Arithmetical abilities and some factors determining them

Table Of Contents

PAGE

Acknowledgments 5

Part I. Purpose and Method of the Study The Purpose of the Study 7

The Method of the Study 7

Schools Tested 7

Explanation of key 9

Method of securing data 9

Purpose and content of tests 9

The tests as given 10

Arrangement of problems 12

Conditions under which given 13

Directions to pupils 14

How the scores were computed 15

Precautions to make scores accurate 18

What the scores measure 19

Part II. Arithmetical Abilities — Their Variability and Relationship The Problem — The nature of the product of the first six years of

arithmetic work 19

Phases of the problem 19

The Data 20

Achievements of the systems as systems 21

Measured by scores made — Tables III- VII with sample

readings 21

Measured by mistakes made — Tables VIII and IX with

sample readings 26

Achievements of pupils as individuals, Tables X and XI. . 29

Variability 33

Of abilities among systems and individual pupils — Table

XII with sample readings 33

Of abilities among boys and among girls — Tables XIII and

XIV with sample readings 35

Relationship 36

Of abilities among systems — Tables XV and XVI with

sample readings 37

Of abilities among individual pupils — Tables XVII and

XVIII with sample readings 40

Summary 42

Factors determining abilities 43

Part III. Arithmetical Abilities and Time Expenditures The Problem — The relation of time expenditure to arithmetical

abilities 46

The problem analyzed 46

GVv* rfii

4 Contents

Part III. Arithmetical Abilities and Time Expenditures — Con. page

Time expenditure 47

Previous related studies 47

Data blank and its uses 50

Amounts and variabilities in twenty-six systems, Tables

XXI to XXIV with sample readings 52

The relation of time expenditure to abilities produced, Tables

XXV to XXVII with sample reading 58

Some factors in time expenditure 62

Distribution among grades 62

Use within a grade 65

Summary 69

Suggestions for determining time allotment 69

Part IV. Arithmetical Abilities and the Course of Study

The Problem — What is the relation between arithmetical abilities and the Course of Study 70

Steps toward solution of the problem 70

Securing Courses of Study 70

Securing ratings of Courses of Study 71

Relation of excellences of courses and excellences of results . 73

Drill and Concreteness in Courses of Study 74

The Courses of the nineteen Public Schools in relation to abilities 78

Phases of excellences of Courses in relation to each other 78

Courses of Study and Time Cost 79

Samples from Courses of Study 79

The Course of Study Situation 86

General Conclusions 89

Appendix 94

ACKNOWLEDGMENTS

It is a pleasure to acknowledge here the wealth of assistance that I have received in connection with this study. Any piece of research work must necessarily be co-operative, but one that attempts to take account of existing practice must be peculiarly so.

For assistance in conceiving and delimiting the subject, I am indebted particularly to Professor Strayer and Professor Mc- Murry; for guidance in executing the statistical phases to Professor Thorndike; for assistance in formulating the tests to Professor Smith; for suggestions and criticisms to Mr. Manny and Professor Snedden; and for sympathetic supervision of the entire study to Professor Strayer and Professor Suzzallo.

For assistance in locating school systems doing distijnpUve wOj;k; in arithmetic, I am indebted particularly to pr,""j6hn^T.^ PrJTjc*?^ and Superintendent F. E. Parlin for schools , of ^MariSachits^etri and Rhode Island, to Mr. Frank H. Hall for schools of Illinois, to Dr. Daniel A. Tear for schools of Chicago.

For assistance in evaluating the Courses of Study, I am indebted to the professors and graduate students who generously gave of their time and experience ; for invaluable assistance in handling the six thousand test papers, executing much of the statistical work, and for her constant encouragement and suggestive criticism, to my wife, Kate King Stone.

But my greatest debt is to those working in the field. Besides the seventy-nine principals and the one hundred fifty-two teachers who generously gave of their time and energy, the following superintendents and principals rendered invaluable assistance both by allowing the tests to be made and by furnishing data:

Supt. A. L. Barber, Natick, Massachusetts. Supt. J. A. Barr, Stockton, California.

Miss M. A. Besley, Superintendent, Waukegan, 111. Supt. A. B. Blodgett, Syracuse, N. Y.

Supt. C. F. Carroll, Rochester, N. Y.

Prin. Chas. S. Chapin, Observation School, State Normal, Providence, R. I.

6 Acknowledgments

Prin. L. W. Colwell, Linne School, Chicago.

Miss Flora J. Cooke, Prin. Francis W. Parker School, Chicago.

Miss Caroline Emanuel, Prin. P. S. No. 50, Manhattan, New York City.

Mr. Harry O. Gillet, Chairman Executive Committee, University Elementary School, University of Chicago.

Supt. C. E. Gorton, Yonkers, N. Y.

Supt. C. N. Kendall, Indianapolis, Ind.

Supt. John Kennedy, Batavia, N. Y.

Supt. Charles S. Meek, Elwood, Ind. 1

Supt. C. H. Morse, Medford, Mass.

Supt. R. O. Ogg, Kokomo, Ind.

Supt. W. D. Parkinson, Waltham, Mass.

Prin. H. C. Pearson, Horace Mann Elementary School, Columbia University.

, /Prin;,^: p. 'Reigart, P. S. No. 2 B, Manhattan, New York City. , SwA !(?-.^- ^P^J^^c^s, Muncie, Ind. ,' :i^oS' ?i E. Rugh, Berkeley, Cal.

Miss Jennie Small, Acting Supt., Elwood, Ind.

Supt. Henry Snyder, Jersey City, N. J.

Supt. R. Spaulding, Montclair, N. J.

Mr. W. E. Stark, Head of Mathematics Department, Ethical Culture School, New York City.

Prin. A. M. Stowe, Training School, State Normal, Hyannis, Mass.

Prin. J. K. Van Denburg, School No. 40, Manhattan, New York City.

Miss Charlotte Weber, Ass't Supt., Jersey City, N. J.

Supt. O. I. Woodley, Passaic, N. J.

No school system will appear by name in the body of this study, but any one whose system is represented will be cordially provided with its key, of which three copies will be preserved — one by the author and two at Teachers College, one in the Office of Elementary Education and one in the Brysoji JLilDr^i:^.

Dr. Rice's study of arithmetic and other helpful works are noted and briefly summarized in the appendix, page 96, or throughout the study in connection with related topics.

Part I.^purpose And Method

The central purpose of this study, broadly stated, is to make one more contribution to exact knowledge of the relation between distinctive educational procedures and the resulting products. " What is the relation between the theories and the products of education ?'' is one statement of the question which prompted the author to attempt a research thesis. Specifically, the portions of the larger question on which this study has bearing are : ( i ) What is the nature of the product of the first six] years of arithmetic work? (2) What is the relation between/ distinctive procedures in arithmetic work and the resulting abilities?

The method of this inquiry may be briefly cha,ra»:tf,rized -as tlis application of the statistical method to mental ; measoipcmertts.* The three principal measurements made are the arithmetical abilities of 6A (high 6th) pupils,^ the time expended and the course of study materials used in securing these abilities.

Sources of Data

The sources drawn upon for data with which to answer the questions above stated were |.wenty-six school ^systems. These were selected after a somewhat close examination of the arithmetic courses of study and time allotments of a large number of city and individual schools scattered over practically the entire United States. The base^of^dectii^ were distinctiveness of practice and geographical accessibility.

Through the courtesy of superintendents, principals and teachers the data were gathered in the following twenty-six systems :^

Batavia, New York.

Decatur, Illinois.

Elwood, Indiana.

iPor a ftill exposition, see Thorndike's Mental and Social Measurements.

2In such of the smaller systems as had less than one hundred 6 A's, tests were used from equal numbers of 6 B's and 7 B's.

SThe author personally gathered the data from each system, securing course of study material from each of the twenty-six superintendents, and time expenditure data from each of seventy-five principals, and conducting the tests in each of the one hundred and fifty-two classes. See p. 13, Conditions under which the Tests were given. The single exception is the Ethical Culture School where, because of the author being known to the children. Mr. J. L. Stockton, to whom the thanks of the author are due, gave the tests after careful preparation.

Ethical Culture School, New York City.

Francis W. Parker School, Chicago.

Horace Mann School, Columbia University, New York City.

Indianapolis, Indiana.

Jersey City, New Jersey.

Kokomo, Indiana.

Linne School, Chicago.

Medford, Massachusetts.

Montclair, New Jersey.

Muncie, Indiana.

Natick, Massachusetts.

Observation School, State Normal, Providence, R. I.

Passaic, New Jersey.

Providence, Rhode Island.

Rocliester, New York. , .Schools jlja ;4Q and No. 50, Manhattan, New York City. , ^ ^pe^er , School, Columbia University. .' /Syracuse,, Hew «York.

Training School, State Normal, Hyannis, Massachusetts.

University Elementary School, University of Chicago.

Waltham, Massachusetts.

Waukegan, Illinois.

Yonkers, New York.

It will be noted that of these twenty-six systems there are seventeen city public systems — Batavia, Decatur, Elwood, Indianapolis, Jersey City, Kokomo, Medford, Montclair, Muncie, Natick, Passaic, Providence, Rochester, Syracuse, Waltham, Waukegan and Yonkers ; three schools in city systems — Linne, School No. 40 and School No. 50 ; four training schools — Observation School, Speyer School, Hyannis Training School and University Elementary School; and three private schools — Ethical Culture, Francis W. Parker and Horace Mann.

According to geographical location these schools may be grouped as follows:

Six in New England, namely Medford, Natick, Observation School, Providence, Training School and Waltham; eleven in the Middle East, namely Batavia, Ethical Culture, Horace Mann, Jersey City, Montclair, Passaic, Rochester, Schools No. 40 and No. 50, Speyer School, Syracuse and Yonkers ; and nine in the

I -

Middle West, namely Decatur, Elwood, Francis W. Parker, Indianapolis, Kokomo, Linne School, Muncie, University Elementary School and Waukegan.

Readers who know these schools will recognize that they represent a diversity^of practice as to the relative reliance placed on the various factors supposed to produce arithmetical abilities. There are among these systems (jtoin^Jive-^^Cfidures in regard to at least the following — the time devoted to arithmetic and its distribution among grades; me supervision of the teaching of arithmetic^ and the nature and arrangement of the subject matter taught..

As stated under acknowledgments, no system will appear by name in connection with its relative standing, each system having been assigned a Roman numeral in a key. The number of each respective system bears no relation whatever to any rating of that system, the numbers having been determined by a chance distribution of the twenty-six systems among the first twenty-six Roman numerals.

Method of Securing Data

In one word the method of securing data may be characterized as personal. Tests to measure arithmetical abilities were personally conducted by the author; blanks calling for the time expenditure and other information were filled out by the principals and teachers at the author's personal request during the time the tests were being given; and after an interview with the author, the courses of study were rated by twenty-one unbiased judges each of whom had had practical experience with the workings of various courses of study and had acquired a knowledge of the principles underlying their construction.

Purpose and Content of the Tests

In Fundmn^ntals^

The main purpose of the test in fundamental operations was the determination of the ability of Grade VI A (high 6th) pupils

iThe word fundamental and the words fundamental operations are used throughout this study to designate the more formal aspects of arithmetic work, viz., addition, subtraction, mtdtiplication and division of whole numbers. It will be recognized that in the sense of priority of time these so-called fundamentals are not more fundamental than certain aspects of reasoning.

in addition, subtraction, multiplication and division. To this end \hhe test was meant to embody all the difficulties of the four fun- Aiamental operations in sufficient quantities to thoroughly test

any VI A Grade. The t^est is purposely too. long, for any pupil

to finish entirely in the twelve minute limit.

In Reasoning

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 requirement.

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.

The Tests as Given

The following are reproductions of the tests. They were printed separately and each pupil was furnished with a copy.

Work as many of these problems as you have time for; work them in order as numbered.

Arithmetical Abilities ii

2. Multiply 3265 by 20.

428 94

75 302

645 984 897

5. Multiply 768 by 604.

Multiply 976 by 87.

Divide 2782542 by 679.

Multiply 5489 by 9876.

Divide 5099941 by 749.

Multiply 876 by 79.

Multiply 96879 by 896.

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 J^ the money and with the other V2 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 2,200 pupils; J^ are in the primary grades, ^A in the grammar grades, Vs in the High School and the rest in the night school. How many pupils are there in the night school ?

7. If SV2 tons of coal cost $21, what will sVz 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 Vs 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 button-holes. 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 V2 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 Arrangement of Problems and the Determination of the Time Limits.

In Fundamentals

The test in fundamental operations was so arranged as to enable the pupils to meet the main difficulties in the first six problems. These were meant to increase in difficulty as were the next three ; and problems ten to fourteen were included so as to furnish enough work for the most rapid pupils. After several preliminary trials, the time limit was set at twelvejniaiites as the time best suited to allow the average of the slower classes to encounter the main difficulties and to keep all of the most rapid pupils from getting out of work.

In Reasoning

The arrangement of the problems in reasoning was determined by a preliminary test. After embodying the desired content (see page 10) the problems were given in chance order to one hundred representative high 6th pupils, to determine their relative difficulty. As the pupils were given as much time as they needed in order to work each problem as well as they could, the per cent of problems worked showed their relative difficulty. On this basis the problems were arranged with the less difficult first and printed as seen on page 11. After several preliminary trials the time limit was set at fifteen^minutes as the time in which a majority of pupils worked through the first six or seven problems and in which practically no pupil completed the test.

Conditions under Which the Tests Were Given

I. Personally Conducted by the Author.

1. All directions given orally by the author, except as

printed on test slips. See page 14 for directions to pupils.

2. Time limits kept exactly.

Twelve minutes for fundamentals.

Fifteen minutes for reasoning.

3. Each pupil provided with a printed copy of each test.

II. Superintendents were asked :

1. To select the four or five buildings containing high 6th

pupils in which the approved course of study had been best carried out for the past five and one-half, or more years.

2. To refrain from announcing the tests to principals or

teachers.

3. Not to be in and not to go into the room that was tested.

4. To furnish a note to principals and 6 A (high 6th) teach-

ers, similar to the following :

Mr. Stone is giving the tests in several cities, and, in order to make the work as accurate as possible, he is anxious to have all children follow exactly the same directions under exactly the same conditions. To this end he asks that principals and teachers refrain from making any announcement of the tests to the pupils, and that they have as little as possible to do with the pupils during the tests.

III. Principals were asked :

1. To introduce the author to the teacher — ^preferably by

note.

2. Not to make any announcement to the pupils.

3. Not to be in the room at all, during or immediately pre-

vious to the tests.

4. To fill out the outline calling for data during the time of

the tests. (A few asked to keep it longer, that they might make it more accurate.)

IV. Teachers were asked:

1. To allow the author to give all the directions to the

pupils.

2. To fill out the outHne calling for data during the tests.

V. Other conditions observed:

1. All the 6 A's in a building tested.

2. About two or three schools tested in the forenoon and

two in the afternoon.^

3. The fundamental operation test was given first in all cases,

immediately followed by the reasoning test; twelve minutes for fundamental operations, fifteen minutes for reasoning.

4. In case a school selected by the superintendent proved to

be abnormal, the results were not used, e. g., a school having a large percentage of Hebrews of foreign parents would not be a fair measure.

5. Nothing was said to the pupils about scratch-paper, work-

ing in steps, amount of work put down, etc.

6. No mention was made to pupils of the time limit.

DIRECTIONS TO PUPILS

[As given orally to each class]

Fundamental Operations

I. Take the materials that you usually take for an arithmetic test. Prepare two sheets of paper — headings and all. Have two sheets ready in case you may need them. Use pencils. Keep slips with printing turned down until we are ready to begin.

^See conclusion of Miss King's Study, p. 14-

II. Now do you have everything? In order for you to do your best in this test, you will need to do just as all the other boys and girls who have taken this test have done. So pay close attention and then do just as I ask you to do.

III. You will not need to mark these (slips) papers at all. You will find directions at the top of these (slips) papers, telling you just what to do, so you will not need to ask any questions, and — (I do not think I need to say this to you, but I will just because I have said it to all the other boys and girls) — be especially careful not to see anybody else's work. It is not ■easy not to see, but if you pay close attention to your own work only, the test will be th« best

IV. Begin.

[Exactly twelve minutes allowed]

I. Have two sheets prepared again. You may not need both but hav-e them ready.

II. Keep the printing turned down imtil we are ready to begin — ^the same as before. Now are you all provided?

III. Begin.

[Exactly fifteen minutes allowed]

How the Scores were Computed

In Fundamentals

As stated under Arrangement of Problems, page 12 the problems in fundamentals were meant to increase in difficulty. With the purpose of determining their relative difficulty and thus securing a basis for scoring, the test was given to one hundred 6th grade pupils. As the pupils were given all the time they needed to do the problems as well as they could, the j>er cent doing €;ach correctly would indicate the relative difficulty. The results of this test are shown in table I below.

i6

TABLE I Preliminary Test Fundamentals — Unlimited Time Per cent of loo Grade VI Pupils Doing Correct Work

Numbers

of Problems

Addition

I

1 1

■2 c;

lOO

Q lo

This table shows the data from which the scoring of the fundamentals was determined. In this table, as in the entire study, the results were tabulated by steps, e. g., in the addition problems a score was given for each column added correctly; in the multiplication problems a score was given for each partial product and for each column in addition.

It was expected that a less number of children would be able to add the long columns than the short columns correctly; and that scores could be assigned to the respective problems according to difficulty as was done in the reasoning test ;^ but as is seen in table I, in this test, one step of a problem in fundamentals is about equal to another, be the step long or short — e. g., 96% of the children did the first column of problem one correctly, and exactly the same per cent did the longer and presumably harder column one of problem four correctly. Similarly, it is seen by examining table I that about as many pupils failed in the very

»See table II, p. i8.

short additions of the partial products in the multipHcation problems as failed in the long columns of the addition problems. And table I also shows that the number of mistakes in addition somewhat exceeded those for any one of the other operations.^ This fact is again brought out in the coefficients of correlation of standings of systems in fundamentals — see table XVI, page 39. Therefore the method of scoring adopted is that of assigning an arbitrary seore of x>ii.^ to~ each step of each problern, e. g., each pupil who added the first column of problem one correctly made one score for his school; each pupil who added all four columns of problem one correctly made a score of four for his school; each pupil who did problem five correctly made a score of five for his school, three in multiplication and two in addition, etc.

In Reasoning

The scores for the reasoning problems were determined from the results of two preliminary tests — one, giving one hundred 6th grade pupils all the time they needed to do the problems as well as they could in the order as printed (see page 11); and another, giving another one hundred 6th grade pupils all the time they needed to do the problems as well as they could in the reverse order from that as printed. The results as tabulated below in table II show that scores for reasoning problems of grade VI pupils can be very definitely ^irranged in a scale on the basis of relative difficulty. Just what the scale should be can\ only be determined by determining the form of distribution and the location of the zero point. From what is known of these the scale of weighting shown in the last column of table II is believed to be the best, and this is the one employed in the computations qi this study. However, in order to enable the reader to satisfy himself as to which is the best method, the scores of the twenty-six systems were calculated on each of three other bases — (i) counting each problem reasoned correctly a score of I ; (2) counting each problem reasoned correctly a score based on the ratio of its difficulty as shown in the next to the last column of table II; and (3) counting the scores made on only the first six problems for which presumably all pupils of all systems had ample time. See appendix, page 98.

iThe detailed study of mistakes among the five hundred individual pupils shows this even more clearly. See table at top of p. 29. 2 i8

TABLE II

Preliminary Tests

Reasoning — Unlimited Time

loo different pupils tested each time

Number

% reasoned

% reasoned

Weight

Weight

of

correctly

correctly ,

reasoned

according

used as

Problems.

as printed

as reversed

correctly

to Ave. % correct

probably best

I

92.6

93-8

I

82 . 2

84.1

I . I

I

81.5

1-15

I.I

57-4

63.2

1-5

1.4

1 .2

36.5

2.6

1.6

15-5

17.2

5-45 lO

27.4

25-7

3-6

II 7-5

12.3

7.6

16.4

II. 7

In both reasoning and fundamentals the scores used as a measure of the achievement of a system were computed by combining the scores of one hundred pupils. Where more than one hundred pupils were tested the papers used were drawn at random, the number drawn from each class being determined by the ratio of its number to the total number tested in the system. Where less than one hundred pupils were tested, the combined scores made were raised to the basis of one hundred pupils.

Precautions Observed to Make the Scoring Accurate

The simplicity of the tests made the scoring comparatively easy ; and with the observance of the following p^recautions it is believed that a high degree of accuracy was attained, (i) In so far as practicable, all the papers were scored by a single judge — only two persons being employed on any phase of the work for the entire twenty-six systems; (2) each problem was scored through one hundred or more papers, then the next followed through, etc. ; (3) the score for each part of each problem, the errors, etc., were entered on a blank provided with a separate column for each item ; (4) where there was doubt as to how the score should be made, the scorer made a written memorandum of how the case was finally decided and this memorandum served as the guide for all future similar cases.

What the Scores Measure

As used in this study the words achjeyements, products, abili- / ties, except where otherwise qualified, must necessarily refer to the results of the particular tests employed in this investigation. That some systems may achieve other and possibly quite as worth while results from their arithmetic work is not denied; but what^^is^eriied is that any system can safely fail to, attain good^resiilts.in the work covered by these particular tests. Whatever else the arithmetic work may produce, it seems safe to say that by^he.,.eii4^of^ the^ixth^^^s^ it should result in at

least good ability in, 4be,fQRi:^fo operations ^q,d th?

§imple^^«v^ryday kind of reasoning called for in these problems. It does not then seem unreasonable, in view of the precautions previously enumerated, to claim that the scores made by the respective systems afford a reliable measure of the products of tiieir respective procedures in arithmetic.

PART II.— ARITHMETICAL ABILITIES— THEIR VARI- ABILITY AND RELATIONSHIP The Problem

The first large question prompted by a preliminary survey of the achievements of the pupils in the twenty-six systems is, — JVhat is the trntiire of the produQt of the first six years, of arithmetic work? Part II is concerned primarily with the answer to this question as found in the results of the tests.

Some Phases of the Problem

Some of the phases of the question that a closer study of the data seems to promise help on are :

I. Is the net result of the arithmetic work of the first six years g product, or is it s^ral products?

2. To what extent are these products uniform (a) among systems, (b) among individual pupils in the systems, (c) among boys, (d) among girls?

3. In how far does the possession of one ability imply the possession of others?

Among those who still believe in the efficacy of formal discipline^ the possibility of there being a plurality of abilities required in the mastery of a subject such as arithmetic is scarcely entertained. Only recently has Educational Psychology shown the probability of the cornpar^tive independence and consequent lack of uniformity of capacities and products. Today, well founded doubt is entertained as to whether the possession of a given degree of ability in one phase of a subject necessarily implies the possession of anything like an equal degree of ability in another. Professor Thorndike concludes from a study of the arithmetical abilities of high school pupils^ and related researches that " ability m axithmetic. is but an abstract name for a number of partially independent abilities."^ This part of the present study addresses itself to the furnishing of more evidence along these lines.

The Data

The source of the data used to help answer the above questions is some §ix thoiisand test pap^r^ gathered from twenty-six representative school systems. Copies of the tests may be found in Part i, pages 10 and 11; as may also a statement of conditions under which the tests were personally given by the author, page 13; and the method of scoring, pages 15 to 18.