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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
Propaedeutic 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 Cooeperation 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 25c," 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 x 8 x 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."