Digitized by the Internet Archive in 2019 with funding from Princeton Theological Seminary Library https://archive.org/details/interrelationofsOOhaug Vol. XXX N«. 6 PSYCHOLOGICAL REVIEW PUBLICATIONS Wkole No. 139 1921 THE Psychological Monographs EDITED BY JAMES ROWLAND ANGELL, Yale University HOWARD C. WARREN, Princeton University ( Review ) JOHN B. WATSON, New York (7. of Exp. Psychol.) SHEPHERD I. FRANZ, Govt. Hosp. for Insane ( Bulletin ) and MADISON BENTLEY, University of Illinois (Index) The Interrelation of Some Higher Learning Processes I BY V B. F. HAUGHT, Ph.D. Associate Professor of Psychology, State University of New Mexico PSYCHOLOGICAL REVIEW COMPANY PRINCETON, N. J. and LANCASTER, PA. Agents: G. E. STECHERT & CO., London (2 Star Yard, Carey St., W.C.) Paris (16 rue de Conde) FOREWORD The writer is greatly indebted to Dr. Joseph Peterson whose guidance and personal interest have been of inestimable value throughout the conduct of this investigation and whose scientific attitude has been a source of great inspiration. He also wishes to take this opportunity to express his thanks and appreciation to the students who served patiently as subjects to make the investigation possible, and especially to his wife who assisted greatly in making the mathematical calculations. ERRATA Attention is directed to the following list of corrections of er¬ rors, which unfortunately were not remarked in time to be set right in the body of the text : In Table VIII, p. 21, “100-19," lower left corner, should read “0-19.” “50-64" in top row should read “60-64.” In Table X, p. 23, “0-10” in top row should read “0-19." In Table XV, p. 31, the omitted spaces in the base line should be 6 and 8, respectively, from left to right; and “20-25” in the top row should read “20-24." In Table XVI, p. 32, “(Modified)” should be added at the end of the line just over the figure, beginning “X = .” In Table XVII, p. 33, numbers omitted in lower row should be, from left to right : 3, 2, 3, 5, 7, 8, 9, 9, 7, 6, 7, 3, 3, and 2. In Table XXII, p. 41, “54-60" in first column should read “64-60;" and “35-35 in the top row should read “35-39.” In Table XXV, p. 47, 7 and 17 in the 9th and the 10th squares of the bottom row should be 11 and 13, respectively. TABLE OF CONTENTS SECTION PAGE I. The Problem . i II. The Method . 2 1. Scores reduced to percentiles . 2 2. Selection of a criterion . 3 3. Scoring tests in the light of the criterion . 6 III. Subjects . 8 1. Manner of selecting . 8 2. College class and social status . . . 8 IV. The Binet-Simon Tests . 8 1. Order of giving tests . 8 2. Method of finding intelligence quotient . 10 V. The Rational Learning Test . 11 1. Description of test . 11 2. Method of scoring . 14 3. Analysis of data . 18 VI. Rational Learning (Modified) . 23 1. Description of test . 23 2. Method of scoring . 25 3. Analysis of data . 28 VII. The Checker Puzzle Test . 34 1. Description of test . . . . 34 2. Method of scoring . 36 3. Analysis of data . 36 VIII. The Tait Labyrinth Puzzle . 37 1. Description of test . 37 2. Method of scoring . 38 3. Analysis of data . 38 v VI TABLE OF CONTENTS SECTION PAGE IX. Intercorrelations . 40 1. Tests analyzed by comparison with the criterion 40 a. Rational learning analyzed . 42 b. Rational learning (modified) analyzed ... 43 c. Checker puzzle test analyzed . 46 d. Tait labyrinth puzzle analyzed . 48 2. Intercorrelations of tests scored in the light of the criterion . 49 3. Intercorrelation of tests scored by combining the factors equally . 54 X. Discussion of Learning and Intelligence . 58 1. Spearman's two factor theory . 59 2. The multiple factor theory . 62 XI. Summary and Conclusions . . 65 1. Method . 65 2. Results . 67 Bibliography . 70 THE INTERRELATION OF SOME HIGHER LEARNING PROCESSES I. Problem The purpose of this investigation is to analyze and to study the interrelations of some higher mental processes. Each of the experiments used involves two or more factors,1 such as time, repetitions, solutions, errors, etc. Each problem requires con¬ siderable time for the learning and in this respect is different from the individual parts of most intelligence tests. The pro¬ blems are also of such a nature that the subject may solve them either by the hit-and-miss method or by a reflective method. In every case it is possible to record all the responses of the subject, and thus provide objective data for the analysis. The purpose is to compare each factor in the tests with a criterion and with every other factor. The tests will be scored by combining the significant factors and they will then be further compared with the criterion and with each other. In this way it is believed that a detailed analysis can be made that will throw light on methods of learning and on the characteristics of tests involving the higher mental processes. This procedure will analyze not only the tests used, but also the criterion. An attempt will be made to answer such questions as the following : Does time measure any elements in learning that are not meas¬ ured by the criterion ? Does time measure any elements in learn¬ ing that are not measured by repetitions or errors? Does one rational learning experiment involve the same functions as any other? Is there a general rational learning function, or are such types of learning simply operations of various factors in different sorts of combinations. 1 Factor is used in this investigation to designate one kind of data, such as time, repetitions, solutions, errors, etc. 2 B. F. H AUGHT II. Method The raw scores in each experiment are first put into percentiles by use of Rugg’s table.2 In order to shorten the work a table based on seventy-four cases, the number used in this investiga¬ tion, was constructed. Three steps are involved in making such a table. First, the numbers from i to 74 are divided by 74, giv¬ ing the percent of subjects making each score. Second, since Rugg’s table provides for the percent failing, it is necessary to subtract each of these percents from 100, getting the percent below each score. Third, the percentile3 corresponding to each number obtained in the second step is taken from the table. An illustration will serve to make the method clearer. We shall take the subject making the highest score. He ranks number 1. This number divided by 74 gives 1.35 percent. If we subtract 1.35 from 100, we get 98.65, the percent of subjects below the best one. The percentile in the table corresponding to 98.65 is 86. Then 86 is the percentile rank of the subject having the highest rank in any test or factor of a test. The corresponding percentiles are found in this manner for each rank and then it is necessary only to rank the subjects by the usual method and read off the per¬ centiles from the table. There may be a slight objection to this method of assigning percentiles. The question may well be asked as to why the scores run down to o and yet up only to 86. Why should they not go up to 100? By this method the upper score will approach 100 as the number of cases is increased. If we think of the scale as a continuous one, we may regard o as extending from o to 14 and 86 as extending from 86 to 100. It would probably have been a little more correct to have moved each score up a half step or to have designated its middle position in a continuous 2 Rugg, H. O., Statistical Method Applied to Education, 1917, 396 ff. 3 Scores are assumed to fit the probability curve and percentages of sub¬ jects who make various scores correspond to percentages of area under the curve from the o point to a point on the base line. This point on base line is measured in units of 52> 53> 54> 57> 59> 62, 66, 67, 68, 69, and 74 used schedule i.16 Subjects 1, 2, 5, 6, 7, 8, 10, 13, 14, 17, 18, 19, 28, 29, 30, 36, 38> 5°> 55> 56, 60, 63, 70, 72, and 73 used schedule 2. 17 Sub¬ jects 3, 4, 9, 11, 16, 23, 24, 26, 27, 32, 33, 41, 42, 43, 45, 46, 48, 51, 58, 61, 64, 65 and 71 used schedule 3. 18 The numbers were arranged according to no special method. Schedule 2 is that used by Dr. Peterson19 in some of his work. Schedules 1 and 3 were made with the idea of having them equal in difficulty to 2. This, of course, makes the statement as to random numbering in the directions to the subject slightly in¬ correct, but each schedule could be obtained by a random selection. Therefore the statement should put the subject to no disadvan¬ tage. From observation there is no difference in the difficulty of the schedules. The orders 3-2, 5-4, and 6-5 in schedules 1, 2, and 3 respectively present especial difficulty to many subjects. The purpose in having different schedules was to reduce probability of coaching. There was no evidence of coaching in this test. Each subject was asked after the test was performed not to tell any other member of the class anything that might assist in the learning. The column headed Uc. in this table gives the unclassified er¬ rors, or the total number of errors regardless of kind. Errors marked f are called logical errors. They are errors which consist in guessing a number that has already been used for an earlier letter of the series, one that could, therefore, not possibly be right. Errors marked * are called perseverative errors. They are errors which consist in repeating a wrong guess while reacting to a single letter. 16 Schedule 1 A B C D E F G H I J 6 4 9 I 8 10 3 2 7 5 17 Schedule 2 A B C D E F G H I J 9 6 2 10 8 1 5 4 7 3 18 Schedule 3 A B C D E F G H I J 4 8 3 1 9 7 10 6 5 2 19 Peterson, Joseph, Experiments in Rational Learning, Psychol. Rev., 1918, 25, 433 ff- 14 B. F. H AUGHT In Table V are given the raw scores of all the subjects, and also, on the right hand side of the table, the scores converted into percentile rank and the final combined score. This combined score for the test is determined by combining the percentile rank in repetitions and in perseverative errors and then converting into percentile score from Rugg’s table, as will presently be ex¬ plained. Tables VI and VII show, respectively, the total and the partial correlations of each factor in the Rational Learning Test with the criterion, the Binet-Simon tests, and the total and the partial correlations of the factors in the Rational Learning Test. Table V. Showing the Number of Minutest, the Number of Repetitions, the Number of Each Kind of Errors, and the Percentile Rank for Each Kind of Data in Rational Learning. Subject Raw Score in Percentile Rank in | Time | Rep. Uc.E. L.E. P.E. Time Rep. Uc.E. 1 L.E. 1 P.E. 1 ScoreJ I 3 4 27 0 0 86 76 72 82 71 78 2 1 7 8 124 64 3 42 46 35 30 55 52 3 14 7 96 34 13 46 53 42 45 35 40 4 18 8 71 8 3 40 46 5o 63 55 52 5 5 4 56 23 0 74 76 57 49 7i 78 6 6 3 23 0 0 72 85 76 82 7i 84 7 17 11 64 13 1 42 35 53 57 63 49 8 21 14 196 78 38 3i 25 21 21 23 18 9 12 7 21 2 0 53 53 79 73 7i 65 IO 19 11 113 34 15 36 35 37 45 28 28 ii 18 9 47 10 3 40 42 61 59 55 48 12 20 14 240 143 12 33 25 18 18 38 28 13 7 5 59 13 4 69 68 55 57 5i 61 14 20 11 168 66 3 33 35 28 28 65 44 15 16 6 53 28 3 44 61 59 47 55 59 16 7 4 44 6 4 69 76 65 66 5i 68 17 25 15 169 72 11 25 16 27 23 40 22 18 14 9 81 28 5 46 42 46 47 48 44 19 9 7 93 38 7 61 53 43 42 45 49 20 13 10 62 18 5 49 38 54 53 48 38 21 12 9 79 42 3 53 42 46 38 55 48 22 8 6 105 44 6 65 61 40 35 46 56 23 18 12 130 58 0 40 31 3i 32 7i 53 24 8 8 43 9 2 65 46 65 61 59 56 25 17 14 109 31 3 42 25 39 46 55 36 26 16 10 88 34 17 24 38 45 45 25 28 27 12 7 88 35 4 53 53 45 43 5i 54 28 12 7 53 16 0 53 53 59 54 7i 65 29 14 7 75 21 6 46 53 48 50 46 51 30 10 6 35 1 4 59 61 69 76 5i 58 31 33 14 no 36 13 0 25 38 43 35 25 32 12 7 72 40 2 53 53 49 40 59 59 INTERRELATION OF HIGHER LEARNING PROCESSES i5 Table V (Continued) Subject .1 Raw Score in Percentile Rank in | Time 1 Rep. Uc.E. L.E. P.E. Time Rep. Uc.E. L.E. P.E. Scoret 1 33 18 12 1 16 38 II 40 31 36 42 40 34 34 13 13 129 41 2 49 28 32 38 59 40 35 10 5 45 5 2 59 68 63 68 58 68 36 19 7 87 36 13 36 53 45 43 35 40 37 8 8 47 9 O 65 46 61 61 7i 60 38 30 11 178 7 1 13 18 35 23 25 35 33 39 5 3 23 4 0 74 85 76 7 1 7 1 84 40 11 6 75 23 10 57 61 48 49 40 52 4i 9 5 4i 7 0 61 68 67 65 7 1 73 42 20 12 174 54 13 33 3i 25 33 35 32 43 11 6 38 10 I 57 61 68 59 63 65 44 17 8 106 20 10 42 46 40 5i 40 38 45 19 6 65 19 12 36 61 52 52 38 51 46 13 5 48 13 3 49 68 60 57 55 62 47 22 9 126 47 28 29 42 33 35 23 3i 48 12 9 62 7 5 53 42 54 65 48 44 49 9 5 44 14 2 61 68 65 56 59 68 50 19 10 146 43 35 36 38 30 36 18 22 5i 24 9 hi 42 14 27 42 38 38 3i 35 52 12 7 55 10 4 53 53 58 59 5i 54 53 11 9 90 40 8 57 42 43 40 44 38 54 13 8 56 12 5 49 46 57 58 48 45 55 19 9 98 59 5 36 42 4i 3i 48 44 56 32 15 463 267 73 14 16 0 0 0 0 57 26 21 398 206 28 23 0 14 14 23 14 58 10 7 59 18 9 59 53 55 53 42 46 59 15 8 97 48 9 45 46 42 34 42 40 60 7 8 24 5 0 69 46 73 68 7 1 60 61 14 5 72 23 9 46 68 49 49 42 57 62 8 6 45 7 1 65 61 63 65 63 65 63 4 6 29 5 0 82 61 70 68 7 1 70 64 8 7 4i 4 0 65 53 67 7i 7 1 65 65 5 5 1 7 0 0 74 68 86 82 7 1 73 66 11 6 69 19 2 57 61 5i 52 59 61 67 14 11 124 68 9 46 35 35 27 42 35 68 8 8 65 18 2 65 46 52 53 59 56 69 12 7 70 14 8 53 53 5i 56 44 48 70 5 5 20 1 0 74 68 82 76 7 1 73 7i 9 5 58 10 13 61 68 56 59 35 53 72 28 6 76 20 15 21 61 47 5i 28 42 73 22 7 1 16 24 14 29 53 36 48 3i 37 74 13 7 54 8 8 49 53 58 63 44 48 t The time is given to the nearest minute. t Scores are found by adding the percentile rank in repetitions to the per¬ centile rank in perseverative errors and then again reducing to absolute percentiles by Rugg’s table. The reason for this will appear later. i6 B. F. H AUGHT Table VI. Showing Correlations! and Partial Correlations of Each Factor! of the Rational Learning Test with the Binet-Simon Tests. The symbol for correlation has been omitted. The table should read ri2 = -27’ ri2.3 = •°8> etC- 12 .27 13 .31 14 .25 15 .23 16 .27 12-3 .08 132 .14 14-2 .05 15-2 .04 16-2 .15 124 .14 13-4 • 17 14-3 .02 15-3 .02 16-3 .18 12-5 .16 13-5 .20 14-5 .11 15-4 —.02 16-4 • 17 12-6 .10 13-6 .18 14-6 •03 156 .05 16-5 .19 12-34 .09 13-24 .14 14-23 — .02 15-23 — .02 16-23 .16 12-35 .08 1325 .14 14-25 .04 14-24 — .01 16-24 .14 12-36 — .02 13-26 .15 14-26 —.03 15-26 .01 16-25 • 14 12-45 .14 13-45 .17 14-35 .00 15-34 .00 16-34 .22 12-46 .10 13-46 .22 14-36 —.14 15-36 —.07 i6-35 .18 12-56 .09 13-56 .19 14-56 — .02 15-46 .04 i6-45 • 17 12-345 .09 13-245 .14 14235 — .01 15-234 .00 16-234 .20 12-346 .01 13-246 .20 14-236 —.14 15-236 —.07 16-235 .16 12-356 .00 13-256 .16 14-256 —.03 15-246 .04 16-245 .14 12-456 .10 I3-456 .24 I4-356 —.15 I5-346 .08 16-345 .23 12-3456 .01 13-2456 .21 142356 —.14 15-2346 .08 16-2345 .21 f All correlations are worked by the product-moment method. $ For the sake of brevity the factors are designated by numbers as follows : 1. Binet-Simon Tests 4. Unclassified Errors 2. Time 5. Logical Errors 3. Repetitions 6. Perseverative Errors Table VII. Showing Correlations and Partial Correlations of the Factors in the Rational Learning Test. 23 .72 24 .78 25 .72 26 .69 34 •78 23-4 .28 243 •50 25-3 •43 26-3 •55 34-2 •50 235 •43 24-5 .46 25-4 —.07 26-4 .26 34-5 .48 23-6 .60 24-6 •55 25-6 .26 26-5 •43 34-6 .72 23-45 .27 24-35 .30 25-34 —.04 26-34 •34 34-25 •34 23-46 •37 24-36 .22 25-36 .24 26-35 •44 34-26 •57 23-56 •44 24-56 .28 25-46 .00 26-45 .23 34-56 •55 23 456 •37 24-356 •03 25-346 .10 26-345 •34 34-256 •50 35 •71 36 •49 45 •94 46 •75 56 .64 35-2 •40 36-2 .00 45-2 .87 46-2 •47 56-2 •29 35-4 — .12 36-4 —.23 45-3 .88 46-3 .67 56-3 •45 35-6 •59 36-5 .07 45-6 .91 46-5 •59 56-4 —•30 35-24 — .10 36-24 —•32 45-23 .84 46-23 •55 56-23 •30 35-26 .41 36-25 —.14 45-26 .88 46-25 •47 56-24 —.28 35-46 —.24 36-45 —.30 45-36 .87 46-35 .64 56-34 —•33 35-246 — .22 36-245 —.36 45-236 .85 46-235 •58 56-234 —•34 If we consider the size of the probable error,20 it seems safe to infer from the data in Table VI that repetitions and persever- 20 The probable error for 74 cases is .075 for zero correlation and .063 for a correlation of .40. The probable errors are not given in each instance, since they are available in tables. INTERRELATION OF HIGHER LEARNING PROCESSES 17 ative errors have elements in common with the criterion that are not common to the other factors or to each other. This con¬ clusion is based on the correlations, r13.2456 and r16.2345, which are .21 in each case. This is three times the probable error and may be regarded as significant. Then if it is desired to score the Rational Learning Test so as to have it correlate highest with the Binet-Simon tests, these two factors must be included. It is evident that time, unclassified errors, and logical errors, have nothing in common with the criterion that is not included in the other four factors, if we hold that a correlation less than three times the probable error is not significant. It must also be kept in mind in this case that linearity is assumed. To be more specific, the expression, r12.3456=.oi, means that everything common to 1 and 2 is contained in 3, 4, 5 and 6. Then factor 2, time, may be discarded if the remaining four are used. Further, since r15.346=.o8, factor 5, logical errors, may be discarded. In like manner, since r14.36=-.i4, it is fairly safe to discard factor 4, unclassified errors. It is true that a correlation of -.14 may be slightly significant. It may mean that, if the elements in re¬ petitions and perseverative errors are removed, the more intelli¬ gent the subject the more errors he will make. Let the question be pushed further by the use of multiple correlation. By the use of formula (1), it is found that ■^■1(23456) *33 But R,(se) = -32 The difference between the correlation with the criterion when all five factors are used and when repetitions and perseverative errors only are used is small enough to be entirely neglected. This is further evidence that factors 3 and 6 are sufficient to use in the final scores. The correlation would be lowered, if any¬ thing were lost by discarding the others. The next step is to find the proper combination of these two factors to give the best correlation. That is, must they be com¬ bined equally or in some other proportion? To determine this, formula (2) will be used. Repetitions will be designated as the i8 B. F. H AUGHT major factor and perseverative errors as the minor factor. The fundamental constants for determining C are : rIM = -3i rlm = .27 rMm = 49 <7 M = 16.35 C J m = I5-56 When these values are substituted in formula (2), the result is 1.05, or for all practical purposes 1. This means that the two combined equally will give the best correlation. To determine what the correlation will be when the two factors are combined equally, formula (3) will be used, in which the letters have the same meaning as in formula (2). The correlation is found to be .33. The actual correlation when the scores in repetitions and perseverative errors are combined equally is .324. The final scores are found by adding the percentile rank in repetitions to the percentile rank in perseverative errors and then again using Rugg’s table for reducing to absolute percentiles as was done with the raw scores. Multiple correlation indicated that a correlation of .32 could be obtained by combining repetitions and perseverative errors. Formula (2) indicated that the best combination is that in which they are combined equally. Formula (3) indicated that a cor¬ relation of .33 should be obtained when they are combined equally. When the actual combination is made and the correlation is worked out, the result is .324. The difference between these two numbers is small enough to be accounted for by the way fractions are carried out, and by using 1 instead of 1.05 as a ratio for combining. (2) Analysis of Data. There is a “present but low”21 positive correlation in this test 21 Correlations below .20 will be considered negligible; from .20 to .40, present but low ; from .40 to .60, marked ; above .60, high. See Rugg, H. O., Statistical Methods Applied to Education, 1917, 256. INTERRELATION OF HIGHER LEARNING PROCESSES 19 between the criterion and each of the factors. In other words, there is slight evidence that ability in the criterion and ability in each factor of the test accompany each other. Subjects above the average in the former will be above the average in each of the latter. Since the variability22 is made the same in each kind of data by reducing to absolute percentiles, a change in one test or factor will be accompanied by a like change in the other test or factor, and equal to the correlation of the two tests or factors. Thus, time and the criterion have a correlation of .27. Therefore, every unit-change in one will be accompanied by a like change of .27 of a unit in the other. In like manner, each of the other factors may be compared with the criterion by refer¬ ence to the correlations in Table VI. When the standard devia¬ tions are equal, the regression line for the columns takes the form y = rx and that for the rows takes the form x = ry. Each factor in this test has something common to the criterion. Repetitions have most and unclassified errors least. There is, moreover, a great overlapping of common elements. Thus re¬ petitions contain all that is common to time and the criterion, all that is common to unclassified errors and the criterion, and all that is common to logical errors and the criterion. Time has all that is in logical or unclassified errors with respect to the criterion. Unclassified and logical errors have practically the same elements as far as the criterion is concerned. Repetitions and perseverative errors have something, however, not found in any one of the other factors or in any combination of them. These two contain everything in all the factors needed to get the highest correlation with the criterion. We shall use Blakeman’s criterion23 for linearity. When the proper values substituted in the formula give a result greater than 2.5, non-linearity will be said to exist and a table will be con¬ structed showing the lines of the means of the rows and the 22 The standard deviations for the Binet-Simon Tests, time, repetitions, unclassified errors, logical errors and perseverative errors are respectively 16.62, 1647, 16.36, 16.68, 16.56, and 15.56. 23 For formula, see Rugg, Statistical Methods Applied to Education, 1917, 283. 20 B. F. H AUGHT means of the columns. If the result is less than 2.5, the correl¬ ation is for all practical purposes linear and no table of regression lines will be constructed. The correlation-ratios for each of the factors with the criterion are as follows: 12 = .46 and .52 13 = .41 and .47 14 = .47 and .52 15 = .44 and .49 16 = .37 and .40 These values substituted in the Blakeman formula give results as follows : Tests 1 and 2, Tests 1 and 3, Tests 1 and 4, Tests 1 and 5, Tests 1 and 6, 2.37 and 2.83 1. 71 and 2.25 2.54 and 2.90 2.39 and 2. 76 1. 61 and 1.88 According to Blakeman’s criterion, the correlations of repetitions and perseverative errors with The Binet-Simon tests are linear, but the correlations of time, unclassified, and logical errors with The Binet-Simon tests are non-linear. The regression lines for the last three correlations are now ready to be constructed. The regression lines showing the correspondence between scores in time and the criterion are shown in Table VIII. Each asterisk indicates the position of one person as determined by both tests. The circles through which the broken line passes represent the means24 of the columns and those through which the whole line passes represent the means of the rows. This table shows that the two sets of relationships are in very close agree¬ ment. That is, the regression of the x-values on y is very nearly the same as the y-values on x. The x-values show a tendency to in¬ crease constantly with an increase in y-values from the fortieth percentile up. Below this point there is little relation between x and y-values. The y-values have a tendency to increase constantly 24 The means for the columns or rows are found by adding the exact per¬ centile ranks in each row1 or column and dividing by the number of cases. INTERRELATION OF HIGHER LEARNING PROCESSES 21 Table VIII. Showing the Distribution of Subjects on the Basis of Time in Rational Learning and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. with an increase in x-values from the fiftieth percentile up. Be¬ low this point there is little relation between y and x-values.25 The relation between unclassified errors and the criterion is shown in Table IX. The agreement between the regression of the x-values on y and the y-values on x is not close. In neither case is the tendency for one value to increase with an increase in the other constant. The x-values increase with an increase in y-values from the fortieth percentile up to the sixtieth. The other regression line is almost horizontal from the thirtieth to the sixtieth percentile. An increase in Binet-Simon scores in¬ dicates nothing with respect to unclassified errors until the for¬ tieth percentile is reached. An increase in Binet-Simon scores 25 Reasons for this lack of relation in the lower quartile will be suggested later in comparing the final scores with the criterion. 22 B. F. H AUGHT Table IX. Showing the Distribution of Subjects on the Basis of Unclassi¬ fied Errors in Rational Learning and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. from the fortieth to the sixtieth percentile means a rather rapid increase in unclassified errors. Above the sixtieth percentile an increase in Binet-Simon scores means no change in unclassified errors. From the beginning to the end, an increase in unclassified errors means nothing with respect to intelligence as measured by the Binet-Simon Tests. Table X shows the relation existing between logical errors and the criterion. The agreement between the two sets of values is not close. There is a fairly constant increase in y-values with an increase in x-values from the fortieth percentile up. A smoothed curve will show an increase from the very be¬ ginning. In the other regression line there is no tendency for the x-values to increase constantly. Up to the fortieth percentile an increase in Binet-Simon scores means no change in score in INTERRELATION OF HIGHER LEARNING PROCESSES 23 Table X. Showing the Distribution of Subjects on the Basis of Logical Errors in Rational Learning and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. logical errors, but from here on to the sixty- fifth percentile, it means a constant increase. Above this point there are only five cases, and the curve has no significance. The skewness of each distribution is almost zero, since the median and the mean almost coincide when the scores are reduced to percentiles. This assumes that skewness is measured in terms of the median, the mean, and the standard deviation. All medians and means are approximately 50, and all standard deviations are approximately 16.66. VI. The Rational Learning Test ( Modified ) (1) Description of Test and Method of Scoring. This test is very similar to Rational Learning. The apparatus consists of a board about twenty inches sguare, through which 24 B. F. H AUGHT are put one hundred bolts arranged in ten rows with ten bolts in a row. The rows are lettered from A to J and the bolts in each are numbered I to io. One bolt, and only one, in each row is connected in a circuit with an electric bell so that when this bolt is touched with a stylus the bell will ring. Figure I shows the apparatus. Fig. I. Showing Apparatus for Rational Learning (Modified). J * * * * * * * * * * I * * ♦ * * * * * * * H * * * * * * * * * * G * * * * * * * * * * F * * ♦ * * * * * * * E * * * * * * * * * * D * * * * * * * * ♦ * C * * * * * * * * ♦ * B * * * ♦ * * * * * * A * * * * * * * * * * i 2 3 4 5 6 7 8 9 IO Each asterisk represents a bolt. The bolts are actually num¬ bered in each row just as indicated in row A in the diagram. The method of recording the data is exactly as in Rational Learn¬ ing. The instructions to the subject follow : “You have in the apparatus before you ten rows of bolts with ten bolts in each row. The rows are lettered from A to J and the bolts in each row are numbered from i to io. One bolt in each row, and only one, is connected in such a way that the bell will ring when the circuit is made. That is, each letter is assigned a number in a random order from i to io. This num¬ ber is the one that will cause the bell to ring when the bolt is touched. No two letters have the same number. Your problem is to begin with row A and find the bolt that will ring the bell. Then go on to row B and find the bolt. Con¬ tinue until you have reached row J. Now go back to row A and repeat the process. Continue repeating the process until you go from A to J twice in succession without making any mistake. Then you are through. INTERRELATION OF HIGHER LEARNING PROCESSES 25 You are to ask no questions after you start, but are to use all the mental powers at your command in order to complete the learning as soon as possible. You will be judged by (1) the total time you take, (2) the number of errors or wrong guesses you make (every bolt you touch being a guess), (3) the number of repetitions from A to J that you require for the learning. Re-read these instructions carefully, if necessary, to understand what you are to do. The meaning will be clearer as we go on with the experiment/' The subject stood facing the apparatus so that row A was to¬ ward him. The experimenter recorded every response just as was done in Rational Learning. When the test was finished the total time was recorded and the subject was asked to write as much as he could about the method he used in learning the problem. The same three schedules of numbers were used in this test as in Rational Learning.26 Those having schedule 1 in Rational Learning had schedule 2 in this one; those having schedule 3, had 1 ; and those having schedule 2, had 3. The same precau¬ tion was taken to prevent coaching as in Rational Learning. Each subject was asked not to tell any other member of the class anything that might assist in the learning. Results are given in Table XI. Table XII shows that time has something in common with the criterion that is not common to the other factors. Perseve- rative errors seem to be least significant and may be dropped as far as the final scores are concerned. If we consider only the four factors and partial correlations of the third order, logical errors appear useless. In like manner, when the three remain¬ ing factors are considered alone, repetitions appear insignificant. If we push the analysis further, however, it is very evident that unclassified errors must be retained with time to make up the final score. We may make the analysis more logical by first considering ri6 2345 = .01, which means that there is nothing com¬ mon to 1 and 6 that is not contained in 2, 3, 4, and 5. Factor 6 may therefore be dropped. In like manner, since ^5.234 *03 26 This test was given before Rational Learning. 26 B. F. H AUGHT Table XI. Showing the Number of Minutes, the Number of Repetitions, the Number of each Kind of Errors, and the Percentile Rank for Each Kind of Data in Rational Learning (Modified). Subject Raw Score in | Time | Rep. j Uc.E. L.E. P.E. I 3 4 47 19 O 2 7 10 138 72 3 3 7 4 42 18 3 4 10 7 55 17 3 5 2 3 45 l6 0 6 11 7 70 24 3 7 9 10 102 l6 13 8 18 12 182 67 21 9 27 10 124 60 8 IO 15 9 130 6l 5 ii 13 13 H5 58 9 12 10 9 69 23 0 13 10 12 164 90 5 14 11 10 142 58 11 15 10 7 67 37 2 16 4 4 59 23 0 1 7 1 7 18 261 94 16 18 10 7 66 27 15 19 7 10 145 66 4 20 16 16 176 72 22 21 12 8 128 79 6 22 7 7 95 55 1 23 1 7 15 167 83 2 24 6 3 48 22 3 25 15 12 114 38 11 26 6 6 73 22 2 27 9 9 92 48 6 28 9 7 1 16 67 5 29 i n ! i° 124 50 8 30 7 7 43 26 2 31 28 12 165 83 24 32 19 12 123 55 10 33 14 16 207 141 15 34 12 10 118 65 3 35 23 14 217 122 47 36 29 14 236 no 14 37 10 9 106 49 2 38 18 14 314 152 30 39 5 3 25 4 0 40 13 10 127 58 10 41 11 10 74 26 4 42 9 12 73 23 0 43 12 11 no 54 13 44 23 15 292 153 24 45 19 6 78 34 6 46 11 7 85 49 3 47 18 16 231 109 11 48 15 8 7i 29 8 49 12 15 109 44 18 Percentile Rank in Time Rep. Uc.E. j L.E. j P.E. | ScoreJ 82 75 75 69 76 79 68 5i 43 41 58 56 68 75 82 71 58 77 57 65 69 73 58 67 87 82 77 76 76 85 53 65 63 63 58 59 61 51 53 7 6 38 58 37 42 35 43 28 35 18 5i 46 1 46 46 1 32 43 56 44 1 46 53 43 46 39 49 48 45 48 57 56 64 65 76 61 57 42 40 33 53 49 53 5i 42 48 4i 48 57 65 65 58 63 64 79 75 68 65 76 73 40 16 21 32 34 29 57 65 65 61 36 61 68 5i 42 43 55 55 4i 27 35 4i 25 38 49 59 45 38 5i 46 68 65 54 49 68 64 40 33 38 35 63 40 74 82 73 67 58 73 43 42 50 57 4i 46 74 70 61 67 63 70 61 56 55 54 5i 59 61 65 49 43 53 55 53 5i 46 52 46 49 68 65 79 62 63 73 14 42 39 35 21 23 33 42 47 49 43 40 45 27 32 18 36 39 49 5i 48 44 58 49 24 37 30 21 0 25 0 37 25 26 37 0 57 56 52 53 63 54 37 37 0 14 14 14 77 82 87 87 76 85 46 5i 45 48 43 45 53 5i 5i , 62 55 53 61 | 42 | 61 | 65 1 76 1 64 49 46 51 50 38 5i 24 33 14 0 21 18 33 70 59 59 5i 45 53 65 58 53 58 56 37 27 28 28 4i 33 43 59 62 60 46 53 49 33 51 54 3i 5i INTERRELATION OF HIGHER LEARNING PROCESSES 27 Table XI (Continued) Subject Rav\ 1 Score in Percentile Rank in Time Rep. Uc.E. ■ ■ 1 L.E. P.E. Time Rep. Uc.E. L.E. P.E. ScoreJ 50 7 8 95 52 6 68 59 54 52 51 64 51 24 12 250 no 17 21 42 23 26 32 21 52 21 17 189 7 1 24 28 21 33 42 21 29 53 12 10 154 87 12 49 5i 40 34 39 44 54 12 8 48 8 2 49 59 73 79 63 64 55 19 11 235 95 21 33 46 27 30 28 27 56 17 14 172 78 7 40 37 36 38 49 38 57 9 8 131 62 0 61 59 43 45 76 53 53 13 18 269 11 7 16 46 16 18 23 34 32 59 19 15 151 83 2 33 33 4i 35 63 36 60 15 16 207 74 19 43 27 32 40 30 37 61 13 13 59 5 16 46 39 68 82 34 58 62 9 8 100 43 7 61 59 54 55 49 58 63 19 15 166 81 11 33 33 38 37 4i 34 64 19 12 91 29 7 33 42 56 60 49 44 65 11 11 122 76 1 53 46 48 39 68 5i 66 19 9 105 4i 2 33 56 52 56 63 42 67 14 16 167 64 9 45 27 38 45 45 42 68 7 11 88 37 6 68 46 58 58 5i 67 69 12 11 104 53 11 49 46 53 5i 4i 52 70 7 10 61 20 2 68 5i 66 68 63 69 7 1 1 7 24 188 94 9 40 0 34 32 45 36 72 22 7 9i 41 4 27 65 56 56 55 42 73 8 10 88 52 1 64 5i 58 52 68 64 74 8 6 52 18 0 64 70 70 7i 76 69 f Scores are found by adding, the percentile rank in time to the percentile rank in unclassified errors and then again reducing to absolute percentiles by Rugg’s table. The reason for this will appear later. Table XII. Showing Correlations and Partial Correlations of Each Factor]* of Rational Learning (Modified) with the Binet-Simon Tests. 12 .42 13 •3i 14 •44 15 42 16 •35 12-3 .31 132 .08 14-2 .24 15-2 •23 16-2 .11 124 .19 13-4 —.05 14-3 •33 15-3 •30 16-3 .21 12-5 .23 13 5 .04 14-5 -15 15-4 .04 164 .07 12-6 .27 13-6 -13 14-6 .29 15-6 .28 16-5 .12 12-34 .20 13-24 .08 14-23 .24 15-23 .21 16-23 .08 12.35 •23 1325 —.04 14-25 .08 15-24 .04 16-24 .00 1236 .24 13-26 .04 14-26 .20 15-26 .20 16-25 .04 12-45 • 19 13-45 — 05 14-35 -15 15-34 .04 i6-34 .07 12-46 .17 13-46 — .06 I4-36 .27 15-36 •25 16-35 .12 12-56 -19 I3-56 .00 I4-56 .10 I5-46 -05 16-45 .07 12-345 .20 13-245 — .08 14-235 .11 15-234 •03 16-234 .01 12-346 .19 13-246 — .09 14236 .23 15-236 .20 16-235 .04 12-356 .19 13-256 —.05 14-256 .07 15-246 .04 16-245 .01 12-456 •17 I3-456 —.05 I4-356 .11 I5-346 .04 16 -345 .08 12-3456 .19 13-2456 — .08 I4-2356 .10 15-2346 .03 16-2345 .01 fFor the sake of brevity the factors are designated by numbers as follows : 1. Binet-Simon Tests 4. Unclassified Errors 2. Time 5. Logical Errors 3. Repetitions 6. Perseverative Errors 28 B. F. H AUGHT factor 5 may be discarded. Since r13.24 = .08, factor 3 may be discarded. But since r12.4 = .19, and r14.2 = .24, it is evident that factors 2 and 4 must be retained. This conclusion may be further verified by using multiple correlation. The following results are found : R-i (23456) ~ -479 RiG*) = 473 It is seen from the above that very little is lost by excluding all the factors except time and unclassified errors. We shall use for our final scores, then, these two factors which appear to give everything that is necessary to get the highest correlation with the criterion. Our next problem is to find the combination of time and un¬ classified errors that will give this best correlation. The same formulae will be used as were used in Rational Learning. Desig¬ nating unclassified errors by M and time by m, we may use the following data for finding the best value of C and the resulting correlation : r IM r Im r Mm = 44 = .42 = .66 When these values are substituted in formula (2), the value of C is found to be .79. This means that the best combination of time and unclassified errors is to add .79 of the time to the un¬ classified errors. This gives a correlation of .47; but if 1 is used in the formula instead of .79, the correlation is still .47. For our final score in this test, we shall use time and unclassified errors combined equally. The exact method of getting them will be to add together the scores in time and unclassified errors and then reduce to absolute percentiles by the same method as was used in Rational Learning. (2) Analysis of Data. There is a “present but low” positive correlation between the INTERRELATION OF HIGHER LEARNING PROCESSES 29 criterion and repetitions and between the criterion and per- severative errors. The correlation of the criterion with each of the other factors is “marked/ Subjects above the average in the former will tend to be above the average in each of the latter. Since the variability^' is made the same in each kind of data by reducing to absolute percentiles, a unit-change in one test or fac¬ tor will be accompanied by a like change in the other test or factor equal to the correlation of the two factors. Thus time and the criterion have a correlation of .42. Therefore every unit- change in one will be accompanied by .42 of a unit-change in the other. In like manner, each of the other factors may be com¬ pared with the criterion by reference to the correlations in Table XII. Each factor in this test has much in common with the criterion. Unclassified errors have most and repetitions least. Table XII Table XIII. Showing Correlations and Partial Correlations of the Factors in the Rational Learning Test (Modified). 23 .60 24 .66 25 .61 26 -65 34 •77 23-4 .19 24-3 -39 25-3 .36 26 3 •45 34-2 .62 23-5 •33 24-5 •32 254 .01 26-4 •35 34-5 •53 23-6 •34 24-6 .38 25-6 •36 26-5 ■45 34-6 .61 23-45 .20 24-35 .18 25-34 .05 26-34 •33 34-25 •47 23-46 .15 24-36 .23 2536 .24 26.35 .38 34-26 •55 23 56 .21 24-56 .14 25-46 .07 26.45 •36 34-56 •45 23-456 .16 24-356 .06 25-346 .09 26-345 •34 34-256 .42 35 .67 36 .61 45 .92 46 .70 56 .60 35-2 .48 36-2 ■36 45-2 .87 46-2 .48 56-2 •34 35-4 — .16 36-4 .16 45-3 -85 46-3 •45 56-3 •33 35-6 .48 36-5 -35 45-6 .87 46-5 •47 56-4 —•15 35-24 — .16 36-24 .10 45-23 .83 46-23 •34 56-23 .20 35-26 .41 36-25 .24 45-26 .85 46-25 •39 56-24 —.17 35-46 —.13 36-45 .14 45-36 .83 46-35 •35 56-34 —.13 35-246 —.14 36-245 .07 45-236 .82 46-235 •32 56-234 — .16 shows that time contains all that is common to repetitions and the criterion and nearly all that is common to perseverative errors and the criterion. Unclassified errors contain all that is common to the criterion and any one of the three factors, repetitions, logical errors and perseverative errors. Uogical errors contain all that is common to repetitions and the criterion. 27 The standard deviations for time, repetitions, unclassified errors, logical errors, and perseverative errors are 16.65, 16.48, 16.59, 16.68, and 16.35 re¬ spectively. 30 B. F. H AUGHT Table XIII shows that the factors have a “high" correlation with each other. Unclassified and logical errors have the highest. Since the correlation of these two factors is so high, neither of them can have very much that is not in the other. Table XIV. Showing the Distribution of Subjects on the Basis of Time in Rational Learning (Modified) and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns, and those through which the continuous line passes repre¬ sent the means of the rows. Each asterisk represents a subject. The several correlation-ratios for each of the factors with the criterion are as follows : 12 = .56 and .66 13 = .47 and .59 14 = .53 and .61 15 = .58 and .60 16 = .49 and .49 INTERRELATION OF HIGHER LEARNING PROCESSES 3i The Blakeman formula gives results as follows: Tests 1 and 2, Tests 1 and 3, Tests 1 and 4, Tests 1 and 5, Tests 1 and 6, 2.36 and 3.24 2.25 and 3.20 1.87 and 2.69 2.55 and 2.73 2.19 and 2.19 All the correlations are non-linear except the last one, the crit¬ erion and perseverative errors. Table XIV shows the correspondence between scores in time and the criterion. The Blakeman formula indicates that each of the regression lines is non-linear. The curve of the means of the rows shows very little change in x-values with an increase in y-values up to the fifty-fifth percentile, where the change is accelerated, probably, on account of the fact that the Binet-Simon Table XV. Showing the Distribution of subjects on the Basis of Repe¬ titions in Rational Learning (Modified) and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns, and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. 32 B. F. H AUGHT tests do not actually test the brightest subjects. The curve of the means of the columns has four distinct parts. As the x-values increase, the y-values decrease rapidly from the twenty-fifth to the thirty-fifth percentile, increase rapidly from the thirty-fifth to the forty-fifth percentile, remain about the same from the forty-fifth to the sixty-fifth percentile, and increase rapidly from the sixty-fifth percentile. Table XV shows the correspondence between the scores in repetitions and the criterion. The regression line of the means of the columns may be considered linear according to the Blake- man test. The regression line of the means of the rows, how¬ ever, is non-linear. The non-linearity is caused by the four cases in the second and third rows from the botton and the one case in the top row. The removal of these five cases will reduce the Table XVI. Showing the Distribution of Subjects on the Basis of Un¬ classified Errors in Rational Learning (Modified) and the Binet- Simon Tests. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. x s Unclassified Errors in Rational Learning 0- /? 20- 24 25- Zf 30- 31- 35- ?? 40- 44 45- 49 5°- 5+ 55- ?,? 60- 65- 6? 70- 74- 75- 7? 80- \00 100- 60 -O* 1 75- 75 ■# *#■ * 4 74- 70 0 % » ♦ *» * / Q* *> A 8 % * •» P » „ 6 59- 55 * »» o »• n / - \ / $ i i k > % 6 54- 50 « * * /§r / k / V i »* \ • '6 6 10 49- 45 ** #■ * A- * 'ft/ ** k *■ * 11 44- 40 * ft* ♦ * 5 35- 55 V \ *\ 1 / / $ » * C * *♦ 7 34- 30 -o;' -O * * 7 29- 25 JPT 2 24- 20 -oc < - 2 ,si * 3 3 2 3 5 7 8 9 n 7 5 6 3 3 2 74 «« £ s: o •$ INTERRELATION OF HIGHER LEARNING PROCESSES 33 Table XVII. Showing the Distribution of Subjects on the Basis of Logical Errors in Rational Learning (Modified) and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. Logical Errors in Rational L earning (Modified) correlation-ratio from .59 to -34> provided the mean and standard deviation are not changed. Such a reduction in correlation-ratio will destroy the non-linearity. Table XVI shows the correspondence between the scores in unclassified errors and the criterion. The regression line of the means of the columns is linear according to the test. The other regression line, however, is non-linear. The non-linearity can be eliminated by removing the two cases in the second row from the bottom. The same result can be obtained by assuming that these two cases fall on the median. Table XVII shows the correspondence between the scores in logical errors and the criterion. Here both regression lines are slightly non-linear. The non-linearity, however, is due to a few cases and for that reason has no significance. 34 B. F. H AUGHT VII. Checker Puzzle (i) Description of Test and Method of Scoring. This may be called a checker puzzle test because of its similar¬ ity to the game of checkers. As far as the writer knows it was first used as a psychological experiment by Dr. Strong in the Jesup Psychological Laboratory of George Peabody College for Teachers. He also gives a suggestion for its use in the Psychological Bulletin.28 The instructions to the subject and the method of scoring are different in many respects from those used in the Jesup Psychological Laboratory. They were de¬ vised by the writer, but, of course, reflect to a considerable de¬ gree those with which he was already acquainted. Fig. II. Showing Apparatus Used in the Checker Puzzle Test. The subject is given a card on which are seven circles as shown in Figure II. He is also given three red and three black checkers. The instructions to the subject are as follows: “You are here given a row of seven circles. The three at the left are red, the three at the right are black, and the middle one is black on the right half and red on the left half. You are also given three red and three black checkers. You are to place the three red checkers on the three black circles and the three black checkers on the three red circles. “Your problem is to get the three red checkers on the three red circles and the three black checkers on the three black circles. You must move but one checker at a time, jump only one at a time, and never move backwards. When you are blocked so that you cannot move farther, set the checkers back to the starting point and begin anew. When you can go through the problem three times in succession without any errors, we will consider 28 Strong, E. K., Jr., The Learning Process, “ Psychol . Bull., 1918, XV, 328 ff. INTERRELATION OF HIGHER LEARNING PROCESSES 35 it learned. Keep in mind that your results are being judged (i) by the time spent, (2) by the number of attempts (each time you begin counting an attempt, whether you are successful or not), (3) by the number of successful solutions required for the learn¬ ing.” The subject sat at one side of the table and the experimenter at the other. The latter kept an accurate account of the time, the number of attempts, and the number of successful solutions. The results are found in Table XVIII. Table XVIII. Showing the Number of Minutes, the Number of Attempts, the Number of Solutions, and the Percentile Rank for Each Kind of Data in the Checker Puzzle Test. Subject Score in Percen Rank tile in Subject Score in Percentile Rank in T A 1 s T A S T A S T A S 1 9 11 4 74 15 78 38 19 30 12 53 39 25 2 16 15 6 58 63 59 39 8 7 5 81 86 69 3 17 23 9 56 5i 40 40 9 8 6 74 82 59 4 26 14 6 4i 65 59 4i 14 16 6 61 61 59 5 12 20 8 67 54 46 42 13 17 6 64 59 59 6 22 18 6 49 58 59 43 18 16 10 55 61 35 7 26 14 3 41 65 86 44 19 30 6 53 39 59 8 3i 28 9 35 43 40 45 13 12 8 64 70 46 9 16 18 8 58 58 46 46 25 27 8 45 44 46 10 29 24 7 38 48 52 47 30 25 7 37 46 52 11 29 29 6 38 42 59 48 58 57 9 14 18 40 12 11 14 5 69 65 69 49 24 30 6 45 39 59 13 19 32 6 53 36 59 50 52 49 11 19 25 3i 14 10 12 6 70 70 59 51 52 49 11 19 25 3i 15 21 25 15 5i 46 52 52 12 19 7 67 56 52 16 29 23 6 38 5i 59 53 21 30 8 5i 39 46 17 23 24 9 47 48 40 54 24 25 5 45 46 69 18 20 10 8 52 77 46 55 26 37 11 4i 34 31 19 19 23 7 53 5i 52 56 4i 52 14 23 21 0 20 35 45 9 33 28 40 57 8 13 5 81 67 69 21 13 9 6 64 79 59 58 14 16 5 61 61 69 22 22 30 12 49 39 25 59 25 33 13 43 35 16 23 9 23 6 74 5i 59 60 35 4i 10 33 30 35 24 35 23 8 33 5i 46 61 25 21 8 43 53 46 25 36 46 13 29 27 16 62 16 19 5 58 56 69 26 6 12 8 86 70 46 63 37 29 8 25 42 46 27 12 19 4 67 56 78 64 19 16 4 53 61 78 28 17 28 8 56 43 46 65 15 39 8 60 32 46 29 30 23 6 37 5i 59 66 9 21 5 69 53 69 30 13 12 8 64 70 46 67 28 12 10 40 70 35 3i 36 31 10 29 37 35 68 12 18 8 67 58 46 32 24 25 6 45 46 59 69 36 62 11 29 14 3i 33 23 20 9 47 54 40 70 23 32 9 47 36 40 34 36 38 7 29 33 52 7i 62 80 12 0 0 25 35 21 23 9 51 5i 40 72 15 13 4 60 67 78 36 23 27 10 47 44 35 73 16 17 7 58 59 52 27 21 40 12 51 3i 25 74 33 23 6 35 5i 59 36 B. F. H AUGHT Table XIX. Showing Correlations and Partial Correlations of Each Factor* of the Checker Puzzle with the Binet-Simon Test and with Each Other. 12 .26 13 .18 14 .20 12-3 .18 13-2 •03 14-3 .06 12-4 .19 13-4 .05 14-2 .08 12-34 .18 13-24 —.03 14-23 .06 23 .62 24 •5i 34 .76 23-4 .41 24-3 .08 342 •65 *For the sake of brevity the factors will be designated as follows : 1. Binet-Simon Tests 3. Attempts 2. Solutions 4. Time. Since r14.23 = .06, factor 4 has nothing in common with the criterion that is not contained in factors 2 and 3. In like man¬ ner, since r13.2 = .03, factor 3 has nothing in common with the criterion that is not contained in factor 2. Nothing will be lost, therefore, as far as the criterion is concerned, by discarding time and attempts from the final score. For the present purpose, then, we shall use only the number of solutions as the final score. The formula for multiple correlation shows that it is not possible by combining the factors to get a higher correlation with the criterion than .27. The range of solutions is so small that the scores are bunched considerably. This probably affects the cor¬ relation somewhat, but there is no way to remedy it. It cannot be raised by using the other factors in any way. The present method has for its purpose to get the highest correlation with the criterion. (2) Analysis of Data. There is a “present but low” positive correlation between the criterion and time and between the criterion and solutions. The correlation is negligible, however, between the criterion and the number of attempts. Since the variability29 is the same in each kind of data, each factor may be directly compared with the criterion by reference to Table XIX. Thus, for every unit-change in solutions there will be a like change of .26 of a unit in the criterion, and vice versa. 29 The standard deviations for time, attempts, and solutions are 16.57, 16.54, and 16. 1 respectively. INTERRELATION OF HIGHER LEARNING PROCESSES 37 The factors, solutions and time, have something in common with the criterion. Solutions have most and contain elements not contained in either of the other factors. Time contains nothing with respect to the criterion that is not contained also in solutions. Each of the factors is composed largely of elements not found in the criterion. Solutions and attempts have a high correlation with each other as have time and attempts. The correlation of solutions and time is marked. The correlation-ratios for each of the factors with the criterion are as follows : 12 = .49 and .54 13 — *3° and .42 14 = .40 and .44 These values substituted in the Blakeman formula give results as follows : Tests 1 and 2, 2.71 and 3.05 Tests 1 and 3, 1.52 and 2.42 Tests 1 and 4, 2.21 and 2.50 According to our test, the correlation of the criterion and solutions is non-linear. The other two correlations are linear. A table will be constructed to show the regression lines in the correlation of solutions and the Binet-Simon tests. This will be postponed, however, until the final scores are analyzed. VIII. The Tait Labyrinth Puzzle (1) Description of Test and Method of Scoring. In this test the subject was given a figure of the Tait Labyrinth Puzzle and a copy of the instructions. Freeman30 has given sug¬ gestions as to its use. Lindley31 also used it in his “Study of Puzzles.” The figure and instructions are here given. “You have before you a figure that can be drawn without lift¬ ing the pencil from the paper and without retracing. Your pro¬ blem is to draw the figure without lifting the pencil from the paper and without retracing. As soon as you are ready you may begin on this blank sheet of paper. You may keep the figure 3° Freeman, F. N., Experimental Education , IQ1^, 36 ff. 3i Lindley, E. H., Study of Puzzles, Amer. J. of Psychol, 8, 430 ff. 38 B. F. H AUGHT before you and refer to it during the drawing if you wish. No attention will be given to the technical excellence of the drawing. If you fail in the first attempt, take another sheet of paper and try it again. Continue until you have made the figure three times Fig. III. Showing the Tait Labyrinth Puzzle. in succession without any errors. You are to be judged by the number of trials required for the learning and by the number of minutes used.” When the subject started, the time was noted and then noted again when the problem was complete. This time included that used in reading the directions as well as that used in solving the problem. It was thought necessary to include the time used in reading the directions, since so many will trace the pencil in the air over the figure before trying to draw it on paper. If we designate the number of trials by 2 and the number of minutes by 3, the correlations are as given in Table XXI. The best possible combination of time and trials gives a cor¬ relation of .30 with the Binet-Simon tests. Since r13.2 gives a correlation of .01, there is nothing common to time and the criterion that is not contained in trials. Therefore the final scores for comparison with the criterion will consist of the percentile ranks in number of trials. (2) Analysis of Data. There is a “present but low” positive correlation between the criterion and number of trials. The correlation of the criterion with the number of minutes, however, is “negligible.” Since the INTERRELATION OF HIGHER LEARNING PROCESSES 39 Table XX. Showing the Number of Trials, the Number of minutes, and the Percentile Rank in Each Kind of Data in Tait Labyrinth Puzzle. Subject Score in Per. Rank in Subject Score in Per. Rank in Trials Time Trials 1 Time Trials Time Trials Time i 11 6 43 66 38 34 52 14 14 2 19 24 31 33 39 5 11 67 50 3 6 8 59 59 40 17 10 35 54 4 5 10 67 54 4i 6 11 59 50 5 6 6 59 66 42 5 7 67 63 6 12 13 42 46 43 6 10 59 54 7 14 14 37 45 44 25 3i 23 21 8 22 21 27 38 45 7 12 57 48 9 5 6 67 66 46 10 11 46 50 10 25 24 23 33 47 9 7 50 63 n 5 22 67 37 48 17 29 35 25 12 9 10 50 54 49 9 18 50 21 13 3 13 82 46 50 14 26 37 28 14 7 7 57 63 5i 5 9 67 57 15 25 24 23 33 52 13 23 40 35 16 13 25 40 30 53 10 8 46 59 17 17 30 35 23 54 12 17 42 42 18 11 4 43 72 55 18 13 33 46 19 6 8 59 59 56 18 67 33 0 20 10 11 46 50 57 25 17 23 42 21 22 19 28 40 58 10 10 46 54 22 9 4 50 72 59 13 15 40 43 23 11 19 43 40 60 8 7 53 63 24 8 7 53 63 61 5 11 67 50 25 8 10 53 54 62 4 4 76 72 26 20 26 3i 28 63 5 15 67 43 27 5 8 67 59 64 7 3 57 78 28 8 2 53 84 65 8 23 53 35 29 3 5 82 68 66 8 33 53 18 30 9 4 50 72 67 10 12 46 48 31 13 21 40 38 68 5 3 67 78 32 3 8 82 59 69 10 11 46 50 33 14 23 37 35 70 6 2 59 84 34 9 24 50 33 7i 38 14 0 45 35 5 10 67 54 72 4 10 76 54 36 6 9 59 57 73 5 4 67 72 37 13 8 40 59 74 8 19 53 40 Table XXI. Showing the Correlations and Partial Correlations of Each Factor in the Tait Labyrinth Puzzle with the Binet-Simon Tests. 12 •30 123 .26 13 • 17 132 .01 23 •55 40 B. F. H AUGHT variability32 is the same in each kind of data, each factor may be compared directly with the criterion by reference to Table XXI. Thus, for every unit change in number of trials there will be a like change of .30 of a unit in the criterion and vice versa. It has already been noted that time contains nothing with re¬ spect to the criterion that is not contained in number of repeti¬ tions. Trials and minutes show a “marked” correlation with each other. The correlation-ratios of the factors of this test with the criterion are : 12 = .46 and .54 13 = -33 and .44 When these values are substituted in Blakeman’s formula, re¬ sults as follows are obtained : For tests 1 and 2, 2.22 and 2.86 For tests 1 and 3, 1.80 and 2.59 One of the regression lines in each correlation is linear and the other non-linear. Table XXII shows the correspondence be¬ tween time and the criterion. The regression line of the means of the columns is relatively linear and shows an increase in y-values with an increase in x-values from the lowest to the high¬ est percentile. The regression line of the means of the rows is relatively non-linear. It shows a rapid increase in x-values with an increase in y-values from the twentieth to the fortieth per¬ centile. The x-values change very little until the sixtieth per¬ centile is reached and then the increase is again rapid. Since the percentile ranks in trials are used as the final scores, the table showing the correspondence between this factor and the criterion will be postponed until the next section. IX. Intercorrelations (i)Tests Analyzed in the Light of the Criterion. We shall first analyze the scores in the light of the criterion. The final scores are obtained in Rational Learning by combining repetitions and perseverative errors equally, in Rational Learn- 32 The standard deviations for trials and time are 16.33 and 16.58 re¬ spectively. INTERRELATION OF HIGHER LEARNING PROCESSES 4i Table XXII. Showing the Distribution of Subjects on the Basis of Time in the Tait Labyrinth Puzzle and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. Table XXIII. Showing the Correlations and Partial Correlations for the Final Scores* with the Binet-Simon Tests.* 12 •33 13 •47 14 .26 15 .30 12-3 .20 13-2 .40 14-2 • 17 152 .19 12-4 .27 13-4 43 14-3 • 13 15-3 .25 12-5 .24 13-5 •45 14-5 .19 15-4 •25 12-34 .18 13-24 .38 14-23 .09 15-23 .19 12-35 .11 13-25 .40 14-25 • 14 15-24 • 17 12-45 .21 13-45 42 14-35 .07 15-34 .22 12-345 .11 13-245 .38 14-235 .06 15-234 .18 ♦The numbers have the following meaning: 1. Binet-Simon Tests. 2. Rational Learning. 3. Rational Learning (Modified). 4. Checker Puzzle. 5. Tait Labyrinth Puzzle. 42 B. F. H AUGHT ing (Modified) by combining minutes and unclassified errors equally, in the Checker Puzzle by taking the solutions, and in the Tait Labyrinth Puzzle by taking the number of trials. (a) Rational Learning. There is a “present but low” positive correlation between the final scores in Rational Learning and the criterion. The correlation is partly due to elements found also in Rational Learning (Modified), found to a less extent in the Tait Labyrinth Puzzle, and to a still less extent in the Checker Puzzle. The correlation is significant when the common ele¬ ments in any one of the other three tests are removed, but when the common elements found also in all the other three tests are removed, the correlation is no longer significant. In other words, everything common to the criterion and Rational Learning is found in the other three tests. The correlation ratios for The Binet-Simon tests and Rational Learning are .47 and .51. These values substituted in Blake- man’s formula give 2.13 and 2.47, which indicate that for all practical purposes the correlation is linear. Rational Learning seems to measure some mental functions not detected by the Binet-Simon tests. The first that may be mentioned is that of being able to attack and solve a problem without getting confused. In support of this statement some special cases are cited. Subject 6 scores “high” in each factor of Rational Learning, but “low” in the criterion. She grasped the situation quickly and completed the learning with only 23 errors. Subject 9 scores “high” in unclassified, logical, and per- severative errors, but “low” in the criterion. She made only 17 errors in the first repetition and finished with a total of 21. We have altogether nine subjects who score “low” in the criterion and “high” in one or more factors in Rational Learning. An examination of the individual records shows that in every case the learning was completed without confusion or distraction. A second mental function that Rational Learning seems to test better than does the criterion is the ability to give attention longer and to more elements than is usually required in the latter tests. Subject 26 illustrates this point fairly well. She scores INTERRELATION OF HIGHER LEARNING PROCESSES 43 “high’ in the criterion and “low" in repetitions and perseverative errors. She has 88 unclassified errors altogether and 70 of these were made in the first two repetitions. The other 18 are distrib¬ uted from the third to the eighth inclusive. The record indicates that the learning was almost complete in the third repetition, in which only two errors were made, yet five more repetitions were required. Subjects 34 and 42 have records similar to that of 26; that is, they have a few errors distributed over several repetitions, a condition which indicates a lack of attention. A third mental function or process measured better by Rational Learning than by the criterion is the kind of attack. Some sub¬ jects read the instructions and make sure that every point is un¬ derstood before beginning. Others read them in a careless way and jump into the problem without knowing just what is to be done. Subject 2 illustrates the later method. She made 77 un¬ classified and 44 logical errors in the first two repetitions. Sub¬ ject 72 made 5 of her 8 perseverative errors in the first repetition. This indicates that the instructions were not fully understood. The fourth and last mental function that seems to be especially well brought out by Rational Learning is the speed of the subject. This may be illustrated by subject 72. She scores “high” in the criterion and “low” in time and perseverative errors. She is a mature woman who goes at everything slowly and deliberately. Subject 40 also is a good example. She worked very slowly and deliberately, thus making a “high” score in repetitions. (b) Rational Learning (Modified). The correlation of Ra¬ tional Learning (Modified) and the criterion is “marked.’" The elements common to the two tests are found to a slight extent in each of the other three tests. The correlation of the third order shows that Rational Learning (Modified) has elements common to the criterion not found in any one of the other three tests or in all of them combined. This means that Rational Learning (Modified) contains elements not found in the other tests. The correlation -ratios for Rational Learning (Modified) and the criterion are .60 and .63. These values substituted in the 44 B. F. H AUGHT Blakeman formula give 2.37 and 2.69, indicating that one re¬ gression line is linear and the other non-linear. Table XXIV shows the actual regression lines. The line joining the means of the rows is relatively linear and the line joining the means of the columns is relatively non-linear. The non-linearity would be eliminated if the average of the seven cases in the fifth column from the left were 50 instead of 26. The number of cases in each row and column is too small for the non-linearity to have any significance when the curve does not take any well defined shape. Table XXIV. Showing the Distribution of Subjects on the Basis of Ra¬ tional learning (Modified) and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. Rational Learning (Modified) 0- 19 20- 24 25- 29 30- 34 35- 39 40- 44 45- 49 50- 54- 55- 59 60- 64 «§- 69 70- 74 75- 79 00- 100 100 00 1 79- 75 «►# # A 4 74- 70 9 1 # 0 69- 65 ♦ * #» ♦ » 7 1 8 64- 60 # # ♦# -O * » * rO 1 # 1 1 * 0 59- 55 # ♦ Oc£ A \ \ P'’ / \ \ t # Y, 6 54- 50 O ♦ # ✓ > ♦ ♦ 10 49- 45 9 1 ' > \ 1 *# / /\* \ # <>• 1 t # jV * U *» • II 44- 40 l 1 f ♦ V \ \ \ # 1 > l+Q « • 5 39- 35 1 */ i \ 1 */ 1 s## J # * ♦ 7 34- 30 1 t 6 \ ' « 1 * ' , r/t * #- * 7 29- 25 'V# 2 24- 20 # * 2 19- 0 > ** 3 3 2 4 4 7 8 9 6 9 8 4 , '4 2 2 74- 42 <0 £ c o CO I "4* CQ ti In Rational Learning, repetitions and perseverative errors are the significant factors. In Rational Learning (Modified), how¬ ever, the significant factors are time and unclassified errors. The cause of this difference is interesting and can be stated only INTERRELATION OF HIGHER LEARNING PROCESSES 45 on a priori grounds. In the former, repetitions are more sig¬ nificant than time and contain everything in time with respect to the criterion. The reverse, however, is the case with Ra¬ tional Learning (Modified). Time includes all that is in re¬ petitions. The writer is of the opinion that the experimenter, in calling out the numbers, controls the speed of the subject to some extent. He enters into the situation in a different way from what he does when he stands back and records the responses. The writer has found that when a subject is naming words, the speed is checked if the words are recorded in plain view of the subject. The next problem is to try to answer why perseverative errors in Rational Learning and unclassified errors in Rational Learn¬ ing (Modified) are the significant factor. This also can be stated only on a priori grounds. It is probable that in the latter experiment space perception makes it easier to avoid persevera¬ tive errors than in the former. The tendency seems to be to go from one end of a row to the other and to skip about here and there less than in Rational Learning. An analysis of individual cases indicates that Rational Learn¬ ing (Modified) tests the same factors as Rational Learning. First, it tests the subject’s ability to work for a period of time without confusion or distraction better than the criterion does. In support of this some special cases are cited. Subject 6 scores “low” in the criterion and “high” in time, repetitions, and un¬ classified errors. The data shows that she was able to con¬ centrate her attention and learn the problem without confusion. Subject 22 scores “low” in the criterion and “high” in time, repetitions, and perseverative errors. She learned this test quickly and was able to avoid confusion and distraction. Second, Rational Learning (Modified) tests a subject’s ability to give attention longer and to a more complex situation than that usually required by the criterion. Subject 11 illustrates this point. She scores “high” in the criterion and “low” in repetitions. In five of the repetitions only one error was made for each. Certainly close attention would have cut down the num- B. F. H AUGHT ber. Subject 49 has. a “high” score in the criterion and a “low” one in repetitions and perseverative errors. The “low” score in repetitions of this subject also is caused by a lack of attention, as was that of subject 11. This conclusion is based on the fact that she made from zero to three errors in each repetition from the sixth to the thirteenth. Subject 61 scored “high” in the criterion and “low” in repetitions and perseverative errors. She made only one error in the third repetition, yet she required thirteen repetitions to complete the learning. The greatest num¬ ber of errors made in any repetition after the second is three. This too probably shows a lack of attention to the correct numbers. In the third place, Rational Learning (Modified) is better for detecting the kind of attack than is the criterion. It is possible to determine whether the subject approaches the problem with that deliberate method which indicates that he is sure of what is to be done, or approaches it in that method characteristic of the person who gets an inkling of what is to be done and then begins in a kind of hit-and-miss sort of way. The fourth mental function that is revealed in this test is the speed of the subject. Here we have reference to the procedure after the instructions have been read and the subject has begun. This is illustrated by subject 66. He scores “high” in the criterion and “low” in time. The ‘low” score in the latter is clearly due to the slow, deliberate method of work. Subject 72 also scores “high” in the criterion and low in time. Her scores are almost identical with those of subject 66. She is a mature woman who worked very slowly and deliberately. (c) Checker Puzzle. There is a “present but low” positive correlation between the scores in the Checker Puzzle and the criterion. The correlation is partly explained by elements found also in each of the other tests. This test as far as the criterion is concerned is most like Rational Learning (Modified and least like the Tait Labyrinth Puzzle. The correlation is not significant when the common elements found also in the other tests are re¬ moved. INTERRELATION OF HIGHER LEARNING PROCESSES 47 The correlation-ratios for the criterion and the Checker Puz¬ zle are .49 and .54. These values substituted in Blakeman’s formula give 2.71 and 3.05, indicating that both of the regression lines are non-linear. Table XXV shows the two regression lines. If they were smoothed they would be nearly straight and show very little correlation. In other words the high correlation-ratio is to a great extent due to the fluctuation of the means of the rows and the means of the columns. Table XXV. Showing the Distribution of Subjects on the Basis of the Checker Puzzle and the Binet-Simon Tests. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. Checker Puxz/e Test Analysis of the individual cases shows no mental functions tested by the Checker Puzzle that are not also tested by the criterion. This may be due to the fact that the responses of the subjects cannot be recorded so exactly as in the other tests. Partial cor¬ relations of the third order show that this test contains nothing 48 B. F. H AUGHT with respect to the criterion that is not contained in the two Rational Learning Tests. (d) Tait Labyrinth Puzzle. The correlation between the criterion and the Tait Labyrinth Puzzle is positive and “present but low.” The correlation is partly due to elements found also in Rational Learning, and to a less extent to elements found in Rational Learning (Modified) and the Checker Puzzle. The correlation is barely significant when the elements common to all three tests are removed. The individual cases reveal no mental functions tested by the Tait Labyrinth Puzzle that are not tested by the criterion. The correlation-ratios for the criterion and the Tait Labyrinth Puzzle are .46 and .54. These values substituted in the Blake- man formula give 2.22 and 2.86, indicating that one regression Table XXVI. Showing the Distribution of Subjects on the Basis of the Tait Labyrinth Puzzle and the Binet-Simon Test. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. INTERRELATION OF HIGHER LEARNING PROCESSES 49 line is linear and the other non-linear. The line joining the means of the rows is linear and the other is non-linear. Table XXVI shows the two regression lines. *(2) Interrelation of Tests Scored in the Light of the Criterion. The four tests are here compared with each other as they are scored in the light of the criterion. Table XXVII. Showing the Correlations and Partial Correlations of the Four Tests, When Scored in the Light of the Criterion.* 23 .36 24 .32 25 .41 23-4 .29 24-3 .23 25-3 .38 23-5 .32 24-5 .24 25-4 .36 23 45 .27 24-35 .17 25-34 •35 34 .32 35 .18 45 .26 34-2 .23 35-2 .04 45-2 U5 34-5 .28 35-4 .11 45-3 .22 34-25 .22 35-24 .01 45-23 .14 * The numbers have the same meanings as in Table XXIII. Rational Learning has a “present but low” positive correlation with Rational Learning (Modified) and with the Checker Puzzle. It has a “marked” correlation with the Tait Labyrinth Puzzle. There are elements common to Rational Learning and Rational Learning (Modified) that are not found in the Checker Puzzle and the Tait Labyrinth Puzzle. In like manner there are elements common to Rational Learning and the Tait Labyrinth Puzzle that are not found in the other two tests. The correlation of Rational Learning with the other three tests combined is .53. The correlation of Rational Learning (Modified) and the Checker Puzzle is “present but low.” The correlation with the Tait Labyrinth Puzzle is barely significant. There are elements common to Rational Learning (Modified) and the Checker Puzzle not found in the other two tests. The correlation of this test with the other three combined is .48. The Checker Puzzle has a “present but low” positive correla¬ tion with the Tait Labyrinth Puzzle. The correlation is mostly due to elements found also in Rational Learning and Rational Learning (Modified). The correlation with the other three tests combined is .41. The Tait Labyrinth Puzzle has a correlation with the other 50 B. F. H AUGHT Table XXVIII. Showing the Distribution of Subjects on the Basis of Ra¬ tional Learning and Rational Learning (Modified). Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. x* Rational L earning (Modified) three tests combined of .43. The multiple correlations may be summarized as follows: ^•2(345) ~ *53 -^•3(245) ~ 48 R4G35) = -41 ■^5(234) = -43 After allowance is made for errors and non-linearity, it seems safe to conclude that each of these tests contains elements that are not found in any of the others. The correlation-ratios for the inter¬ relations are as follows : INTERRELATION OF HIGHER LEARNING PROCESSES 5i 23 — .51 and .57 24 = .46 and .60 25 = 47 and .51 34 = .51 and .58 35 = -35 and .51 45 = 43 and .46 When these values are substituted in the Blakeman formula, re¬ sults obtained are as follows : Tests 2 and 3, Tests 2 and 4, Tests 2 and 5, Tests 3 and 4, Tests 3 and 5, Tests 4 and 5, 2.30 and 2.82 2.02 and 3.23 1.46 and 1.93 2.01 and 3.08 1 .9 1 and 3.04 2.18 and 2.42 Table XXIX. Showing the Distribution of Subjects on the Basis of Ra¬ tional Learning and the Checker Puzzle. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. *a Checker Puzzle Test 52 B. F. H AUGHT Four of the six correlations are non-linear according to the Blake- man test. The correlation of tests 2 and 5 and of tests 4 and 5 may be considered linear. Tables showing the lines of the means of the columns and the means of the rows will now be con¬ structed for the non-linear correlations. Table XXVIII shows the correspondence between the scores in Rational Learning and Rational Learning (Modified). The lines appear linear from the thirteenth to the sixteenth percentiles Table XXX. Showing the Distribution of Subjects on the Basis of Ra¬ tional Learning (Modified) and the Checker Puzzle. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. x* Checker Puzzle Test and show a high correlation within these limits. Outside of these limits, however, the fluctuations are marked and conse¬ quently the correlation-ratio becomes larger than the correlation. Table XXIX shows the correspondence between the scores in Rational Learning and Checker Puzzle. These regression lines when smoothed will be approximately straight and will show a INTERRELATION OF HIGHER LEARNING PROCESSES 53 Table XXXI. Showing the Distribution of Subjects on the Basis of Ra¬ tional Learning (Modified) and the Tait Labyrinth Puzzle. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. x * Toil Labyrinth 'Puzzle low positive correlation. An increase in ability to do the Checker Puzzle shows an increase in ability to do Rational Learning. The reverse, however, is not so true. An increase in ability to perform Rational Learning does not show very much change in ability to perform the Checker Puzzle. Table XXX shows the correspondence between the scores in Rational Learning (Modified) and the Checker Puzzle. The line of the means of the rows smoothed will be approximately straight, but the other regression line will be far from straight. There seems to be closer agreement between the two sets of scores in the upper quartiles than in the lower. Table XXXI shows the agreement of scores in Rational Learn¬ ing (Modified) and the Tait Labyrinth Puzzle. An increase in ability in Rational Learning (Modified) is accompanied by an 54 B. F. H AUGHT increase in ability in the Tait Labyrinth Puzzle. Here again, the reverse is not true. After the fiftieth percentile, an increase in ability in the Tait Labyrinth Puzzle is accompanied by a de¬ crease in ability in Rational Learning (Modified). (3) Interrelation of Tests Scored by Combining the Factors Equally. The next step will be to analyze the four tests when scored by combining all the factors equally. The final scores in Rational Learning are obtained by adding together the percentile ranks in time, repetitions, unclassified, logical and perseverative errors and then reducing to absolute percentiles by use of Rugg’s table. The final scores in the other tests are found in a similar manner. Table XXXII. Showing the Correlations and Partial Correlations, When Scored by Combining All the Factors Equally.* 23 •33 24 .18 25 .44 23-4 .29 24*3 .06 25-3 •37 23*5 .21 24-5 .08 25-4 .41 23-45 .20 24-35 .01 25-34 •37 34 .38 35 •33 45 .26 34*2 •34 35-2 .21 45-2 .20 34-5 •32 35-4 .26 45-3 •15 34-25 •31 35-24 •15 45-23 .14 * The numbers have the same meaning as in Table XXIII. Rational Learning has something in common with each of the other tests. It is most like the Tait Labyrinth Puzzle and least like the Checker Puzzle. The correlation with the former is “marked” and with the latter is barely significant. Every¬ thing in the Checker Puzzle common to Rational Learning is also found in the Tait Labyrinth Puzzle, and in Rational Learn¬ ing (Modified). There are elements common to Rational Learn¬ ing and Rational Learning (Modified) that are not found in the Checker Puzzle and the Tait Labyrinth Puzzle. In like manner there are elements common to Rational Learning and the Tait Labyrinth Puzzle that are not found in the other two tests. The correlation of Rational Learning with the other three tests is .48. Rational Learning (Modified) has a “present but low” positive correlation with the Checker Puzzle and with the Tait Laby- INTERRELATION OF HIGHER LEARNING PROCESSES 55 rinth Puzzle. The correlation with the other three tests com¬ bined is .48. This means that there is much in this test common to the other three tests and much that is not found in them. The Checker Puzzle test has a low correlation with the Tait Labyrinth Puzzle. It correlates with the other three tests com¬ bined to the extent of .40. The Tait Labyrinth Puzzle has a correlation with the three remaining tests of .49. The multiple correlations may be summarized as follows : ■^•2(345) =::: *4-8 R3G45) = *4& ^-4(235) -4° ■^5(234) ~ -49 We may safely conclude that each of these tests contains much that is not found in any of the other tests. The correlation-ratios are as follows: 23 = -45 and -49 24 = .33 and .54 25 = .50 and .61 34 = -53 and -59 35 — -47 and .49 45 = .40 and .47 When these values are substituted in Blakeman’s formula, the following results are obtained : For tests 2 and 3, For tests 2 and 4, For tests 2 and 5, For tests 3 and 4, For tests 3 and 5, For tests 4 and 5, 1.95 and 2.30 1.76 and 3.24 1. 51 and 2.61 2.35 and 2.88 2.13 and 2.30 1.96 and 2.49 According to the Blakeman test three of the correlations are linear and three are non-linear. The correlation of Rational Learning with Rational Learning (Modified) is linear, but with the Checker Puzzle and the Tait Labyrinth Puzzle it is non¬ linear. The correlation of Rational Learning (Modified) with the Checker Puzzle is non-linear, but with the Tait Labyrinth 56 B. F. H AUGHT Puzzle it is linear. The correlation of the Checker Puzzle with the Tait Labyrinth Puzzle is linear. Table XXXIII shows the correspondence between scores in Ra¬ tional Learning and the Checker Puzzle. The line joining the means of the rows shows that as ability in Rational Learning Table XXXIII. Showing Distribution of Subjects on the Basis of Ra¬ tional Learning and the Checker Puzzle. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. increases there is not much change in ability in the Checker Puzzle until the seventieth percentile is reached and then the increase is rapid. A smoothed curve through the means of the columns will show that as ability in the Checker Puzzle in¬ creases the ability in Rational Learning slowly increases until the fifty-fifth percentile is reached and then there is a decrease in ability up to the seventy-fifth percentile. Table XXXIV shows that the agreement between the scores in Rational Learning and the Tait Labyrinth Puzzle is not very INTERRELATION OF HIGHER LEARNING PROCESSES 57 Table XXXIV. Showing the Distribution of Subjects on the Basis of Ra¬ tional Learning and the Tait Labyrinth Puzzle. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. close. A smoothed curve through the means of the rows in¬ dicates that as ability in Rational Learning increases, there is a slight increase in ability to solve the Tait Labyrinth Puzzle up to the sixty-fifth percentile and then a slight decrease in ability from this point on. The line joining the means of the columns is very irregular. Beginning with the twenty-fifth percentile, as ability to solve the Tait Labyrinth Puzzle increases, there is a rapid increase in ability in Rational Learning. From this point on, there is little relation between the two sets of abilities. Table XXXV shows the agreement between the scores in Rational Learning (Modified) and the Checker Puzzle. The line joining the means of the columns is linear according to the Blakeman test. It shows that as ability to solve the Checker Puzzle increases, there is also a constant but slow increase in 58 B. F. H AUGHT Table XXXV. Showing the Distribution of Subjects on the Basis of Ra¬ tional Learning (Modified) and the Checker Puzzle. Circles through which the broken line passes represent the means of the columns and those through which the continuous line passes represent the means of the rows. Each asterisk represents a subject. x-Che chet' Puzzle Test ability in Rational Learning ( Modified). The line joining the means of the rows is non-linear. From the twentieth to the thirtieth percentiles, ability in the Checker Puzzle decreases with an increase in ability in Rational Learning (Modified). From the thirtieth percentile on, there is a slight increase in ability to solve the Checker Puzzle as the ability in Rational Learning (Modified) increases. X. A Discussion of Learning and Intelligence Learning of the reflective or problem solving kind has often been looked upon as involving, among other factors, one general mental function or process. It has been assumed that the per¬ son who has good reasoning ability in one problem will be good in all others of this same general type. This conception implies INTERRELATION OF HIGHER LEARNING PROCESSES 59 that there is always a high correlation between any two such problems. In this investigation, however, the correlations are not high. In fact, they are all comparatively low. The lowest is .18 and the highest .44. The conclusion is that there is not one general rational learning process, but a number of processes. Two tests as similar as Rational Learning and Rational Learning (Modified), although they have something in common, are to a large degree independent of each other, since they have a cor¬ relation of only .36. This conception that any two tests have something in common and something that is not common has been explained in two ways. The first is the Two Factor theory of intelligence set forth by Spearman. The second assumes that each activity, such as a mental test or learning problem, involves a specific number of factors combined in a specific way. These two theories will now be treated in order. Spearman set forth his Two Factor theory of intelligence in 1904. His first statement of the theory was as follows: “All branches of intellectual activity have in common one funda¬ mental function (or group of functions), whereas the remain¬ ing or specific elements of the activity seem in every case to be wholly different from that in all the others.”33 This statement should be supplemented by the following explanation : “It was never asserted, then, that the general factor prevails exclusively in the case of performances too alike : it was only said that when this likeness is diminished (or when the resem¬ bling performances are pooled together), a point is soon reached where the correlations are still of a considerable magnitude, but now indicate no common factor except the General one. The latter, it was urged, produces the basal correlation, while the similarities merely superpose something more or less adventi¬ tious.”34 His most recent statement of the theory is: “The purport of this theory is that the cognitive performances 33 Spearman, C., General Intelligence, Objectively Determined and Meas¬ ured, Amer. J. Psychol., 1904, 15, 201 ff. 34 Hart, B., and Spearman, C., General Ability, Its Existence and Nature, Brit. J. of Psychol, 1912, 5, 51 ff. Co B. F. H AUGHT of any person depend upon : (a) A general factor entering more or less into them all; and (b) a specific factor not entering ap¬ preciably into any two, so long as these have a certain quite moderate degree of unlikeness to one another.”55 Spearman's method was to measure a number of mental abilities in a number of persons and then calculate the correlation coefficients of each of these abilities with each of the others. He then noticed that these correlation coefficients had a certain ielationship among themselves, which he called a hierarchial order. By this he means that if the coefficients of correlation of a number of mental functions are arranged in a descending cider from left to right and from top to bottom as is usualy done in a correlation table, in every row the figures will be in a descending order as they are in the top row, and in every column the figures will be in a descending order as they are in the left column. This also means that in any table of correlations as ordinarily arranged, every column will have a perfect correlation with every other one. Spearman has also reduced this prin¬ ciple to the following exact mathematical equation : r /r = r /r , ap aq bp bq in which a, b, p and q indicate any of the tests and r is the cor- i elation between them. It seems evident that the presence of such a general factor will always produce this hierarchy. In fact, if it can be shown that all correlations arrange themselves in such an order, it might be difficult to formulate any other theory to account for the facts. One exception, however, is enough to disprove the theory, since, if there is such a general factor, all correlations must take this hierarchical form. Thompson36 has shown that it is possible with dice throws to get a set of correlation coeffi¬ cients in excellent hierarchical order. He says that these imita¬ tion mental tests contain no general factor. Spearman, on the other hand, claims that Thompson let in a general factor at the 85 Spearman, C., Manifold Sub-Theories of the “The Two Factors,” Psychol. Rev., 1920, 27 , 159 ft. 80 Thompson, Godfrey H., General versus Group Factors in Mental Activ¬ ities, Psychol. Rev., 1920, 27 , 173 ft. INTERRELA TION OF HIGHER LEARNING PROCESSES 61 back door. It seems to the writer that Thompson has proved nothing more than that it is possible occasionally to get the hier- archial order of correlation coefficients when there is no general factor present. He has not weakened Spearman’s argument in the least, provided Spearman can always get this order. Thomp¬ son further claims in this same article that the hierarchical order is the natural relationship among correlation coefficients. The writer is unable to see, however, just how his argument bears upon the question. He is willing, of course, to admit that this inability may be due to his lack of insight. Spearman, in order to prove his theory, must show that every group of correlation coefficients of intellectual functions has, within the limits of experimental accuracy, this hierarchical order. Then his theory will hold only until it has been shown that this same order can be obtained consistently when there is no common factor present. The data of this investigation and their bearing upon the question are here presented. Table XXXVI. Showing the Correlation Coefficients, when the Tests are Scored in the Light of the Criterion I 3 2 5 4 I 47 33 30 26 3 47 36 18 32 2 33 36 4i 32 5 30 18 4i 26 4 26 32 32 26 In no column, except the first where it was deliberately ar¬ ranged, does the hierarchy exist. Spearman would probably say that the mental functions here tested are too much alike for the criterion to hold. The correlations are so low, however, that this claim can hardly have weight. Our findings are indeed adverse to the Two Factor theory. If we examine the correla¬ tions and partial correlations in Table XXVII, it will be evident that no factor of any size whatever is common to all the mental 62 B. F. H AUGHT functions tested. For instance the correlation of tests 2 and 5 is .41. When the elements in test 3, common to 2 and 5, are removed, the correlation is still .38. This indicates that there is almost nothing common to the three tests. This same con¬ clusion can be deduced from other cases in this same table. The correlation of tests 2 and 3 with the common elements in 5 re¬ moved, of tests 3 and 4 with the common elements in 5 removed, of tests 4 and 5 with the common elements in 3 removed, leads to the conclusion that there is no common factor large enough to account for all the correlations. Table XXXII shows the same conditions. The correlation of tests 2 and 3 with the common elements in 4 removed, of tests 2 and 5 with the com¬ mon elements in 4 removed, of tests 3 and 4 with the common elements in 2 removed, shows that there are no elements common to all the tests sufficient to account for all the correlations. Our data indicate that there is no common factor of any size running through all the tests. This amounts to saying that there is no such thing as general intelligence. What then is the nature of intelligence? One other theory will be considered. This theory assumes that in carrying out any activity, such as a mental test, a number of factors are at play. Each activity involves a specific number of factors combined in a specific way. The specific factors combined will differ with different individuals and with the same individual at different times. It will some¬ times happen that a number of elements will run through sev¬ eral mental activities. In this case there may be said to be an element common to all the activities. In other cases there will be no element or elements common to more than two or three of the mental functions. For instance, tests 1 and 2 may cor¬ relate because of element a, tests 1 and 3 because of element b, tests 2 and 3 because of element c, etc. This theory seems to be in harmony with the data of this in¬ vestigation. According to the Two Factor theory, the correla¬ tions in Table XXXIII show that test 3, Rational Learning (Modified), must have more of the general factor than any other test; yet when the elements in this test common to the criterion INTERRELATION OF HIGHER LEARNING PROCESSES 63 and test 2, to the criterion and test 4, or to the criterion and test 5, are removed, the correlation is still nearly three times the prob¬ able error. If test 3 contains more of the general factor than any other test and all correlation is due to this factor, then it should be reduced nearly to zero when the common elements in this factor are removed. On the other hand, the theory of various elements variously combined can easily account for all correlations and partial correlations. That is, tests 1 and 2 have common elements, some of which are found in each of the other three tests and some of which are not found in any of the other three tests. The same conclusion may be drawn from tests 1 and 3 and tests 1 and 5. Probably everything common to tests 1 and 4 is found in tests 2 and 3 or tests 3 and 5. If we now turn to Table XXVII, it is evident from the view¬ point of the Two Factor theory that test 5 or test 2 has more of the general factor than any of the other tests; yet when the elements in 5 common to 3 and 4 are removed, the correlation is .28. This is an impossible result if the Two Factor theory is true. Table XXXII will also show similar conditions. Test 2 or test 5 must have enough of the general factor to make the correlation of these two tests .44. The other test may have more of this factor, but cannot have less if Spearman’s theory is true. Now since the amount of the general factor involved in either of the two remaining tests, must be less than that involved in test 2 or test 5, the correlation of these tests, 3 and 4, should be reduced to zero, when the common elements in 2 or 5 are re¬ moved. Yet the correlation remains .31 when the common elements in both are removed. These data, which cannot be explained at all by the Two Fac¬ tor theory, are easily explained by the theory that intelligence consists of a large number of factors variously grouped and' combined. Suppose that the correlation of tests 2 and 3 in table XXXII is due to elements a, b, c, d, e, f, g, h, i, and j. Now suppose that element a is the only one of these ten that is found in test 4, and that elements b, c, d, and e are the only ones of the ten found in test 5. When the element a is removed from the 64 B. F. H AUGHT ten common ones, the correlation is reduced from .33 to .29. In like manner, when the elements b, c, d, and e are removed, the correlation is reduced from .33 to .21. When the elements a, b, c, d, and e are removed, the correlation is reduced from .33 to .20. Thus, we have ten elements common to tests 2 and 3, one element common to tests 2, 3, and 4, four elements common to tests 2, 3, and 5. This analysis is not literally correct. There is undoubtedly a common factor of very small importance running through all four tests. This is evident from the fact that r23.45 is not much less than r23.4 or r23.5. That is, most of the elements in test 4 common to tests 2 and 3 are contained in the elements m test 5 common to tests 2 and 3. An examination of any of the correlations and the accompanying partial correlations will show that a very small factor runs through all four tests. This factor, however, is not sufficient to account for the correlations. It seems that most of the investigations, when interpreted by Spearman, are in harmony with the Two Factor theory, but when interpreted by others, are adverse to this theory and more in harmony with the other theory here discussed. Thorndike37 has recently made a study, using the Army Tests and a large number of subjects. His data are not in harmony with the Two Factor theory. He says in this article : “We have considered the correlations obtained from time to time in various studies at Teachers College from the point of view of the Spearman theory, and have in general not been able to corroborate it. The most extensive data at our disposal (McCall, T6) seemed decidedly adverse.” Thorndike in this same article further says: “We must, it appears, turn back with open mind to the details on intercorrelations and experimental analysis to work out the organization of intellect. Especially needed seem studies of the ‘partial’ inter-correlations with one after another of the factors equalized.” 37Thorndike, Edward L., On the Organization of Intellect, Psychol. Rev., 1921, 28 , 141 ff. INTERRELATION OF HIGHER LEARNING PROCESSES 65 Simpson38 discussed general intelligence and the bearings of his study upon the Two Factor theory. He says: “We find justification for the common assumption that there is close inter-relation among certain mental abilities, and cotv sequently a something that may be called ‘general mental ability' or ‘general intelligence’ ; and that on the other hand certain capacities are relatively specialized, and do not imply other abilities except to a very limited extent.” He says again: “We find no justification for the view that ‘general intelligence' is to be explained on the basis of a hier¬ archy of mental functions, the amount of correlation in each case being due to the degree of connection with a common cen¬ tral factor." Peterson39 makes the following statement as to the nature of general intelligence: “General intelligence, if it is a reality at all, is probably not a separate constant factor, but a composite of many different abilities, and probably means different things in unlike situations, as different abilities are stressed. Such factors as energy and perseverance, degree of disturbance by emotions and self-consciousness, and many others that play their role in one’s success in life, have not yet been successfully brought into the field of measurement by tests, especially by group tests." XI. Summary and Conclusions A. Method. The essential features of the method used in this investigation are: ( 1 ) Four tests of the problem solving or rational learning type are used. Two of these tests have five kinds of data — time, repetitions, and three kinds of errors. One has three kinds of data — time, attempts and solutions. One has two kinds of data — time and number of trials. A criterion, the Stanford Revision of the Binet-Simon tests, is used in finding the best method of scor- 38 Simpson, Benj. R., Correlations of Mental Abilities, Teachers’ College Contributions to Education , 1912 (No. 53). 39 Peterson, Joseph, Intelligence and Its Measurement, J. of Educ. Psychol , 1921, 12 , 198#. 66 B. F. H AUGHT ing or combining the different kinds of data and in analyzing the tests. (2) The raw scores in the criterion and in each factor of each test, and the final scores are transmuted into percentiles. This has been found very helpful in calculating correlations. For instance, all standard deviations as well as all the means are made approximately equal. The analysis of the curves through the means of the rows and columns is also simpler when the standard deviations are equal. (3) Every factor in a test is correlated with the criterion and with every other factor. Then a complete set of partial cor¬ relations is worked out. This makes it possible to determine which factors must be used in scoring in order not to discard any elements in common with the criterion. For instance, it was found in Rational Learning that repetitions and perseverative errors contain everything in all the factors in common with the criterion. That is, everything in the other three factors is a duplication of these common elements in repetitions and per¬ severative errors. (4) To determine the best combination of the factors that need to be retained, formula (2) is used. Formula (3) is then used to determine what this correlation is. As a check on the work, formula ( 1 ) is used. This gives the highest possible correlation of all the factors with the criterion. If this result agrees closely with that obtained from formula (2), it is evi¬ dence that the analysis and work are correct. It so happened that not more than two factors needed to be combined in any of the tests; but if it had been necessary to combine three or more factors, two would have been combined in the best way and then this result with the third factor. The writer has com¬ bined as many as five factors in this way and found it very satis¬ factory. (5) For every correlation, Blakeman’s criterion for linearity is applied. If non-linearity exists according to this criterion, the actual curves of the means of the rows and columns are con¬ structed. The writer regards the correlation-ratio and the Blake- INTERRELATION OF HIGHER LEARNING PROCESSES 67 man criterion of very little value unless the number of cases is large enough to eliminate most of the fluctuations in the curves through the means of the rows and columns. These fluctuations are often sufficient to produce a high correlation-ratio when there is no correlation, and, of course, non-linearity is indicated when the criterion is applied. The actual curves through the means of the rows and columns are of more significance, since it is possible to determine the general direction of such curves in making analyses. (6) The final scores of each test are analyzed by comparing them with the criterion through the use of partial correlations and multiple correlation. It does not seem possible to do much with partial correlations in analysis unless a criterion is used. (7) The final scores as obtained in the light of the criterion are correlated with each other and partial correlations worked out. In comparing the tests with each other, the multiple cor¬ relation method is found very valuable. Especially is this the case in determining how much each test has in common with all the others. (8) As a final step in the technique, the tests are scored by combining all the factors in a test equally. This was thought best, since there was a possibility of accentuating certain elements in the tests by scoring them in the light of the criterion. B. Results. The results indicated by the data are as follows : (1) In scoring Rational Learning in the light of the criterion, repetitions and perseverative errors are the significant factors. Time, unclassified errors, and logical errors only duplicate the elements in these two factors. Time and unclassified errors are the significant factors in Rational Learning (Modified) ; the other three factors may be discarded. The number of solutions is the significant factor in the Checker Puzzle. Time and num¬ ber of attempts add nothing. In the Tait Labyrinth Puzzle the number of trials is the significant factor. Time is unessential. It might be interesting to note here that in three of the four tests, time is an unessential factor. This does not mean, however, that the same results would have been obtained if the subject 68 B. F. H AUGHT had been told that time was not being considered. The cor¬ relations would probably have been very different. Time is probably not important in the Checker Puzzle and the Tait Laby¬ rinth Puzzle. In Rational Learning the subject is controlled somewhat by the experimenter but in Rational Learning (Modified) he is free to go as fast as he wishes. This probably accounts for the difference in the value of the time factor in the two tests. In Rational Learning (Modified) the space percep¬ tion makes it easier to avoid perseverative errors, and for that reason this factor becomes unessential. It is thought best to make no comparison of the difficulty of the two rational learn¬ ing tests, since each subject had already taken Rational Learning (Modified), when he took Rational Learning. The similarity was usually recognized at once. It was not uncommon to have the subject say while he was reading the directions for Rational Learning, “This is just like that bell-ringing thing.” (2) Rational Learning and Rational Learning (Modified) seem to test or measure mental functions not detected by the intelligence tests. These may be summarized as follows : first, the ability to attack and solve a problem without getting con¬ fused; second, the ability to give attention longer than that usually required in mental tests; third, the type of attack made by the subject; and fourth, the speed of the subject. The ob¬ jective data do not reveal any mental functions tested by the Checker Puzzle and the Tait Labyrinth Puzzle over and above those tested by the criterion. ( 3 ) Rational Learning ( Modified ) correlates considerably higher with the criterion than does any of the other tests. This may be partly due to the fact that this test was given first. The Checker Puzzle has the lowest correlation with the criterion. Rational Learning (Modified) has elements in common with the criterion that are not found in the other three tests. The same is probably true of the Tait Labyrinth Puzle, but to a less ex¬ tent. Rational Learning and the Checker Puzzle have nothing in common with the criterion that is not found in the other tests. (4) When the tests are scored in the light of the criterion, INTERRELATION OF HIGHER LEARNING PROCESSES 69 every correlation indicates something in common between the two tests correlated. The correlations of the second order also indicate that each pair of tests, except 3 and 5, have something in common that is not contained in the other tests. The factor running through all four tests is almost zero. Multiple correla¬ tion shows that each test has much that is not contained in the other three tests. The Checker Puzzle has most, the Tait Laby¬ rinth comes second and Rational Learning has least. The dif¬ ferences are so small that they are probably not significant. (5) When the tests are scored by combining all the factors in a test equally, the same general results are obtained as in the other method of scoring. There are some differences, however, in specific correlations. These are evident when tables XXVI and XXXII are examined. The correlations of test 2 with test 3 and test 5 are not changed much, but the correlation of test 2 with test 4 is reduced about half. The correlation of test 3 with test 4 is not changed much, but that of test 3 with test 5 is about doubled. The relations of test 4 with test 5 remain exactly the same. The multiple correlations show very little change in general by the two methods of scoring. (6) There is nothing in the data of this investigation to justify the Two Factor theory of intelligence. In fact, every¬ thing is adverse to this theory. If the testing and the calcula¬ tions are absolutely free from errors, the results obtained are impossible on the basis of the Two Factor theory. The correla¬ tions and partial correlations can be accounted for, however, by the theory that intelligence consists of various factors variously grouped for different situations. BIBLIOGRAPHY 1. Freeman, F. N., Experimental Education , 1916. 2. Hart, B., and Spearman, C., General Ability, Its Existence and Nature, Brit. J. of Psychol., 1912, 5, 5 iff. 3. Lindley, E. H., Study of Puzzles, Amer. J. of Psychol., 8, 43off. 4. McCall, W. A., Correlation of Some Psychological and Educational Measurements, Teachers College Contribu¬ tions to Education, 1916, (No. 79). 5. Peterson, Joseph, Experiments in Rational Learning, Psychol. Rev., 1918, 25, 433ft . 6. Peterson, Joseph, The Rational Learning Test Applied to Eighty-one College Students, J. of Educ. Psychol., 1920, 11, 13 7ff. 7. Peterson, Joseph, Intelligence and Its Measurement, /. of Educ. Psychol., 1921, 12, i98ff. 8. Rosenow, Curt, The Analysis of Mental Functions. Psychol. Monog., 1917, 24, (No. 106). 9. Rugg, H. 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