem helps the pupil to learn arithmetic is as
important as the assurance that arithmetic is used to help the pupil
solve his personal problems.

Much ingenuity and experimentation will be required to find the order
that is satisfactory in both quality and quantity of interest or motive
and helpfulness of the bonds one to another. The difficulty of
organizing arithmetic around attractive problems is much increased by
the fact of class instruction. For any one pupil vital, personal
problems or projects could be found to provide for many arithmetical
abilities; and any necessary knowledge and technique which these
projects did not develop could be somehow fitted in along with them. But
thirty children, half boys and half girls, varying by five years in age,
coming from different homes, with different native capacities, will not,
in September, 1920, unanimously feel a vital need to solve any one
problem, and then conveniently feel another on, say, October 15! In the
mechanical laws of learning children are much alike, and the gain we
may hope to make from reducing inhibitions and increasing facilitations
is, for ordinary class-teaching, probably greater than that to be made
from the discovery of attractive central problems. We should, however,
get as much as possible of both.


GENERAL PRINCIPLES

The reader may by now feel rather helpless before the problem of the
arrangement of arithmetical subject matter. "Sometimes you complete a
topic, sometimes you take it piecemeal months or years apart, often you
make queer twists and shifts to get a strategic advantage over the
enemy," he may think, "but are there no guiding principles, no general
rules?" There is only one that is absolutely general, to _take the order
that works best for arithmetical learning_. There are particular rules,
but there are so many and they are so limited by an 'other things being
equal' clause, that probably a general eagerness to think out the _pros_
and _cons_ for any given proposal is better than a stiff attempt to
adhere to these rules. I will state and illustrate some of them, and let
the reader judge.

_Other things being equal, one new sort of bonds should not be started
until the previous set is fairly established, and two different sets
should not be started at once._ Thus, multiplication of two- and
three-place numbers by 2, 3, 4, and 5 will first use numbers such that
no carrying is required, and no zero difficulties are encountered, then
introduce carrying, then introduce multiplicands like 206 and 320.
If other things were equal, the carrying would be split into two
steps--first drills with (4 × 6) + 2, (3 × 7) + 3, (5 × 4) + 1, and the
like, and second the actual use of these habits in the multiplication.
The objection to this separation of the double habit is that the first
part of it in isolation is too artificial--that it may be better to
suffer the extra difficulty of forming the two together than to teach so
rarely used habits as the (_a_ × _b_) + _c_ series. Experimental tests
are needed to decide this point.

_Other things being equal, bonds should be formed in such order that
none will have to be broken later._ For example, there is a strong
argument for teaching long division first, or very early, with
remainders, letting the case of zero remainder come in as one of many.
If the pupils have been familiarized with the remainder notion by the
drills recommended as preparation for short division,[9] the use of
remainders in long division will offer little difficulty. The exclusive
use of examples without remainders may form the habit of not being exact
in computation, of trusting to 'coming out even' as a sole check, and
even of writing down a number to fit the final number to be divided
instead of obtaining it by honest multiplication.

  [9] See page 76.

For similar reasons additions with 2 and 3 as well as 1 to be 'carried'
have much to recommend them in the very first stages of column addition
with carrying. There is here the added advantage that a pupil will be
more likely to remember to carry if he has to think _what_ to carry. The
present common practice of using small numbers for ease in the addition
itself teaches many children to think of carrying as adding one.

_Other things being equal, arrange to have variety._ Thus it is
probably, though not surely, wise to interrupt the monotony of learning
the multiplication and division tables, by teaching the fundamentals of
'short' multiplication and perhaps of division after the 5s, 2s, 3s, and
4s are learned. This makes a break of several weeks. The facts for the
6s, 7s, 8s, and 9s can then be put to varied use as fast as learned. It
is almost certainly wise to interrupt the first half-year's work with
addition and subtraction, by teaching 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2,
2 × 5, later by 2 × 10, 3 × 10, 4 × 10, 5 × 10, later by 1/2 + 1/2,
1-1/2 + 1/2, 1/2 of 2, 1/2 of 4, 1/2 of 6, and at some time by certain
profitable exercises wherein a pupil tells all he knows about certain
numbers which may be made nuclei of important facts (say, 5, 8, 10, 12,
15, and 20).

_Other things being equal, use objective aids to verify an arithmetical
process or inference after it is made, as well as to provoke it._ It is
well at times to let pupils do everything that they can with relations
abstractly conceived, testing their results by objective counting,
measuring, adding, and the like. For example, an early step in adding
should be to show three things, put them under a book, show two more,
put these under the book, and then ask how many there are under the
book, letting the objective counting come later as the test of the
correctness of the addition.

_Other things being equal, reserve all explanations of why a process
must be right until the pupils can use the process accurately, and have
verified the fact that it is right._ Except for the very gifted pupils,
the ordinary preliminary deductive explanations of what must be done are
probably useless as means of teaching the pupils what to do. They use up
much time and are of so little permanent effect that, as we have seen,
the very arithmeticians who advocate making them, admit that after a
pupil has mastered the process he may be allowed to forget the reasons
for it. I am not sure that the deductive proofs of why we place the
decimal point as we do in division by a decimal, or invert and multiply
in dividing by a fraction, and the like, are worth teaching at all. If
they are to be taught at all, the time to teach them is (except for the
very gifted) after the pupil has mastered the process and has confidence
in it. He then at least knows what process he is to prove is right, and
that it is right, and has had some chance of seeing _why_ it is right
from his experience with it.

One more principle may be mentioned without illustration. _Arrange the
order of bonds with due regard for the aims of the other studies of the
curriculum and the practical needs of the pupil outside of school._
Arithmetic is not a book or a closed system of exercises. It is the
quantitative work of the pupils in the elementary school. No narrower
view of it is adequate.




CHAPTER VIII

THE DISTRIBUTION OF PRACTICE


THE PROBLEM

The same amount of practice may be distributed in various ways. Figures
7 to 10, for example, show 200 practices with division by a fraction
distributed over three and a half years of 10 months in four different
ways. In Fig. 7, practice is somewhat equally distributed over the whole
period. In Fig. 8 the practice is distributed at haphazard. In Fig. 9
there is a first main learning period, a review after about ten weeks, a
review at the beginning of the seventh grade, another review at the
beginning of the eighth grade, and some casual practice rather at
random. In Fig. 10 there is a main learning period, with reviews
diminishing in length and separated by wider and wider intervals, with
occasional practice thereafter to keep the ability alive and healthy.

Plans I and II are obviously inferior to Plans III and IV; and Plan IV
gives promise of being more effective than Plan III, since there seems
danger that the pupil working by Plan III might in the ten weeks lose
too much of what he had gained in the initial practice, and so again in
the next ten weeks.

It is not wise, however, to try now to make close decisions in the case
of practice with division by a fraction; or to determine what the best
distribution of practice is for that or any other ability to be
improved. The facts of psychology are as yet not adequate for very close
decisions, nor are the types of distribution of practice that are best
adapted to different abilities even approximately worked out.

    [Illustration: FIG. 7.--Plan I. 200 practices distributed somewhat
    evenly over 3-1/2 years of 10 months. In Figs. 7, 8, 9, and 10,
    each tenth of an inch along the base line represents one month.
    Each hundredth of a square inch represents four practices, a
    little square 1/20 of an inch wide and 1/20 inch high representing
    one practice.]

    [Illustration: FIG. 8.--Plan II. 200 practices distributed
    haphazard over 3-1/2 years of 10 months.]

    [Illustration: FIG. 9.--Plan III. A learning period, three reviews,
    and incidental practice.]

    [Illustration: FIG. 10.--Plan IV. A learning period with reviews
    of decreasing length at increasing intervals.]


SAMPLE DISTRIBUTIONS

Let us rather examine some actual cases of distribution of practice
found in school work and consider, not the attainment of the best
possible distribution, but simply the avoidance of gross blunders and
the attainment of reasonable, defensible procedures in this regard.

Figures 11 to 18 show the distribution of examples in multiplication
with multipliers of various sorts. _X_ stands for any digit except zero.
_O_ stands for 0. _XXO_ thus means a multiplier like 350 or 270 or 160;
_XOX_ means multipliers like 407, 905, or 206; _XX_ means multipliers
like 25, 17, 38. Each of these diagrams covers approximately 3-1/2 years
of school work, or from about the middle of grade 3 to the end of grade
6. They are made from counts of four textbooks (A, B, C, and D), the
count being taken for each successive 8 pages.[10] Each tenth of an inch
along the base line equals 8 pages of the text in question. Each .01 sq.
in. equals one example. The books, it will be observed, differ in the
amount of practice given, as well as in the way in which it is
distributed.

  [10] At the end of a volume or part, the count may be from as
       few as 5 or as many as 12 pages.

These distributions are worthy of careful study; we shall note only a
few salient facts about them here. Of the distributions of
multiplications with multipliers of the _XX_ type, that of book D (Fig.
14) is perhaps the best. A (Fig. 11) has too much of the practice too
late; B (Fig. 12) gives too little practice in the first learning; C
(Fig. 13) gives too much in the first learning and in grade 6. Among the
distributions of multiplication with multipliers of the _XOX_ type, that
of book D (Fig. 18) is again probably the best. A, B, and C (Figs. 15,
16, and 17) have too much practice early and too long intervals between
reviews. Book C (Fig. 17) by a careless oversight has one case of this
very difficult process, without any explanation, weeks before the
process is taught!

    [Illustration: FIG. 11.--Distribution of practise with multipliers
    of the _XX_ type in the first two books of the three-book text A.]

    [Illustration: FIG. 12.--Same as Fig. 11, but for text B. Following
    this period come certain pages of computation to be used by the
    teacher at her discretion, containing 24 _XX_ multiplications.]

    [Illustration: FIG. 13.--Same as Fig. 11, but for text C.]

    [Illustration: FIG. 14.--Same as Fig. 11, but for text D.]

    [Illustration: FIG. 15.--Distribution of practice with multipliers
    of the _XOX_ type in the first two books of the three-book text
    A.]

    [Illustration: FIG. 16.--Same as Fig. 15, but for text B. Following
    this period come certain pages of computation to be used by the
    teacher at her discretion, containing 17 _XOX_ multiplications.]

    [Illustration: FIG. 17.--Same as Fig. 16, but for text C.]

    [Illustration: FIG. 18.--Same as Fig. 16, but for text D.]

Figures 19, 20, 21, 22, and 23 all concern the first two books of the
three-book text E.

Figure 19 shows the distribution of practice on 5 × 5 in the first two
books of text E. The plan is the same as in Figs. 11 to 18, except that
each tenth of an inch along the base line represents ten pages. Figure
20 shows the distribution of practice on 7 × 7; Fig. 21 shows it for
6 × 7 and 7 × 6 together. In Figs. 20 and 21 also, 0.1 inch along the
base line equals ten pages.

Figures 22 and 23 show the distribution of practice on the divisions of
72, 73, 74, 75, 76, 77, 78, and 79 by either 8 or 9, and on the
divisions of 81, 82 ... 89 by 9. Each tenth of an inch along the base
line represents ten pages here also.

Figures 19 to 23 show no consistent plan for distributing practice.
With 5 × 5 (Fig. 19) the amount of practice increases from the first
treatment in grade 3 to the end of grade 6, so that the distribution
would be better if the pupil began at the end and went backward! With
7 × 7 (Fig. 20) the practice is distributed rather evenly and in small
doses. With 6 × 7 and 7 × 6 (Fig. 21) much of it is in very large doses.
With the divisions (Figs. 22 and 23) the practice is distributed more
suitably, though in Fig. 23 there is too much of it given at one time
in the middle of the period.

    [Illustration: FIG. 19.--Distribution of practice with 5 × 5 in
    the first two books of the three-book text E.]

    [Illustration: FIG. 20.--Distribution of practice with 7 × 7 in
    the first two books of text E.]

    [Illustration: FIG. 21.--Distribution of practice with 6 × 7
    or 7 × 6 in the first two books of text E.]

    [Illustration: FIG. 22.--Distribution of practice with
    72, 73 ... 79 ÷ 8 or 9 in the first two books of text E.]

    [Illustration: FIG. 23.--Distribution of practice with
    81, 82 ... 89 ÷ 9 in the first two books of text E.]


POSSIBLE IMPROVEMENTS

Even if we knew what the best distribution of practice was for each
ability of the many to be inculcated by arithmetical instruction, we
could perhaps not provide it for all of them. For, in the first place,
the allotments for some of them might interfere with those for others.
In the second place, there are many other considerations of importance
in the ordering of topics besides giving the optimal distribution of
practice to each ability. Such are considerations of interest, of
welding separate abilities into an integrated total ability, and of the
limitations due to the school schedule with its Saturdays, Sundays,
holidays, and vacations.

Improvement can, however, be made over present practice in many
respects. A scientific examination of the teaching of almost any class
for a year, or of many of our standard instruments of instruction, will
reveal opportunities for improving the distribution of practice with no
sacrifice of interest, and with an actual gain in integrated functioning
arithmetical power. In particular it will reveal cases where an ability
is given practice and then, never being used again, left to die of
inactivity. It will reveal cases where an ability is given practice and
then left so long without practice that the first effect is nearly lost.
There will be cases where practice is given and reviews are given, but
all in such isolation from everything else in arithmetic that the
ability, though existent, does not become a part of the pupil's general
working equipment. There will be cases where more practice is given in
the late than the earlier periods for no apparent extrinsic advantage;
and cases where the practice is put where it is for no reason that is
observable save that the teacher or author in question has decided to
have some drill work at that time!

Each ability has its peculiar needs in this matter, and no set rules are
at present of much value. It will be enough for the present if we are
aroused to the problem of distribution, avoid obvious follies like those
just noted, and exercise what ingenuity we have.




CHAPTER IX

THE PSYCHOLOGY OF THINKING: ABSTRACT IDEAS AND GENERAL NOTIONS IN
ARITHMETIC[11]

  [11] Certain paragraphs in this and the following chapter are
       taken from the author's _Educational Psychology_, with
       slight modifications.


RESPONSES TO ELEMENTS AND CLASSES

The plate which you see, the egg before you at the breakfast table, and
this page are concrete things, but whiteness, whether of plate, egg, or
paper, is, we say, an abstract quality. To be able to think of whiteness
irrespective of any concrete white object is to be able to have an
abstract idea or notion of white; to be able to respond to whiteness,
irrespective of whether it is a part of china, eggshell, paper or
whatever object, is to be able to respond to the abstract element of
whiteness.

Learning arithmetic involves the formation of very many such ideas, the
acquisition of very many such powers of response to elements regardless
of the gross total situations in which they appear. To appreciate the
fiveness of five boys, five pencils, five inches, five rings of a bell;
to understand the division into eight equal parts of 40 cents, 32 feet,
64 minutes, or 16 ones; to respond correctly to the fraction relation in
2/3, 5/6, 3/4, 7/12, 1/8, or any other; to be sensitive to the common
element of 9 = 3 × 3, 16 = 4 × 4, 625 = 25 × 25, .04 = .2 × .2, 1/4 =
1/2 × 1/2,--these are obvious illustrations. All the numbers which the
pupil learns to understand and manipulate are in fact abstractions; all
the operations are abstractions; percent, discount, interest, height,
length, area, volume, are abstractions; sum, difference, product,
quotient, remainder, average, are facts that concern elements or aspects
which may appear with countless different concrete surroundings or
concomitants.

Towser is a particular dog; your house lot on Elm Street is a particular
rectangle; Mr. and Mrs. I.S. Peterson and their daughter Louise are a
particular family of three. In contrast to these particulars, we mean
by a dog, a rectangle, and a family of three, _any_ specimens of these
classes of facts. The idea of a dog, of rectangles in general, of any
family of three is a general notion, a concept or idea of a class or
species. The ability to respond to any dog, or rectangle, or family of
three, regardless of which particular one it may be, is the general
notion in action.

Learning arithmetic involves the formation of very many such general
notions, such powers of response to any member of a certain class. Thus
a hundred different sized lots may all be responded to as rectangles;
9/18, 12/27, 15/24, and 27/36 may all be responded to as members of the
class, 'both members divisible by 3.' The same fact may be responded to
in different ways according to the class to which it is assigned. Thus 4
in 3/4, 4/5, 45, 54, and 405 is classed respectively as 'a certain sized
part of unity,' 'a certain number of parts of the size shown by the 5,'
'a certain number of tens,' 'a certain number of ones,' and 'a certain
number of hundreds.' Each abstract quality may become the basis of a
class of facts. So fourness as a quality corresponds to the class
'things four in number or size'; the fractional quality or relation
corresponds to the class 'fractions.' The bonds formed with classes of
facts and with elements or features by which one whole class of facts is
distinguished from another, are in fact, a chief concern of arithmetical
learning.[12]

  [12] It should be noted that just as concretes give rise to
       abstractions, so these in turn give rise to still more
       abstract abstractions. Thus fourness, fiveness, twentyness,
       and the like give rise to 'integral-number-ness.' Similarly
       just as individuals are grouped into general classes, so
       classes are grouped into still more general classes.
       Half, quarter, sixth, and tenth are general notions, but
       'one ...th' is more general; and 'fraction' is still more
       general.


FACILITATING THE ANALYSIS OF ELEMENTS

Abstractions and generalizations then depend upon analysis and upon
bonds formed with more or less subtle elements rather than with gross
total concrete situations. The process involved is most easily
understood by considering the means employed to facilitate it.

The first of these is having the learner respond to the total situations
containing the element in question with the attitude of piecemeal
examination, and with attentiveness to one element after another,
especially to so near an approximation to the element in question as he
can already select for attentive examination. This attentiveness to one
element after another serves to emphasize whatever appropriate minor
bonds from the element in question the learner already possesses. Thus,
in teaching children to respond to the 'fiveness' of various
collections, we show five boys or five girls or five pencils, and say,
"See how many boys are standing up. Is Jack the only boy that is
standing here? Are there more than two boys standing? Name the boys
while I point at them and count them. (Jack) is one, and (Fred) is one
more, and (Henry) is one more. Jack and Fred make (two) boys. Jack and
Fred and Henry make (three) boys." (And so on with the attentive
counting.) The mental set or attitude is directed toward favoring the
partial and predominant activity of 'how-many-ness' as far as may be;
and the useful bonds that the 'fiveness,' the 'one and one and one and
one and one-ness,' already have, are emphasized as far as may be.

The second of the means used to facilitate analysis is having the
learner respond to many situations each containing the element in
question (call it A), but with varying concomitants (call these V. C.)
his response being so directed as, so far as may be, to separate each
total response into an element bound to the A and an element bound to
the V. C.

Thus the child is led to associate the responses--'Five boys,' 'Five
girls,' 'Five pencils,' 'Five inches,' 'Five feet,' 'Five books,' 'He
walked five steps,' 'I hit my desk five times,' and the like--each with
its appropriate situation. The 'Five' element of the response is thus
bound over and over again to the 'fiveness' element of the situation,
the mental set being 'How many?,' but is bound only once to any one of
the concomitants. These concomitants are also such as have preferred
minor bonds of their own (the sight of a row of boys _per se_ tends
strongly to call up the 'Boys' element of the response). The other
elements of the responses (boys, girls, pencils, etc.) have each only a
slight connection with the 'fiveness' element of the situations. These
slight connections also in large part[13] counteract each other, leaving
the field clear for whatever uninhibited bond the 'fiveness' has.

  [13] They may, of course, also result in a fusion or an alternation
       of responses, but only rarely.

The third means used to facilitate analysis is having the learner
respond to situations which, pair by pair, present the element in a
certain context and present that same context with _the opposite of the
element in question_, or with something at least very unlike the
element. Thus, a child who is being taught to respond to 'one fifth' is
not only led to respond to 'one fifth of a cake,' 'one fifth of a pie,'
'one fifth of an apple,' 'one fifth of ten inches,' 'one fifth of an
army of twenty soldiers,' and the like; he is also led to respond to
each of these _in contrast with_ 'five cakes,' 'five pies,' 'five
apples,' 'five times ten inches,' 'five armies of twenty soldiers.'
Similarly the 'place values' of tenths, hundredths, and the rest are
taught by contrast with the tens, hundreds, and thousands.

These means utilize the laws of connection-forming to disengage a
response element from gross total responses and attach it to some
situation element. The forces of use, disuse, satisfaction, and
discomfort are so maneuvered that an element which never exists by
itself in nature can influence man almost as if it did so exist, bonds
being formed with it that act almost or quite irrespective of the gross
total situation in which it inheres. What happens can be most
conveniently put in a general statement by using symbols.

Denote by _a_ + _b_, _a_ + _g_, _a_ + _l_, _a_ + _q_, _a_ + _v_, and
_a_ + _B_ certain situations alike in the element _a_ and different in
all else. Suppose that, by original nature or training, a child responds
to these situations respectively by r_{1} + r_{2}, r_{1} + r_{7},
r_{1} + r_{12}, r_{1} + r_{17}, r_{1} + r_{22}, r_{1} + r_{27}. Suppose
that man's neurones are capable of such action that r_{1}, r_{2}, r_{7},
r_{12}, r_{22}, and r_{27}, can each be made singly.


Case I. Varying Concomitants

Suppose that _a_ + _b_, _a_ + _g_, _a_ + _l_, etc., occur once each.

  We have _a_ + _b_ responded to by r_{1} + r_{2},
          _a_ + _g_      "       "  r_{1} + r_{7},
          _a_ + _l_      "       "  r_{1} + r_{12},
          _a_ + _q_      "       "  r_{1} + r_{17},
          _a_ + _v_      "       "  r_{1} + r_{22}, and
          _a_ + _B_      "       "  r_{1} + r_{27}, as shown in
                                      Scheme I.

Scheme I

              _a_   _b_   _g_   _l_   _q_   _v_   _B_
  r_{1}        6     1     1     1     1     1     1
  r_{2}        1     1
  r_{7}        1           1
  r_{12}       1                 1
  r_{17}       1                       1
  r_{22}       1                             1
  r_{27}       1                                   1

_a_ is thus responded to by r_{1} (that is, connected with r_{1}) each
time, or six in all, but only once each with _b_, _g_, _l_, _q_, _v_,
and _B_. _b_, _g_, _l_, _q_, _v_, and _B_ are connected once each with
r_{1} and once respectively with r_{2}, r_{7}, r_{12}, etc. The bond
from _a_ to r_{1}, has had six times as much exercise as the bond from
_a_ to r_{2}, or from _a_ to r_{7}, etc. In any new gross situation, _a_
0, _a_ will be more predominant in determining response than it would
otherwise have been; and r_{1} will be more likely to be made than
r_{2}, r_{7}, r_{12}, etc., the other previous associates in the
response to a situation containing _a_. That is, the bond from the
element _a_ to the response r_{1} has been notably strengthened.


Case II. Contrasting Concomitants

Now suppose that _b_ and _g_ are very dissimilar elements (_e.g._, white
and black), that _l_ and _q_ are very dissimilar (_e.g._, long and
short), and that _v_ and _B_ are also very dissimilar. To be very
dissimilar means to be responded to very differently, so that r_{7}, the
response to _g_, will be very unlike r_{2}, the response to _b_. So
r_{7} may be thought of as r_{not 2} or r_{-2}. In the same way r_{12}
may be thought of as r_{not 12} or r_{-12}, and r_{27} may be called
r_{not 22} or r_{-22}.

Then, if the situations _a_ _b_, _a _g_, _a _l_, _a _q_, _a _v_, and
_a_ _B_ are responded to, each once, we have:--

  _a_ + _b_ responded to by r_{1} + r_{2},
  _a_ + _g_     "        "  r_{1} + r_{not 2},
  _a_ + _l_     "        "  r_{1} + r_{12},
  _a_ + _q_     "        "  r_{1} + r_{not 12},
  _a_ + _v_     "        "  r_{1} + r_{22}, and
  _a_ + _B_     "        "  r_{1} + r_{not 22}, as shown in Scheme II.

Scheme II

              _a_  _b_     _g_       _l_     _q_       _v_     _B_
                       (opp. of _b_)     (opp. of _l_)     (opp. of _v_)
  r_{1}        6    1       1         1       1         1       1
  r_{not 1}
  r_{2}        1    1
  r_{not 2}    1            1
  r_{12}       1                      1
  r_{not 12}   1                               1
  r_{22}       1                                        1
  r_{not 22}   1                                                1

r_{1} is connected to _a_ by 6 repetitions. r_{2} and r_{not 2} are each
connected to _a_ by 1 repetition, but since they interfere, canceling
each other so to speak, the net result is for _a_ to have zero tendency
to call up r_{2} or r_{not 2}. r_{12} and r_{not 12} are each connected
to _a_ by 1 repetition, but they interfere with or cancel each other
with the net result that _a_ has zero tendency to call up r_{12} or
r_{not 12}. So with r_{22} and r_{not 22}. Here then the net result of
the six connections of _a_ _b_, _a_ _g_, _a_ _l_, _a_ _q_, _a_ _v_, and
_a_ _B_ is to connect _a_ with _r_, and with nothing else.


Case III. Contrasting Concomitants and Contrasting Element

Suppose now that the facts are as in Case II, but with the addition of
six experiences where a certain element which is the opposite of, or
very dissimilar to, _a_ is connected with the response r_{not 1}, or
r_{-1} which is opposite to, or very dissimilar to r_{1}. Call this
opposite of _a_, - _a_.

That is, we have not only

  _a_ + _b_ responded to by r_{1} + r_{2},
  _a_ + _g_     "       "   r_{1} + r_{not 2},
  _a_ + _l_     "       "   r_{1} + r_{12},
  _a_ + _q_     "       "   r_{1} + r_{not 12},
  _a_ + _v_     "       "   r_{1} + r_{22}, and
  _a_ + _B_     "       "   r_{1} + r_{not 22},

but also

  - _a_ + _b_ responded to by r_{not 1} + r_{2},
  - _a_ + _g_     "       "   r_{not 1} + r_{not 2},
  - _a_ + _l_     "       "   r_{not 1} + r_{12},
  - _a_ + _q_     "       "   r_{not 1} + r_{not 12},
  - _a_ + _v_     "       "   r_{not 1} + r_{22}, and
  - _a_ + _B_     "       "   r_{not 1} + r_{not 22}, as shown in
                                    Scheme III.

Scheme III

            _a_    opp.     _b_     _g_     _l_     _q_     _v_     _B_
                  of _a_   (opp. of _b_)   (opp. of _l_)   (opp. of _v_)
  r_{1}       6              1       1       1       1       1       1
  r_{not 1}         6        1       1       1       1       1       1
  r_{2}       1     1        2
  r_{not 2}   1     1                2
  r_{12}      1     1                        2
  r_{not 12}  1     1                                2
  r_{22}      1     1                                        2
  r_{not 22}  1     1                                                2

In this series of twelve experiences _a_ connects with r_{1} six times
and the opposite of _a_ connects with r_{not 1} six times. _a_ connects
equally often with three pairs of mutual destructives r_{2} and
r_{not 2}, r_{12} and r_{not 12}, r_{22} and r_{not 22}, and so has zero
tendency to call them up. - _a_ has also zero tendency to call up any of
these responses except its opposite, r_{not 1}. _b_, _g_, _l_, _q_, _v_,
and _B_ are made to connect equally often with r_{1} and r_{not 1}. So,
of these elements, _a_ is the only one left with a tendency to call up
r_{1}.

Thus, by the mere action of frequency of connection, r_{1} is connected
with _a_; the bonds from _a_ to anything except r_{1} are being
counteracted, and the slight bonds from anything except _a_ to r_{1} are
being counteracted. The element _a_ becomes predominant in situations
containing it; and its bond toward r_{1} becomes relatively enormously
strengthened and freed from competition.

These three processes occur in a similar, but more complicated,
form if the situations _a_ + _b_, _a_ + _g_, etc., are replaced by
_a_ + _b_ + _c_ + _d_ + _e_ + _f_, _a_ + _g_ + _h_ + _i_ + _j_ + _k_,
etc., and the responses r_{1} + r_{2}, r_{1} + r_{7}, r_{1} + r_{12},
etc., are replaced by r_{1} + r_{2} + r_{3} + r_{4} + r_{5} + r_{6},
r_{1} + r_{7} + r_{8} + r_{9} + r_{10} + r_{11}, etc.--_provided the_
r_{1}, r_{2}, r_{3}, r_{4}, etc., _can be made singly_. In so far as any
one of the responses is necessarily co-active with any one of the others
(so that, for example, r_{13} always brings r_{26} with it and _vice
versa_), the exact relations of the numbers recorded in schemes like
schemes I, II, and III on pages 172 to 174 will change; but, unless
r_{1} has such an inevitable co-actor, the general results of schemes I,
II, and III will hold good. If r_{1} does have such an inseparable
co-actor, say r_{2}, then, of course, _a_ can never acquire bonds with
r_{1} alone, but everywhere that r_{1} or r_{2} appears in the preceding
schemes the other element must appear also. r_{1} r_{2} would then have
to be used as a unit in analysis.

The '_a_ + _b_,' '_a_ + _g_,' '_a_ + _l_,' ... '_a_ + _B_' situations
may occur unequal numbers of times, altering the exact numerical
relations of the connections formed and presented in schemes I, II,
and III; but the process in general remains the same.

So much for the effect of use and disuse in attaching appropriate
response elements to certain subtle elements of situations. There are
three main series of effects of satisfaction and discomfort. They
serve, first, to emphasize, from the start, the desired bonds leading to
the responses r_{1} + r_{2}, r_{1} + r_{7}, etc., to the total
situations, and to weed out the undesirable ones. They also act to
emphasize, in such comparisons and contrasts as have been described,
every action of the bond from _a_ to r_{1}; and to eliminate every
tendency of _a_ to connect with aught save r_{1}, and of aught save _a_
to connect with r_{1}. Their third service is to strengthen the bonds
produced of appropriate responses to _a_ wherever it occurs, whether or
not any formal comparisons and contrasts take place.

The process of learning to respond to the difference of pitch in tones
from whatever instrument, to the 'square-root-ness' of whatever number,
to triangularity in whatever size or combination of lines, to equality
of whatever pairs, or to honesty in whatever person or instance, is thus
a consequence of associative learning, requiring no other forces than
those of use, disuse, satisfaction, and discomfort. "What happens in
such cases is that the response, by being connected with many situations
alike in the presence of the element in question and different in other
respects, is bound firmly to that element and loosely to each of its
concomitants. Conversely any element is bound firmly to any one response
that is made to all situations containing it and very, very loosely to
each of those responses that are made to only a few of the situations
containing it. The element of triangularity, for example, is bound
firmly to the response of saying or thinking 'triangle' but only very
loosely to the response of saying or thinking white, red, blue, large,
small, iron, steel, wood, paper, and the like. A situation thus acquires
bonds not only with some response to it as a gross total, but also with
responses to any of its elements that have appeared in any other gross
totals. Appropriate response to an element regardless of its
concomitants is a necessary consequence of the laws of exercise and
effect if an animal learns to make that response to the gross total
situations that contain the element and not to make it to those that do
not. Such prepotent determination of the response by one or another
element of the situation is no transcendental mystery, but, given the
circumstances, a general rule of all learning." Such are at bottom only
extreme cases of the same learning as a cat exhibits that depresses a
platform in a certain box whether it faces north or south, whether the
temperature is 50 or 80 degrees, whether one or two persons are in
sight, whether she is exceedingly or moderately hungry, whether fish or
milk is outside the box. All learning is analytic, representing the
activity of elements within a total situation. In man, by virtue of
certain instincts and the course of his training, very subtle elements
of situations can so operate.

       *       *       *       *       *

Learning by analysis does not often proceed in the carefully organized
way represented by the most ingenious marshaling of comparing and
contrasting activities. The associations with gross totals, whereby in
the end an element is elevated to independent power to determine
response, may come in a haphazard order over a long interval of time.
Thus a gifted three-year-old boy will have the response element of
'saying or thinking _two_,' bound to the 'two-ness' element of very many
situations in connection with the 'how-many' mental set; and he will
have made this analysis without any formal, systematic training. An
imperfect and inadequate analysis already made is indeed usually the
starting point for whatever systematic abstraction the schools direct.
Thus the kindergarten exercises in analyzing out number, color, size,
and shape commonly assume that 'one-ness' _versus_ 'more-than-one-ness,'
black and white, big and little, round and not round are, at least
vaguely, active as elements responded to in some independence of their
contexts. Moreover, the tests of actual trial and success in further
undirected exercises usually coöperate to confirm and extend and refine
what the systematic drills have given. Thus the ordinary child in school
is left, by the drills on decimal notation, with only imperfect power
of response to the 'place-values.' He continues to learn to respond
properly to them by finding that 4 × 40 = 160, 4 × 400 = 1600,
800 - 80 = 720, 800 - 8 = 792, 800-800 = 0, 42 × 48 = 2016,
24 × 48 = 1152, and the like, are satisfying; while 4 × 40 = 16,
23 × 48 = 832, 800 - 8 = 0, and the like, are not. The process of
analysis is the same in such casual, unsystematized formation of
connections with elements as in the deliberately managed, piecemeal
inspection, comparison, and contrast described above.


SYSTEMATIC AND OPPORTUNISTIC STIMULI TO ANALYSIS

The arrangement of a pupil's experiences so as to direct his attention
to an element, vary its concomitants instructively, stimulate
comparison, and throw the element into relief by contrast may be by
fixed, formal, systematic exercises. Or it may be by much less formal
exercises, spread over a longer time, and done more or less incidentally
in other connections. We may call these two extremes the 'systematic'
and 'opportunistic,' since the chief feature of the former is that it
systematically provides experiences designed to build up the power of
correct response to the element, whereas the chief feature of the latter
is that it uses especially such opportunities as occur by reason of the
pupil's activities and interests.

Each method has its advantages and disadvantages. The systematic method
chooses experiences that are specially designed to stimulate the
analysis; it provides these at a certain fixed time so that they may
work together; it can then and there test the pupils to ascertain
whether they really have the power to respond to the element or aspect
or feature in question. Its disadvantages are, first, that many of the
pupils will feel no need for and attach no interest or motive to these
formal exercises; second, that some of the pupils may memorize the
answers as a verbal task instead of acquiring insight into the facts;
third, that the ability to respond to the element may remain restricted
to the special cases devised for the systematic training, and not be
available for the genuine uses of arithmetic.

The opportunistic method is strong just where the systematic is weak.
Since it seizes upon opportunities created by the pupil's abilities and
interests, it has the attitude of interest more often. Since it builds
up the experiences less formally and over a wider space of time, the
pupils are less likely to learn verbal answers. Since its material comes
more from the genuine uses of life, the power acquired is more likely to
be applicable to life.

Its disadvantage is that it is harder to manage. More thought and
experimentation are required to find the best experiences; greater care
is required to keep track of the development of an abstraction which is
taught not in two days, but over two months; and one may forget to test
the pupils at the end. In so far as the textbook and teacher are able to
overcome these disadvantages by ingenuity and care, the opportunistic
method is better.


ADAPTATIONS TO ELEMENTARY SCHOOL PUPILS

We may expect much improvement in the formation of abstract and general
ideas in arithmetic from the application of three principles in addition
to those already described. They are: (1) Provide enough actual
experiences before asking the pupil to understand and use an abstract or
general idea. (2) Develop such ideas gradually, not attempting to give
complete and perfect ideas all at once. (3) Develop such ideas so far as
possible from experiences which will be valuable to the pupil in and of
themselves, quite apart from their merit as aids in developing the
abstraction or general notion. Consider these three principles in order.

Children, especially the less gifted intellectually, need more
experiences as a basis for and as applications of an arithmetical
abstraction or concept than are usually given them. For example, in
paving the way for the principle, "Any number times 0 equals 0," it is
not safe to say, "John worked 8 days for 0 minutes per day. How many
minutes did he work?" and "How much is 0 times 4 cents?" It will be
much better to spend ten or fifteen minutes as follows:[14] "What does
zero mean? (Not any. No.) How many feet are there in eight yards?
In 5 yards? In 3 yards? In 2 yards? In 1 yard? In 0 yard? How many
inches are there in 4 ft.? In 2 ft.? In 0 ft.? 7 pk. = .... qt.
5 pk. = .... qt. 0 pk. = .... qt. A boy receives 60 cents an hour
when he works. How much does he receive when he works 3 hr.? 8 hr.?
6 hr.? 0 hr.? A boy received 60 cents a day for 0 days. How much did he
receive? How much is 0 times $600? How much is 0 times $5000? How much
is 0 times a million dollars? 0 times any number equals....

   232  (At the blackboard.)  0 time 232 equals what?
    30  I write 0 under the 0.[15]  3 times 232 equals what?
  ----
  6960  Continue at the blackboard with

        734         321         312          41
         20          40          30          60  etc."
        ---         ---         ---          --

  [14] The more gifted children may be put to work using the principle
       after the first minute or two.

  [15] 232
        30   If desired this form may be used, with the appropriate
       ---      difference in the form of the questions and statements.
       000
      696
      ----
      6960

Pupils in the elementary school, except the most gifted, should not be
expected to gain mastery over such concepts as _common fraction_,
_decimal fraction_, _factor_, and _root_ quickly. They can learn a
definition quickly and learn to use it in very easy cases, where even a
vague and imperfect understanding of it will guide response correctly.
But complete and exact understanding commonly requires them to take,
not one intellectual step, but many; and mastery in use commonly comes
only as a slow growth. For example, suppose that pupils are taught
that .1, .2, .3, etc., mean 1/10, 2/10, 3/10, etc., that .01, .02, .03,
etc., mean 1/100, 2/100, 3/100, etc., that .001, .002, .003, etc., mean
1/1000, 2/1000, 3/1000, etc., and that .1, .02, .001, etc., are decimal
fractions. They may then respond correctly when asked to write a decimal
fraction, or to state which of these,--1/4, .4, 3/8, .07, .002,
5/6,--are common fractions and which are decimal fractions. They may
be able, though by no means all of them will be, to write decimal
fractions which equal 1/2 and 1/5, and the common fractions which
equal .1 and .09. Most of them will not, however, be able to respond
correctly to "Write a decimal mixed number"; or to state which of
these,--1/100, .4-1/2, .007/350, $.25,--are common fractions, and which
are decimals; or to write the decimal fractions which equal 3/4 and 1/3.

If now the teacher had given all at once the additional experiences
needed to provide the ability to handle these more intricate and subtle
features of decimal-fraction-ness, the result would have been confusion
for most pupils. The general meaning of .32, .14, .99, and the like
requires some understanding of .30, .10, .90, and .02, .04, .08; but it
is not desirable to disturb the child with .30 while he is trying to
master 2.3, 4.3, 6.3, and the like. Decimals in general require
connection with place value and the contrasts of .41 with 41, 410, 4.1,
and the like, but if the relation to place values in general is taught
in the same lesson with the relation to /10s, /100s, /1000s, the mind
will suffer from violent indigestion.

A wise pedagogy in fact will break up the process of learning the
meaning and use of decimal fractions into many teaching units, for
example, as follows:--

(1) Such familiarity with fractions with large denominators as is
desirable for pupils to have, as by an exercise in reducing to lowest
terms, 8/10, 36/64, 20/25, 18/24, 24/32, 21/30, 25/100, 40/100, and the
like. This is good as a review of cancellation, and as an extension of
the idea of a fraction.

(2) Objective work, showing 1/10 sq. ft., 1/50 sq. ft., 1/100 sq. ft.,
and 1/1000 sq. ft., and having these identified and the forms 1/10 sq.
ft., 1/100 sq. ft., and 1/1000 sq. ft. learned. Finding how many
feet = 1/10 mile and 1/100 mile.

(3) Familiarity with /100s and /1000s by reductions of 750/1000, 50/100,
etc., to lowest terms and by writing the missing numerators in
500/1000 = /100 = /10 and the like, and by finding 1/10, 1/100, and
1/1000 of 3000, 6000, 9000, etc.

(4) Writing 1/10 as .1 and 1/100 as .01, 11/100, 12/100, 13/100, etc.,
as .11, .12, .13. United States money is used as the introduction.
Application is made to miles.

(5) Mixed numbers with a first decimal place. The cyclometer or
speedometer. Adding numbers like 9.1, 14.7, 11.4, etc.

(6) Place value in general from thousands to hundredths.

(7) Review of (1) to (6).

(8) Tenths and hundredths of a mile, subtraction when both numbers
extend to hundredths, using a railroad table of distances.

(9) Thousandths. The names 'decimal fractions or decimals,' and 'decimal
mixed numbers or decimals.' Drill in reading any number to thousandths.
The work will continue with gradual extension and refinement of the
understanding of decimals by learning how to operate with them in
various ways.

Such may seem a slow progress, but in fact it is not, and many of these
exercises whereby the pupil acquires his mastery of decimals are useful
as organizations and applications of other arithmetical facts.

That, it will be remembered, was the third principle:--"Develop abstract
and general ideas by experiences which will be intrinsically valuable."
The reason is that, even with the best of teaching, some pupils will
not, within any reasonable limits of time expended, acquire ideas that
are fully complete, rigorous when they should be, flexible when they
should be, and absolutely exact. Many children (and adults, for that
matter) could not within any reasonable limits of time be so taught the
nature of a fraction that they could decide unerringly in original
exercises like:--

Is 2.75/25 a common fraction?

Is $.25 a decimal fraction?

Is one _x_th of _y_ a fraction?

Can the same words mean both a common fraction and a decimal fraction?

Express 1 as a common fraction.

Express 1 as a decimal fraction.

These same children can, however, be taught to operate correctly with
fractions in the ordinary uses thereof. And that is the chief value of
arithmetic to them. They should not be deprived of it because they
cannot master its subtler principles. So we seek to provide experiences
that will teach all pupils something of value, while stimulating in
those who have the ability the growth of abstract ideas and general
principles.

Finally, we should bear in mind that working with qualities and
relations that are only partly understood or even misunderstood does
under certain conditions give control over them. The general process of
analytic learning in life is to respond as well as one can; to get a
clearer idea thereby; to respond better the next time; and so on. For
instance, one gets some sort of notion of what 1/5 means; he then
answers such questions as 1/5 of 10 = ? 1/5 of 5 = ? 1/5 of 20 = ?;
by being told when he is right and when he is wrong, he gets from
these experiences a better idea of 1/5; again he does his best with
1/5 = _/10, 1/5 = _/15, etc., and as before refines and enlarges his
concept of 1/5. He adds 1/5 to 2/5, etc., 1/5 to 3/10, etc., 1/5 to 1/2,
etc., and thereby gains still further, and so on.

What begins as a blind habit of manipulation started by imitation may
thus grow into the power of correct response to the essential element.
The pupil who has at the start no notion at all of 'multiplying' may
learn what multiplying is by his experience that '4 6 multiplying
gives 24'; '3 9 multiplying gives 27,' etc. If the pupil keeps on doing
something with numbers and differentiates right results, he will often
reach in the end the abstractions which he is supposed to need in the
beginning. It may even be the case with some of the abstractions
required in arithmetic that elaborate provision for comprehension
beforehand is not so efficient as the same amount of energy devoted
partly to provision for analysis itself beforehand and partly to
practice in response to the element in question without full
comprehension.

It certainly is not the best psychology and not the best educational
theory to think that the pupil first masters a principle and then merely
applies it--first does some thinking and then computes by mere routine.
On the contrary, the applications should help to establish, extend, and
refine the principle--the work a pupil does with numbers should be a
main means of increasing his understanding of the principles of
arithmetic as a science.




CHAPTER X

THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC


THE ESSENTIALS OF ARITHMETICAL REASONING

We distinguish aimless reverie, as when a child dreams of a vacation
trip, from purposive thinking, as when he tries to work out the answer
to "How many weeks of vacation can a family have for $120 if the cost is
$22 a week for board, $2.25 a week for laundry, and $1.75 a week for
incidental expenses, and if the railroad fares for the round trip are
$12?" We distinguish the process of response to familiar situations,
such as five integral numbers to be added, from the process of response
to novel situations, such as (for a child who has not been trained with
similar problems):--"A man has four pieces of wire. The lengths are 120
yd., 132 meters, 160 feet, and 1/8 mile. How much more does he need to
have 1000 yd. in all?" We distinguish 'thinking things together,' as
when a diagram or problem or proof is understood, from thinking of one
thing after another as when a number of words are spelled or a poem in
an unknown tongue is learned. In proportion as thinking is purposive,
with selection from the ideas that come up, and in proportion as it
deals with novel problems for which no ready-made habitual response is
available, and in proportion as many bonds act together in an organized
way to produce response, we call it reasoning.

When the conclusion is reached as the effect of many particular
experiences, the reasoning is called inductive. When some principle
already established leads to another principle or to a conclusion about
some particular fact, the reasoning is called deductive. In both cases
the process involves the analysis of facts into their elements, the
selection of the elements that are deemed significant for the question
at hand, the attachment of a certain amount of importance or weight to
each of them, and their use in the right relations. Thought may fail
because it has not suitable facts, or does not select from them the
right ones, or does not attach the right amount of weight to each, or
does not put them together properly.

In the world at large, many of our failures in thinking are due to not
having suitable facts. Some of my readers, for example, cannot solve the
problem--"What are the chances that in drawing a card from an ordinary
pack of playing-cards four times in succession, the same card will be
drawn each time?" And it will be probably because they do not know
certain facts about the theory of probabilities. The good thinkers among
such would look the matter up in a suitable book. Similarly, if a person
did not happen to know that there were fifty-two cards in all and that
no two were alike, he could not reason out the answer, no matter what
his mastery of the theory of probabilities. If a competent thinker, he
would first ask about the size and nature of the pack. In the actual
practice of reasoning, that is, we have to survey our facts to see if we
lack any that are necessary. If we do, the first task of reasoning is to
acquire those facts.

This is specially true of the reasoning about arithmetical facts in
life. "Will 3-1/2 yards of this be enough for a dress?" Reason directs
you to learn how wide it is, what style of dress you intend to make of
it, how much material that style normally calls for, whether you are a
careful or a wasteful cutter, and how big the person is for whom the
dress is to be made. "How much cheaper as a diet is bread alone, than
bread with butter added to the extent of 10% of the weight of the
bread?" Reason directs you to learn the cost of bread, the cost of
butter, the nutritive value of bread, and the nutritive value of butter.

In the arithmetic of the school this feature of reasoning appears in
cases where some fact about common measures must be brought to bear, or
some table of prices or discounts must be consulted, or some business
custom must be remembered or looked up.

Thus "How many badges, each 9 inches long, can be made from 2-1/2 yd.
ribbon?" cannot be solved without getting into mind 1 yd. = 36 inches.
"At Jones' prices, which costs more, 3-3/4 lb. butter or 6-1/2 lb. lard?
How much more?" is a problem which directs the thinker to ascertain
Jones' prices.

It may be noted that such problems are, other things being equal,
somewhat better training in thinking than problems where all the data
are given in the problem itself (_e.g._, "Which costs more, 3-3/4 lb.
butter at 48¢ per lb. or 6-1/2 lb. lard at 27¢ per lb.? How much
more?"). At least it is unwise to have so many problems of the latter
sort that the pupil may come to think of a problem in applied arithmetic
as a problem where everything is given and he has only to manipulate the
data. Life does not present its problems so.

The process of selecting the right elements and attaching proper weight
to them may be illustrated by the following problem:--"Which of these
offers would you take, supposing that you wish a D.C.K. upright piano,
have $50 saved, can save a little over $20 per month, and can borrow
from your father at 6% interest?"

  A

  A Reliable Piano. The Famous D.C.K. Upright. You pay $50 cash down
  and $21 a month for only a year and a half. _No interest_ to pay.
  We ask you to pay only for the piano and allow you plenty of time.

  B

  We offer the well-known D.C.K. Piano for $390. $50 cash and $20 a
  month thereafter. Regular interest at 6%. The interest soon is
  reduced to less than $1 a month.

  C

  The D.C.K. Piano. Special Offer, $375, cash. Compare our prices
  with those of any reliable firm.

If you consider chiefly the "only," "No interest to pay," "only," and
"plenty of time" in offer A, attaching much weight to them and little to
the thought, "How much will $50 plus (18 × $21) be?", you will probably
decide wrongly.

The situations of life are often complicated by many elements of little
or even of no relevance to the correct solution. The offerer of A may
belong to your church; your dearest friend may urge you to accept offer
B; you may dislike to talk with the dealer who makes offer C; you may
have a prejudice against owing money to a relative; that prejudice may
be wise or foolish; you may have a suspicion that the B piano is
shopworn; that suspicion may be well-founded or groundless; the salesman
for C says, "You don't want your friends to say that you bought on the
installment plan. Only low-class persons do that," etc. The statement of
arithmetical problems in school usually assists the pupil to the extent
of ruling out all save definitely quantitative elements, and of ruling
out all quantitative elements except those which should be considered.
The first of the two simplifications is very beneficial, on the whole,
since otherwise there might be different correct solutions to a problem
according to the nature and circumstances of the persons involved. The
second simplification is often desirable, since it will often produce
greater improvement in the pupils, per hour of time spent, than would be
produced by the problems requiring more selection. It should not,
however, be a universal custom; for in that case the pupils are tempted
to think that in every problem they must use all the quantities given,
as one must use all the pieces in a puzzle picture.

It is obvious that the elements selected must not only be right but also
be in the right relations to one another. For example, in the problems
below, the 6 must be thought of in relation to a dozen and as being half
of a dozen, and also as being 6 times 1. 1 must be mentally tied to
"each." The 6 as half of a dozen must be related to the $1.00, $1.60,
etc. The 6 as 6 times 1 must be related to the $.09, $.14, etc.

  Buying in Quantity

  These are a grocer's prices for certain things by the dozen and
  for a single one. He sells a half dozen at half the price of a
  dozen. Find out how much you save by buying 6 all at one time
  instead of buying them one at a time.

                              Doz.  Each
  1. Evaporated Milk         $1.00  $.09
  2. Puffed Rice              1.60   .14
  3. Puffed Wheat             1.10   .10
  4. Canned Soup              1.90   .17
  5. Sardines                 1.80   .16
  6. Beans (No. 2 cans)       1.50   .13
  7. Pork and Beans           1.70   .15
  8. Peas (No. 2 cans)        1.40   .12
  9. Tomatoes (extra cans)    3.20   .28
  10. Ripe olives (qt. cans)  7.20   .65

It is obvious also that in such arithmetical work as we have been
describing, the pupil, to be successful, must 'think things together.'
Many bonds must coöperate to determine his final response.

As a preface to reasoning about a problem we often have the discovery of
the problem and the classification of just what it is, and as a
postscript we have the critical inspection of the answer obtained to
make sure that it is verified by experiment or is consistent with known
facts. During the process of searching for, selecting, and weighting
facts, there may be similar inspection and validation, item by item.


REASONING AS THE COÖPERATION OF ORGANIZED HABITS

The pedagogy of the past made two notable errors in practice based on
two errors about the psychology of reasoning. It considered reasoning as
a somewhat magical power or essence which acted to counteract and
overrule the ordinary laws of habit in man; and it separated too sharply
the 'understanding of principles' by reasoning from the 'mechanical'
work of computation, reading problems, remembering facts and the like,
done by 'mere' habit and memory.

Reasoning or selective, inferential thinking is not at all opposed to,
or independent of, the laws of habit, but really is their necessary
result under the conditions imposed by man's nature and training. A
closer examination of selective thinking will show that no principles
beyond the laws of readiness, exercise, and effect are needed to explain
it; that it is only an extreme case of what goes on in associative
learning as described under the 'piecemeal' activity of situations; and
that attributing certain features of learning to mysterious faculties of
abstraction or reasoning gives no real help toward understanding or
controlling them.

It is true that man's behavior in meeting novel problems goes beyond, or
even against, the habits represented by bonds leading from gross total
situations and customarily abstracted elements thereof. One of the two
reasons therefor, however, is simply that the finer, subtle,
preferential bonds with subtler and less often abstracted elements go
beyond, and at times against, the grosser and more usual bonds. One set
is as much due to exercise and effect as the other. The other reason is
that in meeting novel problems the mental set or attitude is likely to
be one which rejects one after another response as their unfitness to
satisfy a certain desideratum appears. What remains as the apparent
course of thought includes only a few of the many bonds which did
operate, but which, for the most part, were unsatisfying to the ruling
attitude or adjustment.

Successful responses to novel data, associations by similarity and
purposive behavior are in only apparent opposition to the fundamental
laws of associative learning. Really they are beautiful examples of it.
Man's successful responses to novel data--as when he argues that the
diagonal on a right triangle of 796.278 mm. base and 137.294 mm.
altitude will be 808.022 mm., or that Mary Jones, born this morning,
will sometime die--are due to habits, notably the habits of response to
certain elements or features, under the laws of piecemeal activity and
assimilation.

Nothing is less like the mysterious operations of a faculty of reasoning
transcending the laws of connection-forming, than the behavior of men in
response to novel situations. Let children who have hitherto confronted
only such arithmetical tasks, in addition and subtraction with one- and
two-place numbers and multiplication with one-place numbers, as those
exemplified in the first line below, be told to do the examples shown in
the second line.

  ADD    ADD     ADD     SUBT.   SUBT.  MULTIPLY   MULTIPLY   MULTIPLY
   8      37      35       8      37       8          9           6
   5      24      68       5      24       5          7           3
  --      --      23      --      --      --         --          --
                  19
                  --

          MULTIPLY                MULTIPLY                MULTIPLY
             32                      43                      34
             23                      22                      26
             --                      --                      --

They will add the numbers, or subtract the lower from the upper number,
or multiply 3 × 2 and 2 × 3, etc., getting 66, 86, and 624, or respond
to the element of 'Multiply' attached to the two-place numbers by "I
can't" or "I don't know what to do," or the like; or, if one is a child
of great ability, he may consider the 'Multiply' element and the bigness
of the numbers, be reminded by these two aspects of the situation of the
fact that

  '9
   9 multiply'
  --

gave only 81, and that

  '10
   10 multiply'
   --

gave only 100, or the like; and so may report an intelligent and
justified "I can't," or reject the plan of 3 × 2 and 2 × 3, with 66, 86,
and 624 for answers, as unsatisfactory. What the children will do will,
in every case, be a product of the elements in the situation that are
potent with them, the responses which these evoke, and the further
associates which these responses in turn evoke. If the child were one of
sufficient genius, he might infer the procedure to be followed as a
result of his knowledge of the principles of decimal notation and the
meaning of 'Multiply,' responding correctly to the 'place-value' element
of each digit and adding his 6 tens and 9 tens, 20 twos and 3 thirties;
but if he did thus invent the shorthand addition of a collection of
twenty-three collections, each of 32 units, he would still do it by the
operation of bonds, subtle but real.

Association by similarity is, as James showed long ago, simply the
tendency of an element to provoke the responses which have been bound to
it. _abcde_ leads to _vwxyz_ because _a_ has been bound to _vwxyz_ by
original nature, exercise, or effect.

Purposive behavior is the most important case of the influence of the
attitude or set or adjustment of an organism in determining (1) which
bonds shall act, and (2) which results shall satisfy. James early
described the former fact, showing that the mechanism of habit can give
the directedness or purposefulness in thought's products, provided that
mechanism includes something paralleling the problem, the aim, or need,
in question.

The second fact, that the set or attitude of the man helps to determine
which bonds shall satisfy, and which shall annoy, has commonly been
somewhat obscured by vague assertions that the selection and retention
is of what is "in point," or is "the right one," or is "appropriate," or
the like. It is thus asserted, or at least hinted, that "the will," "the
voluntary attention," "the consciousness of the problem," and other such
entities are endowed with magic power to decide what is the "right" or
"useful" bond and to kill off the others. The facts are that in
purposive thinking and action, as everywhere else, bonds are selected
and retained by the satisfyingness, and are killed off by the
discomfort, which they produce; and that the potency of the man's set or
attitude to make this satisfy and that annoy--to put certain
conduction-units in readiness to act and others in unreadiness--is in
every way as important as its potency to set certain conduction-units in
actual operation.

Reasoning is not a radically different sort of force operating against
habit but the organization and coöperation of many habits, thinking
facts together. Reasoning is not the negation of ordinary bonds, but
the action of many of them, especially of bonds with subtle elements of
the situation. Some outside power does not enter to select and
criticize; the pupil's own total repertory of bonds relevant to the
problem is what selects and rejects. An unsuitable idea is not killed
off by some _actus purus_ of intellect, but by the ideas which it itself
calls up, in connection with the total set of mind of the pupil, and
which show it to be inadequate.

Almost nothing in arithmetic need be taught as a matter of mere
unreasoning habit or memory, nor need anything, first taught as a
principle, ever become a matter of mere habit or memory. 5 × 4 = 20
should not be learned as an isolated fact, nor remembered as we remember
that Jones' telephone number is 648 J 2. Almost everything in arithmetic
should be taught as a habit that has connections with habits already
acquired and will work in an organization with other habits to come. The
use of this organized hierarchy of habits to solve novel problems is
reasoning.




CHAPTER XI

ORIGINAL TENDENCIES AND ACQUISITIONS BEFORE SCHOOL


THE UTILIZATION OF INSTINCTIVE INTERESTS

The activities essential to acquiring ability in arithmetic can rely on
little in man's instinctive equipment beyond the purely intellectual
tendencies of curiosity and the satisfyingness of thought for thought's
sake, and the general enjoyment of success rather than failure in an
enterprise to which one sets oneself. It is only by a certain amount of
artifice that we can enlist other vehement inborn interests of childhood
in the service of arithmetical knowledge and skill. When this can be
done at no cost the gain is great. For example, marching in files of
two, in files of three, in files of four, etc., raising the arms once,
two times, three times, showing a foot, a yard, an inch with the hands,
and the like are admirable because learning the meanings of numbers thus
acquires some of the zest of the passion for physical action. Even in
late grades chances to make pictures showing the relations of fractional
parts, to cut strips, to fold paper, and the like will be useful.

Various social instincts can be utilized in matches after the pattern of
the spelling match, contests between rows, certain number games, and the
like. The scoring of both the play and the work of the classroom is a
useful field for control by the teacher of arithmetic.

Hunt ['12] has noted the more important games which have some
considerable amount of arithmetical training as a by-product and which
are more or less suitable for class use. Flynn ['12] has described
games, most of them for home use, which give very definite arithmetical
drill, though in many cases the drills are rather behind the needs of
children old enough to understand and like the game itself.

It is possible to utilize the interests in mystery, tricks, and puzzles
so as to arouse a certain form of respect for arithmetic and also to get
computational work done. I quote one simple case from Miss Selkin's
admirable collection ['12, p. 69 f.]:--

  I. ADDITION

  "We must admit that there is nothing particularly interesting in
  a long column of numbers to be added. Let the teacher, however,
  suggest that he can write the answer at sight, and the task will
  assume a totally different aspect.

  "A very simple number trick of this kind can be performed by
  making use of the principle of complementary addition. The
  arithmetical complement of a number with respect to a larger
  number is the difference between these two numbers. Most
  interesting results can be obtained by using complements with
  respect to 9.

  "The children may be called upon to suggest several numbers of
  two, three, or more digits. Below these write an equal number of
  addends and immediately announce the answer. The children,
  impressed by this apparently rapid addition, will set to work
  enthusiastically to test the results of this lightning
  calculation.

  "Example:--     357 }                          999
                  682 }  A                       × 3
                  793 }                         ----
                                                2997

                  642 }
                  317 }  B
                  206 }

  "Explanation:--The addends in group A are written down at
  random or suggested by the class. Those in group B are their
  complements. To write the first number in group B we look at the
  first number in group A and, starting at the left write 6, the
  complement of 3 with respect to 9; 4, the complement of 5; 2, the
  complement of 7. The second and third addends in group B are
  derived in the same way. Since we have three addends in each
  group, the problem reduces itself to multiplying 999 by 3, or to
  taking 3000 - 3. Any number of addends may be used and each addend
  may consist of any number of digits."

Respect for arithmetic as a source of tricks and magic is very much less
important than respect for its everyday services; and computation to
test such tricks is likely to be undertaken zealously only by the abler
pupils. Consequently this source of interest should probably be used
only sparingly, and perhaps the teacher should give such exhibitions
only as a reward for efficiency in the regular work. For example, if the
work for a week is well done in four days the fifth day might be given
up to some semi-arithmetical entertainment, such as the demonstration of
an adding machine, the story of primitive methods of counting, team
races in computation, an exhibition of lightning calculation and
intellectual sleight-of-hand by the teacher, or the voluntary study of
arithmetical puzzles.

The interest in achievement, in success, mentioned above is stronger in
children than is often realized and makes advisable the systematic use
of the practice experiment as a method of teaching much of arithmetic.
Children who thus compete with their own past records, keeping an exact
score from week to week, make notable progress and enjoy hard work in
making it.


THE ORDER OF DEVELOPMENT OF ORIGINAL TENDENCIES

Negatively the difficulty of the work that pupils should be expected to
do is conditioned by the gradual maturing of their capacities. Other
things being equal, the common custom of reserving hard things for late
in the elementary school course is, of course, sound. It seems probable
that little is gained by using any of the child's time for arithmetic
before grade 2, though there are many arithmetical facts that he can
learn in grade 1. Postponement of systematic work in arithmetic to grade
3 or even grade 4 is allowable if better things are offered. With proper
textbooks and oral and written exercises, however, a child in grades 2
and 3 can spend time profitably on arithmetical work. When all children
can be held in school through the eighth grade it does not much matter
whether arithmetic is begun early or late. If, however, many children
are to leave in grades 5 and 6 as now, we may think it wise to provide
somehow that certain minima of arithmetical ability be given them.

There are, so far as is known, no special times and seasons at which the
human animal by inner growth is specially ripe for one or another
section or aspect of arithmetic, except in so far as the general inner
growth of intellectual powers makes the more abstruse and complex tasks
suitable to later and later years.

Indeed, very few of even the most enthusiastic devotees of the
recapitulation theory or culture-epoch theory have attempted to apply
either to the learning of arithmetic, and Branford is the only
mathematician, so far as I know, who has advocated such application,
even tempered by elaborate shiftings and reversals of the racial order.
He says:--

  "Thus, for each age of the individual life--infancy, childhood,
  school, college--may be selected from the racial history
  the most appropriate form in which mathematical experience
  can be assimilated. Thus the capacity of the infant and early
  childhood is comparable with the capacity of animal consciousness
  and primitive man. The mathematics suitable to later childhood
  and boyhood (and, of course, girlhood) is comparable with Archæan
  mathematics passing on through Greek and Hindu to mediæval
  European mathematics; while the student is become sufficiently
  mature to begin the assimilation of modern and highly abstract
  European thought. The filling in of details must necessarily
  be left to the individual teacher, and also, within some such
  broadly marked limits, the precise order of the marshalling of the
  material for each age. For, though, on the whole, mathematical
  development has gone forward, yet there have been lapses from
  advances already made. Witness the practical world-loss of much
  valuable Hindu thought, and, for long centuries, the neglect of
  Greek thought: witness the world-loss of the invention by the
  Babylonians of the Zero, until re-invented by the Hindus, passed
  on by them to the Arabs, and by these to Europe.

  "Moreover, many blunders and false starts and false principles
  have marked the whole course of development. In a phrase, rivers
  have their backwaters. But it is precisely the teacher's function
  to avoid such racial mistakes, to take short cuts ultimately
  discovered, and to guide the young along the road ultimately found
  most accessible with such halts and retracings--returns up
  side-cuts--as the mental peculiarities of the pupils demand.

  "All this, the practical realization of the spirit of the principle,
  is to be wisely left to the mathematical teacher, familiar with the
  history of mathematical science and with the particular limitations
  of his pupils and himself." ['08, p. 245.]

The latitude of modification suggested by Branford reduces the guidance
to be derived from racial history to almost _nil_. Also it is apparent
that the racial history in the case of arithmetical achievement is
entirely a matter of acquisition and social transmission. Man's original
nature is destitute of all arithmetical ideas. The human germs do not
know even that one and one make two!


INVENTORIES OF ARITHMETICAL KNOWLEDGE AND SKILL

A scientific plan for teaching arithmetic would begin with an exact
inventory of the knowledge and skill which the pupils already possessed.
Our ordinary notions of what a child knows at entrance to grade 1, or
grade 2, or grade 3, and of what a first-grade child or second-grade
child can do, are not adequate. If they were, we should not find
reputable textbooks arranging to teach elaborately facts already
sufficiently well known to over three quarters of the pupils when they
enter school. Nor should we find other textbooks presupposing in their
first fifty pages a knowledge of words which not half of the children
can read even at the end of the 2 B grade.

We do find just such evidence that ordinary ideas about the abilities of
children at the beginning of systematic school training in arithmetic
may be in gross error. For example, a reputable and in many ways
admirable recent book has fourteen pages of exercises to teach the
meaning of two and the fact that one and one make two! As an example of
the reverse error, consider putting all these words in the first
twenty-five pages of a beginner's book:--_absentees, attendance, blanks,
continue, copy, during, examples, grouped, memorize, perfect, similar,
splints, therefore, total_!

Little, almost nothing, has been done toward providing an exact
inventory compared with what needs to be done. We may note here (1) the
facts relevant to arithmetic found by Stanley Hall, Hartmann, and others
in their general investigations of the knowledge possessed by children
at entrance to school, (2) the facts concerning the power of children to
perceive differences in length, area, size of collection, and
organization within a collection such as is shown in Fig. 24, and
certain facts and theories about early awareness of number.

In the Berlin inquiry of 1869, knowledge of the meaning of two, three,
and four appeared in 74, 74, and 73 percent of the children upon
entrance to school. Some of those recorded as ignorant probably really
knew, but failed to understand that they were expected to reply or were
shy. Only 85 percent were recorded as knowing their fathers' names.
Seven eighths as many children knew the meanings of two, three, and four
as knew their fathers' names. In a similar but more careful experiment
with Boston children in September, 1880, Stanley Hall found that 92
percent knew three, 83 percent knew four, and 71-1/2 percent knew five.
Three was known about as well as the color red; four was known about as
well as the color blue or yellow or green. Hartmann ['90] found that two
thirds of the children entering school in Annaberg could count from one
to ten. This is about as many as knew money, or the familiar objects of
the town, or could repeat words spoken to them.

    [Illustration: FIG. 24.--Objective presentation.]

In the Stanford form of the Binet tests counting four pennies is given
as an ability of the typical four-year-old. Counting 13 pennies
correctly in at least one out of two trials, and knowing three of the
four coins,--penny, nickel, dime, and quarter,--are given as abilities
of the typical six-year-old.


THE PERCEPTION OF NUMBER AND QUANTITY

We know that educated adults can tell how many lines or dots, etc., they
see in a single glance (with an exposure too short for the eye to move)
up to four or more, according to the clearness of the objects and their
grouping. For example, Nanu ['04] reports that when a number of bright
circles on a dark background are shown to educated adults for only .033
second, ten can be counted when arranged to form a parallelogram, but
only five when arranged in a row. With certain groupings, of course,
their 'perception' involves much inference, even conscious addition and
multiplication. Similarly they can tell, up to twenty and beyond, the
number of taps, notes, or other sounds in a series too rapid for single
counting if the sounds are grouped in a convenient rhythm.

These abilities are, however, the product of a long and elaborate
learning, including the learning of arithmetic itself. Elementary
psychology and common experience teach us that the mere observation of
groups or quantities, no matter how clear their number quality appears
to the person who already knows the meanings of numbers, does not of
itself create the knowledge of the meanings of numbers in one who does
not. The experiments of Messenger ['03] and Burnett ['06] showed that
there is no direct intuitive apprehension even of two as distinct from
one. We have to _learn_ to feel the two touches or see the two dots or
lines as two.

We do not know by exact measurements the growth in children of this
ability to count or infer the number of elements in a collection seen or
series heard. Still less do we know what the growth would be without
the influence of school training in counting, grouping, adding, and
multiplying. Many textbooks and teachers seem to overestimate it
greatly. Not all educated adults can, apart from measurement, decide
with surety which of these lines is the longer, or which of these areas
is the larger, or whether this is a ninth or a tenth or an eleventh of a
circle.

    [Illustration]

Children upon entering school have not been tested carefully in respect
to judgments of length and area, but we know from such studies as
Gilbert's ['94] that the difference required in their case is probably
over twice that required for children of 13 or 14. In judging weights,
for example, a difference of 6 is perceived as easily by children 13 to
15 years of age as a difference of 15 by six-year-olds.

A teacher who has adult powers of estimating length or area or weight
and who also knows already which of the two is longer or larger or
heavier, may use two lines to illustrate a difference which they really
hide from the child. It is unlikely, for example, that the first of
these lines ______________ ________________  would be recognized as
shorter than the second by every child in a fourth-grade class, and it
is extremely unlikely that it would be recognized as being 7/8 of the
length of the latter, rather than 3/4 of it or 5/6 of it or 9/10 of it
or 11/12 of it. If the two were shown to a second grade, with the
question, "The first line is 7. How long is the other line?" there would
be very many answers of 7 or 9; and these might be entirely correct
arithmetically, the pupils' errors being all due to their inability to
compare the lengths accurately.

  _A_   ______________     ________________

        ______________     ________________
  _B_  |______________|   |________________|


  _C_  |-|-|-|-|-|-|-|

       |-|-|-|-|-|-|-|-|

        __ __ __ __ __ __ __
  _D_  |__|__|__|__|__|__|__|
        __ __ __ __ __ __ __ __
       |__|__|__|__|__|__|__|__|


  _E_   .'\##|##/`.      .'\##|##/`.
       /###\#|#/   \    /###\#|#/###\
       |-----------|    |-----------|
       \###/#|#\###/    \###/#|#\###/
        `./##|##\.'      `./##|##\.'

The quantities used should be such that their mere discrimination offers
no difficulty even to a child of blunted sense powers. If 7/8 and 1 are
to be compared, _A_ and _B_ are not allowable. _C_, _D_, and _E_ are
much better.

Teachers probably often underestimate or neglect the sensory
difficulties of the tasks they assign and of the material they use to
illustrate absolute and relative magnitudes. The result may be more
pernicious when the pupils answer correctly than when they fail. For
their correct answering may be due to their divination of what the
teacher wants; and they may call a thing an inch larger to suit her
which does not really seem larger to them at all. This, of course, is
utterly destructive of their respect for arithmetic as an exact and
matter-of-fact instrument. For example, if a teacher drew a series of
lines 20, 21, 22, 23, 24, and 25 inches long on the blackboard in this
form--____ ______ and asked, "This is 20 inches long, how long is
this?" she might, after some errors and correction thereof, finally
secure successful response to all the lines by all the children. But
their appreciation of the numbers 20, 21, 22, 23, 24, and 25 would be
actually damaged by the exercise.


THE EARLY AWARENESS OF NUMBER

There has been some disagreement concerning the origin of awareness of
number in the individual, in particular concerning the relative
importance of the perception of how-many-ness and that of how-much-ness,
of the perception of a defined aggregate and the perception of a defined
ratio. (See McLellan and Dewey ['95], Phillips ['97 and '98], and
Decroly and Degand ['12].)

The chief facts of significance for practice seem to be these: (1)
Children with rare exceptions hear the names _one_, _two_, _three_,
_four_, _half_, _twice_, _two times_, _more_, _less_, _as many as_,
_again_, _first_, _second_, and _third_, long before they have analyzed
out the qualities and relations to which these words refer so as to feel
them at all clearly. (2) Their knowledge of the qualities and relations
is developed in the main in close association with the use of these
words to the child and by the child. (3) The ordinary experiences of the
first five years so develop in the child awareness of the 'how many
somethings' in various groups, of the relative magnitudes of two groups
or quantities of any sort, and of groups and magnitudes as related to
others in a series. For instance, if fairly gifted, a child comes, by
the age of five, to see that a row of four cakes is an aggregate of
four, seeing each cake as a part of the four and the four as the sum of
its parts, to know that two of them are as many as the other two, that
half of them would be two, and to think, when it is useful for him to do
so, of four as a step beyond three on the way to five, or to think of
hot as a step from warm on the way to very hot. The degree of
development of these abilities depends upon the activity of the law of
analysis in the individual and the character of his experiences.

(4) He gets certain bad habits of response from the ambiguity of common
usage of 2, 3, 4, etc., for second, third, fourth. Thus he sees or hears
his parents or older children or others count pennies or rolls or eggs
by saying one, two, three, four, and so on. He himself is perhaps misled
into so counting. Thus the names properly belonging to a series of
aggregations varying in amount come to be to him the names of the
positions of the parts in a counted whole. This happens especially with
numbers above 3 or 4, where the correct experience of the number as a
name for the group has rarely been present. This attaching to the
cardinal numbers above three or four the meanings of the ordinal numbers
seems to affect many children on entrance to school. The numbering of
pages in books, houses, streets, etc., and bad teaching of counting
often prolong this error.

(5) He also gets the habit, not necessarily bad, but often indirectly
so, of using many names such as eight, nine, ten, eleven, fifteen, a
hundred, a million, without any meaning.

(6) The experiences of half, twice, three times as many, three times as
long, etc., are rarer; even if they were not, they would still be less
easily productive of the analysis of the proper abstract element than
are the experiences of two, three, four, etc., in connection with
aggregates of things each of which is usually called one, such as boys,
girls, balls, apples. Experiences of the names, two, three, and four, in
connection with two twos, two threes, two fours, are very rare.

Hence, the names, two, three, etc., mean to these children in the main,
"one something and one something," "one something usually called one,
and one something usually called one, and another something usually
called one," and more rarely and imperfectly "two times anything,"
"three times anything," etc.

With respect to Mr. Phillips' emphasis of the importance of the
series-idea in children's minds, the matters of importance are: first,
that the knowledge of a series of number names in order is of very
little consequence to the teaching of arithmetic and of still less to
the origin of awareness of number. Second, the habit of applying this
series of words in counting in such a way that 8 is associated with the
eighth thing, 9 with the ninth thing, etc., is of consequence because it
does so much mischief. Third, the really valuable idea of the number
series, the idea of a series of groups or of magnitudes varying by
steps, is acquired later, as a result, not a cause, of awareness of
numbers.

With respect to the McLellan-Dewey doctrine, the ratio aspect of numbers
should be emphasized in schools, not because it is the main origin of
the child's awareness of number, but because it is _not_, and because
the ordinary practical issues of child life do _not_ adequately
stimulate its action. It also seems both more economical and more
scientific to introduce it through multiplication, division, and
fractions rather than to insist that 4 and 5 shall from the start mean 4
or 5 times anything that is called 1, for instance, that 8 inches shall
be called 4 two-inches, or 10 cents, 5 two-cents. If I interpret
Professor Dewey's writings correctly, he would agree that the use of
inch, foot, yard, pint, quart, ounce, pound, glassful, cupful, handful,
spoonful, cent, nickel, dime, and dollar gives a sufficient range of
units for the first two school years. Teaching the meanings of 1/2 of 4,
1/2 of 6, 1/2 of 8, 1/2 of 10, 1/2 of 20, 1/3 of 6, 1/3 of 9, 1/3 of 30,
1/4 of 8, two 2s, five 2s, and the like, in early grades, each in
connection with many different units of measure, provides a sufficient
assurance that numbers will connect with relationships as well as with
collections.




CHAPTER XII

INTEREST IN ARITHMETIC


CENSUSES OF PUPILS' INTERESTS

Arithmetic, although it makes little or no appeal to collecting,
muscular manipulation, sensory curiosity, or the potent original
interests in things and their mechanisms and people and their passions,
is fairly well liked by children. The censuses of pupils' likes and
dislikes that have been made are not models of scientific investigation,
and the resulting percentages should not be used uncritically. They are,
however, probably not on the average over-favorable to arithmetic in any
unfair way. Some of their results are summarized below. In general they
show arithmetic to be surpassed in interest clearly by only the manual
arts (shopwork and manual training for boys, cooking and sewing for
girls), drawing, certain forms of gymnastics, and history. It is about
on a level with reading and science. It clearly surpasses grammar,
language, spelling, geography, and religion.

Lobsien ['03], who asked one hundred children in each of the first five
grades (_Stufen_) of the elementary schools of Kiel, "Which part of the
school work (literally, 'which instruction period') do you like best?"
found arithmetic led only by drawing and gymnastics in the case of the
boys, and only by handwork in the case of the girls.

This is an exaggerated picture of the facts, since no count is made of
those who especially dislike arithmetic. Arithmetic is as unpopular with
some as it is popular with others. When full allowance is made for this,
arithmetic still has popularity above the average. Stern ['05] asked,
"Which subject do you like most?" and "Which subject do you like least?"
The balance was greatly in favor of gymnastics for boys (28--1),
handwork for girls (32--1-1/2), and drawing for both (16-1/2--6).
Writing (6-1/2--4), arithmetic (14-1/2--13), history (9--6-1/2),
reading (8-1/2--8), and singing (6--7-1/2) come next. Religion,
nature study, physiology, geography, geometry, chemistry, language,
and grammar are low.

McKnight ['07] found with boys and girls in grades 7 and 8 of certain
American cities that arithmetic was liked better than any of the school
subjects except gymnastics and manual training. The vote as compared
with history was:--

  Arithmetic    327 liked greatly,     96 disliked greatly.
  History       164 liked greatly,    113 disliked greatly.

In a later study Lobsien ['09] had 6248 pupils from 9 to 15 years old
representing all grades of the elementary school report, so far as they
could, the subject most disliked, the subject most liked, the subject
next most liked, and the subject next in order. No child was forced to
report all of these four judgments, or even any of them. Lobsien counts
the likes and the dislikes for each subject. Gymnastics, handwork, and
cooking are by far the most popular. History and drawing are next,
followed by arithmetic and reading. Below these are geography, writing,
singing, nature study, biblical history, catechism, and three minor
subjects.

Lewis ['13] secured records from English children in elementary schools
of the order of preference of all the studies listed below. He reports
the results in the following table of percents:

  ===================================================================
                  | TOP THIRD OF | MIDDLE THIRD OF | LOWEST THIRD OF
                  | STUDIES FOR  | STUDIES FOR     | STUDIES FOR
                  | INTEREST     | INTEREST        | INTEREST
  ----------------+--------------+-----------------+-----------------
  Drawing         |      78      |       20        |        2
  Manual Subjects |      66      |       26        |        8
  History         |      64      |       24        |       12
  Reading         |      53      |       38        |        9
  Singing         |      32      |       48        |       20
                  |              |                 |
  Drill           |      20      |       55        |       25
  Arithmetic      |      16      |       53        |       31
  Science         |      23      |       37        |       40
  Nature Study    |      16      |       36        |       48
  Dictation       |       4      |       57        |       39
                  |              |                 |
  Composition     |      18      |       28        |       54
  Scripture       |       4      |       38        |       58
  Recitation      |       9      |       23        |       68
  Geography       |       4      |       24        |       72
  Grammar         |      --      |        6        |       94
  ===================================================================

Brandell ['13] obtained data from 2137 Swedish children in Stockholm
(327), Norrköping (870), and Gothenburg (940).

In general he found, as others have, that handwork, shopwork for boys
and household work for girls, and drawing were reported as much better
liked than arithmetic. So also was history, and (in this he differs from
most students of this matter) so were reading and nature study.
Gymnastics he finds less liked than arithmetic. Religion, geography,
language, spelling, and writing are, as in other studies, much less
popular than arithmetic.

Other studies are by Lilius ['11] in Finland, Walsemann ['07],
Wiederkehr ['07], Pommer ['14], Seekel ['14], and Stern ['13 and '14],
in Germany. They confirm the general results stated.

The reasons for the good showing that arithmetic makes are probably the
strength of its appeal to the interest in definite achievement, success,
doing what one attempts to do; and of its appeal, in grades 5 to 8, to
the practical interest of getting on in the world, acquiring abilities
that the world pays for. Of these, the former is in my opinion much the
more potent interest. Arithmetic satisfies it especially well, because,
more than any other of the 'intellectual' studies of the elementary
school, it permits the pupil to see his own progress and determine his
own success or failure.

The most important applications of the psychology of satisfiers and
annoyers to arithmetic will therefore be in the direction of utilizing
still more effectively this interest in achievement. Next in importance
come the plans to attach to arithmetical learning the satisfyingness of
bodily action, play, sociability, cheerfulness, and the like, and of
significance as a means of securing other desired ends than arithmetical
abilities themselves. Next come plans to relieve arithmetical learning
from certain discomforts such as the eyestrain of some computations and
excessive copying of figures. These will be discussed here in the
inverse order.


RELIEVING EYESTRAIN

At present arithmetical work is, hour for hour, probably more of a tax
upon the eyes than reading. The task of copying numbers from a book to a
sheet of paper is one of the very hardest tasks that the eyes of a pupil
in the elementary schools have to perform. A certain amount of such
work is desirable to teach a child to write numbers, to copy exactly,
and to organize material in shape for computation. But beyond that,
there is no more reason for a pupil to copy every number with which he
is to compute than for him to copy every word he is to read. The
meaningless drudgery of copying figures should be mitigated by arranging
much work in the form of exercises like those shown on pages 216, 217,
and 218, and by having many of the textbook examples in addition,
subtraction, and multiplication done with a slip of paper laid below the
numbers, the answers being written on it. There is not only a resulting
gain in interest, but also a very great saving of time for the pupil
(very often copying an example more than quadruples the time required to
get its answer), and a much greater efficiency in supervision.
Arithmetical errors are not confused with errors of copying,[16] and the
teacher's task of following a pupil's work on the page is reduced to a
minimum, each pupil having put the same part of the day's work in just
the same place. The use of well-printed and well-spaced pages of
exercises relieves the eyestrain of working with badly made gray
figures, unevenly and too closely or too widely spaced. I reproduce in
Fig. 25 specimens taken at random from one hundred random samples of
arithmetical work by pupils in grade 8. Contrast the task of the eyes in
working with these and their task in working with pages 216 to 218. The
customary method of always copying the numbers to be used in computation
from blackboard or book to a sheet of paper is an utterly unjustifiable
cruelty and waste.

  [16] Courtis finds in the case of addition that "of all the
       individuals making mistakes at any given time in a class,
       at least one third, and usually two thirds, will be making
       mistakes in carrying or copying."

    [Illustration: FIG. 25_a_.--Specimens taken at random from the
    computation work of eighth-grade pupils. This computation
    occurred in a genuine test. In the original gray of the pencil
    marks the work is still harder to make out.]

    [Illustration: FIG. 25_b_.--Specimens taken at random from the
    computation work of eighth-grade pupils. This computation
    occurred in a genuine test. In the original gray of the pencil
    marks the work is still harder to make out.]

Write the products:--

 A. 3   4s=           B. 5   7s=           C. 9   2s=
    5   2s=              8   3s=              4   4s=
    7   2s=              4   2s=              2   7s=
    1   6 =              4   5s=              6   4s=
    1   3 =              4   7s=              5   5s=
    3   7s=              5   9s=              3   6s=
    4   1s=              7   5s=              3   2s=
    6   8s=              7   1s=              3   9s=
    9   8s=              6   3s=              5   1s=
    4   3s=              4   9s=              8   6s=
    2   4s=              3   5s=              8   4s=
    2   2s=              9   6s=              8   5s=
    8   7s=              2   5s=              7   9s=
    5   8s=              5   4s=              6   2s=
    7   6s=              8   2s=              7   4s=
    7   3s=              8   9s=              9   3s=

 D. 4   20s =     E.    9   60s =       F.    40 × 2 = 80
    4  200s =           9  600s =             20 × 2 =
    6   30s =           5   30s =             30 × 2 =
    6  300s =           5  300s =             40 × 2 =
    7 ×  50 =           8 ×  20 =             20 × 3 =
    7 × 500 =           8 × 200 =             30 × 3 =
    3 ×  40 =           2 ×  70 =            300 × 3 = 900
    3 × 400 =           2 × 700 =            300 × 2 =

Write the missing numbers: (_r_ stands for remainder.)

  25 = .... 3s and .... _r_.
  25 = .... 4s  "  .... _r_.
  25 = .... 5s  "  .... _r_.
  25 = .... 6s  "  .... _r_.
  25 = .... 7s  "  .... _r_.
  25 = .... 8s  "  .... _r_.
  25 = .... 9s  "  .... _r_.

  26 = .... 3s and .... _r_.
  26 = .... 4s  "  .... _r_.
  26 = .... 5s  "  .... _r_.
  26 = .... 6s  "  .... _r_.
  26 = .... 7s  "  .... _r_.
  26 = .... 8s  "  .... _r_.
  26 = .... 9s  "  .... _r_.

  30 = .... 4s and .... _r_.
  30 = .... 5s  "  .... _r_.
  30 = .... 6s  "  .... _r_.
  30 = .... 7s  "  .... _r_.
  30 = .... 8s  "  .... _r_.
  30 = .... 9s  "  .... _r_.

  31 = .... 4s and .... _r_.
  31 = .... 5s  "  .... _r_.
  31 = .... 6s  "  .... _r_.
  31 = .... 7s  "  .... _r_.
  31 = .... 8s  "  .... _r_.
  31 = .... 9s  "  .... _r_.

Write the whole numbers or mixed numbers which these fractions equal:--

      5      4      9      4      7
      -      -      -      -      -
      4      3      5      2      3

      7      5      11     3      8
      -      -      --     -      -
      4      3      8      2      8

      8      6      9      9      16
      -      -      -      -      --
      4      3      8      4      8

      11     7      13     8      6
      --     -      --     -      -
      4      5      8      5      6

Write the missing figures:--

  6         2         8          1          2
  - = -     - = -     -- = -     - = --     - = -
  8   4     4   2     10   5     5   10     3   6

Write the missing numerators:--

  1
  - = --  -  --  -  --  -  --
  2   12  8  10  4  16  6  14

  1
  - = --  -  --  -  --  --  --
  3   12  9  18  6  15  24  21

  1
  - = --  --  -  --  --  --  --
  4   12  16  8  24  20  28  32

  1
  - = --  --  --  --  --  --  --
  5   10  20  15  25  40  35  30

  2
  - = --  --  --  -  --  --  -
  3   12  18  21  6  15  24  9

  3
  - = -  --  --  --  --  --  --
  4   8  16  12  20  24  32  28

Find the products. Cancel when you can:--

  5            11           2
  -- × 4 =     -- × 3 =     - × 5 =
  16           12           3

  7            8            1
  -- × 8 =     - × 15 =     - × 8 =
  12           5            6


SIGNIFICANCE FOR RELATED ACTIVITIES

The use of bodily action, social games, and the like was discussed in
the section on original tendencies. "Significance as a means of securing
other desired ends than arithmetical learning itself" is therefore our
next topic. Such significance can be given to arithmetical work by using
that work as a means to present and future success in problems of
sports, housekeeping, shopwork, dressmaking, self-management, other
school studies than arithmetic, and general school life and affairs.
Significance as a means to future ends alone can also be more clearly
and extensively attached to it than it now is.

Whatever is done to supply greater strength of motive in studying
arithmetic must be carefully devised so as not to get a strong but wrong
motive, so as not to get abundant interest but in something other than
arithmetic, and so as not to kill the goose that after all lays the
golden eggs--the interest in intellectual activity and achievement
itself. It is easy to secure an interest in laying out a baseball
diamond, measuring ingredients for a cake, making a balloon of a certain
capacity, or deciding the added cost of an extra trimming of ribbon for
one's dress. The problem is to _attach_ that interest to arithmetical
learning. Nor should a teacher be satisfied with attaching the interest
as a mere tail that steers the kite, so long as it stays on, or as a
sugar-coating that deceives the pupil into swallowing the pill, or as an
anodyne whose dose must be increased and increased if it is to retain
its power. Until the interest permeates the arithmetical activity itself
our task is only partly done, and perhaps is made harder for the next
time.

One important means of really interfusing the arithmetical learning
itself with these derived interests is to lead the pupil to seek the
help of arithmetic himself--to lead him, in Dewey's phrase, to 'feel the
need'--to take the 'problem' attitude--and thus appreciate the
technique which he actively hunts for to satisfy the need. In so far as
arithmetical learning is organized to satisfy the practical demands of
the pupil's life at the time, he should, so to speak, come part way to
get its help.

Even if we do not make the most skillful use possible of these interests
derived from the quantitative problems of sports, housekeeping,
shopwork, dressmaking, self-management, other school studies, and school
life and affairs, the gain will still be considerable. To have them in
mind will certainly preserve us from giving to children of grades 3 and
4 problems so devoid of relation to their interests as those shown
below, all found (in 1910) in thirty successive pages of a book of
excellent repute:--

  A chair has 4 legs. How many legs have 8 chairs? 5 chairs?

  A fly has 6 legs. How many legs have 3 flies? 9 flies? 7 flies?

              (Eight more of the same sort.)

  In 1890 New York had 1,513,501 inhabitants, Milwaukee had
  206,308, Boston had 447,720, San Francisco 297,990. How many
  had these cities together?

              (Five more of the same sort.)

  Milton was born in 1608 and died in 1674. How many years
  did he live?

             (Several others of the same sort.)

  The population of a certain city was 35,629 in 1880 and 106,670
  in 1890. Find the increase.

              (Several others of this sort.)

  A number of others about the words in various inaugural addresses
  and the Psalms in the Bible.

It also seems probable that with enough care other systematic plans of
textbooks can be much improved in this respect. From every point of
view, for example, the early work in arithmetic should be adapted to
some extent to the healthy childish interests in home affairs, the
behavior of other children, and the activities of material things,
animals, and plants.

TABLE 9

FREQUENCY OF APPEARANCE OF CERTAIN WORDS ABOUT FAMILY LIFE, PLAY, AND
ACTION IN EIGHT ELEMENTARY TEXTBOOKS IN ARITHMETIC, pp. 1-50.

  ================================================================
                  |  A  |  B  |  C  |  D  |  E  |  F  |  G  |  H
  ----------------+-----+-----+-----+-----+-----+-----+-----+-----
  baby            |     |     |     |  2  |     |  4  |     |
  brother         |  2  |     |  6  |  1  |  1  |     |  1  |
  family          |     |     |  2  |     |  2  |     |  4  |
  father          |  1  |     |  3  |  5  |     |  2  |  1  |
  help            |     |     |     |     |     |     |     |
  home            |  2  |     |  4  |  4  |  2  |  2  |  7  |  1
  mother          |  4  |  2  |  9  |  5  |     |  5  |  1  |  7
  sister          |     |     |  1  |  2  |  2  |  9  |  1  |  1
                  |     |     |     |     |     |     |     |
  fork            |     |     |     |     |     |     |     |
  knife           |     |     |     |     |     |     |     |
  plate           |  4  |  2  |     |  2  |     |  1  |     |
  spoon           |     |     |     |     |     |     |     |
                  |     |     |     |     |     |     |     |
  doll            | 10  |  1  | 10  |  6  |     | 10  |     |  9
  game            |  1  |     |     |  3  |     |     |  5  |  5
  jump            |     |     |     |     |     |     |     |  4
  marbles         | 10  |  4  | 10  |     | 10  |     |  1  |
  play            |     |     |  1  |     |     |  3  |     |
  run             |     |     |     |     |     |  1  |     |  3
  sing            |     |     |     |     |     |     |     |
  tag             |     |     |     |     |     |     |     |
  toy             |     |     |     |     |     |     |     |  1
                  |     |     |     |     |     |     |     |
  car             |     |     |  2  |  4  |     |  2  |  3  |  1
  cut             |     |     | 10  |     |  6  |  2  |     |  8
  dig             |     |     |     |     |     |     |  2  |
  flower          |  1  |     |     |  4  |  1  |  1  |  2  |
  grow            |     |     |     |  1  |     |     |     |
  plant           |     |     |  2  |     |     |     |     |
  seed            |     |     |     |  3  |     |     |  1  |
  string          |     |     |     |     |  1  | 10  |  1  |  1
  wheel           |  5  |     |     |     |     | 10  |     |
  ================================================================

The words used by textbooks give some indication of how far this aim is
being realized, or rather of how far short we are of realizing it.
Consider, for example, the words home, mother, father, brother, sister,
help, plate, knife, fork, spoon, play, game, toy, tag, marbles, doll,
run, jump, sing, plant, seed, grow, flower, car, wheel, string, cut,
dig. The frequency of appearance in the first fifty pages of eight
beginners' arithmetics was as shown in Table 9. The eight columns refer
to the eight books (the first fifty pages of each). The numbers refer to
the number of times the word in question appeared, the number 10 meaning
10 _or more_ times in the fifty pages. Plurals, past tenses, and the
like were counted. _Help_, _fork_, _knife_, _spoon_, _jump_, _sing_, and
_tag_ did not appear at all! _Toy_ and _grow_ appeared each once in the
400 pages! _Play_, _run_, _dig_, _plant_, and _seed_ appeared once in a
hundred or more pages. _Baby_ did not appear as often as _buggy_.
_Family_ appeared no oftener than _fence_ or _Friday_. _Father_ appears
about a third as often as _farmer_.

Book A shows only 10 of these thirty words in the fifty pages; book B
only 4; book C only 12; and books D, E, F, G, and H only 13, 8, 14, 13,
10, respectively. The total number of appearances (counting the 10s as
only 10 in each case) is 40 for A, 9 for B, 60 for C, 42 for D, 25 for
E, 62 for F, 30 for G, and 37 for H. The five words--apple, egg, Mary,
milk, and orange--are used oftener than all these thirty together.

If it appeared that this apparent neglect of childish affairs and
interests was deliberate to provide for a more systematic treatment of
pure arithmetic, a better gradation of problems, and a better
preparation for later genuine use than could be attained if the author
of the textbook were tied to the child's apron strings, the neglect
could be defended. It is not at all certain that children in grade 2 get
much more enjoyment or ability from adding the costs of purchases for
Christmas or Fourth of July, or multiplying the number of cakes each
child is to have at a party by the number of children who are to be
there, than from adding gravestones or multiplying the number of hairs
of bald-headed men. When, however, there is nothing gained by
substituting remote facts for those of familiar concern to children, the
safe policy is surely to favor the latter. In general, the neglect of
childish data does not seem to be due to provision for some other end,
but to the same inertia of tradition which has carried over the problems
of laying walls and digging wells into city schools whose children never
saw a stone wall or dug well.

       *       *       *       *       *

I shall not go into details concerning the arrangement of courses of
study, textbooks, and lesson-plans to make desirable connections between
arithmetical learning and sports, housework, shopwork, and the rest. It
may be worth while, however, to explain the term _self-management_,
since this source of genuine problems of real concern to the pupils has
been overlooked by most writers.

By self-management is meant the pupil's use of his time, his abilities,
his knowledge, and the like. By the time he reaches grade 5, and to some
extent before then, a boy should keep some account of himself, of how
long it takes him to do specified tasks, of how much he gets done in a
specified time at a certain sort of work and with how many errors, of
how much improvement he makes month by month, of which things he can do
best, and the like. Such objective, matter-of-fact, quantitative study
of one's behavior is not a stimulus to morbid introspection or egotism;
it is one of the best preventives of these. To treat oneself
impersonally is one of the essential elements of mental balance and
health. It need not, and should not, encourage priggishness. On the
contrary, this matter-of-fact study of what one is and does may well
replace a certain amount of the exhortations and admonitions concerning
what one ought to do and be. All this is still truer for a girl.

The demands which such an accounting of one's own activities make of
arithmetic have the special value of connecting directly with the
advanced work in computation. They involve the use of large numbers,
decimals, averaging, percentages, approximations, and other facts and
processes which the pupil has to learn for later life, but to which his
childish activities as wage-earner, buyer and seller, or shopworker from
10 to 14 do not lead. Children have little money, but they have time in
thousands of units! They do not get discounts or bonuses from commercial
houses, but they can discount their quantity of examples done for the
errors made, and credit themselves with bonuses of all sorts for extra
achievements.


INTRINSIC INTEREST IN ARITHMETICAL LEARNING

There remains the most important increase of interest in arithmetical
learning--an increase in the interest directly bound to achievement and
success in arithmetic itself. "Arithmetic," says David Eugene Smith, "is
a game and all boys and girls are players." It should not be a _mere_
game for them and they should not _merely_ play, but their unpractical
interest in doing it because they can do it and can see how well they do
do it is one of the school's most precious assets. Any healthy means to
give this interest more and better stimulus should therefore be eagerly
sought and cherished.

Two such means have been suggested in other connections. The first is
the extension of training in checking and verifying work so that the
pupil may work to a standard of approximately 100% success, and may
know how nearly he is attaining it. The second is the use of
standardized practice material and tests, whereby the pupil may measure
himself against his own past, and have a clear, vivid, and trustworthy
idea of just how much better or faster he can do the same tasks than he
could do a month or a year ago, and of just how much harder things he
can do now than then.

Another means of stimulating the essential interest in quantitative
thinking itself is the arrangement of the work so that real arithmetical
thinking is encouraged more than mere imitation and assiduity. This
means the avoidance of long series of applied problems all of one type
to be solved in the same way, the avoidance of miscellaneous series and
review series which are almost verbatim repetitions of past problems,
and in general the avoidance of excessive repetition of any one
problem-situation. Stimulation to real arithmetical thinking is weak
when a whole day's problem work requires no choice of methods, or when a
review simply repeats without any step of organization or progress, or
when a pupil meets a situation (say the 'buy _x_ things at _y_ per
thing, how much pay' situation) for the five-hundredth time.

Another matter worthy of attention in this connection is the unwise
tendency to omit or present in diluted form some of the topics that
appeal most to real intellectual interests, just because they are hard.
The best illustration, perhaps, is the problem of ratio or "How many
times as large (long, heavy, expensive, etc.) as _x_ is _y_?" Mastery of
the 'times as' relation is hard to acquire, but it is well worth
acquiring, not only because of its strong intellectual appeal, but also
because of its prime importance in the applications of arithmetic to
science. In the older arithmetics it was confused by pedantries and
verbal difficulties and penalized by unreal problems about fractions of
men doing parts of a job in strange and devious times. Freed from these,
it should be reinstated, beginning as early as grade 5 with such simple
exercises as those shown below and progressing to the problems of food
values, nutritive ratios, gears, speeds, and the like in grade 8.

  John is 4 years old.
  Fred is 6 years old.
  Mary is 8 years old.
  Nell is 10 years old.
  Alice is 12 years old.
  Bert is 15 years old.

  Who is twice as old as John?
  Who is half as old as Alice?
  Who is three times as old as John?
  Who is one and one half times as old as Nell?
  Who is two thirds as old as Fred?
      etc., etc., etc.

  Alice is .... times as old as John.
  John is .... as old as Mary.
  Fred is .... times as old as John.
  Alice is .... times as old as Fred.
  Fred is .... as old as Mary.
      etc., etc., etc.

Finally it should be remembered that all improvements in making
arithmetic worth learning and helping the pupil to learn it will in the
long run add to its interest. Pupils like to learn, to achieve, to gain
mastery. Success is interesting. If the measures recommended in the
previous chapters are carried out, there will be little need to entice
pupils to take arithmetic or to sugar-coat it with illegitimate
attractions.




CHAPTER XIII

THE CONDITIONS OF LEARNING


We shall consider in this chapter the influence of time of day, size of
class, and amount of time devoted to arithmetic in the school program,
the hygiene of the eyes in arithmetical work, the use of concrete
objects, and the use of sounds, sights, and thoughts as situations and
of speech and writing and thought as responses.[17]

  [17] Facts concerning the conditions of learning in general will
       be found in the author's _Educational Psychology_, Vol. 2,
       Chapter 8, or in the _Educational Psychology, Briefer Course_,
       Chapter 15.


EXTERNAL CONDITIONS

Computation of one or another sort has been used by several
investigators as a test of efficiency at different times in the day.
When freed from the effects of practice on the one hand and lack of
interest due to repetition on the other, the results uniformly show an
increase in speed late in the school session with a falling off in
accuracy that about balances it.[18] There is no wisdom in putting
arithmetic early in the session because of its _difficulty_. Lively and
sociable exercises in mental arithmetic with oral answers in fact seem
to be admirably fitted for use late in the session. Except for the
general principles (1) of starting the day with work that will set a
good standard of cheerful, efficient production and (2) of getting the
least interesting features of the day's work done fairly early in the
day, psychology permits practical exigencies to rule the program, so far
as present knowledge extends. Adequate measurements of the effect of
time of day on _improvement_ have not been made, but there is no reason
to believe that any one time between 9 A.M. and 4 P.M. is appreciably
more favorable to arithmetical learning than to learning geography,
history, spelling, and the like.

  [18] See Thorndike ['00], King ['07], and Heck ['13].

The influence of size of class upon progress in school studies is very
difficult to measure because (1) within the same city system the average
of the six (or more) sizes of class that a pupil has experienced will
tend to approximate closely to the corresponding average for any other
child; because further (2) there may be a tendency of supervisory
officers to assign more pupils to the better teachers; and because
(3) separate systems which differ in respect to size of class probably
differ in other respects also so that their differences in achievement
may be referable to totally different differences.

Elliott ['14] has made a beginning by noting size of class during the
year of test in connection with his own measures of the achievements of
seventeen hundred pupils, supplemented by records from over four hundred
other classes. As might be expected from the facts just stated, he finds
no appreciable difference between classes of different sizes within the
same school system, the effect of the few months in a small class being
swamped by the antecedents or concomitants thereof.

The effect of the amount of time devoted to arithmetic in the school
program has been studied extensively by Rice ['02 and '03] and Stone
['08].

Dr. Rice ['02] measured the arithmetical ability of some 6000 children
in 18 different schools in 7 different cities. The results of these
measurements are summarized in Table 10. This table "gives two averages
for each grade as well as for each school as a whole. Thus, the school
at the top shows averages of 80.0 and 83.1, and the one at the bottom,
25.3 and 31.5. The first represents the percentage of answers which were
absolutely correct; the second shows what per cent of the problems were
correct in principle, _i.e._ the average that would have been received
if no mechanical errors had been made."

The facts of Dr. Rice's table show that there is a positive relation
between the general standing of a school system in the tests and the
amount of time devoted to arithmetic by its program. The relation is
not close, however, being that expressed by a correlation coefficient
of .36-1/2. Within any one school system there is no relation between
the standing of a particular school and the amount of time devoted to
arithmetic in that school's program. It must be kept in mind that the
amount of time given in the school program may be counterbalanced by
emphasizing work at home and during study periods, or, on the other
hand, may be a symptom of correspondingly small or great emphasis on
arithmetic in work set for the study periods at home.

A still more elaborate investigation of this same topic was made by
Stone ['08]. I quote somewhat fully from it, since it is an instructive
sample of the sort of studies that will doubtless soon be made in the
case of every elementary school subject. He found that school systems
differed notably in the achievements made by their sixth-grade pupils in
his tests of computation (the so-called 'fundamentals') and of the
solution of verbally described problems (the so-called 'reasoning'). The
facts were as shown in Table 11.

TABLE  10

AVERAGES FOR INDIVIDUAL SCHOOLS IN ARITHMETIC

  KEY    A: CITY
         B: SCHOOL
         C: Result
         D: Principle
         E: Percent of Mechanical Errors
         F: Minutes Daily

  ===========================================================
     |   |6TH YEAR |7TH YEAR |8TH YEAR |SCHOOL AVERAGE |
     |   |----+----+----+----+----+----+----+----+-----+-----
   A | B | C  | D  | C  | D  | C  | D  | C  | D  |  E  |  F
  ---+---+----+----+----+----+----+----+----+----+-----+-----
  III| 1 |79.3|80.3|81.1|82.3|91.7|93.9|80.0|83.1| 3.7 |  53
    I| 1 |80.4|81.5|64.2|67.2|80.9|82.8|76.6|80.3| 4.6 |  60
    I| 2 |80.9|83.4|43.5|50.9|72.7|79.1|69.3|75.1| 7.7 |  25
    I| 3 |72.2|74.0|63.5|66.2|74.5|76.6|67.8|72.2| 6.1 |  45
    I| 4 |69.9|72.2|54.6|57.8|66.5|69.1|64.3|70.3| 8.5 |  45
   II| 1 |71.2|75.3|33.6|35.7|36.8|40.0|60.2|64.8| 7.1 |  60
  III| 2 |43.7|45.0|53.9|56.7|51.1|53.1|54.5|58.9| 7.4 |  60
   IV| 1 |58.9|60.4|31.2|34.1|41.6|43.5|55.1|58.4| 5.6 |  60
   IV| 2 |59.8|63.1| -- | -- |22.5|22.5|53.9|58.8| 8.3 |  --
   IV| 3 |54.9|58.1|35.2|38.6|43.5|45.0|51.5|57.6|10.5 |  60
   IV| 4 |42.3|45.1|16.1|19.2|48.7|48.7|42.8|48.2|11.2 |  --
    V| 1 |44.1|48.7|29.2|32.5|51.1|58.3|45.9|51.3|10.5 |  40
   VI| 1 |68.3|71.3|33.5|36.6|26.9|30.7|39.0|42.9| 9.0 |  33
   VI| 2 |46.1|49.5|19.5|24.2|30.2|40.6|36.5|43.6|16.2 |  30
   VI| 3 |34.5|36.4|30.5|35.1|23.3|24.1|36.0|42.5|15.2 |  48
  VII| 1 |35.2|37.7|29.1|32.5|25.1|27.2|40.5|45.9|11.7 |  42
  VII| 2 |35.2|38.7|15.0|16.4|19.6|21.2|36.5|40.6|10.1 |  75
  VII| 3 |27.6|33.7| 8.9|10.1|11.3|11.3|25.3|31.5|19.6 |  45
  ===========================================================

High achievement by a system in computation went with high achievement
in solving the problems, the correlation being about .50; and the
system that scored high in addition or subtraction or multiplication or
division usually showed closely similar excellence in the other three,
the correlations being about .90.

TABLE 11

SCORES MADE BY THE SIXTH-GRADE PUPILS OF EACH OF TWENTY-SIX SCHOOL
SYSTEMS

  =================================================
  SYSTEM | SCORE IN TESTS WITH | SCORE IN TESTS IN
         |      PROBLEMS       |     COMPUTING
  -------+---------------------+-------------------
    23   |         356         |       1841
    24   |         429         |       3513
    17   |         444         |       3042
     4   |         464         |       3563
    25   |         464         |       2167
    22   |         468         |       2311
    16   |         469         |       3707
    20   |         491         |       2168
    18   |         509         |       3758
    15   |         532         |       2779
     3   |         533         |       2845
     8   |         538         |       2747
     6   |         550         |       3173
     1   |         552         |       2935
    10   |         601         |       2749
     2   |         615         |       2958
    21   |         627         |       2951
    13   |         636         |       3049
    14   |         661         |       3561
     9   |         691         |       3404
     7   |         734         |       3782
    12   |         736         |       3410
    11   |         759         |       3261
    26   |         791         |       3682
    19   |         848         |       4099
     5   |         914         |       3569
  =================================================

Of the conditions under which arithmetical learning took place, the one
most elaborately studied was the amount of time devoted to arithmetic.
On the basis of replies by principals of schools to certain questions,
he gave each of the twenty-six school systems a measure for the
probable time spent on arithmetic up through grade 6. Leaving home study
out of account, there seems to be little or no correlation between the
amount of time a system devotes to arithmetic and its score in
problem-solving, and not much more between time expenditure and score in
computation. With home study included there is little relation to the
achievement of the system in solving problems, but there is a clear
effect on achievement in computation. The facts as given by Stone are:--

TABLE 12

CORRELATION OF TIME EXPENDITURES WITH ABILITIES

  Without Home Study      { Reasoning and Time Expenditure        -.01
                          { Fundamentals and Time Expenditure      .09

  Including Home Study    { Reasoning and Time Expenditure         .13
                          { Fundamentals and Time Expenditure      .49

These correlations, it should be borne in mind, are for school systems,
not for individual pupils. It might be that, though the system which
devoted the most time to arithmetic did not show corresponding
superiority in the product over the system devoting only half as much
time, the pupils within the system did achieve in exact proportion to
the time they gave to study. Neither correlation would permit inference
concerning the effect of different amounts of time spent by the same
pupil.

Stone considered also the printed announcements of the courses of study
in arithmetic in these twenty-six systems. Nineteen judges rated these
announced courses of study for excellence according to the instructions
quoted below:--

CONCERNING THE RATING OF COURSES OF STUDY

Judges please read before scoring

I. Some Factors Determining Relative Excellence.

(N. B. The following enumeration is meant to be suggestive rather than
complete or exclusive. And each scorer is urged to rely primarily on his
own judgment.)

  1. Helpfulness to the teacher in teaching the subject matter outlined.

  2. Social value or concreteness of sources of problems.

  3. The arrangement of subject matter.

  4. The provision made for adequate drill.

  5. A reasonable minimum requirement with suggestions for valuable
     additional work.

  6. The relative values of any predominating so-called methods--such as
     Speer, Grube, etc.

  7. The place of oral or so-called mental arithmetic.

  8. The merit of textbook references.

II. Cautions and Directions.

(Judges please follow as implicitly as possible.)

  1. Include references to textbooks as parts of the Course of Study.

     This necessitates judging the parts of the texts referred to.

  2. As far as possible become equally familiar with all courses before
     scoring any.

  3. When you are ready to begin to score, (1) arrange in serial
     order according to excellence, (2) starting with the middle one
     score it 50, then score above and below 50 according as courses
     are better or poorer, indicating relative differences in
     excellence by relative differences in scores, _i.e._ in so far
     as you find that the courses differ by about equal steps, score
     those better than the middle one 51, 52, etc., and those poorer
     49, 48, etc., but if you find that the courses differ by
     unequal steps show these inequalities by omitting numbers.

  4. Write ratings on the slip of paper attached to each course.

The systems whose courses of study were thus rated highest did not
manifest any greater achievement in Stone's tests than the rest. The
thirteen with the most approved announcements of courses of study were
in fact a little inferior in achievement to the other thirteen, and the
correlation coefficients were slightly negative.

Stone also compared eighteen systems where there was supervision of the
work by superintendents or supervisors as well as by principals with
four systems where the principals and teachers had no such help. The
scores in his tests were very much lower in the four latter cities.


THE HYGIENE OF THE EYES IN ARITHMETIC

We have already noted that the task of reading and copying numbers is
one of the hardest that the eyes have to perform in the elementary
school, and that it should be alleviated by arranging much of the work
so that only answers need be written by the pupil. The figures to be
read and copied should obviously be in type of suitable size and style,
so arranged and spaced on the page or blackboard as to cause a minimum
of effort and strain.

    [Illustration: FIG. 26.--Type too large.]

    [Illustration: FIG. 27.--12-point, 11-point, and 10-point type.]

_Size._--Type may be too large as well as too small, though the latter
is the commoner error. If it is too large, as in Fig. 26, which is a
duplicate of type actually used in a form of practice pad, the eye has
to make too many fixations to take in a given content. All things
considered, 12-point type in grades 3 and 4, 11-point in grades 5 and 6,
and 10-point in grades 7 and 8 seem the most desirable sizes. These are
shown in Fig. 27. Too small type occurs oftenest in fractions and in the
dimension-numbers or scale numbers of drawings. Figures 28, 29, and 30
are samples from actual school practice. Samples of the desirable size
are shown in Figs. 31 and 32. The technique of modern typesetting makes
it very difficult and expensive to make fractions of the horizontal type

  (1  3  5
   -  -  -
   4, 8, 6)

large enough without making the whole-number figures with which they
are mingled too large or giving an uncouth appearance to the total.
Consequently fractions somewhat smaller than are desirable may have
to be used occasionally in textbooks.[19] There is no valid excuse,
however, for the excessively small fractions which often are made in
blackboard work.

  [19] A special type could be constructed that would use a large
       type body, say 14 point, with integers in 10 or 12 point and
       fractions much larger than now.

    [Illustration: FIG. 28.--Type of measurements too small.

        This is a picture of Mary's garden. How many feet is it
        around the garden?]

    [Illustration: FIG. 29.--Type too small.]

    [Illustration: FIG. 30.--Numbers too small and badly designed.]

    [Illustration: FIG. 31.--Figure 28 with suitable numbers.]

    [Illustration: FIG. 32.--Figure 30 with suitable numbers.]

_Style._--The ordinary type forms often have 3 and 8 so made as to
require strain to distinguish them. 5 is sometimes easily confused with
3 and even with 8. 1, 4, and 7 may be less easily distinguishable than
is desirable. Figure 33 shows a specially good type in which each figure
is represented by its essential[20] features without any distracting
shading or knobs or turns. Figure 34 shows some of the types in common
use. There are no demonstrably great differences amongst these. In
fractions there is a notable gain from using the slant form (2/3, 3/4)
for exercises in addition and subtraction, and for almost all mixed
numbers. This appears clearly to the eye in the comparison of Fig. 35
below, where the same fractions all in 10-point type are displayed in
horizontal and in slant form. The figures in the slant form are in
general larger and the space between them and the fraction-line is
wider. Also the slant form makes it easier for the eye to examine the
denominators to see whether reductions are necessary. Except for a few
cases to show that the operations can be done just as truly with the
horizontal forms, the book and the blackboard should display mixed
numbers and fractions to be added or subtracted in the slant form. The
slant line should be at an angle of approximately 45 degrees. Pupils
should be taught to use this form in their own work of this sort.

  [20] It will be still better if the 4 is replaced by an open-top 4.

When script figures are presented they should be of simple design,
showing clearly the essential features of the figure, the line being
everywhere of equal or nearly equal width (that is, without shading, and
without ornamentation or eccentricity of any sort). The opening of the 3
should be wide to prevent confusion with 8; the top of the 3 should be
curved to aid its differentiation from 5; the down stroke of the 9
should be almost or quite straight; the 1, 4, 7, and 9 should be clearly
distinguishable. There are many ways of distinguishing them clearly, the
best probably being to use the straight line for 1, the open 4 with
clear angularity, a wide top to the 7, and a clearly closed curve for
the top of the 9.

    [Illustration: FIG. 33.--Block type; a very desirable type except
    that it is somewhat too heavy.]

    [Illustration: FIG. 34.--Common styles of printed numbers.]

    [Illustration: FIG. 35.--Diagonal and horizontal fractions
    compared.]

    [Illustration: FIG. 36.--Good vertical spacing.]

    [Illustration: FIG. 37.--Bad vertical spacing.]

    [Illustration: FIGS. 38 (above) and 39 (below).--Good and bad
    left-right spacing.]

The pupil's writing of figures should be clear. He will thereby be saved
eyestrain and errors in his school work as well as given a valuable
ability for life. Handwriting of figures is used enormously in spite of
the development of typewriters; illegible figures are commonly more
harmful than illegible letters or words, since the context far less
often tells what the figure is intended to be; the habit of making clear
figures is not so hard to acquire, since they are written unjoined and
require only the automatic action of ten minor acts of skill. The
schools have missed a great opportunity in this respect. Whereas the
hand writing of words is often better than it needs to be for life's
purposes, the writing of figures is usually much worse. The figures
presented in books on penmanship are also commonly bad, showing neglect
or misunderstanding of the matter on the part of leaders in penmanship.

_Spacing._--Spacing up and down the column is rarely too wide, but very
often too narrow. The specimens shown in Figs. 36 and 37 show good
practice contrasted with the common fault.

Spacing from right to left is generally fairly satisfactory in books,
though there is a bad tendency to adopt some one routine throughout and
so to miss chances to use reductions and increases of spacing so as to
help the eye and the mind in special cases. Specimens of good and bad
spacing are shown in Figs. 38 and 39. In the work of the pupils, the
spacing from right to left is often too narrow. This crowding of
letters, together with unevenness of spacing, adds notably to the task
of eye and mind.

_The composition or make-up of the page._--Other things being equal,
that arrangement of the page is best which helps a child most to keep
his place on a page and to find it after having looked away to work on
the paper on which he computes, or for other good reasons. A good page
and a bad page in this respect are shown in Figs. 40 and 41.

    [Illustration: FIG. 40.--A page well made up to suit the action
    of the eye.]

    [Illustration: FIG. 41.--The same matter as in Fig. 40, much
    less well made up.]

_Objective presentations._--Pictures, diagrams, maps, and other
presentations should not tax the eye unduly,

  (_a_) by requiring too fine distinctions, or

  (_b_) by inconvenient arrangement of the data, preventing easy
        counting, measuring, comparison, or whatever the task is, or

  (_c_) by putting too many facts in one picture so that the eye
        and mind, when trying to make out any one, are confused by the
        others.

Illustrations of bad practices in these respects are shown in Figs. 42
to 52. A few specimens of work well arranged for the eye are shown in
Figs. 53 to 56.

Good rules to remember are:--

Other things being equal, make distinctions by the clearest method, fit
material to the tendency of the eye to see an 'eyeful' at a time
(roughly 1-1/2 inch by 1/2 inch in a book; 1-1/2 ft. by 1/2 ft. on the
blackboard), and let one picture teach only one fact or relation, or
such facts and relations as do not interfere in perception.

The general conditions of seating, illumination, paper, and the like are
even more important when the eyes are used with numbers than when they
are used with words.

    [Illustration: FIG. 42.--Try to count the rungs on the ladder,
    or the shocks in the wagon.]

    [Illustration: FIG. 43.--How many oars do you see? How many
    birds? How many fish?]

    [Illustration: FIG. 44.--Count the birds in each of the three
    flocks of birds.]

    [Illustration: FIG. 45.--Note the lack of clear division of the
    hundreds. Consider the difficulty of counting one of these
    columns of dots.]

    [Illustration: FIG. 46.--What do you suppose these pictures are
    intended to show?]

    [Illustration: FIG. 47.--Would a beginner know that after
    THIRTEEN he was to switch around and begin at the other end?
    Could you read the SIX of TWENTY-SIX if you did not already know
    what it ought to be? What meaning would all the brackets have
    for a little child in grade 2? Does this picture illustrate or
    obfuscate?]

    [Illustration: FIG. 48.--How long did it take you to find out
    what these pictures mean?]

    [Illustration: FIG. 49.--Count the figures in the first row,
    using your eyes alone; have some one make lines of 10, 11, 12,
    13, and more repetitions of this figure spaced closely as here.
    Count 20 or 30 such lines, using the eye unaided by fingers,
    pencil, etc. ]

    [Illustration: FIG. 50.--Can you answer the question without
    measuring? Could a child of seven or eight?]

    [Illustration: FIG. 51.--What are these drawings intended to
    show? Why do they show the facts only obscurely and dubiously?]

    [Illustration: FIG. 52.--What are these drawings intended to
    show? What simple change would make them show the facts much
    more clearly?]

    [Illustration: FIG. 53.--Arranged in convenient "eye-fulls."]

    [Illustration: FIG. 54.--Clear, simple, and easy of comparison.]

    [Illustration: FIG. 55.--Clear, simple, and well spaced.]

    [Illustration: FIG. 56.--Well arranged, though a little wider
    spacing between the squares would make it even better.]


THE USE OF CONCRETE OBJECTS IN ARITHMETIC

We mean by concrete objects actual things, events, and relations
presented to sense, in contrast to words and numbers and symbols which
mean or stand for these objects or for more abstract qualities and
relations. Blocks, tooth-picks, coins, foot rules, squared paper, quart
measures, bank books, and checks are such concrete things. A foot rule
put successively along the three thirds of a yard rule, a bell rung five
times, and a pound weight balancing sixteen ounce weights are such
concrete events. A pint beside a quart, an inch beside a foot, an apple
shown cut in halves display such concrete relations to a pupil who is
attentive to the issue.

Concrete presentations are obviously useful in arithmetic to teach
meanings under the general law that a word or number or sign or symbol
acquires meaning by being connected with actual things, events,
qualities, and relations. We have also noted their usefulness as means
to verifying the results of thinking and computing, as when a pupil,
having solved, "How many badges each 5 inches long can be made from
3-1/3 yd. of ribbon?" by using 10 × 12/5, draws a line 3-1/3 yd. long
and divides it into 5-inch lengths.

Concrete experiences are useful whenever the meaning of a number, like 9
or 7/8 or .004, or of an operation, like multiplying or dividing or
cubing, or of some term, like rectangle or hypothenuse or discount, or
some procedure, like voting or insuring property against fire or
borrowing money from a bank, is absent or incomplete or faulty. Concrete
work thus is by no means confined to the primary grades but may be
appropriate at all stages when new facts, relations, and procedures are
to be taught.

How much concrete material shall be presented will depend upon the fact
or relation or procedure which is to be made intelligible, and the
ability and knowledge of the pupil. Thus 'one half' will in general
require less concrete illustration than 'five sixths'; and five sixths
will require less in the case of a bright child who already knows 2/3,
3/4, 3/8, 5/8, 7/8, 2/5, 3/5, and 4/5 than in the case of a dull child
or one who only knows 2/3 and 3/4. As a general rule the same topic will
require less concrete material the later it appears in the school
course. If the meanings of the numbers are taught in grade 2 instead of
grade 1, there will be less need of blocks, counters, splints, beans,
and the like. If 1-1/2 + 1/2 = 2 is taught early in grade 3, there will
be more gain from the use of 1-1/2 inches and 1/2 inch on the foot rule
than if the same relations were taught in connection with the general
addition of like fractions late in grade 4. Sometimes the understanding
can be had either by connecting the idea with the reality directly, or
by connecting the two indirectly _via_ some other idea. The amount of
concrete material to be used will depend on its relative advantage per
unit of time spent. Thus it might be more economical to connect 5/12,
7/12, and 11/12 with real meanings indirectly by calling up the
resemblance to the 2/3, 3/4, 3/8, 5/8, 7/8, 2/5, 3/5, 4/5, and 5/6
already studied, than by showing 5/12 of an apple, 7/12 of a yard, 11/12
of a foot, and the like.

In general the economical course is to test the understanding of the
matter from time to time, using more concrete material if it is needed,
but being careful to encourage pupils to proceed to the abstract ideas
and general principles as fast as they can. It is wearisome and
debauching to pupils' intellects for them to be put through elaborate
concrete experiences to get a meaning which they could have got
themselves by pure thought. We should also remember that the new idea,
say of the meaning of decimal fractions, will be improved and clarified
by using it (see page 183 f.), so that the attainment of a _perfect_
conception of decimal fractions before doing anything with them is
unnecessary and probably very wasteful.

A few illustrations may make these principles more instructive.

(_a_) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000,
need more concrete aids than are commonly given. Guessing contests about
the value in dollars of the school building and other buildings, the
area of the schoolroom floor and other surfaces in square inches, the
number of minutes in a week, and year, and the like, together with
proper computations and measurements, are very useful to reënforce the
concrete presentations and supply genuine problems in multiplication and
subtraction with large numbers.

(_b_) Numbers very much smaller than one, such as 1/32, 1/64, .04,
and .002, also need some concrete aids. A diagram like that of
Fig. 57 is useful.

(_c_) _Majority_ and _plurality_ should be understood by every citizen.
They can be understood without concrete aid, but an actual vote is well
worth while for the gain in vividness and surety.

    [Illustration: FIG. 57.--Concrete aid to understanding fractions
    with large denominators. A = 1/1000 sq. ft.; B = 1/100 sq. ft.;
    C = 1/50 sq. ft.; D = 1/10 sq. ft.]

(_d_) Insurance against loss by fire can be taught by explanation and
analogy alone, but it will be economical to have some actual insuring
and payment of premiums and a genuine loss which is reimbursed.

(_e_) Four play banks in the corners of the room, receiving deposits,
cashing checks, and later discounting notes will give good educational
value for the time spent.

(_f_) Trade discount, on the contrary, hardly requires more concrete
illustration than is found in the very problems to which it is applied.

(_g_) The process of finding the number of square units in a rectangle
by multiplying with the appropriate numbers representing length and
width is probably rather hindered than helped by the ordinary objective
presentation as an introduction. The usual form of objective
introduction is as follows:--

    [Illustration: FIG. 58.]

    How long is this rectangle? How large is each square? How many
    square inches are there in the top row? How many rows are
    there? How many square inches are there in the whole rectangle?
    Since there are three rows each containing 4 square inches, we
    have 3 × 4 square inches = 12 square inches.

    Draw a rectangle 7 inches long and 2 inches wide. If you divide
    it into inch squares how many rows will there be? How many inch
    squares will there be in each row? How many square inches are
    there in the rectangle?

    [Illustration: FIG. 59.]

It is better actually to hide the individual square units as in Fig. 59.
There are four reasons: (1) The concrete rows and columns rather
distract attention from the essential thing to be learned. This is not
that "_x_ rows one square wide, _y_ squares in a row will make _xy_
squares in all," but that "by using proper units and the proper
operation the area of any rectangle can be found from its length and
width." (2) Children have little difficulty in learning to multiply
rather than add, subtract, or divide when computing area. (3) The habit
so formed holds good for areas like 1-2/3 by 4-1/2, with fractional
dimensions, in which any effort to count up the areas of rows is very
troublesome and confusing. (4) The notion that a square inch is an area
1' by 1' rather than 1/2' by 2' or 1/3 in. by 3 in. or 1-1/2 in. by 2/3
in. is likely to be formed too emphatically if much time is spent upon
the sort of concrete presentation shown above. It is then better to use
concrete counting of rows of small areas as a means of _verification
after_ the procedure is learned, than as a means of deriving it.

There has been, especially in Germany, much argument concerning what
sort of number-pictures (that is, arrangement of dots, lines, or the
like, as shown in Fig. 60) is best for use in connection with the number
names in the early years of the teaching of arithmetic.

Lay ['98 and '07], Walsemann ['07], Freeman ['10], Howell ['14], and
others have measured the accuracy of children in estimating the number
of dots in arrangements of one or more of these different types.[21]
Many writers interpret a difference in favor of estimating, say, the
square arrangements of Born or Lay as meaning that such is the best
arrangement to use in teaching. The inference is, however, unjustified.
That certain number-pictures are easier to estimate numerically does not
necessarily mean that they are more instructive in learning. One set may
be easier to estimate just because they are more familiar, having been
oftener experienced. Even if the favored set was so after equal
experience with all sets, accuracy of estimation would be a sign of
superiority for use in instruction only if all other things were equal
(or in favor of the arrangement in question). Obviously the way to
decide which of these is best to use in teaching is by using them in
teaching and measuring all relevant results, not by merely recording
which of them are most accurately estimated in certain time exposures.

  [21] For an account in English of their main findings see
       Howell ['14], pp. 149-251.

It may be noted that the Born, Lay, and Freeman pictures have claims for
special consideration on grounds of probable instructiveness. Since they
are also superior in the tests in respect to accuracy of estimate,
choice should probably be made from these three by any teacher who
wishes to connect one set of number-pictures systematically with the
number names, as by drills with the blackboard or with cards.

    [Illustration: FIG. 60.--Various proposed arrangements of dots
    for use in teaching the meanings of the numbers 1 to 10.]

Such drills are probably useful if undertaken with zeal, and if kept as
supplementary to more realistic objective work with play money, children
marching, material to be distributed, garden-plot lengths to be
measured, and the like, and if so administered that the pupils soon get
the generalized abstract meaning of the numbers freed from dependence on
an inner picture of any sort. This freedom is so important that it may
make the use of many types of number-pictures advisable rather than the
use of the one which in and of itself is best.

As Meumann says: "Perceptual reckoning can be overdone. It had its chief
significance for the surety and clearness of the first foundation of
arithmetical instruction. If, however, it is continued after the first
operations become familiar to the child, and extended to operations
which develop from these elementary ones, it necessarily works as a
retarding force and holds back the natural development of arithmetic.
This moves on to work with abstract number and with mechanical
association and reproduction." ['07, Vol. 2, p. 357.]

Such drills are commonly overdone by those who make use of them, being
given too often, and continued after their instructiveness has waned,
and used instead of more significant, interesting, and varied work in
counting and estimating and measuring real things. Consequently, there
is now rather a prejudice against them in our better schools. They
should probably be reinstated but to a moderate and judicious use.


ORAL, MENTAL, AND WRITTEN ARITHMETIC

There has been much dispute over the relative merits of oral and written
work in arithmetic--a question which is much confused by the different
meanings of 'oral' and 'written.' _Oral_ has meant (1) work where the
situations are presented orally and the pupil's final responses are
given orally, or (2) work where the situations are presented orally and
the pupils' final responses are written or partly written and partly
oral, or (3) work where the situations are presented in writing or print
and the final responses are oral. _Written_ has meant (1) work where the
situations are presented in writing or print and the final responses are
made in writing, or (2) work where also many of the intermediate
responses are written, or (3) work where the situations are presented
orally but the final responses and a large percentage of the
intermediate computational responses are written. There are other
meanings than these.

It is better to drop these very ambiguous terms and ask clearly what are
the merits and demerits, in the case of any specified arithmetical work,
of auditory and of visual presentation of the situations, and of saying
and of writing each specified step in the response.

The disputes over mental _versus_ written arithmetic are also confused
by ambiguities in the use of 'mental.' Mental has been used to mean
"done without pencil and paper" and also "done with few overt
responses, either written or spoken, between the setting of the task and
the announcement of the answer." In neither case is the word _mental_
specially appropriate as a description of the total fact. As before, we
should ask clearly, "What are the merits and demerits of making certain
specified intermediate responses in inner speech or imaged sounds or
visual images or imageless thought--that is, _without_ actual writing or
overt speech?"

It may be said at the outset that oral, written, and inner presentations
of initial situations, oral, written, and inner announcements of final
responses, and oral, written, and inner management of intermediate
processes have varying degrees of merit according to the particular
arithmetical exercise, pupil, and context. Devotion to oralness or
mentalness as such is simply fanatical. Various combinations, such as
the written presentation of the situation with inner management of the
intermediate responses and oral announcement of the final response have
their special merits for particular cases.

These merits the reader can evaluate for himself for any given sort of
work for a given class by considering: (1) The amount of practice
received by the class per hour spent; (2) the ease of correction of the
work; (3) the ease of understanding the tasks; (4) the prevention of
cheating; (5) the cheerfulness and sociability of the work; (6) the
freedom from eyestrain, and other less important desiderata.

It should be noted that the stock schemes A, B, C, and D below are only
a few of the many that are possible and that schemes E, F, G, and H have
special merits.

    PRESENTATION OF         MANAGEMENT OF           ANNOUNCEMENT OF
    INITIAL SITUATION       INTERMEDIATE PROCESSES  FINAL RESPONSE

  A. Printed or written      Written                 Written

  B.    "          "         Inner                   Oral by one pupil,
                                                     inner by the rest

  C. Oral (by teacher)       Written                 Written

  D.   "         "           Inner                   Oral by one pupil,
                                                     inner by the rest

  E. As in A or C            A mixture, the pupil    As in A or B or H
                            writing what he needs

  F. The real situation      As in E                 As in A or B or H
     itself, in part at
     least

  G. Both read by the pupil  As in E                 As in A or B or H
     and put orally by the
     teacher

  H. As in A or C or G       As in E                 Written by all
                                                     pupils, announced
                                                     orally by one pupil

The common practice of either having no use made of pencil and paper or
having all computations and even much verbal analysis written out
elaborately for examination is unfavorable for learning. The demands
which life itself will make of arithmetical knowledge and skill will
range from tasks done with every percentage of written work from zero up
to the case where every main result obtained by thought is recorded for
later use by further thought. In school the best way is that which, for
the pupils in question, has the best total effect upon quality of
product, speed, and ease of production, reënforcement of training
already given, and preparation for training to be given. There is
nothing intellectually criminal about using a pencil as well as inner
thought; on the other hand there is no magical value in writing out for
the teacher's inspection figures that the pupil does not need in order
to attain, preserve, verify, or correct his result.

The common practice of having the final responses of all _easy_ tasks
given orally has no sure justification. On the contrary, the great
advantage of having all pupils really do the work should be secured in
the easy work more than anywhere else. If the time cost of copying the
figures is eliminated by the simple plan of having them printed, and if
the supervision cost of examining the papers is eliminated by having the
pupils correct each other's work in these easy tasks, written answers
are often superior to oral except for the elements of sociability and
'go' and freedom from eyestrain of the oral exercise. Such written work
provides the gifted pupils with from two to ten times as much practice
as they would get in an oral drill on the same material, supposing them
to give inner answers to every exercise done by the class as a whole; it
makes sure that the dull pupils who would rarely get an inner answer at
the rate demanded by the oral exercise, do as much as they are able to
do.

Two arguments often made for the oral statement of problems by the
teacher are that problems so put are better understood, especially in
the grades up through the fifth, and that the problems are more likely
to be genuine and related to the life the pupils know. When these
statements are true, the first is a still better argument for having the
pupils read the problems _aided by the teacher's oral statement of
them_. For the difficulty is largely that the pupils cannot read well
enough; and it is better to help them to surmount the difficulty rather
than simply evade it. The second is not an argument for oralness
_versus_ writtenness, but for good problems _versus_ bad; the teacher
who makes up such good problems should, in fact, take special care to
write them down for later use, which may be by voice or by the
blackboard or by printed sheet, as is best.




CHAPTER XIV

THE CONDITIONS OF LEARNING: THE PROBLEM ATTITUDE


Dewey, and others following him, have emphasized the desirability of
having pupils do their work as active seekers, conscious of problems
whose solution satisfies some real need of their own natures. Other
things being equal, it is unwise, they argue, for pupils to be led along
blindfold as it were by the teacher and textbook, not knowing where they
are going or why they are going there. They ought rather to have some
living purpose, and be zealous for its attainment.

This doctrine is in general sound, as we shall see, but it is often
misused as a defense of practices which neglect the formation of
fundamental habits, or as a recommendation to practices which are quite
unworkable under ordinary classroom conditions. So it seems probable
that its nature and limitations are not thoroughly known, even to its
followers, and that a rather detailed treatment of it should be given
here.


ILLUSTRATIVE CASES

Consider first some cases where time spent in making pupils understand
the end to be attained before attacking the task by which it is
attained, or care about attaining the end (well or ill understood) is
well spent.

It is well for a pupil who has learned (1) the meanings of the numbers
one to ten, (2) how to count a collection of ten or less, and (3) how to
measure in inches a magnitude of ten, nine, eight inches, etc., to be
confronted with the problem of true adding without counting or
measuring, as in 'hidden' addition and measurement by inference. For
example, the teacher has three pencils counted and put under a book; has
two more counted and put under the book; and asks, "How many pencils are
there under the book?" Answers, when obtained, are verified or refuted
by actual counting and measuring.

The time here is well spent because the children can do the necessary
thinking if the tasks are well chosen; because they are thereby
prevented from beginning their study of addition by the bad habit of
pseudo-adding by looking at the two groups of objects and counting their
number instead of real adding, that is, thinking of the two numbers and
inferring their sum; and further, because facing the problem of adding
as a real problem is in the end more economical for learning arithmetic
and for intellectual training in general than being enticed into adding
by objective or other processes which conceal the difficulty while
helping the pupil to master it.

The manipulation of short multiplication may be introduced by
confronting the pupils with such problems as, "How to tell how many
Uneeda biscuit there are in four boxes, by opening only one box."
Correct solutions by addition should be accepted. Correct solutions by
multiplication, if any gifted children think of this way, should be
accepted, even if the children cannot justify their procedure.
(Inferring the manipulation from the place-values of numbers is beyond
all save the most gifted and probably beyond them.) Correct solution by
multiplication by some child who happens to have learned it elsewhere
should be accepted. Let the main proof of the trustworthiness of the
manipulation be by measurement and by addition. Proof by the stock
arguments from the place-values of numbers may also be used. If no child
hits on the manipulation in question, the problem of finding the length
_without_ adding may be set. If they still fail, the problem may be made
easier by being put as "4 times 22 gives the answer. Write down what you
think 4 times 22 will be." Other reductions of the difficulty of the
problem may be made, or the teacher may give the answer without very
great harm being done. The important requirement is that the pupils
should be aware of the problem and treat the manipulation as a solution
of it, not as a form of educational ceremonial which they learn to
satisfy the whims of parents and teachers. In the case of any particular
class a situation that is more appealing to the pupils' practical
interests than the situation used here can probably be devised.

The time spent in this way is well spent (1) because all but the very
dull pupils can solve the problem in some way, (2) because the
significance of the manipulation as an economy over addition is worth
bringing out, and (3) because there is no way of beginning training in
short multiplication that is much better.

In the same fashion multiplication by two-place numbers may be
introduced by confronting pupils with the problem of the number of
sheets of paper in 72 pads, or pieces of chalk in 24 boxes, or square
inches in 35 square feet, or the number of days in 32 years, or whatever
similar problem can be brought up so as to be felt as a problem.

Suppose that it is the 35 square feet. Solutions by (5 × 144) +
(30 × 144), however arranged, or by (10 × 144) + (10 × 144) +
(10 × 144) + (5 × 144), or by 3500 + (35 × 40) + (35 × 4), or by
7 × (5 × 144), however arranged, should all be listed for verification
or rejection. The pupils need not be required to justify their
procedures by a verbal statement. Answers like 432,720, or 720,432,
or 1152, or 4220, or 3220 should be listed for verification or
rejection. Verification may be by a mixture of short multiplication
and objective work, or by a mixture of short multiplication and
addition, or by addition abbreviated by taking ten 144s as 1440, or
even (for very stupid pupils) by the authority of the teacher. Or the
manipulation in cases like 53 × 9 or 84 × 7 may be verified by the
reverse short multiplication. The deductive proof of the correctness
of the manipulation may be given in whole or in part in connection
with exercises like

  10 ×   2 =          30 ×  14 =
  10 ×   3 =           3 ×  44 =
  10 ×   4 =          30 ×  44 =
  10 ×  14 =           3 × 144 =
  10 ×  44 =          20 × 144 =
  10 × 144 =          40 × 144 =
  20 ×   2 =          30 × 144 =
  20 ×   3 =           5 × 144 =
  30 ×   3 =          35 = 30 + ....
  30 ×   4 =          30 × 144 added to 5 × 144 =
   3 ×  14 =

Certain wrong answers may be shown to be wrong in many ways; _e.g._,
432,720 is too big, for 35 times a thousand square inches is only
35,000; 1152 is too small, for 35 times a hundred square inches would be
3500, or more than 1152.

The time spent in realizing the problem here is fairly well spent
because (1) any successful original manipulation in this case
represents an excellent exercise of thought, because (2) failures show
that it is useless to juggle the figures at random, and because (3) the
previous experience with short multiplication makes it possible for the
pupils to realize the problem in a very few minutes. It may, however, be
still better to give the pupils the right method just as soon as the
problem is realized, without having them spend more time in trying to
solve it. Thus:--

1 square foot has 144 square inches. How many square inches are there in
35 square feet (marked out in chalk on the floor as a piece 10 ft. × 3
ft. plus a piece 5 ft. × 1 ft.)?

1 yard = 36 inches. How many inches long is this wall (found by measure
to be 13 yards)?

Here is a quick way to find the answers:--

   144
    35
  ----
   720
  432
  ----
  5040 sq. inches in 35 sq. ft.

   36
   13
  ---
  108
  36
  ---
  468 inches in 13 yd.

Consider now the following introduction to dividing by a decimal:--

  Dividing by a Decimal

  1. How many minutes will it take a motorcycle, to go 12.675 miles
     at the rate of .75 mi. per minute?

                                      16.9
                                      ______
                                  .75|12.675
                                       7 5
                                       ---
                                       5 17
                                       4 50
                                       ----
                                         675
                                         675
                                         ---

  2. Check by multiplying 16.9 by .75.

  3. How do you know that the quotient cannot be as little as 1.69?

  4. How do you know that the quotient cannot be as large as 169?

  5. Find the quotient for 3.75 ÷ 1.5.

  6. Check your result by multiplying the quotient by the divisor.

  7. How do you know that the quotient cannot be .25 or 25?
                                  ____
  8. Look at this problem.     .25|7.5

     How do you know that 3.0 is wrong for the quotient?

     How do you know that 300 is wrong for the quotient?

  State which quotient is right for each of these:--

      .021 or .21 or 2.1 or 21 or 210
         ______
  9.  1.8|3.78


      .021 or .21 or 21 or 210
         ______
  10. 1.8|37.8


      .03 or .3 or 3 or 30 or 300
          ______
  11. 1.25|37.5


      .03 or .3 or 3 or 30 or 300
          ______
  12. 12.5|37.5


      .05 or .5 or 5 or 50 or 500
          ______
  13. 1.25|6.25


      .05 or .5 or 5 or 50 or 500
          ______
  14. 12.5|6.25


  15. Is this rule true? If it is true, learn it.

        #In a correct result, the number of decimal places in
        the divisor and quotient together equals the number
        of decimal places in the dividend.#

These and similar exercises excite the problem attitude in children _who
have a general interest in getting right answers_. Such a series
carefully arranged is a desirable introduction to a statement of the
rule for placing the decimal point in division with decimals. For it
attracts attention to the general principle (divisor × quotient should
equal dividend), which is more important than the rule for convenient
location of the decimal point, and it gives training in placing the
point by inspection of the divisor, quotient, and dividend, which
suffices for nineteen out of twenty cases that the pupil will ever
encounter outside of school. He is likely to remember this method by
inspection long after he has forgotten the fixed rule.

It is well for the pupil to be introduced to many arithmetical facts by
way of problems about their common uses. The clockface, the railroad
distance table in hundredths of a mile, the cyclometer and speedometer,
the recipe, and the like offer problems which enlist his interest and
energy and also connect the resulting arithmetical learning with the
activities where it is needed. There is no time cost, but a time-saving,
for the learning as a means to the solution of the problems is quicker
than the mere learning of the arithmetical facts by themselves alone. A
few samples of such procedure are shown below:--

  GRADE 3

  To be Done at Home

  Look at a watch. Has it any hands besides the hour hand and the
  minute hand? Find out all that you can about how a watch tells
  seconds, how long a second is, and how many seconds make a minute.


  GRADE 5

  Measuring Rainfall

         =Rainfall per Week=
    (=cu. in. per sq. in. of area=)
    June   1-7      1.056
           8-14     1.103
          15-21     1.040
          22-28      .960
          29-July 5  .915
    July   6-12      .782
          13-19      .790
          20-26      .670
          27-Aug. 2  .503
    Aug.   3-9       .512
          10-16      .240
          17-23      .215
          24-30      .811

  1. In which weeks was the rainfall 1 or more?

  2. Which week of August had the largest rainfall for that month?

  3. Which was the driest week of the summer? (Driest means with
     the least rainfall.)

  4. Which week was the next to the driest?

  5. In which weeks was the rainfall between .800 and 1.000?

  6. Look down the table and estimate whether the average rainfall for
     one week was about .5, or about .6, or about .7, or about .8, or
     about .9.

  Dairy Records

    =Record of Star Elsie=

          Pounds of Milk     Butter-Fat per Pound of Milk
      Jan.    1742                  .0461
      Feb.    1690                  .0485
      Mar.    1574                  .0504
      Apr.    1226                  .0490
      May     1202                  .0466
      June    1251                  .0481

  Read this record of the milk given by the cow Star Elsie. The first
  column tells the number of pounds of milk given by Star Elsie each
  month. The second column tells what fraction of a pound of butter-fat
  each pound of milk contained.

  1. Read the first line, saying, "In January this cow gave 1742 pounds
      of milk. There were 461 ten thousandths of a pound of butter-fat
      per pound of milk." Read the other lines in the same way.

  2. How many pounds of butter-fat did the cow produce in Jan.?
      3. In Feb.? 4. In Mar.? 5. In Apr.? 6. In May? 7. In June?


  GRADE 5 OR LATER

  Using Recipes to Make Larger or Smaller Quantities

  I. State how much you would use of each material in the following
  recipes: (_a_) To make double the quantity. (_b_) To make half the
  quantity. (_c_) To make 1-1/2 times the quantity. You may use pencil
  and paper when you cannot find the right amount mentally.

    1. PEANUT PENUCHE

      1 tablespoon butter
      2 cups brown sugar
      1/3 cup milk or cream
      3/4 cup chopped peanuts
      1/3 teaspoon salt

    2. MOLASSES CANDY

      1/2 cup butter
      2 cups sugar
      1 cup molasses
      1-1/2 cups boiling water

    3. RAISIN OPERA CARAMELS

      2 cups light brown sugar
      7/8 cup thin cream
      1/2 cup raisins

    4. WALNUT MOLASSES SQUARES

      2 tablespoons butter
      1 cup molasses
      1/3 cup sugar
      1/2 cup walnut meats

    5. RECEPTION ROLLS

      1 cup scalded milk
      1-1/2 tablespoons sugar
      1 teaspoon salt
      1/4 cup lard
      1 yeast cake
      1/4 cup lukewarm water
      White of 1 egg
      3-1/2 cups flour

    6. GRAHAM RAISED LOAF

      2 cups milk
      6 tablespoons molasses
      1-1/2 teaspoons salt
      1/3 yeast cake
      1/4 cup lukewarm water
      2 cups sifted Graham flour
      1/2 cup Graham bran
      3/4 cup flour (to knead)

  II. How much would you use of each material in the following recipes:
  (_a_) To make 2/3 as large a quantity? (_b_) To make 1-1/3 times as
  much? (_c_) To make 2-1/2 times as much?

    1. ENGLISH DUMPLINGS

      1/2 pound beef suet
      1-1/4 cups flour
      3 teaspoons baking powder
      1 teaspoon salt
      1/2 teaspoon pepper
      1 teaspoon minced parsley
      1/2 cup cold water

    2. WHITE MOUNTAIN ANGEL CAKE

      1-1/2 cups egg whites
      1-1/2 cups sugar
      1 teaspoon cream of tartar
      1 cup bread flour
      1/4 teaspoon salt
      1 teaspoon vanilla

In many cases arithmetical facts and principles can be well taught in
connection with some problem or project which is not arithmetical, but
which has special potency to arouse an intellectual activity in the
pupil which by some ingenuity can be directed to arithmetical learning.
Playing store is the most fundamental case. Planning for a party, seeing
who wins a game of bean bag, understanding the calendar for a month,
selecting Christmas presents, planning a picnic, arranging a garden, the
clock, the watch with second hand, and drawing very simple maps are
situations suggesting problems which may bring a living purpose into
arithmetical learning in grade 2. These are all available under ordinary
conditions of class instruction. A sample of such problems for a higher
grade (6) is shown below.

  Estimating Areas

  The children in the geography class had a contest in estimating
  the areas of different surfaces. Each child wrote his estimates
  for each of these maps, A, B, C, D, and E. (Only C and D are
  shown here.) In the arithmetic class they learned how to find
  the exact areas. Then they compared their estimates with the
  exact areas to find who came nearest.

    [Illustration]

  Write your estimates for A, B, C, D, and E. Then study the
  next 6 pages and learn how to find the exact areas.

  (The next 6 pages comprise training in the mensuration of
  parallelograms and triangles.)

In some cases the affairs of individual pupils include problems which
may be used to guide the individual in question to a zealous study of
arithmetic as a means of achieving his purpose--of making a canoe,
surveying an island, keeping the accounts of a Girls' Canning Club, or
the like. It requires much time and very great skill to direct the work
of thirty or more pupils each busy with a special type of his own, so as
to make the work instructive for each, but in some cases the expense of
time and skill is justified.


GENERAL PRINCIPLES

In general what should be meant when one says that it is desirable to
have pupils in the problem-attitude when they are studying arithmetic is
substantially as follows:--

_First._--Information that comes as an answer to questions is better
attended to, understood, and remembered than information that just
comes.

_Second._--Similarly, movements that come as a step toward achieving an
end that the pupil has in view are better connected with their
appropriate situations, and such connections are longer retained, than
is the case with movements that just happen.

_Third._--The more the pupil is set toward getting the question answered
or getting the end achieved, the greater is the satisfyingness attached
to the bonds of knowledge or skill which mean progress thereto.

_Fourth._--It is bad policy to rely exclusively on the purely
intellectualistic problems of "How can I do this?" "How can I get the
right answer?" "What is the reason for this?" "Is there a better way to
do that?" and the like. It is bad policy to supplement these
intellectualistic problems by only the remote problems of "How can I be
fitted to earn a higher wage?" "How can I make sure of graduating?" "How
can I please my parents?" and the like. The purely intellectualistic
problems have too weak an appeal for many pupils; the remote problems
are weak so long as they are remote and, what is worse, may be deprived
of the strength that they would have in due time if we attempt to use
them too soon. It is the extreme of bad policy to neglect those personal
and practical problems furnished by life outside the class in arithmetic
the solution of which can really be furthered by arithmetic then and
there. It is good policy to spend time in establishing certain mental
sets--stimulating, or even creating, certain needs--setting up problems
themselves--when the time so spent brings a sufficient improvement in
the quality and quantity of the pupils' interest in arithmetical
learning.

_Fifth._--It would be still worse policy to rely exclusively on
problems arising outside arithmetic. To learn arithmetic is itself a
series of problems of intrinsic interest and worth to healthy-minded
children. The need for ability to multiply with United States money or
to add fractions or to compute percents may be as truly vital and
engaging as the need for skill to make a party dress or for money to buy
it or for time to play baseball. The intellectualistic needs and
problems should be considered along with all others, and given whatever
weight their educational value deserves.


DIFFICULTY AND SUCCESS AS STIMULI

There are certain misconceptions of the doctrine of the
problem-attitude. The most noteworthy is that difficulty--temporary
failure--an inadequacy of already existing bonds--is the essential and
necessary stimulus to thinking and learning. Dewey himself does not, as
I understand him, mean this, but he has been interpreted as meaning it
by some of his followers.[22]

  [22] In his _How We Think_.

Difficulty--temporary failure, inadequacy of existing bonds--on the
contrary does nothing whatsoever in and of itself; and what is done by
the annoying lack of success which sometimes accompanies difficulty
sometimes hinders thinking and learning.

Mere difficulty, mere failure, mere inadequacy of existing bonds, does
nothing. It is hard for me to add three eight-place numbers at a glance;
I have failed to find as effective illustrations for pages 276 to 277 as
I wished; my existing sensori-motor connections are inadequate to
playing a golf course in 65. But these events and conditions have done
nothing to stimulate me in respect to the behavior in question. In the
first of the three there is no annoying lack and no dynamic influence at
all; in the second there was to some degree an annoying lack--a slight
irritation at not getting just what I wanted,--and this might have
impelled me to further thinking (though it did not, and getting one
tiptop illustration would as a rule stimulate me to hunt for others more
than failing to get such). In the third case the lack of the 65 does not
annoy me or have any noteworthy dynamic effect. The lack of 90 instead
of 95-100 is annoying and is at times a stimulus to further learning,
though not nearly so strong a stimulus as the attainment of the 90 would
be! At other times this annoying lack is distinctly inhibitory--a
stimulus to ceasing to learn. In the intellectual life the inhibitory
effect seems far the commoner of the two. Not getting answers seems as a
rule to make us stop trying to get them. The annoying lack of success
with a theoretical problem most often makes us desert it for problems to
whose solution the existing bonds promise to be more adequate.

The real issue in all this concerns the relative strength, in the
pupil's intellectual life, of the "negative reaction" of behavior in
general. An animal whose life processes are interfered with so that an
annoying state of affairs is set up, changes his behavior, making one
after another responses as his instincts and learned tendencies
prescribe, until the annoying state of affairs is terminated, or the
animal dies, or suffers the annoyance as less than the alternatives
which his responses have produced. When the annoying state of affairs is
characterized by the failure of things as they are to minister to a
craving--as in cases of hunger, loneliness, sex-pursuit, and the
like,--we have stimulus to action by an annoying lack or need, with
relief from action by the satisfaction of the need.

Such is in some measure true of man's intellectual life. In recalling a
forgotten name, in solving certain puzzles, or in simplifying an
algebraic complex, there is an annoying lack of the name, solution, or
factor, a trial of one after another response, until the annoyance is
relieved by success or made less potent by fatigue or distraction. Even
here the _difficulty_ does not do anything--but only the annoying
interference with our intellectual peace by the problem. Further,
although for the particular problem, the annoying lack stimulates, and
the successful attainment stops thinking, the later and more important
general effect on thinking is the reverse. Successful attainment stops
our thinking _on that problem_ but makes us more predisposed later to
thinking _in general_.

Overt negative reaction, however, plays a relatively small part in man's
intellectual life. Filling intellectual voids or relieving intellectual
strains in this way is much less frequent than being stimulated
positively by things seen, words read, and past connections acting under
modified circumstances. The notion of thinking as coming to a lack,
filling it, meeting an obstacle, dodging it, being held up by a
difficulty and overcoming it, is so one-sided as to verge on phantasy.
The overt lacks, strains, and difficulties come perhaps once in five
hours of smooth straightforward use and adaptation of existing
connections, and they might as truly be called hindrances to
thought--barriers which past successes help the thinker to surmount.
Problems themselves come more often as cherished issues which new facts
reveal, and whose contemplation the thinker enjoys, than as strains or
lacks or 'problems which I need to solve.' It is just as true that the
thinker gets many of his problems as results from, or bonuses along
with, his information, as that he gets much of his information as
results of his efforts to solve problems.

As between difficulty and success, success is in the long run more
productive of thinking. Necessity is not the mother of invention.
Knowledge of previous inventions is the mother; original ability is the
father. The solutions of previous problems are more potent in producing
both new problems and their solutions than is the mere awareness of
problems and desire to have them solved.

In the case of arithmetic, learning to cancel instead of getting the
product of the dividends and the product of the divisors and dividing
the former by the latter, is a clear case of very valuable learning,
with ease emphasized rather than difficulty, with the adequacy of
existing bonds (when slightly redirected) as the prime feature of the
process rather than their inadequacy, and with no sense of failure or
lack or conflict. It would be absurd to spend time in arousing in the
pupil, before beginning cancellation, a sense of a difficulty--viz.,
that the full multiplying and dividing takes longer than one would like.
A pupil in grade 4 or 5 might well contemplate that difficulty for years
to no advantage. He should at once begin to cancel and prove by checking
that errorless cancellation always gives the right answer. To emphasize
before teaching cancellation the inadequacy of the old full multiplying
and dividing would, moreover, not only be uneconomical as a means to
teaching cancellation; it would amount to casting needless slurs on
valuable past acquisitions, and it would, scientifically, be false.
For, until a pupil has learned to cancel, the old full multiplying is
not inadequate; it is admirable in every respect. The issue of its
inadequacy does not truly appear until the new method is found. It is
the best way until the better way is mastered.

In the same way it is unwise to spend time in making pupils aware of the
annoying lacks to be supplied by the multiplication tables, the division
tables, the casting out of nines, or the use of the product of the
length and breadth of a rectangle as its area, the unit being changed to
the square erected on the linear unit as base. The annoying lack will
be unproductive, while the learning takes place readily as a
modification of existing habits, and is sufficiently appreciated as soon
as it does take place. The multiplication tables come when instead of
merely counting by 7s from 0 up saying "7, 14, 21," etc., the pupil
counts by 7s from 0 up saying "Two sevens make 14, three sevens make 21,
four sevens make 28," etc. The division tables come as easy selections
from the known multiplications; the casting out of nines comes as an
easy device. The computation of the area of a rectangle is best
facilitated, not by awareness of the lack of a process for doing it, but
by awareness of the success of the process as verified objectively.

In all these cases, too, the pupil would be misled if we aroused first a
sense of the inadequacy of counting, adding, and objective division, an
awareness of the difficulties which the multiplication and division
tables and nines device and area theorem relieve. The displaced
processes are admirable and no unnecessary fault should be found with
them, and they are _not_ inadequate until the shorter ways have been
learned.


FALSE INFERENCES

One false inference about the problem-attitude is that the pupil should
always understand the aim or issue before beginning to form the bonds
which give the method or process that provides the solution. On the
contrary, he will often get the process more easily and value it more
highly if he is taught what it is _for_ gradually while he is learning
it. The system of decimal notation, for example, may better be taken
first as a mere fact, just as we teach a child to talk without trying
first to have him understand the value of verbal intercourse, or to keep
clean without trying first to have him understand the bacteriological
consequences of filth.

A second inference--that the pupil should always be taught to care about
an issue and crave a process for managing it before beginning to learn
the process--is equally false. On the contrary, the best way to become
interested in certain issues and the ways of handling them is to learn
the process--even to learn it by sheer habituation--and then note what
it does for us. Such is the case with ".1666-2/3 × = divide by 6,"
".333-1/3 × = divide by 3," "multiply by .875 = divide the number by 8
and subtract the quotient from the number."

A third unwise tendency is to degrade the mere giving of information--to
belittle the value of facts acquired in any other way than in the course
of deliberate effort by the pupil to relieve a problem or conflict or
difficulty. As a protest against merely verbal knowledge, and merely
memoriter knowledge, and neglect of the active, questioning search for
knowledge, this tendency to belittle mere facts has been healthy, but as
a general doctrine it is itself equally one-sided. Mere facts not got by
the pupil's thinking are often of enormous value. They may stimulate to
active thinking just as truly as that may stimulate to the reception of
facts. In arithmetic, for example, the names of the numbers, the use of
the fractional form to signify that the upper number is divided by the
lower number, the early use of the decimal point in U. S. money to
distinguish dollars from cents, and the meanings of "each," "whole,"
"part," "together," "in all," "sum," "difference," "product,"
"quotient," and the like are self-justifying facts.

A fourth false inference is that whatever teaching makes the pupil face
a question and think out its answer is thereby justified. This is not
necessarily so unless the question is a worthy one and the answer that
is thought out an intrinsically valuable one and the process of thinking
used one that is appropriate for that pupil for that question. Merely
to think may be of little value. To rely much on formal discipline is
just as pernicious here as elsewhere. The tendency to emphasize the
methods of learning arithmetic at the expense of what is learned is
likely to lead to abuses different in nature but as bad in effect as
that to which the emphasis on disciplinary rather than content value has
led in the study of languages and grammar, or in the old puzzle problems
of arithmetic.

The last false inference that I shall discuss here is the inference that
most of the problems by which arithmetical learning is stimulated had
better be external to arithmetic itself--problems about Noah's Ark or
Easter Flowers or the Merry Go Round or A Trip down the Rhine.

Outside interests should be kept in mind, as has been abundantly
illustrated in this volume, but it is folly to neglect the power, even
for very young or for very stupid children, of the problem "How can I
get the right answer?" Children do have intellectual interests. They do
like dominoes, checkers, anagrams, and riddles as truly as playing tag,
picking flowers, and baking cake. With carefully graded work that is
within their powers they like to learn to add, subtract, multiply, and
divide with integers, fractions, and decimals, and to work out
quantitative relations.

In some measure, learning arithmetic is like learning to typewrite. The
learner of the latter has little desire to present attractive-looking
excuses for being late, or to save expense for paper. He has no desire
to hoard copies of such and such literary gems. He may gain zeal from
the fact that a school party is to be given and invitations are to be
sent out, but the problem "To typewrite better" is after all his main
problem. Learning arithmetic is in some measure a game whose moves are
motivated by the general set of the mind toward victory--winning right
answers. As a ball-player learns to throw the ball accurately to
first-base, not primarily because of any particular problem concerning
getting rid of the ball, or having the man at first-base possess it, or
putting out an opponent against whom he has a grudge, but because that
skill is required by the game as a whole, so the pupil, in some measure,
learns the technique of arithmetic, not because of particular concrete
problems whose solutions it furnishes, but because that technique is
required by the game of arithmetic--a game that has intrinsic worth and
many general recommendations.




CHAPTER XV

INDIVIDUAL DIFFERENCES


The general facts concerning individual variations in abilities--that
the variations are large, that they are continuous, and that for
children of the same age they usually cluster around one typical or
modal ability, becoming less and less frequent as we pass to very high
or very low degrees of the ability--are all well illustrated by
arithmetical abilities.


NATURE AND AMOUNT

The surfaces of frequency shown in Figs. 61, 62, and 63 are samples. In
these diagrams each space along the baseline represents a certain score
or degree of ability, and the height of the surface above it represents
the number of individuals obtaining that score. Thus in Fig. 61, 63 out
of 1000 soldiers had no correct answer, 36 out of 1000 had one correct
answer, 49 had two, 55 had three, 67 had four, and so on, in a test with
problems (stated in words).

Figure 61 shows that these adults varied from no problems solved
correctly to eighteen, around eight as a central tendency. Figure 62
shows that children of the same year-age (they were also from the same
neighborhood and in the same school) varied from under 40 to over 200
figures correct. Figure 63 shows that even among children who have all
reached the same school grade and so had rather similar educational
opportunities in arithmetic, the variation is still very great. It
requires a range from 15 to over 30 examples right to include even nine
tenths of them.

    [Illustration: FIG. 61.--The scores of 1000 soldiers in the
    National Army born in English-speaking countries, in Test 2 of the
    Army Alpha. The score is the number of correct answers obtained in
    five minutes. Probably 10 to 15 percent of these men were unable to
    read or able to read only very easy sentences at a very slow rate.
    Data furnished by the Division of Psychology in the office of the
    Surgeon General.]

It should, however, be noted that if each individual had been scored by
the average of his work on eight or ten different days instead of by his
work in just one test, the variability would have been somewhat less
than appears in Figs. 61, 62, and 63.

    [Illustration: FIG. 62.--The scores of 100 11-year-old pupils
    in a test of computation. Estimated from the data given by Burt
    ['17, p. 68] for 10-, 11-, and 12-year-olds. The score equals the
    number of correct figures.]

It is also the case that if each individual had been scored, not in
problem-solving alone or division alone, but in an elaborate examination
on the whole field of arithmetic, the variability would have been
somewhat less than appears in Figs. 61, 62, and 63. On the other hand,
if the officers and the soldiers rejected for feeblemindedness had been
included in Fig. 61, if the 11-year-olds in special classes for the very
dull had been included in Fig. 62, and if all children who had been to
school six years had been included in Fig. 63, no matter what grade they
had reached, the effect would have been to _increase_ the variability.

    [Illustration: FIG. 63.--The scores of pupils in grade 6 in city
    schools in the Woody Division Test A. The score is the number of
    correct answers obtained in 20 minutes. From Woody ['16, p. 61].]

In spite of the effort by school officers to collect in any one school
grade those somewhat equal in ability or in achievement or in a mixture
of the two, the population of the same grades in the same school system
shows a very wide range in any arithmetical ability. This is partly
because promotion is on a more general basis than arithmetical ability
so that some very able arithmeticians are deliberately held back on
account of other deficiencies, and some very incompetent arithmeticians
are advanced on account of other excellencies. It is partly because of
general inaccuracy in classifying and promoting pupils.

In a composite score made up of the sum of the scores in Woody
tests,--Add. A, Subt. A, Mult. A, and Div. A, and two tests in
problem-solving (ten and six graded problems, with maximum attainable
credits of 30 and 18), Kruse ['18] found facts from which I compute
those of Table 13, and Figs. 64 to 66, for pupils all having the
training of the same city system, one which sought to grade its pupils
very carefully.

    [Illustration: FIGS. 64, 65, and 66.--The scores of pupils in
    grade 6 (Fig. 64), grade 7 (Fig. 65), and grade 8 (Fig. 66) in a
    composite of tests in computation and problem-solving. The time
    was about 120 minutes. The maximum score attainable was 196.]

The overlapping of grade upon grade should be noted. Of the pupils in
grade 6 about 18 percent do better than the average pupil in grade 7,
and about 7 percent do better than the average pupil in grade 8. Of the
pupils in grade 8 about 33 percent do worse than the average pupil in
grade 7 and about 12 percent do worse than the average pupil in grade 6.

TABLE 13

RELATIVE FREQUENCIES OF SCORES IN AN EXTENSIVE TEAM OF ARITHMETICAL
TESTS.[23] IN PERCENTS

  ==============================================
     SCORE    |  GRADE 6  |  GRADE 7  |  GRADE 8
  ------------+-----------+-----------+---------
    70 to  79 |    1.3    |     .9    |     .4
    80  "  89 |    5.5    |    2.3    |     .4
    90  "  99 |   10.6    |    4.3    |    2.9
   100  " 109 |   19.4    |    5.2    |    4.4
   110  " 119 |   19.8    |   18.5    |    5.8
   120  " 129 |   23.5    |   16.2    |   16.8
   130  " 139 |   12.6    |   17.5    |   16.8
   140  " 149 |    4.6    |   13.9    |   22.9
   150  " 159 |    1.7    |   13.6    |   17.1
   160  " 169 |    1.2    |    4.8    |    9.4
   170  " 179 |           |    2.5    |    3.3
  ==============================================

  [23] Compiled from data on p. 89 of Kruse ['18].


DIFFERENCES WITHIN ONE CLASS

The variation within a single class for which a single teacher has to
provide is great. Even when teaching is departmental and promotion is by
subjects, and when also the school is a large one and classification
within a grade is by ability--there may be a wide range for any given
special component ability. Under ordinary circumstances the range is so
great as to be one of the chief limiting conditions for the teaching of
arithmetic. Many methods appropriate to the top quarter of the class
will be almost useless for the bottom quarter, and _vice versa_.

    [Illustration: FIGS. 67 and 68.--The scores of ten 6 B classes in
    a 12-minute test in computation with integers (the Courtis Test 7).
    The score is the number of units done. Certain long tasks are
    counted as two units.]

Figures 67 and 68 show the scores of ten classes taken at random from
ninety 6 B classes in one city by Courtis ['13, p. 64] in amount of
computation done in 12 minutes. Observe the very wide variation present
in the case of every class. The variation within a class would be
somewhat reduced if each pupil were measured by his average in eight or
ten such tests given on different days. If a rather generous allowance
is made for this we still have a variation in speed as great as that
shown in Fig. 69, as the fact to be expected for a class of thirty-two 6
B pupils.

    [Illustration: FIG. 69.--A conservative estimate of the amount of
    variation to be expected within a single class of 32 pupils in
    grade 6, in the number of units done in Courtis Test 7 when all
    chance variations are eliminated.]

The variations within a class in respect to what processes are
understood so as to be done with only occasional errors may be
illustrated further as follows:--A teacher in grade 4 at or near the
middle of the year in a city doing the customary work in arithmetic will
probably find some pupil in her class who cannot do column addition even
without carrying, or the easiest written subtraction

  (8   9      78)
  (5   3  or  37)
  (-   -      --),

who does not know his multiplication tables or how to derive them, or
understand the meanings of + - × and ÷, or have any useful ideas
whatever about division.

There will probably be some child in the class who can do such work as
that shown below, and with very few errors.

  Add       3/8 + 5/8 + 7/8 + 1/8       2-1/2       1/6 + 3/8
                                        6-3/8
                                        3-3/4
                                        -----

  Subtract     10.00                4 yd.    1 ft.   6 in.
                3.49                2 yd.    2 ft.   3 in.
               -----                ----------------------

  Multiply    1-1/4 × 8           16              145
                                   2-5/8          206
                                  ------          ---
                   _______               _____
  Divide          2)13.50              25)9750

The invention of means of teaching thirty so different children at once
with the maximum help and minimum hindrance from their different
capacities and acquisitions is one of the great opportunities for
applied science.

Courtis, emphasizing the social demand for a certain moderate
arithmetical attainment in the case of nearly all elementary school
children of, say, grade 6, has urged that definite special means be
taken to bring the deficient children up to certain standards, without
causing undesirable 'overlearning' by the more gifted children. Certain
experimental work to this end has been carried out by him and others,
but probably much more must be done before an authoritative program for
securing certain minimum standards for all or nearly all pupils can be
arranged.


THE CAUSES OF INDIVIDUAL DIFFERENCES

The differences found among children of the same grade in the same city
are due in large measure to inborn differences in their original
natures. If, by a miracle, the children studied by Courtis, or by Woody,
or by Kruse had all received exactly the same nurture from birth to
date, they would still have varied greatly in arithmetical ability,
perhaps almost as much as they now do vary.

The evidence for this is the general evidence that variation in original
nature is responsible for much of the eventual variation found in
intellectual and moral traits, plus certain special evidence in the case
of arithmetical abilities themselves.

Thorndike found ['05] that in tests with addition and multiplication
twins were very much more alike than siblings[24] two or three years
apart in age, though the resemblance in home and school training in
arithmetic should be nearly as great for the latter as for the former.
Also the young twins (9-11) showed as close a resemblance in addition
and multiplication as the older twins (12-15), although the similarities
of training in arithmetic have had twice as long to operate in the
latter case.

  [24] Siblings is used for children of the same parents.

If the differences found, say among children in grade 6 in addition,
were due to differences in the quantity and quality of training in
addition which they have had, then by giving each of them 200 minutes of
additional identical training the differences should be reduced. For the
200 minutes of identical training is a step toward equalizing training.
It has been found in many investigations of the matter that when we make
training in arithmetic more nearly equal for any group the variation
within the group is not reduced.

On the contrary, equalizing training seems rather to increase
differences. The superior individual seems to have attained his
superiority by his own superiority of nature rather than by superior
past training, for, during a period of equal training for all, he
increases his lead. For example, compare the gains of different
individuals due to about 300 minutes of practice in mental
multiplication of a three-place number by a three-place number shown
in Table 14 below, from data obtained by the author ['08].[25]

  [25] Similar results have been obtained in the case of arithmetical
       and other abilities by Thorndike ['08, '10, '15, '16], Whitley
       ['11], Starch ['11], Wells ['12], Kirby ['13], Donovan and
       Thorndike ['13], Hahn and Thorndike ['14], and on a very
       large scale by Race in a study as yet unpublished.

TABLE 14

THE EFFECT OF EQUAL AMOUNTS OF PRACTICE UPON INDIVIDUAL DIFFERENCE IN
THE MULTIPLICATION OF THREE-PLACE NUMBERS

  ====================================================================
                                     |     AMOUNT     | PERCENTAGE OF
                                     |                |CORRECT FIGURES
                                     |----------------+---------------
                                     | Initial |      | Initial |
                                     |  Score  | Gain |  Score  | Gain
  -----------------------------------+---------+------+---------+-----
  Initially highest five individuals |    85   |  61  |    70   |  18
            next    five     "       |    56   |  51  |    68   |  10
            next    six      "       |    46   |  22  |    74   |   8
            next    six      "       |    38   |   8  |    58   |  12
            next    six      "       |    29   |  24  |    56   |  14
  ====================================================================


THE INTERRELATIONS OF INDIVIDUAL DIFFERENCES

Achievement in arithmetic depends upon a number of different abilities.
For example, accuracy in copying numbers depends upon eyesight, ability
to perceive visual details, and short-term memory for these. Long
column addition depends chiefly upon great strength of the addition
combinations especially in higher decades, 'carrying,' and keeping one's
place in the column. The solution of problems framed in words requires
understanding of language, the analysis of the situation described into
its elements, the selection of the right elements for use at each step
and their use in the right relations.

Since the abilities which together constitute arithmetical ability are
thus specialized, the individual who is the best of a thousand of his
age or grade in respect to, say, adding integers, may occupy different
stations, perhaps from 1st to 600th, in multiplying with integers,
placing the decimal point in division with decimals, solving novel
problems, copying figures, etc., etc. Such specialization is in part due
to his having had, relatively to the others in the thousand, more or
better training in certain of these abilities than in others, and to
various circumstances of life which have caused him to have, relatively
to the others in the thousand, greater interest in certain of these
achievements than in others. The specialization is not wholly due
thereto, however. Certain inborn characteristics of an individual
predispose him to different degrees of superiority or inferiority to
other men in different features of arithmetic.

We measure the extent to which ability of one sort goes with or fails to
go with ability of some other sort by the coefficient of correlation
between the two. If every individual keeps the same rank in the second
ability--if the individual who is the best of the thousand in one is the
best of the group in the other, and so on down the list--the correlation
is 1.00. In proportion as the ranks of individuals vary in the two
abilities the coefficient drops from 1.00, a coefficient of 0 meaning
that the best individual in ability A is no more likely to be in first
place in ability B than to be in any other rank.

The meanings of coefficients of correlation of .90, .70, .50, and 0 are
shown by Tables 15, 16, 17 and 18.[26]

  [26] Unless he has a thorough understanding of the underlying
       theory, the student should be very cautious in making
       inferences from coefficients of correlation.

TABLE 15

 DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .90

  ======================================================================
            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
  1st tenth |     |     |     |     |  .1 |  .4 | 1.8 | 6.6 |22.4 |68.7
  2d tenth  |     |     |  .1 |  .4 | 1.4 | 4.7 |11.5 |23.5 |36.0 |22.4
  3d tenth  |     |  .1 |  .5 | 2.1 | 5.8 |12.8 |21.1 |27.4 |23.5 | 6.6
  4th tenth |     |  .4 | 2.1 | 6.4 |12.8 |20.1 |23.8 |21.2 |11.5 | 1.8
  5th tenth |  .1 | 1.4 | 5.8 |12.8 |19.3 |22.6 |20.1 |12.8 | 4.7 |  .4
  6th tenth |  .4 | 4.7 |12.8 |20.1 |22.6 |19.3 |12.8 | 5.8 | 1.4 |  .1
  7th tenth | 1.8 |11.5 |21.2 |23.8 |20.1 |12.8 | 6.4 | 2.1 |  .4 |
  8th tenth | 6.6 |23.5 |27.4 |21.1 |12.8 | 5.8 | 2.1 |  .5 |  .1 |
  9th tenth |22.4 |36.0 |23.5 |11.5 | 4.7 | 1.4 |  .4 |  .1 |     |
  10th tenth|68.7 |22.4 | 6.6 | 1.8 |  .4 |  .1 |     |     |     |
  ======================================================================

TABLE 16

DISTRIBUTION OF ARRAYS IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .70

  ======================================================================
            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
  1st tenth |     |  .2 |  .7 | 1.5 | 2.8 | 4.8 | 8.0 |13.0 |22.3 |46.7
  2d tenth  |  .2 | 1.2 | 2.6 | 4.5 | 7.0 | 9.8 |13.4 |17.3 |21.7 |22.3
  3d tenth  |  .7 | 2.6 | 5.0 | 7.3 |10.0 |12.5 |14.9 |16.7 |17.3 |13.0
  4th tenth | 1.5 | 4.5 | 7.3 | 9.8 |12.0 |13.7 |14.8 |14.9 |13.4 | 8.0
  5th tenth | 2.8 | 7.0 |10.0 |12.0 |13.4 |14.0 |13.7 |12.5 | 9.8 | 4.8
  6th tenth | 4.8 | 9.8 |12.5 |13.7 |14.0 |13.4 |12.0 |10.0 | 7.0 | 2.8
  7th tenth | 8.0 |13.4 |14.9 |14.8 |13.7 |12.0 | 9.8 | 7.3 | 4.5 | 1.5
  8th tenth |13.0 |17.3 |16.7 |14.9 |12.5 |10.0 | 7.3 | 5.0 | 2.6 |  .7
  9th tenth |22.3 |21.7 |17.3 |13.4 | 9.8 | 7.0 | 4.5 | 2.6 | 1.2 |  .2
  10th tenth|46.7 |22.3 |13.0 | 8.0 | 4.8 | 2.8 | 1.5 |  .7 |  .2 |
  ======================================================================

TABLE 17

DISTRIBUTION OF ARRAYS OF SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .50

  ======================================================================
            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
  1st tenth |  .8 | 2.0 | 3.2 | 4.6 | 6.2 | 8.1 |10.5 |13.9 |18.0 |31.8
  2d tenth  | 2.0 | 4.1 | 5.7 | 7.3 | 8.8 |10.5 |12.2 |14.1 |16.4 |18.9
  3d tenth  | 3.2 | 5.7 | 7.4 | 8.9 |10.0 |11.2 |12.3 |13.3 |14.1 |13.9
  4th tenth | 4.6 | 7.3 | 8.8 | 9.9 |10.8 |11.6 |12.0 |12.3 |12.2 |10.5
  5th tenth | 6.2 | 8.8 |10.0 |10.8 |11.3 |11.5 |11.6 |11.2 |10.5 | 8.1
  6th tenth | 8.1 |10.5 |11.2 |11.6 |11.5 |11.3 |10.8 |10.0 | 8.8 | 6.2
  7th tenth |10.5 |12.2 |12.3 |12.0 |11.6 |10.8 | 9.9 | 8.8 | 7.5 | 4.6
  8th tenth |13.9 |14.1 |13.3 |12.3 |11.2 |10.0 | 8.8 | 7.4 | 5.7 | 3.2
  9th tenth |18.9 |16.4 |14.1 |12.2 |10.5 | 8.8 | 7.3 | 5.7 | 4.1 | 2.0
  10th tenth|31.8 |18.9 |13.9 |10.5 | 8.1 | 6.2 | 4.6 | 3.2 | 2.0 |  .8
  ======================================================================

TABLE 18

DISTRIBUTION OF ARRAYS, IN SUCCESSIVE TENTHS OF THE GROUP WHEN _r_ = .0

  ======================================================================
            |10TH |9TH  |8TH  |7TH  |6TH  |5TH  |4TH  |3D   |2D   |1ST
  ----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----
  1st tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  2d tenth  |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  3d tenth  |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  4th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  5th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  6th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  7th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  8th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  9th tenth |10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  10th tenth|10   |10   |10   |10   |10   |10   |10   |10   |10   |10
  ======================================================================

The significance of any coefficient of correlation depends upon the
group of individuals for which it is determined. A correlation of .40
between computation and problem-solving in eighth-grade pupils of 14
years would mean a much closer real relation than a correlation of .40
in all 14-year-olds, and a very, very much closer relation than a
correlation of .40 for all children 8 to 15.

Unless the individuals concerned are very elaborately tested on several
days, the correlations obtained are "attenuated" toward 0 by the
"accidental" errors in the original measurements. This effect was not
known until 1904; consequently the correlations in the earlier studies
of arithmetic are all too low.

In general, the correlation between ability in any one important feature
of computation and ability in any other important feature of computation
is high. If we make enough tests to measure each individual exactly
in:--

  (_A_) Subtraction with integers and decimals,
  (_B_) Multiplication with integers and decimals,
  (_C_) Division with integers and decimals,
  (_D_) Multiplication and division with common fractions, and
  (_E_) Computing with percents,

we shall probably find the intercorrelations for a thousand 14-year-olds
to be near .90. Addition of integers (_F_) will, however, correlate less
closely with any of the above, being apparently dependent on simpler and
more isolated abilities.

The correlation between problem-solving (_G_) and computation will be
very much less, probably not over .60.

It should be noted that even when the correlation is as high as .90,
there will be some individuals very high in one ability and very low in
the other. Such disparities are to some extent, as Courtis ['13, pp.
67-75] and Cobb ['17] have argued, due to inborn characteristics of the
individual in question which predispose him to very special sorts of
strength and weakness. They are often due, however, to defects in his
learning whereby he has acquired more ability than he needs in one line
of work or has failed to acquire some needed ability which was well
within his capacity.

In general, all correlations between an individual's divergence from the
common type or average of his age for one arithmetical function, and his
divergences from the average for any other arithmetical function, are
positive. The correlation due to original capacity more than
counterbalances the effects that robbing Peter to pay Paul may have.

Speed and accuracy are thus positively correlated. The individuals who
do the most work in ten minutes will be above the average in a test of
accuracy. The common notion that speed is opposed to accuracy is correct
when it means that the same person will tend to make more errors if he
works at too rapid a rate; but it is entirely wrong when it means that
the kind of person who works more rapidly than the average person is
likely to be less accurate than the average person.

Interest in arithmetic and ability at arithmetic are probably correlated
positively in the sense that the pupil who has more interest than other
pupils of his age tends in the long run to have more ability than they.
They are certainly correlated in the sense that the pupil who 'likes'
arithmetic better than geography or history tends to have relatively
more ability in arithmetic, or, in other words, that the pupil who is
more gifted at arithmetic than at drawing or English tends also to like
it better than he likes these. These correlations are high.

It is correct then to think of mathematical ability as, in a sense, a
unitary ability of which any one individual may have much or little,
most individuals possessing a moderate amount of it. This is
consistent, however, with the occasional appearance of individuals
possessed of very great talents for this or that particular feature of
mathematical ability and equally notable deficiencies in other features.

Finally it may be noted that ability in arithmetic, though occasionally
found in men otherwise very stupid, is usually associated with superior
intelligence in dealing with ideas and symbols of all sorts, and is one
of the best early indications thereof.




BIBLIOGRAPHY OF REFERENCES MADE IN THE TEXT


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  Ballou, F. W.; '16
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  Brandell, G.; '13
    Skolbarns intressen.  Translated ['15] by W. Stern as, Das Interesse
    der Schulkinder an den Unterrichtsfächern.

  Brandford, B.; '08
    A Study of Mathematical Education.

  Brown, J. C.; '11, '12
    An Investigation on the Value of Drill Work in the Fundamental
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  Burnett, C. J.; '06
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  Burt, C.; '17
    The Distribution and Relations of Educational Abilities. Report of
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  Chapman, J. C.; '14
    Individual Differences in Ability and Improvement and Their
    Correlations. Teachers College Contributions to Education, No. 63.

  Chapman, J. C.; '17
    The Scientific Measurement of Classroom Products. (With G. P. Rush.)

  Cobb, M. V.; '17
    A Preliminary Study of the Inheritance of Arithmetical Abilities.
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    How to Teach Arithmetic.

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    The Supervision of Arithmetic.

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INDEX


  Abilities, arithmetical, nature of, 1 ff.;
    measurement of, 27 ff.;
    constitution of, 51 ff.;
    organization of, 137 ff.

  Abstract numbers, 85 ff.

  Abstraction, 169 ff.

  Accuracy, in relation to speed, 31;
    in fundamental operations, 102 ff.

  Addition, measurement of, 27 ff., 34;
    constitution of, 52 f.;
    habit in relation to, 71 f.;
    in the higher decades, 75 f.;
    accuracy in, 108 f.;
    amount of practice in, 122 ff.;
    interest in 196 f.

  Aims of the teaching of arithmetic, 23 f.

  AMES, A. F., 89

  Analysis, learning by, 169 ff.;
    systematic and opportunistic stimuli to, 178 f.;
    gradual progress in, 180 ff.

  Area, 257 f., 275

  Arithmetic, sociology of, 24 ff.

  Arithmetical abilities. _See_ Abilities.

  Arithmetical language, 8 f., 19, 89 ff., 94 ff.

  Arithmetical learning, before school, 199 ff.;
    conditions of, 227 ff.;
    in relation to time of day, 227 ff.;
    in relation to time devoted to arithmetic, 228 ff.

  Arithmetical reasoning. _See_ Reasoning.

  Arithmetical terms, 8, 19

  Averages, 40 f.; 135 f.


  BALLOU, F. W., 34, 38

  Banking, 256 f.

  BINET, A., 201

  Bonds, selection of, 70 ff.;
    strength of, 102 ff.;
    for temporary service, 111 ff.;
    order of formation of, 141 ff.
    _See also_ Habits.

  BRANDELL, G., 211

  BRANDFORD, B., 198 f.

  BROWN, J. C., xvi, 103

  BURGERSTEIN, L., 103

  BURNETT, C. J., 202

  BURT, C., 286


  Cardinal and ordinal numbers confused, 206

  Catch problems, 21 ff.

  CHAPMAN, J. C., 49

  Class, size of, in relation to arithmetical learning, 228;
    variation within a, 289 ff.

  COBB, M. V., 299

  COFFMAN, L. D., xvi

  Collection meaning of numbers, 3 ff.

  Computation, measurements of, 33 ff.;
    explanations of the processes in, 60 ff.;
    accuracy in, 102 ff.
    _See also_ Addition, Subtraction, Multiplication, Division,
          Fractions, Decimal numbers, Percents.

  Concomitants, law of varying, 172 ff.;
    law of contrasting, 173 ff.

  Concrete numbers, 85 ff.

  Concrete objects, use of, 253 ff.

  Conditions of arithmetical learning, 227 ff.

  Constitution of arithmetical abilities, 51 ff.

  Copying of numbers, eyestrain due to, 212 f.

  Correlations of arithmetical abilities, 295 ff.

  Courses of study, 232 f.

  COURTIS, S. A., 28 ff., 43 ff., 49, 103, 291, 293, 299

  Crutches, 112 f.

  Culture-epoch theory, 198 f.


  Dairy records, 273

  Decimal numbers, uses of, 24 f.;
    measurement of ability with, 36 ff.;
    learning, 181 ff.;
    division by, 270 f.

  DE CROLY, M., 205

  Deductive reasoning, 60 ff., 185 ff.

  DEGAND, J., 205

  Denominate numbers, 141 f., 147 f.

  Described problems, 10 ff.

  Development of knowledge of number, 205 ff.

  DE VOSS, J. C., 49

  DEWEY, J., 3, 83, 150, 205, 207, 208, 219, 266, 277

  Differences in arithmetical ability, 285 ff.;
    within a class, 289 ff.

  Difficulty as a stimulus, 277 ff.

  Drill, 102 ff.

  Discipline, mental, 20

  Distribution of practice, 156 ff.

  Division, measurement of, 35 f., 37;
    constitution of, 57 ff.;
    deductive explanations of, 63, 64 f.;
    inductive explanations of, 63 f., 65 f.;
    habit in relation to, 72;
    with remainders, 76;
    with fractions, 78 ff.;
    amount of practice in, 122 ff.;
    distribution of practice in, 167;
    use of the problem attitude in teaching, 270 f.

  DONOVAN, M. E., 295


  Elements, responses to, 169 ff.

  Eleven, multiples of, 85

  ELLIOTT, C. H., 228

  Equation form, importance of, 77 f.

  Explanations of the processes of computation, 60 ff.;
    memory of, 115 f.;
    time for giving, 154 ff.

  Eyestrain in arithmetical work, 212 ff.


  Facilitation, 143 ff.

  Figures, printing of, 235 ff.;
    writing of, 214 f., 241

  FLYNN, F. J., 196

  Fractions, uses of, 24 f.;
    measurement of ability with, 36 ff.;
    knowledge of the meaning of, 54 ff.

  FREEMAN, F. N., 259, 261

  FRIEDRICH, J., 103


  Generalization, 169 ff.

  GILBERT, J. A., 203

  Graded tests, 28 ff., 36 ff.

  Greatest common divisor, 88 f.


  Habits, importance of, in arithmetical learning, 70 ff.;
    now neglected, 75 ff.;
    harmful or wasteful, 83 ff.; 91 ff.;
    propædeutic, 117 ff.;
    organization of, 137 ff.;
    arrangement of, 141 ff.

  HAHN, H. H., 295

  HALL, G. S., 200 f.

  HARTMANN, B., 200 f.

  HECK, W. H., 227

  Heredity in arithmetical abilities, 293 ff.

  Highest common factor, 88 f.

  HOKE, K. J., 49

  HOLMES, M. E., 103

  HOWELL, H. B., 259

  HUNT, C. W., 196

  Hygiene of arithmetic, 212 ff., 234 ff.


  Individual differences, 285 ff.

  Inductive reasoning, 60 ff., 169 ff.

  Insurance, 256

  Interest as a principle determining the order of topics, 150 ff.

  Interests, instinctive 195 ff.;
    censuses of, 209 ff.;
    neglect of childish, 226 ff.;
    in self-management, 223 f.;
    intrinsic, 224 ff.

  Interference, 143 ff.

  Inventories of arithmetical knowledge and skill, 199 ff.


  JESSUP, W. A., xvi


  KELLY, F. J., 49

  KING, A. C., 103, 227

  KIRBY, T. J., 76 f., 104, 295

  KLAPPER, P., xvi

  KRUSE, P. J., 289, 293


  Ladder tests, 28 ff., 36 ff.

  Language in arithmetic, 8 f., 19, 89 ff., 94 ff.

  LASER, H., 103

  LAY, W. A., 259, 261

  Learning, nature of arithmetical, 1 ff.

  Least common multiple, 88 f.

  LEWIS, E. O., 210 f.

  LOBSIEN, M., 209 f.


  MCCALL, W. A., 49

  MCDOUGLE, E. C., 85 ff.

  MCKNIGHT, J. A., 210

  MCLELLAN, J. A., 3, 83, 89, 205, 207

  Manipulation of numbers, 60 ff.

  Meaning, of numbers, 2 ff., 171;
    of a fraction, 54 ff.;
    of decimals, 181 f.

  Measurement of arithmetical abilities, 27 ff.

  Mental arithmetic, 262 ff.

  MESSENGER, J. F., 202

  Metric system, 147

  MEUMANN, E., 261

  MITCHELL, H. E., 24

  MONROE, W. S., 49

  Multiplication, measurement of, 35, 36;
    constitution of, 51;
    deductive explanations of, 61;
    inductive explanations of, 61 f.;
    with fractions, 78 ff.;
    by eleven, 85;
    amount of practice in, 122 ff.;
    order of learning the elementary facts of, 144 f.;
    distribution of practice in, 158 ff.;
    use of the problem attitude in teaching, 267 ff.


  NANU, H. A., 202

  National Intelligence Tests, 49 f.

  Negative reaction in intellectual life, 278 f.

  Number pictures, 259 ff.

  Numbers, meaning of, 2;
    as measures of continuous quantities, 75;
    abstract and concrete, 85 ff.;
    denominate, 141 f., 147 f.;
    use of large, 145 f.;
    perception of, 205 ff.;
    early awareness of, 205 ff.;
    confusion of cardinal and ordinal, 206.
    _See also_ Decimal numbers _and_ Fractions.


  Objective aids, used for verification, 154;
    in general, 243 ff.

  Oral arithmetic, 262 ff.

  Order of topics, 141 ff.

  Ordinal numbers, confused with cardinal, 206

  Original tendencies and arithmetic, 195 ff.

  Overlearning, 134 ff.


  Percents, 80 f.

  Perception of number, 202 ff.

  PHILLIPS, D. E., 3, 4, 205, 207

  Pictures, hygiene of, 246 ff.;
    number, 259 ff.

  POMMER, O., 212

  Practice, amount of, 122 ff.;
    distribution of, 156 ff.

  Precision in fundamental operations, 102 ff.

  Problem attitude, 266 ff.

  Problems, 9 ff.;
    "catch," 21 ff.;
    measurement of ability with, 42 ff.;
    whose answer must be known in order to frame them, 93 f.;
    verbal form of, 111 f.;
    interest in, 220 ff.;
    as introductions to arithmetical learning, 266 ff.

  Propædeutic bonds, 117 ff.

  Purposive thinking, 193 ff.


  Quantity, number and, 85 ff.;
    perception of, 202 ff.


  RACE, H., 295

  Rainfall, 272

  Ratio, 225 f.;
    meaning of numbers, 3 ff.

  Reaction, negative, 278 f.

  Reality, in problems, 9 ff.

  Reasoning, arithmetical, nature of, 19 ff.;
    measurement of ability in, 42 ff.;
    derivation of tables by, 58 f.;
    about the rationale of computations, 60 ff.;
    habit in relation to, 73 f., 190 ff.;
    problems which provoke false, 100 f.;
    the essentials of arithmetical, 185 ff.;
    selection in, 187 ff.;
    as the coöperation of organized habits, 190 ff.

  Recapitulation theory, 198 f.

  Recipes, 273 f.

  Rectangle, area of, 257 f.

  RICE, J. M., 228 ff.

  RUSH, G. P., 49


  SEEKEL, E., 212

  SELKIN, F. B., 196 f.

  Sequence of topics, 141 ff.

  Series meaning of numbers, 2 ff.

  Size of class in relation to arithmetical learning, 228

  SMITH, D. E., xvi, 224

  Social instincts, use of, 195 f.

  Sociology of arithmetic, 24 ff.

  Speed in relation to accuracy, 31, 108

  SPEER, W. W., 3, 5, 83

  Spiral order, 141, 145

  STARCH, D., 49, 295

  STERN, W., 210, 212

  STONE, C. W., 27 ff., 42 ff., 228 ff.

  Subtraction, measurement of, 34 f.;
    constitution of, 57 f.;
    amount of practice in, 122 ff.

  Supervision, 233 f.

  SUZZALLO, H., xvi


  Temporary bonds, 111 ff.

  Terms, 113 f.

  Tests of arithmetical abilities, 27 ff.

  THORNDIKE, E. L., 34, 38 ff., 227, 294

  Time, devoted to arithmetic, 228 ff.;
    of day, in relation to arithmetical learning, 227 f.

  Type, hygiene of, 235 ff.


  Underlearning, 134 ff.

  United States money, 148 ff.

  Units of measure, arbitrary, 5, 83 f.


  Variation, among individuals, 285 ff.

  Variety, in teaching, 153

  Verification, 81 f.;
    aided by greater strength of the fundamental bonds, 107 ff.


  WALSH, J. H., 11

  WELLS, F. L., 295

  WHITE, E. E., 5

  WHITLEY, M. T., 295

  WIEDERKEHR, G., 212

  WILSON, G. M., 24, 49

  WOODY, C., 29 ff., 52, 287, 293

  Words. _See_ Language _and_ Terms.

  Written arithmetic, 262 ff.


  Zero in multiplication, 179 f.




TRANSCRIBER'S NOTES:


1. Passages in italics are surrounded by _underscores_.

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3. Mixed fractions are represented using forward slash and hyphen in
this text version. For example, 5-1/2 represents five and a half.

4. Images and footnotes have been moved from the middle of a paragraph
to the closest paragraph break.

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