CONTENTS

CHAPTER PAGE
[Introduction: The Psychology of the Elementary School Subjects][xi]
[I.]The Nature of Arithmetical Abilities[1]
Knowledge of the Meanings of Numbers
Arithmetical Language
Problem Solving
Arithmetical Reasoning
Summary
The Sociology of Arithmetic
[II.]The Measurement of Arithmetical Abilities[27]
A Sample Measurement of an Arithmetical Ability
Ability to Add Integers
Measurements of Ability in Computation
Measurements of Ability in Applied Arithmetic: the Solution of Problems
[III.]The Constitution of Arithmetical Abilities[51]
The Elementary Functions of Arithmetical Learning
Knowledge of the Meaning of a Fraction
Learning the Processes of Computation
[IV.]The Constitution of Arithmetical Abilities (continued)[70]
The Selection of the Bonds to Be Formed
The Importance of Habit Formation
Desirable Bonds Now Often Neglected
Wasteful and Harmful Bonds
Guiding Principles
[V.]The Psychology of Drill in Arithmetic: The Strength of Bonds[102]
The Need of Stronger Elementary Bonds
Early Mastery
The Strength of Bonds for Temporary Service
The Strength of Bonds with Technical Facts and Terms
The Strength of Bonds Concerning the Reasons for Arithmetical Processes
Propædeutic Bonds
[VI.]The Psychology of Drill in Arithmetic: The Amount of Practice and the Organization of Abilities[122]
The Amount of Practice
Under-learning and Over-learning
The Organization of Abilities
[VII.]The Sequence of Topics: The Order of Formation of Bonds[141]
Conventional versus Effective Orders
Decreasing Interference and Increasing Facilitation
Interest
General Principles
[VIII.]The Distribution of Practice[156]
The Problem
Sample Distributions
Possible Improvements
[IX.]The Psychology of Thinking: Abstract Ideas and General Notions in Arithmetic[169]
Responses to Elements and Classes
Facilitating the Analysis of Elements
Systematic and Opportunistic Stimuli to Analysis
Adaptations to Elementary-school Pupils
[X.]The Psychology of Thinking: Reasoning in Arithmetic[185]
The Essentials of Arithmetical Reasoning
Reasoning as the Coöperation of Organized Habits
[XI.]Original Tendencies and Acquisitions before School[195]
The Utilization of Instinctive Interests
The Order of Development of Original Tendencies
Inventories of Arithmetical Knowledge and Skill
The Perception of Number and Quantity
The Early Awareness of Number
[XII.]Interest in Arithmetic[209]
Censuses of Pupils' Interests
Relieving Eye Strain
Significance for Related Activities
Intrinsic Interest in Arithmetical Learning
[XIII.]The Conditions of Learning[227]
External Conditions
The Hygiene of the Eyes in Arithmetic
The Use of Concrete Objects in Arithmetic
Oral, Mental, and Written Arithmetic
[XIV.]The Conditions of Learning: the Problem Attitude[266]
Illustrative Cases
General Principles
Difficulty and Success as Stimuli
False Inferences
[XV.]Individual Differences[285]
Nature and Amount
Differences within One Class
The Causes of Individual Differences
The Interrelations of Individual Differences
[Bibliography of References][302]
[Index][311]

GENERAL INTRODUCTION

THE PSYCHOLOGY OF THE ELEMENTARY SCHOOL SUBJECTS

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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