The Psychology of Arithmetic
by Edward L. Thorndike · English · Gutenberg #39300 · 545,529 chars · public domain (US)
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Title: The Psychology of Arithmetic
Author: Edward L. Thorndike
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Language: English
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THE MACMILLAN COMPANY
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TORONTO
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."
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 x 7 = 2 x 3 = 4 x 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 x 2 =
2-5/8 59 3.2
--- ---- ----
(33) (35) (37) (38)
2-1/2 x 3-1/2 = 987-3/4 2-1/4 x 4-1/2 x 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 62-1/2c, 66-2/3c, 56-1/4c, 60c, and 62-1/2c.
_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 - 89c 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_. 62-1/2c, 66-2/3c, 40c, 83-1/3c, $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 x 33 32 x 3/14 39 x 2/13 60 x 11/28 77 x 4/11
63 x 2/27 54 x 8/45 65 x 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 | | |
45c. 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 35c. 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 67c. a day carrying papers, was | | |
hired to run on a long errand for which he received | | |
50c. 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 26c. 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 10c. | | |
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 10c 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 x 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 x 5 = .... 9 nickels = .... cents.
8 5s = .... 4 x 5 = .... 6 " = .... "
3 5s = .... 2 x 5 = .... 5 " = .... "
7 5s = .... 9 x 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 .... x 5 = 25 20 cents = .... nickels.
.... 5s = 20 20 = .... 5s .... x 5 = 50 30 cents = .... nickels.
.... 5s = 40 15 = .... 5s .... x 5 = 35 15 cents = .... nickels.
.... 5s = 25 45 = .... 5s .... x 5 = 10 40 cents = .... nickels.
.... 5s = 30 50 = .... 5s .... x 5 = 40
.... 5s = 35 25 = .... 5s .... x 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 x 2," write 8 under the 2 in the ones column._
4 _Think "4 x 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 x 2 or 2 x 32 will give the answer.
3. How many little cakes will they need if each child has three
cakes? 32 x 3 or 3 x 32 will give the answer.
32 3 x 2 = .... _Where do you write the 6?_
3 3 x 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 x 3?
How many are 3 x 32?
How many are 8 x 5?
How many are 4 x 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 x 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 x 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 x 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 x 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 x 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 x
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 x _A_ a result
about twice as large as for 13 x _A_, for 38 or 39 x _A_ a result about
three times as large; for 115 x _A_ a result about ten times as large as
for 11 x _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 x 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 x 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 cooeperation 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 reenforce 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 reenforcing.
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 x 4 = .... 9 x 2 = ....
4 x 4 = .... 6 x 2 = ....
2 x 4 = .... 3 x 2 = ....
1 x 4 = .... 1 x 2 = ....
1/2 x 4 = .... 1/3 x 2 = ....
1/4 x 4 = .... 1/6 x 2 = ....
1/8 x 4 = .... 1/9 x 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 x
12% " = .12 x
3% " = .03 x
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 x 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 x 1.23 is much more than 24.6.
______ It cannot be 2; for 2 x 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 x 2, 12 x 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 x 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 reestablishing 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 x 2
6. 4 x 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
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