Digitized by the Internet Archive in 2011 with funding from The Library of Congress http://www.archive.org/details/highermentalprocOOpete Zbc lUnivcvQit^ of Cbicago The Higher Mental Processes in Learning \S BY JOHN C: PETERSON, Ph.D. Professor of Psychology, State Agricultural College, Manhattan, Kansas A DISSERTATION Submitted to the Faculty OF the Graduate School of Arts and Literature In Candidacy for the Degree of Doctor of Philosophy Department of Psychology Private Edition, Distributed by The University of Chicago Libraries CHICAGO, ILLINOIS " 1920 Reprinted from the Psychological Monographs, Vol. XXVIII, No. 7, Whole No. 129. ACKNOWLEDGMENTS I am much indebted to Professors James R. Angell and Harvey A. Carr for numerous helpful suggestions and for gen- erous a:id stimulating interest at all stages of the work. My thanks are also due to those who kindly gave their time to serve as subjects. CONTENTS I. Introduction i II. Technique and Procedure 3 A. Description and Analysis of Problems Presented to Subjects for Solution 3 B. Apparatus and Recording of Data 10 C Order of Presentation of Problems 11 D. Degree of Learning and Uniformity of Procedure 13 E. Instructions to Subjects 16 F. The Subjects 17 III. Results 19 A. Unit of Measurement 19 B. Changes in the Rate and Character of Progress. . 22 C. Perceptual Solutions 35 D. Analysis 37 1. Types of Elements Abstracted: Direction of Analysis 38 a. Frequency of Repetition as a Factor 41 b. Effect of Nearness to a Goal 47 c. Effect of Speed of Reaction 49 2. Explicitness and Extent of Analysis 53 3. Time Relations of Manipulation and Analysis 56 4. Summary 62 E. Generalization 63 1. Relative Absence of Generalization in Percep- tual Stage 63 2. Development of the Concept of the Critical Number 64 3. Random Hypotheses 73 4. Summary 81 F. Transfer 82 1. Degree of Transfer 83 2. Conditions of Transfer 89 a. Objectively Identical Elements 89 b. Subjectively Identical Elements 90 c. Generalized Methods of Procedure 91 d. Effect of Thoroughness of Learning upon Transfer 93 3. Negative Transfer 95 Summary 98 G. Effect of Age and Education 99 IV. Discussion and Conclusions 103 A. Abstraction of Elements 103 B. Combination of Elements no C. Application of Knowledge 116 D. General Conclusions 120 I. INTRODUCTION The present investigation is an experimental study of the mental processes involved in the solution of certain novel prob- lems and in the utilization of experiences so gained for the sub- sequent mastery of other problems of a similar nature. The problems were chosen with a view to the possibility of accurate measurement of the progress in their mastery. They are ar- ranged roughly in the order of increasing complexity, and are so related that the solutions for later problems are, for the most part, simply more generalized statements of solutions for earlier ones. In the mastery of these problems all forms of mental process, from very simple acts of perception to fairly difficult and complicated acts of abstraction and generalization, were in- volved. It is these latter processes chiefly to which attention is directed in the following pages. In its use of objective methods the study is closely related to the large experimental literature in the field of animal learning and the so-called lower forms of human learning. Its domi- nant interest in the processes of abstraction and generalization brings it into relation with the researches of Kiilpe, Mittenzwei, Griinbaum, Moore, Aveling, and Fisher in this field. ^ Its rela- tion to the more dynamic studies of Cleveland" and Ruger is ^ Kiilpe, Oswald, "Versuche uber Abstraktion," Ber. uber den I Kong. f. exp. Psych, in Geissen, 1904. Mittenzwei, Kuno, "Uber abstrahierende Apperception," Psych. Stud. 1907, 2. Grunbaum, A. A., "Uber die Abstraktion der Gleichheit," Arch. f. d. ges. Psychol. 1908, 12. Moore, Thomas Verner, "The Process of Abstraction: An Experimental Study," University of California Pubs, in Psych., Vol. I. Aveling, Francis, "The Consciousness of the Universal." Fisher, Sara Carolyn, "The Process of Generalizing Abstraction," Psych. Rev., Mon. Sup., Vol. XXI, No. 2, 1916. Fisher summarizes the work in generalization and abstraction prior to 1916. 2 Cleveland, A. A., "The Psychology of Chess and Learning to Play It," Am. Jour. Psych., Vol. 18, 1907. 4 ' JOHN C. PETERSON When a single problem of this sort is solved, the solution is usually couched in more or less specific terms which do not readily function in the subsequent solution of other similar prob- lems. In the solving- of a series of related problems, hov^ever, it is possible to observe the gradual abstraction of common ele- ments and the association of these elements with appropriate terms, leading finally to the formulation of a general principle for the solution of all problems of the series. If the mastery of 14 as the initial number of beads constitutes the first problem, mastery of 15 as the initial number will constitute the second problem, mastery of 16 as the initial number the third, and so on until enough problems have been solved to permit the subject to develop a general formula for his guidance in drawing from any number of beads. ^ In the first problem the subject learned that he could not win if required to draw from 3, 6, 9, or 12 beads. He also learned that when the initial number of beads was 14, he coitld win by drawing so as to compel his opponent to draw from these "criti- cal" numbers. In the later problems of the series he discovers that these numbers are critical (i.e., the one who must draw from them inevitably loses the game) in all problems of the series, regardless of the initial number of beads presented for solution. Here also he generalizes to the effect that all multiples of 3 are critical numbers, and that he can win any number which is not a multiple of 3 by reducing it to such a multiple at his first draw and through successively lower multiples of 3 to o at sub- sequent draws. Aside, then, from the development of specific responses for the solution of individual problems, the learning process here consists in the formation of ( i ) a general concept through which the essential elements of all problems may be represented by a single term and treated as a unit, and (2) a 2 For the sake of uniformity it is necessary to fix upon a definite degree of mastery of problems to be required in all cases. Some difficulty arises here owing to the fact that it is impossible for the subject to win when the initial number of beads presented for solution is a multiple of 3. We have considered such problems "solved" when the subject expressed a con- viction that he could not win and called for a new problem. All other prob- lems have been considered solved when two consecutive trials were won. HIGHER MENTAL PROCESSES IN LEARNING 5 system of draws under the control of this concept, by means of which all non-critical numbers of the series may invariably be won by the subject. The basic concept here developed may be termed the critical-number concept. Its functioning and further development through use may be observed by requiring the sub- ject to solve some additional series of problems somewhat simi- lar to those described above. ^ New series of problems of this sort may be had by varying the numbers of beads which may be taken at a single draw. For example, the extension of the numbers which may be drawn to 1, 2, or 3 (instead of only i or 2 as above) yields a new series of problems in which the critical numbers are multiples of 4. Further extension of the numbers which may be drawn to i, 2, 3, or 4 beads yields another series of problems in which the critical numbers are multiples of 5, etc. By further changes in the numbers which may be drawn an indefinite number of simi- lar series of problems may be obtained. The various series of problems thus obtained will hereafter be designated by the lowest and the highest numbers which may be drawn. Thus Series 1-2 will denote the series of problems in which only i or 2 beads may be taken at a draw; Series 1-3, the series in which i, 2, or 3 beads may be drawn ; Series 2-3, the series in which only 2 or 3 beads may be drawn, etc. It will be worth while to direct our attention to the location of critical numbers in the various series since these numbers furnish the key tO' the solution of the various problems and series of problems. We have observed already that all multiples of 3 are critical numbers in Series 1-2. A clearer insight into the reason why these numbers are critical in this series will facilitate the search for critical numbers in other series of problems. In Series 1-2 any pair of draws may result in the removal of either 2, 3, or 4 beads (the possible combinations of S's and E's draws being i-i, 1-2, 2-1, 2-2, and the sum of these combinations 2, 3, 3 A series of problems was considered solved when the subject gave an adequate general formula for the solution of all problems of the series or in some other manner clearly indicated his ability to solve all new problems of the series at sight. 6 JOHN C. PETERSON 3, and 4 respectively). At any draw either S or E can take away such a number as to make the sum of his own draw and the immediately preceding draw of his opponent equal to 3 by taking i when the opponent takes 2, and 2 when the opponent takes I. Thus either S or E can cause the number of beads to be reduced by 3 with each successive pair of draws, beginning with any draw of his opponent, but he cannot force a reduction by 2's or 4's in this manner for if the opponent's draw is 2, the sum of the pair of draws will necessarily be greater than 2, and if the opponent's draw is i, the sum of the pair will necessarily be less than 4. The only number, therefore, by which successive reductions can thus be forced is 3, and it is clear that 3 is under control in this manner only because it is the sum of the low and the high draw. If, then, we designate the low draw by the let- ter L and the high draw by H, we may say in more general terms that all multiples of the sum of L -\- H are critical num- bers.^ Notice that these numbers are critical not merely because they are multiples of the sum of L + H but because they are greater than o by exact multiples of this sum, and o is critical by definition. If the conditions of the game were so altered as to make i instead of o a critical number, the critical numbers would be represented by the terms of an arithmetical progression in which the common difference is L + H and the first term is * That this formula holds for all series of problems of this sort wherein winning consists in securing the last draw, can be shown by substituting the general terms, L and H, for i and 2 in the foregoing statement. Thus either S or E can take away such a number of beads at any draw as to make the sum of his own draw and the opponent's immediately preceding draw equal to L + H. If one's opponent draws L, one can draw H; if the opponent draws H, one can draw L, and if the opponent draws L -|- x, one can bring the sum of the pair of draws up to L -j- H by taking away H — x beads. Moreover, L -)- H is the only number by which successive reductions can thus regularly be forced for if the opponent's draw is L, one cannot draw enough to raise the sum of the pair of draws above L -|- H, and if the opponent's draw is H, one cannot draw so low a number as to make the sum of the pair of draws less than L -|- H. ^ This condition is actually realized if in Series 1-2, for example, the defi- nition of success is so modified that drawing last constitutes losing instead of winning the game. HIGHER MENTAL PROCESSES IN LEARNING In all series of problems in which L is 2 both o and i may be said to be critical by definition. The critical numbers are here represented by the terms of two arithmetical progressions hav- ing each a common difference of L + H. The first terms of these progressions are o and i respectively. When L is 3, 2 also becomes a critical number. In fact, it will be noticed that all numbers below L, including o, are in every series critical. These numbers may be called the basic critical numbers. Each basic critical number is the first term of an arithmetical pro- gression in which the common difference is L + H and of which all terms are critical numbers. The critical numbers of any series are therefore o, 1,2 L — i and all numbers which are greater than any of these basic critical numbers by exactly a multiple of the sum of L + H. This generalization holds for all series in which all numbers between L and H may be drawn. Series of this sort have been called continuous series. All series of problems described in the foregoing pages are continuous series. Other series of problems requiring more difficult gen- eralizations may be had by restricting the draws to L and H. Series of this sort in which only L and H may be drawn will be known as discontinuous series. Only a limited number of dis- continuous series were solved by our subjects, owing to the diffi- culty of manipulation when the values of L and H are high. To facilitate their analysis these discontinuous series with some of their critical numbers are listed below. Bear in mind that the numbers by which a discontinuous series is indicated are the only numbers which may be drawn in that series. Series (discontinuous). Critical Numbers. I or 3 I or 5 1 or 7 2 or 6 2 or 10 3 or 9 4 or 12 I or 4 I or 6 1 or 8 2 or 5 o, 2, 4, 6, 8, 10, 12, 14, 16, etc. 0, 2, 4, 6, 8, 10, Ji>, 14, 16, etc. 0, 2, 4, 6, 8, 10, 12, 14, 16, etc. 0,1, 4,5, 8,9, 12,13, i6,iy, 20,21, etc. 0,1, 4,5, 8,9, 12,13, 16,17, 20,21, etc. 0,1,2, 6,7,8, 12,13,14, 18,19,20, 24,25,26, etc. 0,1,2,3, 8,9,10,11, 16,17,18,19, 24,25,26,27, etc. 0, 2, 5, 7, 10, 12, 15, 17, 20, 22, etc. o, 2, 4, 7, 9, II, 14, 16, 18, 21, etc. 0, 2, 4, 6, p, II, 13, 15, 18, 20, etc. 0,1, 4,-, 7,8, II,—, 14,15, 18,—, etc. 8 JOHN C. PETERSON 2 or 7 ; 0,1, 4, 5, 9,10, 13,14, 18,19, 22,23, etc. 2 or 8; 0,1, 4,5, /o,/i, 14,15, 20,21, 24, 25, etc. 2 or 9; 0,/, 4,5, 8,-, 1/,/^", IS, 16, 19, — , 22,23, etc. 3 or 7; 0,/,^, 6,-,-, 10,11,12, 16, — , — , 20,21,22, etc. 3 or 8; 0,7,.?, 6,7,-, 11,12,13, 17,18, — , 22,23,24, etc. 4 or 9; o,7,^,j, 8,-,-,-, 13,14,15,16, 21,—, — ,— , etc. The numbers in italics are those obtained by applying our old formula to these series. These numbers will be known as pri- mary critical numbers and groups of them as primary C groups. Within the intervals between primary C groups there are now other critical numbers which have become critical because of the restriction of draws to L and H. Such numbers will be known as secondary critical numbers. The secondary critical numbers fall into order in accordance with a principle already applied to the primary critical numbers; namely, that there are certain basic critical numbers each of which is the first term of an arithmetical progression in which the common difference is L + H and of which all terms are critical numbers. Our task will therefore be confined to finding the basic secondary critical numbers in the series listed above; i.e., those secondary critical numbers which fall in the interval between the first two groups of primary critical numbers. Note that in every series in the list all numbers below H, which may be obtained by adding a multiple of 2L to any nWmber in the primary C group, are critical. Why are these numbers criti- cal ? and why do they not extend beyond H — i above o ? Ob- viously the possibility of using the high draw no longer exists when the number of beads has been reduced below H. Below this point the game must necessarily proceed by successive re- ductions of L beads at each draw, and each pair of draws will result in the reduction of the number by 2L. Now if, after reduction below H, the number of beads remaining is equal to any number in the primary C group plus a multiple of 2L, it will itself be critical, because the one who draws first in the first pair of draws will necessarily draw first in all succeeding pairs of draws, wherefore the last and winning draw will fall to his opponent. Concerning the second question, it is clear that any number which is greater than o, or than any other member of HIGHER MENTAL PROCESSES IN LEARNING 9 the basic primary C group, by exactly the amount of H, cannot be critical; because it can he reduced by one draw to less than L, i.e., to a basic critical number. This is the reason why 4, 6, and 8 are not critical numbers in Series i or 4, i or 6, and i or 8 respectively. This is the reason also why the secondary C groups sometimes consist of less than L critical numbers, as in Series 2 or 5, 2 or 9, 3 or 7, 3 or 8, and 4 or 9. Each of the numbers in these series, which would have been critical but for the limiting effect of the H draw, is indicated in the list by a dash. All numbers that are greater than a basic secondary criti- cal number by an exact multiple of L -|- H, are themselves critical for the same reason that numbers which are greater than a basic primary critical number by that amount, are critical. To recapitulate, we have found certain uniformities in the distribution of critical numbers in our series of problems as fol- lows : (i) There is in ez^ery series a group of critical numbers extending from o to L minus i inclusive, which we have called the basic primary C group. (2) All numbers in any series, which are greater than any number in the basic primary C group by exactly a multiple of the sum of L + H, are critical. (3) In the discontinuous series all numbers below H which are greater than any number in the basic primary C group by exactly a multi- ple of 2L, are critical. These are the basic secondary critical numbers. (4) All numbers in a series, which are greater than a basic secondary critical number by exactly a multiple of the sum of L -]- H, are critical.*^ The primary critical numbers are, found in all series, both continuous and discontinuous ; secondary critical numbers are found only in discontinuous series. To win a game, or trial, beginning with any number which is not criti- cal, one must draw so as to reduce the number of beads to a critical number at his first draw ; thereafter he must draw so as to make the sum of each of his draws and the preceding draw of the opponent, equal to L -f- H.'' 6 Other more difficult series of problems may be obtained by permitting three or more non-consecutive numbers or groups of numbers to be drawn. But the foregoing series furnish enough difficulty for the ordinary gradu- ate student under the conditions of our experiment. 7 In the discontinuous series this procedure may result in the final re- 10 JOHN C. PETERSON B. Apparatus and Recording of Data The apparatus consisted of a stop-watch, a metronome, an Edison telescribe, or dictaphone, and a row of thirty beads strung upon a steel wire the ends of which were inserted into metal cubes of sufficient height to permit the beads to move freely. These beads were substituted for the matches of the original game because they offered less obstruction to free ma- nipulation, and therefore increased the value of the time records of individual reactions. While the experiment progressed the dictaphone was in ac- tion with the horn so adjusted as to secure as clear a record as possible of all the verbal reactions of the subject. The subject was asked to state aloud the number taken at each draw as the "move" was being made. The experimenter also called out the number taken by him at each draw. Thus the dictaphone records contained all the draws and such comments as were made during the experiment. To mark the time consumed by these reactions a metronome beating half seconds was placed near the dicta- phone. The beats were clearly audible in the records. As the experiment progressed the experimenter made a written record of the individual draws of the subject and of the experimenter in separate columns. Incipient movements, indicating that the subject was considering a particular draw, were recorded with a distinctive mark but were not counted in the summing up of data. Upon these written records the dictaphone records were later transcribed. The beats of the metronome were counted, and the number of half seconds from the beginning of the trial to each of its component reactions were recorded. Though no high degree of accuracy is claimed for these time records, they were accurate enough to be exceedingly useful in showing the distribution of attention among the different reactions entering into a trial. They also serve to show differences in the speed of reaction of different subjects and of any one subject in different portions of the experiment. From the dictaphone records were duction of the number of beads to a basic secondary critical number, which will be reduced to a basic primary critical number by a series of L draws. HIGHER MENTAL PROCESSES IN LEARNING ii also obtained the comments of subjects, which were recorded in their proper places in the records. These verbal reactions were often useful in the interpretation of our data. In order that the learning process might go on as naturally and as free from interruption as possible, no introspections were called for and few were given. For various reasons it was found necessary in certain por- tions of the work to dispense with the use of the dictaphone except for the recording of the more important verbal responses. Time measurements were here obtained by means of a stop- watch. In such cases it was of course impossible to record the time of each separate reaction, although the time of complete trials was taken with as great accuracy as was possible by the more cumbersome method. With this procedure it was not possible to distinguish the time consumed by the experimenter's draws from that consumed by the reactions of the subject. To make the records of different subjects comparable in time, there- fore, the experimenter made the time of his draws as uniform as possible. A fair estimate of the degree of their uniformity may be made from the record of a relatively small number of typical reactions. In 206 trials with 11 beads taken from the records of various subjects, it was found that the average time per trial was 41.14 metronome beats, or half seconds. The aver- age time per trial consumed by the experimenter's draws was 9.76 half seconds with an average deviation of 1.99. This A.D. of 20 per cent from the average of the experimenter's time per trial is not a serious matter when it is remembered that his time was on the average less than one-fourth of that consumed by the reactions of the subject. The effect of the inclusion of the time required for the experimenter's draws is to put the subject who draws most rapidly at a slight disadvantage, but the time differences in our records are large enough to make these small errors rather unimportant. C. Order of Presentation of Problems As already stated, each series consists of what are to the subject at the beginning a succession of relatively independent 12 JOHN C. PETERSON problems. Beginning with H + ^ beads in all series in which L is I, and with H + 2L — i beads in all series in which L is greater than i, the numbers (or problems) of each series were presented in the order of their magnitude from lowest to high- est. As soon as a practicable solution was found for one num- ber, the next higher number of the series was presented. Thus successive problems of a series were solved until a satisfactory generalization was formulated for the entire series. A higher series of problems was then presented in the same manner. The order of presentation of series was the order of the magnitude of their L and H draws. First, all series in which L is i were presented, beginning with Series 1-2 and proceeding through Series 1-3, 1-4, 1-5, etc., tmtil a general solution for all series of this order^ was found. Series of the next higher order (i.e., in which L is 2) were next presented, beginning with Series 2-3 and continuing through Series 2-4, 2-5, etc., until a general so- lution for all series in which L is 2, was found. In the same manner successively higher orders of series were presented until a general solution was obtained for all continuous series. The number of continuous series was not the same for all subjects. A wor'kable generalization was made by some subjects after the solution of but a few series in only two or three orders. Others were unable to arrive at a suitable generalization until after the solution of numerous series extending through a number of orders. Only a limited number of discontinuous series was pre- sented, however, and this number was the same for all subjects. These discontinuous series are listed on page 7 in the order in which they were presented. The order of presentation of the discontinuous series may need a word of explanation. The first portion of this group of series, extending from Series i or 3 to Series 4 or 12 inclusive, was given to bring out the fact that secondary critical numbers are found by the addition of multiples of 2L to the primary critical numbers. This is the simplest possible collection of series with a sufficient variety of values for L and H, in which ^ The order of a series refers to the number represented by L. In series of the first order L is i, in the second order L is 2, etc. HIGHER MENTAL PROCESSES IN LEARNING 13 all multiples of 2L above any primary critical number, are criti- cal. Not all subjects succeeded in getting this generalization from this small group of series, but the addition of other series of this character was impracticable owing to the large number of beads that would need to be manipulated. In Series i or 4 to 4 or 9 it was expected that subjects would discover that not all multiples of 2L above a primary critical number, are critical ; and also find some rule by which to locate such multiples of 2L above primary critical numbers as are not, in a given series, critical. After the completion of Series 4 or 9 the beads were put aside, and the subject was asked to name the critical numbers in Series 4 or 1 1 and later in Series 5 or 11. The work in these series was necessarily done mentally and often served to compel the subject to return to his earlier generalizations in the at- tempt to find a principle which would apply here. If at the end of these series no single workable generalization for all series had been given, the subject was permitted to refer to a table of critical numbers for all the discontinuous series.^ Here, as in the preceding portions of the work the time required and such of the subject's observations as were given verbal expres- sion, were recorded. D. Degree of Learning and Uniformity of Procedure In order to make the conditions uniform for all subjects it was necessary that the experimenter govern his draws in ac- cordance with some definite rules. It was, of course, necessary that no rule should prevent the experimenter from taking ad- vantage immediately of any error made by the subject, but aside from this there was a fair latitude within which he might vary his draws without necessarily affecting the outcome of the game. In Series 1-2, for example, so long as the subject took care to reduce the number of beads to a multiple of 3 at each of his draws, it might seem to be a matter of indifference whether the experimenter took i or 2 at a draw. But the indiscriminate changing of his draws would result in a serious lack of uni- ^ This table with a few changes in the italics is given on page 7. 14 JOHN C. PETERSON formity in the condition to be met by the different subjects. If, on the other hand, the experimenter were to take i always or 2 always so long as the subject's draws were correct the latter might, by mere mechanical memory and repetition of previous draws, win a sufficient number of consecutive trials to permit him to pass on to the next higher number in the series, and some mechanical performance of this sort might be repeated indefi- nitely. To prevent such mechanical repetition of accidental suc- cesses while still maintaining uniform conditions for all subjects, the experimenter first drew i at every draw through an entire trial and then 2 at every draw throughout the next trial. This change in the number drawn by the experimenter in alternate trials was continued until the problem was solved. But if S made an error in any trial, E immediately departed from his uniform reactions and, taking advantage of the error, drew so as to win. The change in E's draw in alternate trials often proved very dis- concerting for a time to subjects who tried to repeat accidental successes from memory, but it usually resulted in bringing about some kind of attempt at critical analysis. When two trials had been won in succession by the subject, another bead was added and the process repeated. Thus successive increments of i were made and the solutions for the resulting numbers found until a solution for the entire series was obtained. This changing back and forth of the experimenter's draw in alternate trials, together with the requirement that S win two trials in succession, resulted in bringing out all possible varia- tions in Series 1-2. Two lines of procedure each somewhat in accord with this were possible in the higher series. (i) E might alternate the L draw in one trial with the H draw in the next, neglecting all the intermediate draws except when an error on the part of the subject made it possible for E to win with one of the intermediate draws. (2) He might use all of the possible draws, taking each exclusively for an entire trial in some regular order. The latter procedure was used with some of the subjects in the preliminary experiments. It has the advan- tage of preventing, in a measure, the direction of attention to non-essentials. But, if consistently carried out, it requires that the HIGHER MENTAL PROCESSES IN LEARNING 15 subject win each number in the series as many times as there are possible draws in the series and so interferes seriously with the comparability of data from different series. The requirement that S win H minus L consecutive trials with each number also becomes somewhat monotonous in the higher series. The former procedure has the advantage of avoiding this monotony and keeping the degree of learning more nearly uniform through- out all series. It therefore makes the data from different series more truly comparable. This procedure was followed in the wofk of all of our subjects except a few who served in the preliminary experiments. No attempt is made to compare their work in the later series with that of other subjects. ^° As soon as a subject recognized two successive critical num- bers in a series by inspection, or showed his mastery by predic- tion of the outcome with higher numbers in the series, or gave a correct generalization, he was presented with a considerably higher number in the same series. He did not draw from this higher number, but merely announced his decision as to what would be the correct draw and gave a reason for his decision. In case the correct draw was indicated but no intelligible reason could be given, the subject was taken back to the point previously attained and required to follow the regular procedure until clear evidence was given of at least some sort of a practical solution which would hold for all numbers in the series. This was not a very common occurrence, however, and never resulted in serious departures from the ordinary procedure. In most cases i<> Even this procedure cannot, however, be said to make the data from different series entirely comparable. The fact that the critical and the non- critical numbers are not in the same proportion in. different series, is bound to affect the degree of learning in some measure. Compare Series 1-2 with Series 1-5 in this respect. In the former series the subject encounters five critical numbers in working from the beginning of the series to and in- cluding 15 beads; in the latter he encounters two. In the former series he need not reduce the number of beads to each of these critical numbers more than four times before passing to the next higher critical number. In the latter series he must reduce the beads to each of the critical numbers at least ten times before he is permitted to pass on to the next higher critical number. There is, of course, in the more frequent occurrence of critical numbers in the lower series, some compensation for the higher degree of learning of critical numbers in the higher series. i6 JOHN C. PETERSON a verbal formula was given, and sometimes this was expressed in mathematical form. At the completion of a series the next higher series followed immediately or at the next sitting. The solving of successive series continued until a general solution for an entire order of series was given or until a complete series was solved without a draw. When this occurred, a higher "test" series of the same order was usually but not always given, being omitted only when the subject's statement of his generalization left no doubt that he had found an adequate solution for all series of that order. In case of failure in one of these test series, it need hardly be said, a return was made to the series following the last one which had been solved. If the test series was successfullly solved, the first series of a higher order was presented, and so on through- out the experiment. As already stated, it was impossible to adhere strictly to this uniform degree of learning in case of the discontinuous series. Here the degree of learning differed con- siderably with different subjects, some of whom persistently worked out the relations existing between successive series whereas others were content to generalize for each series in isolation. E. Instructions to Subjects With the apparatus in position and the subject ready to begin work, the experimenter said in substance: "We shall draw alternately from this string of beads. You may take i or 2 beads at a draw and I also may take i or 2 at any draw. (With each statement of the number to be drawn the experimenter il- lustrated by manipulation of the beads, showing the alternation between his and the subject's draws which were made from op- posite ends of the row of beads.) The object is to get the last draw, i.e., you win if you get the last draw. You always draw first. Will you call out the number you take at each draw?" The number was then immediately reduced to 4 and the sub- ject asked to begin. The procedure from this point has already been described. As soon as the subject announced his inability to win with 6 beads as the initial number, the experimenter HIGHER MENTAL PROCESSES IN LEARNING 17 asked : "Hereafter whenever you make such a discovery or get an idea which seems to be signifcant, will you let me know at once?" The subject was also told at this point, provided his questions had not brought out the fact earlier, that he would be expected to find a general rule for the solution of all numbers in the series. The request that S make known any seemingly relevant new ideas was repeated at the beginning of each of the early series and at other times when his actions seemed to in- dicate that he had made a new discovery. Subjects were not permitted to record any data though the experimenter's record for a few of the immediately preceding trials was usually in sight. This was unavoidable owing to the difficulties of manipu- lation and recording. While attempting to recall successful re- actions of earlier portions of the work, subjects sometimes asked permission to see their records. This was not permitted. At the beginning of the second series the subject was told : "The requirements of the game are the same as before but you may now take either i, 2, or 3 beads at a draw." The subjects were of course informed at the beginning of each of the later series as to the numbers which might be drawn. Subjects were asked to refrain from thinking of the experiments during the intervals between their periods of work. F. The Subjects Exclusive of those who served in the preliminary experimen- tation by which the final mode of procedure was determined, 46 subjects served in our experiments. The subjects were divided into groups as follows: (a) Group I consisting of 14 subjects, (b) Group II consisting of 20 subjects, (c) Group III consisting of 12 subjects. The major portion of our study is based upon the work of the subjects in Group I. These subjects will be designated by Roman numerals. Subjects i to x are numbered in the order of the speed of their performance as determined by the number of trials required for the solution of all the problems presented, Subject i being the speediest. Subject ii the next speed- iest, etc. Subjects xi, xii, xiii, and xiv did not finish the entire experiment and could not, therefore, be given their relative i8 JOHN C. PETERSON ranks. Subject iii was a first-year high school boy who was thir- teen years old; Subject vi was a college senior, and the remain- ing ones were all graduate students or instructors in the Uni- versity of Chicago. All of the subjects of this group were males except Subject v. Subject ii solved most of the problems at two sittings in one day and returned three weeks later to finish in one short period. Subjects i, iii, iv, vi, vii, and xii were scheduled to work every day, but failed to live completely up to the schedule. Subjects ix and x worked with a fair degree of regularity at weekly intervals, and the remaining subjects of this group were quite irregular in the distribution of their effort. The work with this group of subjects was done at the University of Chicago during the fall and winter of 1916-17. Group II consists of 20 members of a class of 21 third- and fourth-year college and graduate students at the Kansas State Agricultural College. This group is subdivided into Groups Ila and lib each consisting of 10 subjects. Group Ila solved first Series 1-2 and then Series 1-3. The members of this group are designated by the capital letters A to J in the order of their speed of learning, the speediest first. Group lib solved first Series 1-3 and then Series 1-2. The members of this group are designated by the small letters from a to j, again in the order of their speed of learning. The work with Group II was done at the Kansas State Agricultural College in the spring of 19 18. Group III consists of all of the members of a class of third- and fourth-year college students. The subjects in this group worked out solutions for problems in Series 1-2 from 14 to 20 inclusive. The work with this group of subjects was done at the Kansas State Agricultural College during the fall of 1918. III. RESULTS A. Unit of Measurement Before proceeding with the presentation of data it is neces- sary to attend to the selection of a suitable unit for the measure- ment of progress. To be entirely comparable successive units of effort should be uniform in (a) time consumed, (b) degree and distribution of attention, and (c) number and character of reactions. Perfect uniformity in all of these characteristics is obviously out of the question, if subjects are to be given the requisite freedom for normal reasoning. There is no constant unit of time into which the work of our subjects can be divided so that the distribution of effort in successive units will conform even approximately to (b) and (c) of the foregoing require- ments. Of a number of units of response which suggest them- selves the trial^ conforms most closely to all of these require- ments.^ Measurement and comparison of progress will there- fore be made chiefly in terms of the trial, though the number of errors made during the solution of the various problems, as well as the time required, will also be stated and utilized to some ex- tent in the treatment of some portions of our data. It will be worth while briefly to inquire into the variability of the trial. The characteristics in which variability is greatest and also most susceptible to accurate measurement, are the time and ^A series of draws culminating in the reduction of any initial number of beads to o, constitutes a trial, regardless of who draws last. 2 It might be supposed that the draw would be a better unit of measure- ment, but this unit is altogether too variable to be of service. In time con- sumed it varies from less than a half second to several hundred seconds. In the amount of critical attention called forth, it varies from serious, con- centrated effort to the most mechanical performance. Often, indeed, the draws do not stand out singly as significant to the subject, but are linked up in various combinations into series each of which constitutes a trial and a real unit in the distribution of attention. Nor can the measurement of progress be made in terms of errors or of successful performances, — i.e., of successes in reducing the number of beads to o, — since in drawing from, a critical number no successes of this objective sort and likewise no erro- neous draws, are possible. 20 JOHN C. PETERSON number of draws per trial. The facts regarding the variability of trials in these characteristics, as found in the work of our major group of subjects in Series 1-2, are given in Table I. 3V Table I No. of Subject's Av. No. Draws per Trial T ime per Trial 5Ubj. Draws A.D. V3 Med. Time Q^ V5 iii 3-13 •95 .30 27 7.0 .26 iv 3-33 .71 .21 39 131 •34 i 3-35 .67 .20 41 11.7 .29 V 3.58 .91 •25 58 30.0 .52 ii 3-66 .89 .24 85 34-9 .41 viii 4.20 .68 .16 25 5-4 .22 vii 4.21 1.41 .34 27 10.5 •39 xi 4-25 1.02 .24 54 16.2 •30 ix 431 1.29 .29 26 6.8 .26 xiii S-oi 1.68 •33 25 9-5 •38 xii 5.22 1.62 •31 48 26.5 •55 X 6.84 2.89 .42 43 14.6 •33 vi 7.15 1.68 .23 36 10.3 .29 xiv 8.03 364 •45 33 14-5 •44 A.D. 40 —Z— -Q^ 5\7 — . Q Av. Med. The table is self-explanatory. Note the slight tendency for subjects who rank high in the number of draws per trial to rank low in time per trial. The average of the average numbers of draws per trial for the first seven subjects listed is 3.64 and the average of their median times per trial is 21.5 seconds. The corresponding averages for the last seven subjects listed are 5.83 draws and 16.9 seconds. The correlation between these two lists of averages is — •105,® The variations in time and in number of draws per trial do not, therefore, tend to show any cumulative effect upon the relative values of trials in the work of different subjects. Moreover, the distribution of attention is such as largely to discount the value of variations in time and number of draws per trial — especially the latter. It seldom occurred that a subject attended carefully to all of the draws in a trial when the initial number of beads was higher than 17 or 18. One portion of the trial was usually regarded as crucial and the at- 6 Rank method. HIGHER MENTAL PROCESSES IN LEARNING 21 tention mainly directed upon that point with the result that all but a few of the draws in the trial were executed with only a minimum of attention. If these mechanical draws were here neglected, the variation in number of draws per trial would be extremely slight, and the time variations would actually tend to compensate for the variations in the number of draws per trial/ Again, after the longer pauses, subjects frequently came back tO' the work declaring that they had permitted their minds to wander in the pursuit of irrelevant associations. The comments of subjects indicate that such distractions occurred rather fre- quently. If these distractions are to be regarded as indications of a low degree of attention, we have here another case of an inverse or compensating relationship between variations in dif- ferent aspects of the trial. Such practice value as the mechani- cal draws may have possessed was perhaps completely offset by their distracting effect upon attention to points which were re- garded as crucial. At any rate this was the spontaneous verdict of a number of the subjects. The variability of trials within Series 1-2 is typical of that in all later series in which any considerable difficulty was encount- ered. There was no great variation in the average length and duration of trials from series to series. In the later series — especially those which were presented late in any group of series of the same order^ — there were generally fewer draws per trial, hnd attention tended to be more evenly distributed among the various draws of the trial, though mechanical draws practically always appeared in any series where the number of draws per trial ran high. The average time per trial required by Subjects i to X inclusive in some of the more difficult series, is given in Table II. The series reported in this table represent approxi- mately two-thirds of all trials of these subjects in the entire experiment. '' This compensating relation in the variations of time and number of draws per trial is not, of course, found in the trials upon higher as com- pared with those upon lower numbers of a series in the work of the same subject. Here the time per trial usually varies directly vi^th the number of draws though not in the same proportion. But even here it is doubtful whether the longer trials are of more value than the shorter ones, owing to the presence and distracting effect of the more mechanical draws. 22 JOHN C. PETERSON Table II Series No . Trials Av. Time per Trial^ A.D. V. (in seconds) 1-2 2o6o 24.09 546 .23 2-3 , 647 42.72 10.68 .25 2 or 6 360 43-15 23.69 •55 I or 4 423 38.42 14. 1 1 •37 2 or 7 265 42.19 2.94 .07 8 These averages and deviations are those of all trials in a series without regard to the average time per trial of individual subjects. There is a general tendency for subjects to exercise a little more caution after the solution of one or two series than at first. This accounts for the marked increase in the average time per trial in Series 2-3 as compared with that of Series 1-2. In the last four series listed the uniformity of the average time per trial is fair, but the deviations are large. However, it should not be overlooked that in the more important matter of degree and distribution of attention the variability, though not sus- ceptible to quantitative statement, is undoubtedly much less marked, and that the variations in one aspect of a trial often tend to compensate for these in other aspects. On the whole, it is safe to say, the trial in the present experiment is not less uniform than the units of measurement regularly employed in studies on maze learning. Its uniformity is probably far greater than that of the trials in certain experiments in ball-tossing, from which curves of learning have been plotted and conclusions of far-reaching consequence drawn. B. Changes in the Rate and Character of Progress The progress of the ten subjects of our major group, who solved all of the series of problems, can be traced in Table III. The series of problems are listed at the left in the first column. Following each series are the records of the work of individual subjects upon it. In columns P and E are given the number of problems (i.e. of different numbers within a series) which are solved and the number of errors made in arriving at a general solution of the series. The time is given in seconds. HIGHER MENTAL PROCESSES IN LEARNING 23 Table III Progress of Subjects through all Series of Problems •Subj ect i Subject ii^ Subject iii Series P E Trials Time P E Trials Time P E Trials Time 1-2 10 24 48 1716 12 8 35 1572 12 5 28 464 1-3 4 7 247 9 4 23 585 5 I 10 120 1-4 5 5 14 124 i-S 19 2-3 22 II 44 1661 8 3 17 507 II I 25 427 2-4 10 4 4 215 28 13 84 168s 2-5 38 16 13 60 1115 2-6 32 3-4 165 53 IS 3-5 123 4-5 215 SO 45 I or 3 4 — 4 667 8 — 8 269 7 — II "5 I or 5 12 — 13 1387 2 — 3 118 6 — 13 173 I or 7 — 54 — 36 4 — 5 62 2 or 6 17 — 21 2483 3 — 3 178 15 — 30 618 2 or 10 — 483 4 — 4 20I 12 — 18 289 3 or 9 7 — 6 381 5 — 6 437 — 65 4 or 12 — 120 — 193 — 14 I or 4 4 I 8 492 12 15 38 2701 20 6 47 757 I or 6 46 24 I 47 4076 14 I 29 467 I or 8 23 II I 19 636 193 2 or 5 34 II 14 606 6 9 361 2 or 7 32 12 15 644 16 I 19 835 2 or 8 342 164 2 or 9 78 II 15 647 55 3 or 7 35 80 3 or 8 21 II 15 664 27 4 or 9 38 310 90 4 or II 40 36s 25 5 or II 30 230 62 General solution for all series 155 533 70 Total 85 41 165 11215 147 32 266 15864 172 41 388 8425 9 Series 2 or 8 and 3 or 7 were accidentally left out of the experiment with this subject. 24 JOHN C. PETERSON Table III (Continue d) Subject i\ Subject V Subject vi Series P E Trials Time P E Trials Time P E Trials Time 1-2 II 55 96 2205 13 23 62 2834 27 221 331 7934 1-3 12 13 35 704 8 I 15 363 II 6 30 265 1-4 10 6 27 449 2 3 33 2 4 54 1-5 6 10 87 9 40 1-6 14 24 241 1-7 8 14 157 1-8 o 36 2-3 13 16 44 1800 21 19 71 4566 15 5 30 709 2-4 6 I 13 279 13 7 29 894 I I 268 2-5 o 30 54 3-4 o 25 17 I 205 3-5 o 28 74 4-5 o 26 10 75 I or 3 7 — 17 180 4 — 4 52 10 — 10 237 I or 5 5 — 9 no 10 — 18 377 14 — 18 503 I or 7 2 — 3 33 10 — 14 269 13 — 14 235 2 or 6 27 — 45 1818 33 — 46 1189 24 — 36 821 2 or ID 6 — II 546 25 — 32 916 II — 14 262 3 or 9 4 — 6 151 — 72 13 — 13 258 4 or 12 o — 75 — 165 9 — 8 142 I or 4 12 7 26 430 13 3 21 392 19 I 31 484 I or 6 13 5 24 1006 23 5 47 1657 6 8 147 I or 8 o 58 16 5 28 984 2 2 66 2 or 5 o 152 8 8 427 16 2 21 346 2 or 7 8 14 590 21 2 34 163 1 6 6 145 2 or 8 2 2 333 7 10 527 3 4 75 2 or 9 O 58 10 15 993 66 3 or 7 17 6 33 1526 285 8 10 159 3 or 8 600 192 3 2 58 4 or 9 o 604 40 I6S 4 or II 820 85 65 5 or II 55 290 35 General solution for all ! series 8610 1210* 280 Total 183 109 453 23794 227 65 457 20561 214 235 594 14173 HIGHER MENTAL PROCESSES IN LEARNING 25 Table III (Continue d) Subject vii Subject viii Subj ect i>. Series P E Trial 5 Time P E Trials Time P E Trials Time 1-2 19 182 371 6659 15 90 168 2591 28 187 315 6129 1-3 12 3 23 754 12 24 56 684 6 4 16 244 1-4 5 8 207 10 8 30 294 2 3 23 1-5 15 I 2 23 I 2 52 1-6 15 4 9 115 1-7 6 2-3 18 I 25 1827 69 27 167 5408 42 25 119 5859 2-4 9 13 1047 35 9 71 2679 25 4 53 1956 2-5 28 15 20 500 10 I 15 295 2-6 2 2 59 35 2-7 9 3-4 8 12 301 19 4 41 699 9 10 155 3-5 (^2, 3 4 75 6 9 118 3-6 19 I 30 3115 13 20 3-7 21 4-5 12 21 I or 3 6 — 12 372 24 — 22 1067 17 — 36 IIOI I or 5 8 — II 327 19 — 24 870 5 — II 195 I or 7 — 6 3 — 4 76 I — 2 52 2 or 6 19 — 19 2427 23 — 38 1268 51 — 108 4189 2 or 10 7 — 16 979 29 — 34 755 17 — 20 489 3 or 9 — 60 19 — 20 309 14 — 18 680 4 or 12 — 80 — 52 — 87 I or 4 14 I 22 1583 24 2 39 877 37 41 154 7230 I or 6 12 I 19 4691 12 16 303 14 24 1318 I or 8 30 40 14 21 947 2 or 5 II 2 20 728 24 8 46 1441 II I 19 758 2 or 7 7 12 614 48 12 86 3702 12 3 27 966 2 or 8 7 13 954 9 2 14 263 9 2 13 446 2 or 9 8 10 1 102 12 I IS 271 10 15 660 3 or 7 7- 2 13 293 26 2 34 677 10 21 360 3 or 8 8 8 678 II 14 369 ID 17 369 4 or 9 4 5 828 8 9 278 II I 22 589 4 or II 455 690 240 5 or II 1 105 910 ISO General solution for all series 4980 2420 7740* Total 208 193 662 36341 472 189 987 29708 376 269 1970 43753 26 JOHN C. PETERSON Table III (Continued) Subject X Series P E Trials Time Series P E Trials Time 1-2 47 313 597 17538 2 or 10 7 — 7 233 1-3 16 3 45 1 122 3 or 9 27 — 28 980 1-4 2 136 4 or 12 9 — 8 514 1-5 I or 4 19 II 37 1306 2-3 36 16 105 4881 I or 6 22 14 49 1421 2-4 15 26 728 I or 8 26 5 z(> 1973 2-5 3 2 77 2 or 5 14 3 21 1016 2-6 2 or 7 35 4 50 1922 3-4 37 40 240 2 or 8 9 12 377 3-5 I 230 2 or 9 I 118 z-(> 10 12 AAA 3 or 7 19 I 21 968 317 28 3 or 8 12 I 14 548 4-5 II 10 570 4 or 9 12 2 18 485 5-6 50 4 or nil 327 I or 3 21 10 23 858 5 or II 193 I or 5 8 — 10 466 General solution I or 7 3 — 4 225 for all series 1960= 2 or 6 13 — 14 545 Totals 435 373 1 192 42470 The most striking feature of these figures is the great ir- regularity in the number of trials, the time required, etc., for the solution of various problems. Note the relatively large num- ber of trials required in Series 1-2, and the rapid decrease in the number of trials in successive series of the first order.^^ The same sort of change is found in series of the second order and, to some extent, in all higher orders of the continuous series as well as in those of the discontinuous series. These regular and more or less gradual changes are due in part to the separate mastery of numerous relatively isolated though often poorly defined "elements" of the problematic situation; in part they are due to a gradual definition and development of larger and more complex units in the form of concepts and general principles. ^^ 10 No errors are possible in those series where blanks appear in the error column. "Subjects were required to work Series 4 or 11 and 5 or 11 "mentally." * These subjects failed to find a general solution for all series in the time at their disposal. 12 The order to which a series belongs is determined by the value of L. In series of the first order L is i ; in those of the second order L is 2, etc. 13 Any aspect or isolable portion of the objective situation and any rela- tionship between such portions or aspects is here regarded as an element of the problem. HIGHER MENTAL PROCESSES IN LEARNING 27 These gradual changes are sometimes obscured by irregular and often extremely abrupt changes arising from various causes, such as, distraction of attention, sudden utilization of old con- cepts, monotony, fatigue, rest, elimination of erroneous assump- tions, change of methods, etc. The more striking irregularities in the rate of progress are shown in graphic form in Figure I. The number of trials and the amount of time required as well as the number of errors made by each of the ten subjects in various series and groups of series, is represented by the columns in the figure. The series and groups of series represented in the figure are indicated at the left. The subjects are designated by the numerals at the base of columns. This irregularity is in some degree illusory owing to the failure of some bonds in the early stages of their development to register their effect in terms of those particular overt responses which were taken into account in the score. That bonds do begin to develop long before their effect is apparent in the crude score, is evident from the comment of subjects, the comparative length of delays, incipient reactions, etc., some of which are in- cluded in the detailed records and will be noted later. Further evidence of the inadequacy of any one sort of measure to regis- ter all of the changes in the strength of bonds, is found in the relation between the time, the errors, and the number of trials involved in the solution of successive series of problems. In Figure I the number of errors made and the amount of time consumed in the solution of the problems in each section of the figure are shown by the relative height of the first and third columns respectively. Observe that in every case in Series 1-2 the error column is higher than the time column, and that there is a somewhat gradual shifting in the relative height of these columns so that in the last group of series the time column is in every case higher than the error column.^* This change is to 1* The relation between the trial column and the time and error columns in Series 1-2 and in the last group of series, is worthy of note. In every case but one the height of the trial column is intermediate between that of the other two columns. The measurement of progress in terms of trials there- fore takes special account of the conflicting claims of time and errors where the conflict is greatest. 28 JOHN C. PETERSON Figure I. — The Roman numerals at the base of the columns of the lowest section represent the various subjects of Group I, who solved all series. The height of the middle column for each subject represents the number of trials required by him for the solution of the problems indicated by the num- ber at the left. The trial columns are all directly comparable. The num- ber of errors made and the amount of time required by each subject are represented by his first and third columns respectively. The time and error columns are drawn to such a scale for each subject as to make their total length for all series equal to the total length of the trial columns of that HIGHER MENTAL PROCESSES IN LEARNING 29 some extent influenced by increased caution, as is shown by the lengthened average time per trial in the later series as compared with the first series; but it is due mainly to the fact that per- formance represents a distinctly lower plane of learning here than does formulation. It is a little surprising to find this rela- tion between power of performance and power of formulation here in view of the simplicity and definiteness of the terms which are required for formulation. It is possible that if success were to demand a high speed of reaction, as is generally the case in acts of skill and to a large extent in practical thinking, this re- lationship would tend to disappear. However, with the time allowed for each reaction limited only by his patience, the sub- ject is usually able to respond correctly while he is yet unable to foretell the correct response in the absence of the appropriate concrete situation or to say why the response is correct when made. This disparity in power of performance and power of formulation as well as some of the more striking cases of change in speed of progress, both of the regular and of the irregular sort, will receive more special attention in later sections. The principal features of the progress from series to series are duplicated in the progress from problem to problem in some of the more novel series. There are, however, some important differances. Series 1-2 being the first and most novel series il- lustrates the natural progress from problem to problem better than do any of the later ones. The progress of the fourteen subjects of the major group through the problems of this series, particular subject. They are thus directly comparable with one another and with the trial columns of the same subject. But the error and time columns of different subjects are not directly comparable. The numbers at the left represent various series and groups of series of problems as follows : 1. Series 1-2. 2. All later continuous series in which L is i. 3. Series 2-3. 4. All later continuous series in which L is 2. 5. All continuous series in which L is 3 or greater. 6. All discontinuous series from Series i or 3 to Series 4 or 12 inclusive 7. Series i or 4. 8. All remaining discontinuous series. 30 JOHN C. PETERSON, is shown in Table IV. The problems of the series are listed at the left in the column headed "Initial No. of Beads." The num- ber of trials and of seconds required by each subject for the so- lution of each problem are listed in self-explanatory form. The errors are not included in this table but their number is roughly proportional to the number of trials per problem except in the critical numbers, where no errors are possible. Certain points of difficulty for various subjects are revealed by the large number of trials and the amount of time required for the solution of some of the problems. Such points of unusual difficulty are found in the record of Subject iv at lo beads, in that of Subject nI at II beads, and in that of Subject vi at 25 beads, etc. Some- times the troublesome problems are so distributed as to suggest delnite, regularly recurring types of difficulty. This is illustrated in the record of Subject vi where numbers which are equal to i plus any multiple of 3 offer greater difficulty than those which are equal to 2 plus a multiple of 3. Problems of the former type will be referred to as L-problems and those of the latter type as H-problems. It will be noticed that the L-problems from 7 to 22 inclusive required 72 trials for their solution whereas the H-problems from 8 to 23 inclusive required only 25 trials. The solution of 25 beads which is an L-problem, required but 6 trials. Only in two instances did this subject solve an L-problem with Table IV. — Progress of Subjects through Series 1-2 Initial No. Subj.i Subj. ii Subj. iii Subj. iv Subj. v of Beads Trials Time Trials Time Trials Time Trials Time Trials Time 4 2 21 2 29 3 42 4 32 2 20 5 2 13 2 29 2 24 4 28 2 17 6 4 48 2 35 3 36 3 44 2 32 7 2 25 2 97 3 27 7 117 6 89 8 4 59 3 96 4 56 4 79 4 78 9 5 86 3 70 I 21 9 148 II 240 10 5 104 2 79 3 50 50 1151 7 256 II 18 719 6 375 3 81 4 152 II 577 12 4 102 3 112 2 42 15 2 117 13 4 539 2 143 2 32 8 270 3 106 14 . 7 355 2 29 3 169 7 647 15 I 152 24 2 252 16 3 406 otal 48 1716 35 1572 28 464 96 2205 62 2834 HIGHER MENTAL PROCESSES IN LEARNING 31 Table IV (Continued) Initial No. Subj. x Subj. ix Subj. vi Subj. vii Subj. viii of Beads Trials Time Trials Time Trials Time Trials Time Trials Time 4 4 35 4 22 2 4 4 40 2 15 5 4 42 2 14 2 4 4 68 2 21 6 4 94 9 85 8 33 22 190 7 60 7 4 62 3 25 II 87 74 533 2 16 8 4 77 3 25 5 46 4 35 4 36 9 17 .292 4 35 7 56 12 343 8 88 10 8 108 4 44 5 45 48 664 15 158 II 24 339 68 822 6 68 II 145 32 395 12 28 423 15 192 8 86 26 335 18 193 13 100 2067 6 94 IS 211 12 156 33 559 14 9 204 76 1455 2 30 18 257 24 551 15 44 1229 3 54 7 lOI 36 590 12 264 16 48 1 165 88 1799 ID 151 II 237 3 104 17 48 1220 3 126 7 108 21 428 3 91 18 15 369 3 76 ^ 5 84 16 567 3 40 19 21 687 3 135 26 468 7 202 20 21 446 2 72 3 42 24 538 21 3 58 I A7 10 154 2 265 167 2591 22 38 1054 4 248 5 76 9 1064 22 3 67 2 82 2 28 24 6 154 2 34 7 123 371 6659 25 5 121 2 137 156 5061 26 3 78 2 84 6 126 27 8 264 I 48 10 450 Sub . X (Continued) 28 2 68 2 114 4 100 No. of Beads Trials Time 29 9 262 2 152 2 22 41 2 100 30 5 141 19 ' 157 42 2 114 31 10 337 I 87 43 7 480 32 6 244 44 3 237 33 ID 325 331 7934 45 HID 34 5 181 315 6129 46 18 I2I2 35 2 95 47 2 151 36 5 187 48 19 37 17 696 49 6 467 38 2 77 Tex SO 4 400 39 3 151 40 7 284 597 17538 Table IV (Continued) il No. Subj. xiv Subj. xii Subj. xiii Subj. xi Beads Trials Time Trials Time Trials Time Trials Time 4 4 18 2 17 2 16 2 9 5 4 42 2 9 3 13 3 40 6 3 59 5 58 5 49 I 31 7 3 44 6 69 4 26 2 25 8 5 •86 7 137 3 22 2 51 9 II 124 8 99 6 49 I 33 lO 24 352 17 295 8 63 7 182 II 36 568 4 116 30 338 8 225 12 12 123 9 156 38 481 2 53 13 4 52 II 250 2 31 13 419 14 2 14 28 1016 4 89 2 53 IS 10 131 7 256 6 103 I 8 i6 5 93 39 1622 10 183 2 56 17 14 230 5 393 3 38 2 57 i8 10 150 127 5 125 41 19 4 16 5 253 2 38 3 83 20 9 254 4 211 7 217 2 55 21 12 2X8 63 2 96 60 22 5 78 4 170 5 132 23 7 152 2 97 2 105 — 24 3 51 2 2 108 53 1481 25 13 295 3 114 4 132 26 2 38 2 65 2 70 27 4 68 I 2 216 28 7 146 2 152 3 87 29 2 37 2 58 2 76 30 3 62 88 I 27 31 14 355 2 91 3 136 32 6 214 2 74 33 2 68 22 34 25 758 2 49 166 3066 35 2 74 2 39 36 2 72 5 Subj. xiv (Continued) zy 23 793 2 106 No. Beads Trials Time 38 5 117 50 3 81 39 I 4 2 II 40 7 zn 184 6283 5 2 9 41 3 134 6 2 42 I 97 7 5 84 43 18 ^1^ 8 4 47 44 2 60 9 I 16 45 4 10 3 23 46 4 149 II 3 58 47 4 148 12 3 48 6 13 2 14 49 4 116 14 15 16 17 2 2 2 17 I 9 24 367 8I7I HIGHER MENTAL PROCESSES IN LEARNING 33 fewer trials than were required for the preceding H-problem, A total of 236 trials was required for the solution of 9 L-prob- lems and only 35 for 9 H-problems. Subject x required 296 trials for the solution of the L-problems of this series and only 146 for an equal number of H-problems. Subject xiv experi- enced a change in the type of difficulty at about the middle of the series. The L-problems from 7 to 22 inclusive recjuired only 45 trials whereas the H-problems from 8 to 23 inclusive required 73 trials. But from this point to 44 beads the L-problems be- came the more difficult, requiring 107 trials as compared with the 22 trials required for the solution of an equal number of H-prob- lems in this portion of the series. A similar change in the type of difficulty occurs in the record of subject vii. The L-problems below 12 required 126 trials while the H-problems were solved in 19 trials. Yet the numbers of trials required for the solution of an equal number of L- and H-problems above this point were 30 and 83 respectively. The principal causes of these irregularities are found to be false assumptions; erroneous generalizations, founded usually upon very incomplete analysis; uncritical applications of methods and generalizations from preceding problems, and, in some of the extreme cases, monotony and fatigue. Some of these irregulari- ties will be considered at greater length in later sections on gen- eralization and transfer. Another noteworthy feature of the progress of most of the subjects through the problems of Series 1-2 is the low number of trials upon the early and the late problems of the series as com- pared with the number of trials required for the solution of some of the intermediate problems. The extent to which this feature characterizes the work of various subjects is shown in Figure H. The number of trials is represented on the abscissae and the num- ber of problems solved — or recognized as insoluble in case of the critical numbers — is represented on the ordinates. Note that with the exception of the curve of Subject iii, all curves show a period of relatively rapid advancement followed by one of slower progress, after which a final period of rapid progress terminates with a successful generalization for the entire series. 34 JOHN C. PETERSON O31';OdM0idaQ:]3i]Kl HIGHER MENTAL PROCESSES IN LEARNING 35 The three stages in the learning- process represented by these features of the curves are also clearly indicated by the behavior of the subjects and by their comments. The period of rapid ini- tial rise of the curves was obviously a period of perceptual solu- tions. During the intermediate period of slower advancement the abstraction of significant elements of the problematic situa- tion occupied the greater portion of the energies of the subjects. Some combination among these elements of course occurred im- mediately, but it was not until after considerable experience with the separate elements that most subjects were able to combine and organize them into effective means of control, capable of altering markedly the direction of the curve and leading speedily to a satisfactory generalization for the entire series. The types of mental process which in some degrees dominated these various stages of learning may profitably be treated separately in the three following sections. C. Perceptual Solutions At the beginning of Series 1-2 comparatively few erroneous draws were made, but the percentage of errors increased rapidly for some time with the increase in the number of beads. This increase in the percentage of errors is shown in Table V. This table includes all of the reactions of thirteen subjects upon the non-critical numbers from 4 to 11 inclusive. Table V Number of Beads 4 Total Draws from each Number 222 Erroneous Draws from each Number 14 Percentage of Errors 6.4 Note the rapid increase in the percentage of errors as the num- ber of beads advances from 4 to 8 and the high but fluctuating percentage beyond 8. The unevenness of this increase is also worthy of note. From 5 beads the percentage of erroneous draws is 17.3 per cent higher than from 4; from 8 beads it is 17.6 per cent higher than from 7. But the percentage of erroneous draws from 7 is only 4. i per cent higher than from 5 though the inter- val between 5 and 7 is twice as wide as that between the members 5 7 8 ID II 194 282 262 297 401 46 79 119 92 173 23-7 27.8 454 30.9 431 36 JOHN C. PETERSON of the other pairs of numbers compared. This difference is un- doubtedly due to the nature of the errors which are possible in drawing from the different numbers. The only errors possible in drawing from 4 and from 5 beads result in leaving 2 and 4 beads respectively; from 7 and 8 the only possible errors result in leaving 5 and 7 beads respectively. Thus from the point of view of the number of beads left after an erroneous draw from each of the foregoing numbers, the long intervals occur where the change in the percentage of errors is greatest. The change in the percentage of errors in the early numbers of the series is therefore apparently due to the degree of ease with which the consequences of a given move may be foreseen in a purely per- ceptual manner.^^ The perceptual character of these early solutions is further shown by the behavior of the subjects. Usually i or 2 beads were moved aside tentatively or otherwise marked off so that the possible result of further draws could be directly perceived. ^^ This tentative manipulation was purely a trial and error affair at first, as was indicated by the constant shifting from one of the possible draws to another, as well as by the comments of those subjects who mentioned the matter at all. Sometimes both hands were employed in this sort of manipulation,' — one repre- senting the subject and the other the experimenter, — and so the consequences of all possible draws were figured out on the per- ceptual level. The movements used to mark off, or temporarily exclude some of the beads from attention, were not always of this overt character. Often they became almost imperceptible and occasionally verbal reactions served this function even in the early stages of the game. For some subjects these perceptual judgments were important factors in the solution of problems far beyond the immediate span of attention. Thus some of the 15 The large downward fluctuation in the percentage of errors at 10 offers no serious difficulty to this view. This fluctuation occurs where it might most reasonably be expected; i.e., immediately after one of the long inter- vals and after the range of direct perceptual control has been passed. 16 If the hold upon the beads thus tentatively drawn aside was released the maneuver was counted a draw; therefore the caution, and also the diffi- culty of applying perceptual checks upon the higher numbers. HIGHER MENTAL PROCESSES IN LEARNING 37 subjects, upon discovering late in the game that 6 or 9 could not be won, would mark off and exclude so many beads from at- tention and then proceed to solve the remaining numbers upon the perceptual level. But usually when the number of beads was increased to beyond 7 or 8, the perceptual form of solution more or less completely broke down. The important feature of these perceptual solutions is the fact that all progress was here made through trial and error per- ceptually checked and entirely without the use of symbols except an occasional word used in a very specific way. That is to say, there was a conspicuous absence of generalization in these early perceptual solutions, due to the lack of any necessity for the use of symbols. As will be seen later, the failure to utilize symbols in these early problems seriously limited the transference of con- trol from the lower to the higher problems of the series. D. Analysis Immediately following upon the period of perceptual solutions there was usually a period of evident confusion. Though sub- jects had been instructed at the beginning of the series to look for underlying principles, they generally failed up to this point of the game to see any relation between problems and to remem- ber how or understand why certain numbers had been won. In accordance with James's view that all analysis depends upon the "law of varying concomitants" or upon the elements having somehow previously been brought to attention in isolation, fur- ther progress here would require either that the solutions of earlier numbers be recalled and applied to the problems at hand or that some sort of manipulation be carried on by means of which the significant elements of the various problems could be abstracted and associated with appropriate reactions. Both of these alternatives were tried, the former without success in a single case when attempted early in Series 1-2, the latter with varying degrees of success, depending on the individual who made the attempt and the sort of manipulation resorted to. The sorts of analysis which occurred are classified from the points of view of (i) the specific elements and types of elements 38 JOHN C. PETERSON which were abstracted and employed for generalization, (2) the explicitness and extent of analysis, and (3) the temporal rela- tions of manipulation and ideational analysis/^ I. Types of Elements Abstracted: The Direction of Analysis. Any aspect or isolable portion of the objective situation and any relation between such portions or aspects will hereafter be referred to as an "element" of the general problem. Elements of this sort appeared in considerable variety and exhibited uni- formities of such varied types that different subjects might con- ceivably have arrived at equally valid solutions from quite differ- ent lines of approach. Subjects usually began early in Series 1-2 to count the number of beads from which they were required to draw at each move. This usually led to an early discovery of the fact that certain numbers are especially significant as points of orientation and control in the series. Certain multiples of 3 were usually the first numbers to take on this special significance in Series 1-2. Thus subjects almost invariably came to regard the numbers which they could not win as the important elements to be sought out. Sometimes subjects also tried to remember the numbers which they were able to win, but these numbers were seldom made the objects of special attention and the basis of hypotheses and generalizations, except as they were brought in to complete the formulation after the solution had been practically worked out upon some other basis. Perhaps the next most common type of element to attract special attention was the relationship of the draws made by the experimenter to those of the subject. Occasionally a subject would become so absorbed in the pursuit of this relation that he would utterly fail for a time to notice the significance of the number of beads presented. Most of our subjects, however, noticed some sort of relationship here rather early and divided their attention between it and the number element mentioned above. Other elements of particular interest at times to various sub- 1^ See Ruger, "The Psychology of Efificiency," pp. 10-14. HIGHER MENTAL PROCESSES IN LEARNING 39 jects were the number of beads obtained by the subject or by the experimenter or both, or sometimes the total number of beads drawn, or the number of draws obtained by the experimenter or by the subject, or the relation between the number of draws and the number of beads obtained, etc. Though elements of this sort all exhibit uniformities of such character as might well be- come the basis for successful generalizations, no one succeeded in getting a successful solution from these elements alone, al- though some subjects lost a considerable amount of time in the attempt. Without going further into detail it may be said that the more successful subjects usually began very early to give special at- tention to the number of beads remaining after each draw and often failed entirely to notice the relation between the draws of the subject and those of the experimenter. Often in the first series, and quite generally in the later series, elements of both of these types were constantly taken into account with good re- sults. The attention of the less successful subjects usually fluctu- ated considerably between the different types of elements, but failed to follow up any type consistently enough to discover the uniformities lying beneath the surface. Table VI shows the point in the first series where each subject gave the first evidence of having become definitely aware of the uniformities in the elements of each of the first two types men- tioned above. This, of course, implies a considerable amount of previous attention to the elements underlying the uniformities. For example a subject would often study his own successful draws and those of the experimenter for some time before be- coming aware of the "opposite"^^ relation between them; or he might realize for a considerable time that certain definite num- bers are critical without noticing that they are multiples of 3. It will be noticed that a number of the most successful sub- jects seemingly failed to discover the principle of drawing by ^^ In all trials beginning with a critical number E drew i when S drew 2, and 2 when S drew i. Likewise, in order to win any non-critical number, S had to draw the "opposite" of E's preceding draw throughout the trial after first having reduced the number of beads to a multiple of 3 at his first draw. 40 JOHN C. PETERSON Table VI Point in Series 1-2 where Point in Series 1-2 whe the principle of drawing it was discovered that ibject by "opposites" was the C numbers are discovered multiples of 3 20 I3bi& 50th trial •; 15b 35" ( 11 iii 13b 25 th trial 15b 28" ' iv pb 25" " 15b i6b 28b 96" 63rd 384th i V vi 14b 71st trial ' vii 6b 21 " " i8b 310" ' viii i8b 168" a 17b 162nd ' ' ix 23b 303rd " 31b aisth i X 12b 77th " 40b 2ib 547" 53rd I XI xii 13b 70th trial lob 30th ' xiii 12b 86" " 15b III " ( xiv lib 88" " 40b 293rd « 1^ 13b, 15b, etc., stand for the various problems in the series and indicate the initial number of beads presented at each trial in the problem. 20 The principle of drawing opposites was not discovered in this series by Subjects i, ii, v, and xi. Opposites, and that some of the subjects who found the greatest difficulty in solving the problems discovered this principle very early in the series. By the rank method the correlation between the number of trials required for the discovery of this principle and the number of trials required to find a satisfactory solution for the series, is — .324. There is a positive correlation of .728 between the number of trials required to discover that the critical numbers are multiples of 3 and the number of trials required to find a satisfactory solution for the series. Attention on the part of the subject to the numbers of beads from which he must draw thus leads more directly to fruitful generalizations than does at- tention to the relation of his own draws to those of the experi- menter. What factors, it may be asked, determine which of the various elements shall be abstracted from the total situation in any case? This question cannot be answered fully from the data at hand but it seems worth while to point out that the order of abstrac- HIGHER MENTAL PROCESSES IN LEARNING 41 tion of elements is exactly what might be expected if the prin- cipal selective factors were the relative frequency of occurrence of various situation and response elements, the relative nearness of the various elements to a goal, or end of action, and the speed of the subject's reactions. These three factors will be discussed separately in a later section. Some data concerning them may well be given here. a. Frequency of Repetition. — As already noted the numbers in Series 1-2 which were first to attract special attention and to become objects of active research were almost invariably the critical numbers. This emphasis upon the critical numbers was the result not of sudden, comprehensive insight, but of a gradual and often tedious growth of meaning. There were, to be sure, sudden spurts and long plateaus in the progress of some sub- jects, as a result of the drawing in of old concepts by associa- tion. But the regularity of progress in the abstraction of ele- ments in the first series was not often seriously obscured by these factors. The first number to be isolated and treated as especially significant was almost invariably 6."^ Thereafter 9, 12, and the higher critical numbers followed in the order of their magnitude except as the order was affected by the influx of old concepts in the form of generalizations. The early stages of the abstraction of the various critical numbers progressed with surprising inde- pendence. The first objective signs of the process of abstraction appeared usually in the form of short delays, exclamations, and other indications of critical reaction, which, if we may trust the comments of subjects, were accompanied by occasional fleeting insights into the nature of the situation. Continued repetition of these numbers served to give further emphasis to them and greater depth and stability to the erstwhile fleeting insights into their signifiance for the solution of other numbers, until finally it became possible to formulate their relations into satisfactory principles of control. 21 It is true that prior to the recognition of the critical significance of 6, 3 was usually recognized in the concrete as a losing number. But, owing apparently to the easy perceptual control of 3, it was almost never mentioned explicitly as a losing number, except as an afterthought in rounding out a generalization which had been made upon the basis of higher critical numbers. 42 JOHN C. PETERSON Corresponding to the gradual abstraction of certain critical numbers and the order in which these numbers were affected by the process, is the gradually increasing excess of the subjects' reactions to critical numbers over their reactions to non-critical ones. Not only does this excess of draws from critical numbers increase from problem to problem, but the ratio of draws from critical to those from non-critical numbers also increases rather steadily. The regularity of this increase, both in the excess of draws from critical numbers and in their ratio to the draws from non-critical numbers, may be seen in Table VII. At the left are listed the various problems of the series from the first problem, in which 4 beads are presented, to the twelfth problem, in which 15 beads are presented for solution. The figures in the succeed- ing columns indicate the total number of times ten subjects were required to draw from each of the numbers listed above the columns, in the solution of the problem opposite which the figures occur and all preceding problems. Thus in the solution of the problem, 4b, the subjects drew 12, 14, o, and 28 times from i, 2, 3, and 4 beads respectively. In the solution of 4b and 5b they drew 23, 26, 5, 28, and 26 times from i, 2, 3, 4, and 5 beads respectively, and so on. Or reading down the column, under 3 for example, the ten subjects drew o times from 3 while work- ing upon 4b; 5 times while working upon 4b and 5b; 46 times while working upon 4b, 5b, and 6b, and so on. The ratios of draws from critical numbers to the averages of those from ad- joining non-critical numbers, are given in italics. The frequency of subjects' draws from any critical number does not greatly exceed that of their draws from adjacent non- critical numbers until after the game has progressed to a point considerably beyond the critical number in question. Thus the frequency of draws from 3 beads becomes greater than from 2 or 4 beads only after considerable work upon 6 beads; and the frequency of draws from 6 beads becomes greater than from 5 or 7 beads only after some work upon 9 beads, etc. This is in entire accord with the fact that the status of these numbers (i.e., whether critical or non-critical), was in most cases forgotten shortly after the commencement of work upon higher numbers, HIGHER MENTAL PROCESSES IN LEARNING 43 Table VIII Total Number of Draws of ten Subjects from various Numbers in Solving the first twelve Problems of Series 1-2. I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 4b 12 14 28 5b 23 26 5 .19 28 26 * 6b 23 26 46 1.70 28 26 41 1.58 7b 35 41 61 1-47 42 43 41 ■9^ 44 8b 44 54 77 145 52 58 55 1.08 44 38 9b 44 54 152 2.8T 52 58 130 ^•55 44 38 75 1.97 lob 65 69 241 78 79 195 89 88 75 .69 131 lib 108 78 408 4.08 122 lOI 333 2.66 149 151 172 1.22 131 220 12b 108 78 533 122 lOI 458 149 151 297 2.11 131 220 125 .56 13b 137 98 669 5.J5 152 124 578 3-66 192 186 385 2.03 194 291 125 ■52 186 14b 163 109 758 5-28 178 137 657 226 208 447 2.01 236 339 161 .61 186 126 15b 163 109 845 5.59 178 137 744 4.09 226 208 534 236 339 248 .96 186 126 87 .69 (See preceding paragraph for explanation of table.) and was re-discovered only after a considerable number of trials during which the frequency of repetition of forgotten critical numbers became gradually more preponderant over that of for- gotten non-critical numbers. Thus, after working for a while upon 7 or 8 beads, the subject usually forgot that he had been unable to win 6 and did not often re-discover the fact until after some work upon 9 beads or even upon higher numbers; that is, until the number of draws frorn 6 had come to be greatly in excess of those from 5 or 7 beads. The behavior of subjects toward 9 and 12 beads, and often toward higher numbers, was 44 JOHN C. PETERSON similar in this respect. But, owing to the increasing influence of generaHzation and other less conscious sorts of transfer, the pre- ponderance of draws from the critical numbers became gradually less marked or entirely vanished. Though in the work of these subjects the order in which the various critical numbers acquired their special significance is closely paralleled by the relative frequency of their presentation, it might be questioned whether this order was not determined by the order of their original presentation as problems of the series rather than by the frequency of subjects' reactions to them. That the order of the original presentation of these numbers as separate problems is not the determining factor, however, is evi- dent from the facts presented in Table VIII where the critical numbers were not presented as separate problems prior to their discovery. The discovery of a critical number was usually not a sudden event, but a gradual process of isolation and of growth of meaning. The process will be referred to hereafter as the abstraction of critical numbers. Various stages of the process must be distinguished and defined before quantitative comparison becomes possible. The first stage in the abstraction of a critical number may be regarded as completed when, with this number of beads concretely before him, the subject gives the first clear in- dication of his realization that in the current trial defeat is in- evitable. This recognition of defeat may be expressed in a shak- ing of the head or in some form of exclamation, such as, 'T lose," "No use," "You win," "I cannot win now," or "No matter how I draw now you can win." Such recognition of defeat often comes in the form of a perceptual judgment, without explicit awareness of the exact number of beads remaining or of the im- possibility of ever winning from this particular number. The second stage in the abstraction of a critical number will be re- garded as completed when the subject clearly states the recog- nition of his inability ez'er to win the number in question. The third stage is regarded as completed when the subject announces his conviction that the number in question is also critical for the experimenter; i.e., that the experimenter must inevitably lose if required to draw from that number. In the interpretation of HIGHER MENTAL PROCESSES IN LEARNING 45 reactions indicative of the various stages the general context must be taken into account; our criteria are not, therefore, wholly objective. The individual records of twelve new subjects, reacting to 14 beads as their initial problem, are given in Table VIII. The sub- jects are listed at the left as Si, S2, S3, etc., according to the number of trials required to win two trials in succession from 14 beads. In the / columns of the various A sections (i.e., 3 A, 6A, 9A, and 12 A) of the table are given the numbers qf draws made by individual subjects from each of the critical numbers during the first stage of its abstraction. Similar data for the second and third stages of abstraction are given in the B and C sectons of the table respectively. The totals of draws from the two non-critical numbers adjacent to each critical number, dur- ing the various stages, are given in the s columns of the respec- tive sections. Thus the record of S3 shows that only the first critical number was abstracted. During the first stage, 3A, of this process two draws were made from 3 beads and none from 2 or 4 beads. Prior to the completion of Stage 3B, 39 draws were made from 3 beads and a total of only 6 from 2 and 4 beads. Before 3C was completed 74 draws had been made from 3 beads though the total number of draws from 2 and 4 beads was only 16. In only 4 of the 73 pairs of /- and .y-column figures here pre- sented does the number of draws from a critical number fall short of the sum of all draws from the two adjacent non-critical numbers, and in no case does it fall below the average number of draws from the two adjacent non-critical numbers. The average number of draws from critical numbers at the completion of Stage A in their abstraction, is 31.2; that of draws from adja- cent non-critical numbers is 4.2. At the completion of Stage B the corresponding averages are 51.6 and 9.9 respectively; and at the completion of Stage C they are 100.6 and 21.8 respectively. The total number of draws from all critical numbers in all stages of their abstraction as here reported, is 3787. From non-critical numbers — though there are twice as many of them as of critical numbers — the total number of draws is only 703. That is to 46 JOHN C. PETERSON Table VIII Frequency of Subjects' Draws from the Critical Numbers at the Completion of the various Stages of their Abstraction. 3A 3 B 3C 6A 6B 6C / .$■ / ^ / ^ / J / ^ / J Si I o 55 2 167 31 52 4 57 4 158 40 S2 4 I 7 I S3 2 39 6 74 16 S4 I 28 4 3 14 2 ss I o 4 20 43 14 S6 I II I S7 3 I II 5 2 I 12 8 58 2.2 5 68 26 S9 I 27 2 Sio 2 I 76 15 Sii 7 o 19 ^ 20 Sl2 7 I 7 I 90 9 Total 30 4 122 12 252 52 197 24 292 49 269 80 Av. 2.7 •4 40.7 4.0 84.0 17.3 21.9 2.7 36.5 6.3 89.7 26.7 9A 9B 9C 1 12 A 12B 12 C / .y / .? i .y / J / ^ / ^ Si 57 13 64 13 148 50 104 74 108 82 S4 15 2 21 5 17 8 22 10 S5 24 12 27 12 33 24 32 40 33 42 S6 II 7 80 28 83 31 S7 22 26 52 34 19 35 S8 24 27 38 27 55 39 29 Zl Sio 98 41 103 41 118 48 Sii 36 II 85 70 60 51 S12 105 II 125 II 251 45 164 149 172 149 190 160 Total 359 117 496 205 657 240 505 422 ^17 190 331 284 Av. 51-3 16.7 55-1 22.8 109.5 40.0 63.1 52.8 92.3 63.3 1 10.3 94.0 say, at the time of completion of the various stages of abstraction of the critical numbers, the number of draws from them is on the average more than ten times as great as from adjacent non- critical numbers. The order of abstraction of the various critical numbers is also in practically complete harmony with the order of frequency of subjects' reactions to them individually. Not only are a sub- ject's draws from a lower critical number invariably more nu- merous than from a higher one, but his draws from the lower non-critical numbers are also fewer than from higher ones. Both of these facts tend to give greater emphasis to the lower critical numbers and favor the abstraction of critical numbers from lower to higher in the order of their magnitude. A com- parison of the actual order of abstraction requires that different stages of the process be listed separately. This has been done in Table IX. HIGHER MENTAL PROCESSES IN LEARNING A7 As before the subjects are listed at the left. The critical numbers appear at the top of the columns, the various stages be- ing segregated as indicated by the capitals. The figures in the columns indicate the order in which the critical numbers emerged from the various stages of abstraction. Thus the record of Si shows that 3 was the first number to emerge from Stage A, 6 the second, 9 the third, etc. Points where hesitation and other inconclusive signs showed the probable occurrence of abstraction though no clear indications of the process were observed, are marked with asterisks. No overt indications of abstraction were observed for the blank points in the table. When a given stage in the abstraction of two or more critical numbers was completed simultaneously, they were given the same value in the table. Table IX Order of Abstraction of Critical Numbers. 3A 6A 9A 12A 3B 6B 9B 12B 3C 6C 9C 12C Si I 2 3 4 I 2 3 * I I I I S2 I 2 - - - - - - — - - - S3 I - - I - - - I - - - S4 I 2 3 4 I 2 3 3 - - - - ss I 2 3 4 - I 2 ¥ - I I 2 S6 I * * 2 - I 2 3 - - - - S7 2 I * 3 - I 2 * I - 2 - S8 - * I 2 - I 2 - - I I - S9 I 2 - - - - - - - - - - Sio I 2 3 - - - I - - - I - Sii I 2 3 4 - I 2 - - - - - S12 I 2 3 4 - I 2 3 - - I I That the correspondence between the frequency of reaction to various critical numbers and the order of their abstraction is really very close, is shown by the fact that in only one case (S7, 3 A) in the entire table did a lower number emerge later than a higher one from a given stage of abstraction. There is, however, a strong tendency in this group for all critical numbers to emerge simultaneously from Stage C. This is due to the fact that abstraction has here reached a sufficiently advanced stage to permit generalization to become effective. h. Effect of Nearness to a Goal. — As already noted the fre- quency of reaction to the various critical numbers varies directly with their nearness to o, the immediate goal of every trial. It is therefore impossible to measure separately the effects of fre- 48 JOHN C. PETERSON quency of repetition and those of the nearness of elements to a goal. Logically the relative nearness of elements to a goal re- solves itself into two distinct factors : ( i ) the greater ease with which the consequences of tentative or hypothetical draws can be foreseen from lower than from higher numbers and (2) the closer temporal proximity of the goal to reactions toward lower than to those toward higher critical numbers. Both of these factors were probably operative as will later be pointed out more fully. Some evidence of the effects of (i) is found in the per- centages of erroneous draws from various non-critical numbers as reported in Table VI. In the further presentation of evidence at this point no effort will be made to distinguish between the effects of these two factors. That the critical character of 6 is more easily discovered than that of 9, is shown by the number of trials required by 24 sub- jects from each of these numbers in discovering the impossibility of winning it. The total numbers of trials upon 6b and 9b were 126 and 194 respectively. Seventeen of these subjects required fewer trials upon 6b than upon 9b. Likewise fewer trials were required upon 9 than upon 12 beads, both by the group as a whole and by a majority of the individual members. But, ow- ing to the rapidly increasing effects of previous learning upon the higher numbers of the series, the difference here is not so great as in the former pair. The comparative difficulty of abstraction of various critical numbers is further shown in Table VIII. The averages as they stand show a marked general increase in the number of reactions required to raise successively higher critical numbers out of a given stage of abstraction. But this evidence is open to the objection that the averages do not all represent the work of exactly the same group of subjects. The results are not much modified, however, when the work of those subjects whose rec- ords do not show complete data for the stages compared, is left out of account. In the work of 9 subjects whose records con- tain data for both 3A and 6A there is only one case, that of S7, in which 3A offers more difficulty than 6A. The average num- bers of draws are 3 and 21.9 for 3 A and 6A respectively. Stages HIGHER MENTAL PROCESSES IN LEARNING 49 6A and 9A are both represented in the work of 6 subjects all of whom required more draws from 9 in the attainment of 9A than from 6 in the attainment of 6 A. The average numbers of trials are here 26.7 and 55.8 for 6A and 9A respectively. Data for both 9A and 12 A are contained in the records of 6 subjects all of whom required more draws for the attainment of 12A than of 9A. The average numbers of draws are here 43.5 and 67.7 for 9 A and 12A respectively. Without going into further de- tail it may be said that the results in the B Stages show the same marked tendency for higher critical numbers to require many more reactions for their abstraction than are required by the lower ones. The less marked tendency in this direction in the C Stages is clearly due to the onset of effective generalization and need not concern us here. c. Effect of Speed of Reaction. — The ease with which the principle of drawing by "opposites"^^ is discovered appears to depend largely upon the speed of reaction of the subject. The subjects who reacted rapidly almost invariably discovered the "opposite" relation between their draws and those of the ex- perimenter early in the game, but those who reacted slowly usu- ally became aware of this relation late or not at all. Data re- garding the speed of reaction and the number of trials required by subjects of Groups I and III are presented in Table X.-^ The data for Group I are shown in the upper section of the table, those for Group II in the lower section. Since the extent of the series is not the same for different subjects, early or late dis- covery of the principle of drawing opposites must be determined not by the absolute number of trials required for the discovery, but by the percentage of all reactions upon Series 1-2 which sufficed for the discovery of the principle. These percentages are given in the last column of the table. 22 That is, drawing i when the opponent draws 2, and 2 when the opponent draws i. This procedure obviates the necessity of counting the remaining beads when their number has once been reduced to a multiple of 3. 23 Similar data cannot be given for Group II owing to the fact that no separate records were kept of the time required for the solution of indi- vidual problems of the series. 50 JOHN C. PETERSON Table X Speed of Reaction as Related to Ease of Discovery of the Principle of Drawing by "Opposites." Per cent of total Av. No. of seconds No. of trials re- trials upon Series per draw in dis- quired for dis- 1-2 required for covering the covery of the the discovery of Subjects principle of principle of the principle of opposites-* drawing by drawing by opposites opposites i 10.25 ( 5.2)25 I. GO ii 12.28 (12.8) I. GO iii 5-94 ( 5-2) 25 .89 iv 574 ( 6.0) 25 .26 V 12.76 ( 6.4) 1. 00 vi 2.86 ( 3.4) 71 .21 vii 5-05 ( 40) 21 .06 viii Z-(>7 ( 3-1) 167 1. 00 ix 4.88 ( 4.5) 308 .96 X 4.56 ( 6.1) 77 •13 xi 6.07 (13-5) 1. 00 xii 4-98 (5-0 70 .38 xiii 2.59 ( 3-1 ) 86 .52 xiv 3.18 ( 57) 88 .24 I 6.88' ■(* 5.8) 32 •57 2 7-49 (5-3) ^S 1. 00 3 13.13 (11.6) 1. 00 4 3.96 ( 7-2) 1. 00 5 4-8s ( 4-4) 1. 00 6 2.70 (3.1) 22 •31 7 6.16 ( 5.8) 35 .26 8 6.08 ( 5.6) 107 75 9 5-13 ( .S-S) 1. 00 10 3.83 ( 3.9) 63 .24 II 3-57 ( 4-1) III .39 12 378 ( 31) 84 .24 2* Or until the completion of Series 1-2. 25 In the parentheses beside the second column the average number of sec- onds per draw in the 'first fifty draws of each subject, is given. Note the general correspondence between the ease of discovery, as indicated by the percentage values in the last column, and the speed of reaction as indicated in the second column of the table. The average time per draw for subjects who did not dis- cover the principle of drawing opposites was 10.34 seconds in Group I and 6.77 seconds in Group III. The averages for sub- jects who did discover the principle were 4.35 and 5.06 for Group I and Group III respectively. By the rank method the correlation between the ease of discovery of the principle of op- posites and the speed of reaction is .668 for Group I and .516 for Group III. HIGHER MENTAL PROCESSES IN LEARNING 5i It might be supposed that these correlations are due to an in- crease in the speed of reaction after the discovery of the prin- ciple of opposites but before its announcement. That the cor- relations were perhaps raised somewhat by this tendency is not denied. But the effect of this factor was probably slight since the subjects were urged repeatedly to notify the experimenter of any new discovery or seemingly relevant new idea at the time of its occurrence. Moreover, the records show that subjects who reacted slowly at the beginning of the work usually failed to discover the principle of drawing opposites, whereas subjects who drew rapidly from the beginning were likely to discover the principle early in the game. Thus the average time per draw of the first fifty draws of Subjects i, ii, v, and xi, who did not dis- cover the principle, is 9.47 seconds, whereas the average time of the remaining subjects of this group, all of whom discovered the principle, is only 4.42 seconds. The corresponding averages for subjects of Group III who did and for those who did not discover the principle, are 7.20 and 4.60 seconds respectively. The correlation between ease of discovery of the principle of op- posites and the speed of reaction in the first fifty draws is found to be fairly high. For Group I this correlation is .373 and for Group III it is .629. It may still be argued that the speedy reaction of some sub- jects was merely the result of their attention to the relation be- tween their own draws and those of the experimenter; but such comments of subjects as we have, together with their time rec- ords, show that even while attending to the number of beads remaining or recalling their success or failure with certain num- bers in past trials, these subjects reacted with less deliberation than others. As shown by their comments, the work of these subjects was characterized by incessant fluctuations of attention between relatively isolated elements or by wholly irrelevant re- flections. These subjects appeared to react quickly not so much because of any marked speed of mental activity as because of a lack of those deeper insights into relations and consequences, which might have barred hasty reaction. In striking contrast with these positive correlations between 52 JOHN C. PETERSON speed of reaction and relative ease of discovery of the principle of drawing opposites, are the correlations between speed of re- action and facility in finding a solution for the entire series, as measured by the number of trials rec[uired for the solution of the series. The average speed of reaction in the entire series for all subjects of Groups I and II, together with the number of trials required for the solution of the series, is shown in Table XL'*' ^ Table XI Speed of Reaction as Related to the Number of Trials Required for Solution of Series 1-2. Average No. of sec- No. of trials required Subjects onds per draw in for the solution of Series 1-2 Series 1-2 10.25 50 ii 12.28 35 iii 5-52 28 iv 6.89 96 V 12.76 62 !vi 3-19 331 vii 4.00 371 viii 2.^7 167 ix 4.60 315 X 4-33 597 xi 6.07 53 xii 6.54 184 xiii 3.68 166 xiv 2.77 367 A 12.22 48 B 7.00 62 C 5.61 85 D 11.86 112 E 6.70 220 F 11.67 300 G 4.68 322 H 7.00 389 I 4.86 617 J 3-91 823 The data for Group I are given in the upper portion of the table and those for Group II in the lower portion. The corre- lations between speed of reaction and the number of trials re- quired for the mastery of the series are — .748 for Group I and — .686 for Group II, when the speed of reaction is stated in 26 Similar data are not available for Group III owing to the failure of most of the subjects of the group to obtain a solution for the series while solving the limited number of problems presented to them. HIGHER MENTAL PROCESSES IN LEARNING 53 terms of the average time per draw in the entire series. When the speed of reaction is determined on the basis of the first fifty draws of each subject, the correlation is ■ — .264 for Group I. Thus intelHgence as determined by the ease of solution of Series 1-2, does not affect the speed of reaction so much at the begin- ning of the series as later when there is a greater accumulation of experience which may serve to inhibit hasty reaction. Speed of reaction appears then in the double role of an effect of the depth of insight into the conditions of the problem and as a factor in determining the direction of analysis. 2. Explicitness and Extent of Analysis In the analysis of puzzles Ruger reports a "wide variation of felt clearness from extremely vague to perfectly clear. This range of felt clearness," he finds, "is matched by differences in results." Our results show a similar variation in the explicitness of analysis from cases in which an element of a situation is barely recognized in passing, to those in which clear verbal form- ulations are accompanied by ability to recognize and control the element in question under novel conditions. Numerous cases were observed where analysis was complete enough to be effec- tive for manipulation but not clear enough to be put into words. A great many of the perceptual solutions were of this character. The analysis of the lower numbers practically never reached the explicitness of a clear verbal formulation until generalizations made upon higher numbers were applied to them. Occasionally a subject would declare that he knew how to win a given num- ber but could not express it in words. One subject, after win- ning from 20 beads twice in succession declared that she believed she could do it again though she could not say how. She then proceeded to win two more trials in succession without difficulty. Another subject, having just won twice in succession from 10 beads, was asked whether she had any new ideas. She replied : "I can't state the idea in words but I have it in motor terms." Ruger mentions several different sorts of analysis which af- fected only a part of the situation. A very indefinite type of partial analysis he describes as "picking out the portion of the 54 JOHN C. PETERSON puzzle to be attacked. In many cases," he says, "there was a mere spatial analysis, 'locus analysis' without involving any per- ception of the mechanical necessities concerned." This locus analysis resulted in the confining of random movements to a por- tion of the puzzle. This is very similar to a common experience of our subjects many of whom quickly discovered that the cru- cial point of the trial lay in the early draws, and busied them- selves, accordingly, with the task of determining what the first one or two draws must be. This often resulted in some im- patience with the requirement that every trial must continue un- til the number of beads was reduced to o, and necessitated a few brief departures from the rule. Often the point conceived of as crucial was in the lower and sometimes in the intermediate num- bers rather than in the higher ones, as the following examples show: "I wonder if you often draw 2 at the beginning as I do, merely to hurry and reduce the number of beads" (xiv, 23b). ^^ "No matter what number you are working with you can win if you can get it to 4 or 5 beads. The problem isn't with the larger numbers but with the few at the end" (xii, lib). "It seems to revolve about the start" (xii, i6b). "Well, I would eliminate the beads until I got my first turn on 4 or 5 (xii reacting to the problem of 59 beads). "When I win, the crucial draw is the second, or not later than the second" (xiv, 25b). A slightly more definite type of analysis is described by Ruger as follows : "An important form of partial analysis noticed was that of a single step in the process while the other steps were attained only by random movement. This single step was often the final one. The solution would come accidentally but the sub- ject would notice the last step. In subsequent trials he would know what to do if he chanced to get to that step but not how to get there." The folowing comments from subjects indicate this type of partial analysis : 2^ The Roman numerals in parentheses indicate the subject whose com- ment is quoted and the numbers following show the particular problem in the series, during the solution of which the observation occurred. HIGHER MENTAL PROCESSES IN LEARNING 55 "I can't beat when we get to 6. I never have beaten you on 6" (xiii, I2b). "I can't win this; there are 6 left" (ix, lib). "I could win if it would come out ii there" (ix, 14b). "I know I can't win unless I can make it 13 there . , . after you have drawn" (vi, 25b). "Some of the first moves I win and others I lose. Now on 5 if I have the first move, I win" (vii, 22b). "I was trying to get it so there would be 7 beads left on my draw" (viii, 12b). These partial analyses, especially when concerned with the higher numbers in a series, depended largely upon the possession of some general scheme by means of which attention could be freed from most of the details of the series and concentrated upon some one point. Such schemes usually took the form of drawing uniformly either i or 2 beads throughout the trial, and generally resulted quickly in some sort of appreciation of the principle of drawing by opposites. This principle is one of the best examples found here of what Ruger called "schematic an- alysis." A partial insight into the significance of multiples of 3 sometimes served as a schema of this sort. One subject, for example, expressed a schematic view of the line of approach to a solution of Series 1-2 as follows: "You draw 2 when I draw I, and I when I draw 2, so that we reduce it by 3's each time; that is, we are drawing them out by multiples of 3" (vii, i8b). From this comment it might be thought that the subject had ef- fected a complete analysis of the series, at least up to and in- cluding 18 beads; but it required two additional trials to bring him to realize that he could not win 18 beads, and many more trials to complete the series. During the next trial he observed ; "We are drawing right down on multiples of 3" ; but here again he failed to realize the significance of multiples of 3, for it re- quired still another trial to convince him that he could not win 18 beads, and his conclusion was not based on multiples of 3 at all. "I can't win from 6," he said, "so since this is a multiple of 6, I can't win it." Of the problem presented by 18 beads the subject had probably made a total analysis in the sense that it 56 JOHN C. PETERSON "reached all the elementary steps or movements." That it was far from a complete analysis of the entire series is evident from the fact that it required still 45 trials upon 19, 20, 21, and 22 beads for the development of sufficient insight to permit the sub- ject to give a solution for the entire series. At this stage of the process, however, analysis proceeded far less than formerly by frequency of repetition, nearness to a goal, etc., and far more by means of ideas and principles formulated in early problems and now applied to new problems and verified with a minimum of repetition. In other words, generalization, which has played a role of increasing importance since the earliest stages of analy- sis, has here become the dominant factor. 3. Time Relations of Manipidation and Analysis In describing the time relations of ideational analysis and motor variations Ruger says : "These two types of variation, acts of analysis and motor responses, may be quite varied, espe- cially in their time relations. At the one extreme is the motor variation which, perhaps, brings success but which runs its course unnoticed. At the other extreme the analysis may occur first and only after a considerable interval be followed by the motor response. "^^ This wide variation in the time relations of analy- sis and manipulation is very characteristic of the work of our subjects, except that in the first series we find little ideational analysis preceding manipulation. Our procedure was not of course such as to encourage attempts at analysis prior to manipu- lation in the early problems of the series, though we believe that such efforts were seldom discouraged except by their failure to bring results. Such analysis as occurred early in the series was usually of a perceptual nature, as already explained, and was ac- companied and checked by manipulation of the beads. The ex- tent to which the method of trial and error became operative in the intermediate and higher numbers of the series, is indicated roughly by the percentage of erroneous draws. These per- centages for some of the subjects in the first six problems of the series are given in table V. In approximately the first two- -s Op. cit., p, 12. HIGHER MENTAL PROCESSES IN LEARNING 57 thirds of the trials of a subject upon each of the higher numbers the erroneous first draws usually amounted to about 50 per cent of all of the draws from the number in question. Often, indeed, half of the first draws of all trials in a number of successive problems, were erroneous. Thus Subject viii, in working from 7 to 14 beads in the first series, made 86 errors of this sort out of a possible 175. In the 7 problems from 8 to 14 beads in- clusive the possible errors for this subject were 156 and his actual errors were 86. Subject i required 38 trials in working over the numbers from 7 to 14 inclusive in the first series; i.e., there were 38 possibilities of erroneous first draws. Although this subject was one of the most successful of the lot and made special effort at analysis before manipulation, he made 21 errors in the 38 draws. His comments upon his methods of procedure are of interest here : "I'm doing this by the trial and error method until I get it down to 4 or 5 where I can handle it." Later he says : "I find that any attempt to analyze this series is apt to inhibit action; at least it doesn't conduce to a solution." Repeated comments of this nature were made by many of the subjects. It often happened, while drawing through a trial which had been given up as lost, that success would be accidentally achieved. Frequently the subject would attempt to recall how it was done. Often he succeeded, but even if he failed, the effort was not en- tirely lost, for the arousal of attention to the existence of ne- glected possibilities resulted in a change of attitude and some- times in a fruitful examination of his assumptions. Numerous occurrences of accidental success without ability to recall the successful variation were noted. The following case is repre- sentative : Subject ix had experienced particular difficulty with 11 beads, Series 1-2, and finally concluded that he could not win if the experimenter drew i early in the trial. He was persuaded to continue, however, and finally happened to take 2 at every draw and so won the trial. His comment was : "I just fell on to that by trial and error procedure. It was accident; I don't believe I could do it again." Some time afterwards he met with the same S8 JOHN C. PETERSON difficulty when drawing from 14 beads. After winning with considerable difficulty, he said: "I don't know how I did that." Later on with 17 beads he repeated his old error seemingly none the wiser for his two accidental successes. A striking example of how accidental successes may be seized upon and made the basis for further progress is found in the reactions of Subject vi who, having made good progress up to 25 beads in Series 1-2, became stranded and required 156 trials to win this number twice in succession. This subject soon dis- covered that when his initial draw was i and the experimenter drew 2, he could win by taking i at every draw throughout the trial. But when the experimenter took i at the first draw, the subject was helpless, though he could have won by merely chang- ing his draws to 2 from this point to the end of the trial. Some- how the latter procedure escaped his attention, although he fol- lowed it once by accident near the beginning of the plateau, so that with 57 opportunities to win by this procedure he succeeded only twice. After his 155th trial upon 25 beads, when the suc- cessful variation occurred for the second time, he said : "That's another way I win — certainly I could win that way if I won the other way !" When questioned as to whether he had known from the beginning of the trial that he would win, he replied : "Not until the third to the last draw. In fact, I started drawing 2 without any intention of continuing to draw 2 on down through the trial. Then I later decided to continue (because, as subse- quently brought out in a comment, he did not remember having tried that before) as I started. ... It didn't occur to me until after I had finished, that it didn't make any difference whether I got that one extra bead first or last, if you drew i each time." The last sentence throws some light on the nature of the diffi- culty experienced by him with this combination. If the subject and the experimenter drew i bead each at the first draw, there would be 23 beads remaining. Now, if the subject took i bead at his second draw and the draws proceeded by opposites after that, there would be 2 beads left for him at the last draw; that is, "i extra bead." The subject's reactions clearly show that some such assumption as, that he must provide for that "extra HIGHER MENTAL PROCESSES IN LEARNING 5Q bead" at his first opportunity, was made; for of the 55 errors which he made in drawing i when he should have taken 2, ioriy consisted in drawing i from 23 beads. The other 15 errors of this sort can be accounted for largely on the basis of his efforts to reach certain lower numbers which he was gradually learning to regard as significant. This false assumption explains why the subject made 40 errors in 57 draws from 23 beads. The case shows how an accidental success may serve to clear away false assumptions and so contribute indirectly as well as directly to a more speedy solution of the problem. It is worthy of note that the accidental successes in the fore- going cases, which occurred before a fair acquaintance with the series had been formed, either failed to attract attention or could not be recalled, whereas an accidental success occurring later in the series was quickly seized upon and applied to the solution of the series. This was quite a common occurrence in the work of a majority of our subjects. Having seen the futility of early attempts at analysis without manipulation, and the varying degrees in which subjects profited by accidental success at different stages of their mastery of a series, we turn briefly to the growing capacity for ideational analysis, which usually began to be effective towards the end of Series 1-2 and became increasingly apparent as the experiment progressed. The first successful attempts at ideational analysis of more than perhaps one step in the series usually consisted in a return to lower numbers which were regarded as related to the present situation, with an attempt to determine their forgotten status, or in a return to the very beginning of the series to re- cover orientation. Reasoning about these lower numbers would invariably take the form of ideational manipulation mentioned earlier in the discussion. Usually some numbers remembered as critical served to facilitate the determination of the status of other numbers. An example will show the character of this sort of analysis more clearly : After gaining a fair acquaintance with Series 1-2 in 371 trials upon the first 17 problems of the series, Subject vii took an excursion back to the beginning to check up and get his bearings. 6o JOHN C. PETERSON "Some numbers," he said, "the first (draw) wins and some it loses. Now at 5 beads if I have the first move, I win; at 7 I lose — I take 2 and yon take 2 and I lose. From 7 if I take 2, that leaves you 5 and you can take 2 and win. If I take i and leave 6 — if I have 7 and take i, I win; If I have 9, I lose. If I have 9 and take i, I lose the game; if I have 9 and take 2, I lose, therefore I lose on 9 always. If I have 8 and take i, the other fellow wins; if I take 2, — O yes! / see why it is that I zvin 8, because I leave 6. I can win 5 by taking 2 and I can win 7 by taking i ; 6 I can never win. If I have 8 and take 2, I always win. I can win 5 if I take 2 ; 6 I can never win, 7 I can win if I take i, and 8 I can win if I take 2 ; 9 I can never win — 6 and 9 I can never win; 10 if I take 2, I lose; if I take i, I can always win. 11 if I take i, I lose; if I take 2, I can always win. 12, if I take 2, you can take i and leave 9; if I take i, — I can't win it. 5 I win and 6 I lose — on even numbers I take away i: On 11 if I take 2 — 12 I lose because you can leave either 9 or 8. 6, 9, 12 I can't win. I can't win any multiple of 3. On other multiples of 5' — I can't carry it out. Let's go on." Here the subject was asked how he would draw from 59 beads. "An odd number," he said. "This will be merely a guess. I should judge that since it is an odd number, I could win. I would try to leave it odd all the way. I would take 2 and the other fellow couldn't win if I watched my draws. I think he couldn't win if I watched my draws. I think he couldn't win multiples of 3' — I would reduce him to midtiples of ^!" Even after the solution of an entire series most of the sub- jects showed but slight tendency towards or capacity for antici- patory analysis in the attack upon new series. Twelve of the fourteen subjects in Group I began the attack upon Series 1-3 in exactly the same manner as that upon the first series; that is, they began to draw without delay and only after some time at- tempted to formulate a statement for the series. The effect of their Avork upon the preceding series was very noticeable, how- ever, in the readiness with which some subjects took note of the fact that 4 could not be won, and in their alertness for other critical numbers. HIGHER MENTAL PROCESSES IN LEARNING 6i Subject i said at the beginning of Series 1-3: "I am going to try to reduce this trial and error to a minimum. With i or 2 (i.e., in Series 1-2) the smallest combination was 3; i, 2, or 3 is a more difficult proposition. I imagine I could sit down and work it out." He drew 2 and the experimenter took the remain- ing 3. "Oh, I forgot," he said "that there were three possi- bilities. I will try all the possibilities now and see if you can get me." From this point manipulation and analysis progressed together. This failure of one of the most successful subjects, certainly the one who was most inclined towards anticipatory analysis, shows the difficulty of such analysis in the early stages of acquaintance with the elements of the problem. His progress had been, however, very rapid in the first series so that time had not permitted a very thorough stamping in of the elements, and he had altogether failed to notice some of the important elements of the series. When presented with Series 1-3, Subject xiv began to draw after slight hesitation but said immediately after the first trial : "Now I think I can get this right off. This time you are going to let me win every fourth one; that is, on the fourth you won't let me win and on the fifth you will let me win, provided I start with I. On 6 I should have to start with 2, and with 3 on 7. (And on the 8th?) I wouldn't win 8." This subject quickly solved two more series of this order without a draw. This exceptional case may well be due to the fact that Subject xiv made a more thorough analysis of Series 1-2 than was ef- fected by any other subject. Also by a great deal of shifting of attention from one set of elements to another during 367 trials occupying 8173 seconds, as compared with an average of 204 trials in 4943 seconds for all subjects of Group I, he had prob- ably succeeded in stamping in these elements more thoroughly than had been possible for most of the subjects. With less than half as many trials as xiv required for Series 1-2, Subject i solved in the neighborhood of thirty series of problems and gave satisfactory generalizations for all problems included in the experiment. It is clear, therefore, that the insight shown here b}^ Subject xiv cannot be regarded as a case of ideational analy- I 62 JOHN C. PETERSON sis without a fair acquaintance with the elements of the situation. It required the solution of only one or two additional series of problems to give this control of all series of the first order (i.e., all series in which L is equal to i) to nearly all of the sub- jects, and all succeeded in getting a general solution after only a few more series. But no subject solved Series 2-3 without manipulation of the beads. The solution of fewer series was re- quired, however, for the mastery of the second order of series than for the first order. Likewise the mastery of all series of the third order required actual work on fewer series than that of either of the preceding orders, and usually resulted in the mastery of all series in which the numbers between H and L might be drawn. This transfer of power from series to series was due largely to the easy recognition of certain elements of the new series, which were identical with familiar elements of old ones, thus obviating in part the necessity for new analytic activity. The transfer of power was further facilitated by gen- eralizations upon familiar elements, which were easily applicable to new series. These factors of familiarization and generaliza- tion became more prominent as the work progressed, but in series where some new elements were introduced there was still a ncessity for genuine analytic learning. Much of this analysis continued to follow or accompany manipulation, but a gradually increasing proportion was performed in ideational terms, largely by the sort of ideational trial and error already described. Thus with increasing acquaintance with the materials we find a con- tinual receding of activity from overt trial-and-error manipula- tion to a very similar sort of process carried on in ideational terms, which in turn gives way to general ideas that have evolved gradually by the bringing together of similar elements in asso- ciation with general symbols. In the later series all three of these factors were almost constantly in operation. 4. Summary After the range of perceptual solutions was passed various elements of the problematic situation were gradually abstracted and associated with verbal symbols. The type or combination HIGHER MENTAL PROCESSES IN LEARNING 63 of types of elements so abstracted differed largely for different subjects and in different periods of the work of the same sub- ject. The type of elements abstracted was shown to vary in ac- cordance with the speed of reaction. Frequency of reaction to various elements and their nearness to a goal, were shown to be closely correlated with the order of their abstraction. Large differences in the degree of explicitness of analysis were ob- served, and there was found to be a somewhat gradual develop- ment from vague to clear and explicit states of analysis. The extent of analysis varied with different subjects and at different stages of the learning process, but analysis usually occurred first in isolated spots and spread from these to other portions of the series. The time relations of ideational analysis and manipula- tion varied greatly. In general, manipulation preceded analysis in the first few series, and persisted throughout the greater por- tion of the learning process as an important method of procedure. Gradually, however, as acquaintance with the elements of the situation grew, overt trial and error was replaced by a very similar sort of ideational manipulation which, in turn, tended to give way to general ideas. E. Generalization In the preceding section attention was directed principally to the abstraction of the elements of the problems presented for so- lution. As was shown in several instances, however, such ele- ments as are abstracted do not remain in their early state of rela- tive isolation during the entire course of analysis, but tend to combine into higher units which become associated with appro- priate symbols and take on general meanings. This process of generalization and the resulting general ideas will be the subject of discussion in the present section. I. Relative Absence of Generalisation in the Perceptual Stage Mention has already been made of the apparent lack of gener- alization during the period of perceptual solutions. Practically no comments were made during this period, which would seem to indicate any attempts at generalization. Moreover, when a 64 JOHN C. PETERSON subject did finally attempt to generalize, those numbers which are well within the range of perceptual control were rarely taken into account, though they had been more often repeated than any of the higher numbers. Only two of the fourteen subjects of Group I noticed, with sufficient clearness to mention the fact, that 3 is a critical number, until attention was directed to it by their final generalizations which were based upon higher num- bers. Some of the subjects were very much surprised that this fact had so long escaped their attention. The first important landmark mentioned by eleven of the subjects of this group was 6, and this was generally mentioned not earlier than the last trial on g beads, sometimes not until much later. But this failure to recognize the status of lower numbers in the abstract does not preclude recognition of the critical or non- critical character of the concrete situations for which they stand. In fact, the situation which is represented by the critial number 3, was almost invariably the first to be reognized in the concrete as critical. This fact is shown in Table VIII where the realiza- tion of the impossibility of winning 3 is shown to have been the first step taken by ten of the twelve subjects of Group III in the abstraction of critical numbers. Table IX, however, shows that 3 was explicitly mentioned as a critical number before 6 was so mentioned, by only 2 of the twelve subjects. The failure of sub- jects to utilize numbers below 6 along with higher numbers as a basis for generalization, is not, therefore, due to the difificulty of recognizing their status in the concrete but rather to the ease of such recognition, which made it unnecessary to associate these situations with verbal symbols. 2. Development of the Concept of the Critical Number Though the concept of the critical number seems to be a very simple affair, its development required the expenditure of a con- siderable amount of time and effort on the part of the subject. To facilitate the description of this development we may divide it into the following seven stages : Stage A. — ^The first appearance in explicit form of the criti- cal-number idea was at the point where the subject discovered HIGHER MENTAL PROCESSES IN LEARNING 65 that he could not win 6 beads. But this was straightway forgot- ten in most cases, or at any rate failed to function in slightly new situations. Similarly, the impossibility of winning 9, 12, etc. was later discovered without a full realization of the signi- ficance of these numbers at the time. This first discovery of the impossibility of winning a number constitutes the first stage in the growth of the concept of the critical number. Critical num- bers in this stage stand out in comparative isolation. Stage B. — Since the subject was required to draw first in every trial, it was impossible to compel him to draw from a given critical number in any trial in which the inital number of beads was only i greater than the crtical number in question. There- fore during the solution of the next higher number in the series the recently discovered critical number was usually forgotten. Its re-discovery after more or less delay constitutes the second stage in the development of the concept. Here the individual critical number is recognized as an element of all problems of the series; e.g., 6 is recognized as critical regardless of what may have been the initial number of beads presented in the trial. Stage C. — At this stage the relationship between two or more critical numbers in Stage B, or perhaps in Stages A and B, was discovered; as, 'T cannot win 9 because you can always reduce me to 6"; or, 'T cannot win 12 because I lose 6, and 12 is com- posed of two 6's," etc. Here the critical numbers are not only recognized as critical in all problems of the series, but they are associated with each other in more or less definite relations. Stage D. — The fourth stage consists in the discovery of the fact that the subject cannot win any multiple of 3. All critical numbers are recognized as critical and all are associated with a common symbol. Stage E. — The foregoing stages all deal with the critical num- bers as related to the subject's recognition of the possibility of success. A somewhat similar development may be observed in his utilization of the critical-number idea as a means of control in the achievement of success. Thus it often happened that a subject discovered the possibility of winning by reducing the number of beads to 6 and so forcing the experimenter to draw 66 JOHN C. PETERSON from it, long" before learning that the same procedure applied to the higher critical numbers would bring the same results. In this stage one particular number is recognized as critical for the one zi'ho draws first. Stage F.- -In Stage F two or more critical numbers are recog- nized as having- this broader significance. Stage G. -Here all critical numbers are recognized as critical for the one who draws first. It must not be supposed that learning- progressed with perfect regularity from stage to stage in exactly the foregoing- order, or that all of the stages were distinguishable in the progress of every subject. They do, however, occur separately in the ma- jority of cases, and in some instances in the exact order named. The following comments of subjects will serve to illustrate this development. They are all taken from Series 1-2. The stage of development of the concept, as inferred from each comment, is indicated in parentheses. The portion of the series in which the comment occurred is in each case indicated at the left. 12b, 12, for example, signifies that the first comment occurred during the twelfth trial on 12 beads. Subject vii 12b, 12: "You alwa3^s manage to leave me 6." (B) " 26: "I can't win 12 because you can always reduce it to 6; 12 is two 6's." (C) 15b, 11: "You reduce it to 6 every time." (B) " 22: "You always manage to get to 6; I don't know how I am going to prevent you." (B) " 24: "There are 12 left; I can't do much with that." (B) " 24: "There are 9 left; I haven't been able to win any 9's." (B) i8b, 8: "Evidently this is one that you don't expect me to take because you throw it into the 6's." (C) " 9: "Your idea is to balance my moves so as to make it 6; and 9 is just as good for you." (B) " IS : "You draw 2 when I draw i and i when I draw 2, so that we reduce it by 3's ; i.e., we are drawing them out by multiples of 3." (C) " 16: "We are drawing right down the multiples of 3." (C) " 17: "This is a multiple of 6; I can't win from 6 so I can't win it." (C) 2ob, I : "The subject draws two and is asked why. "I have no reason," he replies. E then draws 2 and the subject takes i, explaining: "I draw that way to keep it on multiples of 5 rather than on multiples of 3, and so to avoid 12, 9, and 6." (C) HIGHER MENTAL PROCESSES IN LEARNING 67 " 2: After drawing 2 S says: ''I can't win with 18." (B) (E?) 22b, 9: "If I have 8 beads and take i, the other fellow wins. If I take 2 — Oh yes, I see why I win: because I leave 6. (E) ... I can't win .any multiple of 3. ... I can't carry it out; let us go on." (D) The subject is here asked how he would draw from 59 beads. After some irrelevant speculations he concludes : "I would take 2 and the other fellow couldn't win if I watched my draws. I think I would reduce him to multiples of 3." (G) Subject xiii "It depends on which draws first." (E?) "I can't win when we get to 6. I never have beaten on 6." (B) "I couldn't w^in 6 ; I don't believe I can win 12. I have been trying to see how to prevent you from leaving me 6. Double it and it is the same. It would be the same on all multiples of 6." (C) "I can't beat you on this ; it goes by 3's. / have forgotten whether I beat you on 3. 6 is the first I fell down on, isn't it? I don't know about 9." (D) "I can't beat you on 16 — Oh no, it was 18 I lost. (B) If it is 20 and I draw first, I win; but if you draw first, I lose. Let me see, I didn't beat you on 18. If you draw first on 18, I win." (F) "I don't believe I can beat 18, 12, or 9, or 6. It must be multiples of 6 I can't win." (C) (D?) "When I get 26, I see that by drawing 2 first I can give you 24, and I can therefore win." (F) "I don't believe I can win. You make me draw on 24 no matter how I move. It must be multiples of 3, i.e., 3, 6, 9, 12, 15, 18, etc. That's what it is! (D) It depends on who draws first. There are certain numbers that the one who draws first can win and others that his opponent can win. (G) ... After your first draw, i.e., after you get it to that number, draw opposites." The subject is here asked how he would draw from 59 beads and replies : "I would take i and, provided you take 2 every time — let me see, there may be another element there. I must draw i first on the odd numbers which I can win and 2 on the even numbers that I can win. But I am not certain yet." 28b, 3 : "It looks like it is even numbers that I draw i on ; and on odd numbers I draw 2 perhaps." 29b, i: "That's the way it is. 3's I can't win, and multiples of 3. (D) 4 and 5 I win ; 7 and 8 I win ; 10 and 12 I win, etc. So on even numbers I draw i and on odd numbers I draw 2, and the oppo- site after the first draw." 31b, i: "My scheme didn't work!" " 2 : "I have missed something there." " 3 : "It is a question of making you draw first on the multiples of 3. I didn't draw 2 here so that was wrong (i.e., his idea that he must draw 2 from odd numbers, etc.). It is a question of what to draw so as to make you draw first on a multiple of 3." (G) lob. 6 12b, 30 36 15b, 6 20b, 2 24b, 2 26b, 2 27b, 2 68 JOHN C. PETERSON Subject xii gb, o: "The first I couldn't win was 6, wasn't it? This is 9." " 8: "Is it possible that with 6 or multiples of 6, such as 9 and 12, one can't get any results? that is, on all multiples of 3? (D) ... In all cases of multiples of 3 I am going to assume that I can't get any results." 12b, 9: "I am positive I can't get this. I always begin with 6 (B). . . . I came back to the original hypothesis and thought of 6 as a multiple of 3." (D) S is here asked how he would draw from 59 beads. His reply shows that he has failed utterly to realize the significance of his idea of multiples of 3 for the achieve- ment of success: "Well, I would eliminate the beads until I got my first draw on 4 or 5." 14b, 6: "When there are 6 and it is my turn to draw, you can always win." (B) The subject does not realize yet that 9 and 12 are also critical numbers. "You can always make up the deficiency so as to take away an even number and leave 6." (B) "I am going to try out that hypothesis again, that any number which is a multiple of 3 precludes any possibility of success for me." (D) "Whenever a number is a multiple of 3, it is useless for me to try it." (D) "When it gets to 9, I lose. (B). In this case I am working on the first 7 out of 16." "It seems to revolve about the start." "No matter how I move you can always bring me to 9. I lose 12 too." (B) "I'll bet a cow I can't win 18." S is here asked how he would draw from 59 beads, and replies : "I couldn't get it." "I lose multiples of 3." (D). When asked how he would draw from 61 beads, he says: "I would draw 2. (Why 2?) To get to 59 and bridge over 60. (E?) 22b, I : "Well, I lose and I thought I had it cinched." The subject later stated that he took 2 beads at the first draw in this trial in order to "bridge over" 21. 22b, 2 : "I see my mistake. I took 2 and you 2 leaving 18, another multi- ple of 3-" (D) 24b, o: "I'll pass it up." (D) 25b, I : "That was foolish. I took 2 and allowed you to reduce it 21, a multiple of 3, and I lose." (D) 27b, o: "I'll pass it up." (D) 30b, o: "I'll pass it up." (D). The subject was asked how he would draw from 82 beads. He replied: "I would take i. No, I would take 2, because that would throw me off the multiple of 3 again." (E?) From 67 beads he said he would take i "to avoid 66, a multiple of 3." (D). This subject did not advance 14b, 9 15b, I 15b, 7 i6b. 4 (( 16 *' 37 i8b, 2ib, I 9b, 5 nb, " 4 11 6 HIGHER MENTAL PROCESSES IN LEARNING 6g beyond Stage D in Series i-2, although he seemed at the point of making the larger generalizations at several times. Subject i "I cannot beat you at g. No matter how I draw you can take such a number as to reduce me to 6 which I can't win." (B) "Now if I knew what my winning numbers were for 7, 8, 9, and 4, and 5, I think I could win." "Did I win 6 before? I don't think so." (B) "You reduce it to 6 no matter how I draw, and 6 is impossi- ble." (B) " 7: "The first party can't solve 11 if his opponent reduces the num- ber to 6 (E). I have it. . . . If I can prevent you from re- ducing the number to 6, I can win." 12b, 3 : "It reduces itself to this : Can I prevent you from reducing it to 6? You can leave 6 irrespective of what my move is. Twelve is composed of two 6's." (C) 13b, I : "Now I want to prevent you from getting 6 or 12, so I must necessarily start with 2 (E?). (S draws 2 and E i) Nine beads. There you have got me because I take i and you 2 and leave me 6, or I take 2 and you i and leave me 6." (C) " 2: "I'll take the other chance and start with i. No there is no use because I would be leaving you 12. Well, as far as I can see my only chance is to start with 2, but even then you can reduce me to 6 if I am not mistaken. " 3: "It seems to me that I lose. If I take i, you have 12 and win; if I take 2, you can bring it down to 9 and hence to 6 (C). If you can reduce me to 6 or 12 no matter how I draw, why I lose, that is certain. Let's see if I can beat you on 6. . . . No I can't beat you on 6, that is evident. I have an idea that it is multiples of 3 you win." (D) When asked how he would draw from 59 beads, he replies : "56 is the nearest multiple of 3 to 59. If you could reduce it to 57, you would have me. There again I can't move so as to prevent you. If I took i, you would take I and leave me 57. If I took 2 — Oh yes ! If I moved 2 and followed it out consistently, I think I would have you beaten." (F, G?) The subject here stated that he had a glimpse of the right procedure at 11 beads, but that some distraction or other caused him to lose sight of it for a time. Subject viii lib, 20: "When there are 4 and it is E's draw, I can get it. My problem is to get it reduced to 4 when it is his draw. (E?) 12b, 18 : "The idea strikes me that this cannot be gotten, and also that it is a multiple of 6 that cannot be beaten." (C) 13b, 18: "First I wanted to get 4 beads with my draw; I later found that I would have to work not from 4 but from 5. . . . Later I 70 JOHN C. PETERSON changed it to 7. I thought I could work down to 4 from 7, and then I began to work from 9. I then found that I would have to begin to check off from the total; that is, if there are 13 beads, I draw i and then count 12. . . . My object was to get g beads when it mas my draw." (E?) " ?,2'- "I counted back to 11 and worked it out from 11. Here / tried to get my opponent down to 6 instead of myself. (E) 14b, 12 : "I forgot that I was trying to keep from getting 6, and conse- quently for a good while I was trying to get 6, and not being able to win caused me a good deal of confusion. I have had this mixed up for some time." 15b, 7: "I am now working on 9 instead of 6." (F) i6b, 4: "I think that the point of the game is that the one who wins has to have the other one draw on some multiple of 3." (G) The foregoing comments of subjects are merely milestones in their progress through Series 1-2, yet if the reader will refer to the position in the series where each comment belongs, he will appreciate the extreme slowness with which the simple meanings often developed and the lack of uniformity in their attachment to various elements of the series. It has already been shown that the lower critical numbers are usually wholly neglected for a time after their first discovery and that they are later re-dis- covered in about the same order as that of their first discovery. In general, each new acquisition of meaning — each advance to a higher stage in the foregoing classification^ — began with those critical numbers above the range of purely perceptual solutions, which had been most often repeated, and spread to the higher critical numbers much in the order of their frequency of occur- rence in the work of the subject. Thus 6 was the first critical number, usually, to be re-discovered, and also the first to be definitely associated with a higher critical number. So also 6 and the immediately following critical numbers were usually the elements between which the multiple-of-three relationship was first apprehended though higher critical numbers were at the time known to the subject. Likewise 6 was usually the first critical number to advance to Stage E, and the higher critical numbers followed roughly in the order of their magnitude. It has already been shown that the reactions of any subject to a lower critical number were invariably more frequent than his reactions to a higher one. Thus in the elevation of critical num- HIGHER MENTAL PROCESSES IN LEARNING 71 bers to higher levels of meaning frequency of repetition and nearness to a goal appear to be important factors, just as in the preceding section they were shown to be factors in the direc- tion of attention to the critical numbers. In the elevation of critical numbers to higher levels of mean- ing there was, however, considerable deviation from the order of their frequency of repetition. Thus 12 was often taken up into Stage C upon its first discovery long before 9 was re- discovered. This was clearly due to the simple multiple-relation existing between 12 and 6, by which the associations already built up about the latter were made to attach immediately to the former. ^'^ This spread of meaning through the agency of pre- viously acquired associations might well be expected since the elements of our problems were not only familiar to the subjects, but were also closely related in past experience. But this calling up of familiar concepts by association is itself perhaps dependent in large measure on the frequency of their previous repetition in association with some element of the present situation. These irregularities in the process of learning did not greatly alter the general character of the learner's progress in the early stages of learning. The generalization of the formula for Series 1-2 into a suitable formula for all series of the first order^*^ required from I to 6 further series for its completion; and the generalization of the latter formula so as to make it cover all continuous series"^ required from 2 to 10 additional series. It is therefore evident, notwithstanding frequent rapid advances by means of familiar concepts which were brought in by means of association, that the development of meaning was a rather slow and fluctuating affair, presenting some striking parallels to the formation of sensori-motor co-ordinations. Another fact of interest, which is exemplified in some of the comments listed above, is the fleeting and unstable character o£ meanings in the early stages of their development. Individual critical numbers and generalizations based upon them were given 29 For example, see the foregoing comments under Subject i, 12b, 3; Subject vii, 12b, 26; and Subject xiii, 12b, 36. 30 Series in which the value of L is i. 31 Series in which all numbers between L and H may be drawn. 72 JOHN C. PETERSON to slipping- out of mind and to various distortions of meaning, which clearly reveal the weakness or utter lack of association with other elements of the situation. Thus Subject i, while working upon his seventh trial with 1 1 beads, appears to have had a fleeting insight into the significance of critical numbers as a means of actual control. Several other subjects spoke of a similar tendency for promising ideas to vanish during the at- tempt to formulate or apply them. Sometimes a critical number would be recognized and remembered well enough, but with an inversion of meaning somewhat comparable to the reversals of perspective in ambiguous drawings. Thus Subject xiv (lib, 29) said: "Whenever I win, I always have to leave 3 beads; that is, there must be 3 beads before I take my last draw." This sort of inversion of meaning often occurred shortly before the appearance of Stage E in the development of the concept of the critical number. After trying for some time at this point to get to draw from 6 beads subject viii said: "I forgot I was try- ing to keep from getting 6, and consequently for a good while I was trying to get 6." Another instance is found in the com- ments of Subject i quoted above (13b, i). Similarly the idea of the significance of multiples of 3 often dawned and was lost, later to reappear, perhaps several times, before finally being ap- plied in a thoroughgoing way with a full realization of its sig- nificance. Examples of the early instability of this idea are found in the comments listed above as follows: vii, i8b-22b; xii, 9b-3ob, and xiii, I5b-27b. In tracing the further consolidation of the various elements of the concept of the critical number through succeeding series of problems, it is found that they are often thrown out of co-ordination by the appearance of some new element which requires adjustment and further generalization, or by the recall of some old hypothesis which has not been satis- factorily disposed of. The interference of old and erroneous hypotheses with the formation and stability of correct generalizations, is worthy of further attention. The comments of Subject xiii, quoted above (27b, 2), show how the idea that odd and even numbers were possessed of special significance, came in to confuse him after HIGHER MENTAL PROCESSES IN LEARNING 73 he had made a perfectly good generalization. This subject had played somewhat with the idea of odd and even numbers much earlier in the series, but during the greater portion of the work he seems entirely to have dismissed the idea, making no mention of it at all. The very common interference of erroneous hy- potheses and irrelevant ideas which have once received attention without being definitely settled, will be further illustrated in the following paragraphs on trial and error in generalization. 3. Random Hypotheses Having observed the slow growth of meanings and their in- stability in the early stages of development, we may turn briefly to the manner in which the elements of a situation were brought together and combined into adequate generalizations. It rarely happened in the early members of a group of series that the essential elements were directly sought out and held in mind without attempts at generalization until sufficient data were at hand to insure correct inference at once. This type of procedure was followed apparently by one of our subjects in Series 1-2, but was not carried out consistently in all later series. The work of all other subjects in the first series showed some loss or omis- sion of essential data with consequent erroneous generalizations. These random hypotheses were often numerous and far afield. The rate of progress depended very much upon the thoroughness with which they were followed up and tested. Because of the unusual fullness of his comments the work of Subject xiv will be taken to illustrate the random efforts at generalization in the first series. The place in the series at which each comment oc- curred will be indicated in the manner already familiar to the reader. 9b, 3 : "You can always win if you draw tlie same as I." lib, 20: "If, with an odd number, you take an even number, I can always beat you if I work it right." " 36: "Whenever I get a combination by which I can win, thereafter I must reverse when you reverse in order to win." 15b, 6 : "I win some odds and some evens." " 7: "I draw another conclusion, that my principle is not to follow you." 74 « 8 " 9 I7b, I (( 9 i8b, 2 2ib, 7 22b, 2 25b, I " 12 13: 31b, 12: 32b i: 34b, 21 : 36b, I : 37b, 4: 40b, I : 40b, 5 : 41b, 4: 42b, I : 43b, 4: JOHN C. PETERSON "Neither is it to draw opposite from you." "Neither is it to alternate with you." "I can always beat you on even numbers ; I don't think I can beat you on any odds." "There must always be 3 beads when you draw last." "As long as you draw opposite to me I can't win 18." "You win odd numbers by taking the opposite of my draw." "The principle of alternation will not work with the even numbers." "Even and odd numbers is not the criterion." "There must be some relation between the total number of beads and the number of draws." "When I win, the critical draw is the second or no later." For some time after this the subject's attention was absorbed in the number of beads he obtained in each trial. "I haven't kept track of the numbers I won and those I lost." "To win an odd number of beads I draw an odd number." Here again the subject's attention is absorbed for some time in the number of draws and of beads obtained in each trial. "2 from 34 leaves 32. Now if I draw i always and you 2, I would leave even numbers." ( !) "It seems that I can't win every third time." "Of course I know you are going to take 2 all the time and I'll take I, but I don't know how it will come out." "I will beat you but I can't say how with any degree of certainty. I'll beat you twice and then you will beat me once. I notice that the numbers upon vi^hich you beat me are always divisible by 3." The subject is asked how he would draw from 59 beads and re- plies that he cannot tell. He then figures slowly but correctly to 59, after which he is asked how he would draw from 1004 beads. "I couldn't figure that up," he said, "I may win even numbers and you odd numbers. No, I won 27- It may be numbers that are divisible by 2. No, I don't get it. . . . It might be that you allow me to beat you on all numbers that are divisible by a certain number, but it isn't by 2 and it isn't by 3 or 4. ... I have never figured out this drawing business ; I do it wholly ac- cording to my feeling. But I watch the way you draw on your second draw and then take the opposite. ... I am not sure what is the criterion, whether it is your first two draws or my first two." "I ought to have noticed that 3-matter long before because 1 noticed so often that we drew i and 2." "I knew I could beat; I had 3 beads after my second draw." "I can't beat you on 42, it is divisible by 3. I know what you can beat me on but I don't know why." "They go in pairs. I can beat you on 43 if I take 2 the first time (He loses). You haven't arbitrarily made up your mind not to let me beat you the first time, have you?" HIGHER MENTAL PROCESSES IN LEARNING 75 " 8 : "When I win I guess I must draw 2 the first time ; but I can't win that way, because I've tried it and failed. . . . But if you hap- pen to be drawing the opposite of what I draw the second time, I'll beat you. . . . I'll take 2 and then you'll let me beat you." " g: "Well, I thought all I had to do to beat you the second time was to reverse what I did the first. I see, evidently I have to start out with the same number in order to beat you on a given num- ber." " 10 : "Three from 43 leaves 40. All right, I have it. I have an even number; I'll take 2." " 12: "I had 15 draws — Oh yes, that is all right. I was thinking I could get out by leaving an odd number of beads." 44b, i: The subject takes 2 and wins. "That confirms my rule," he says. 4Sb, o : "I can't win 45." 46b, I : "I can win 46. I'll have to start with 2. Yes, I must or I can't win, I believe." " 3 : S draws i and wins. "On the numbers on which I can beat you," he says, "I have to start out with i." 47b, I : "I can beat you on this. . . . Well, that rule didn't work (i.e., that S must always draw i first to win). Well, maybe the rule is that one time I draw i in order to win and the next time I draw 2 to win." 48b, : "Did I win 47 by drawing i ? No, I won it by taking 2. Then I can't win 48, of course." 49b, I : "I'll draw here on the hypothesis that on the first one after the number I lose, I can win by taking i, and on the second after it I must start with 2 to win." . " 2: "It worked. Now this time I'll start with 2 (S loses). No, that isn't right." "Now I'll start with 2 and beat you." Here the subject is taken back to the beginning of the series. (S takes 2 and loses) "Now that's funny." "I'll take I to start with." S takes 2 and loses. "I've got to start out with i on 7," he ob- serves, "and there is a lot of other numbers en which I have found that I must start with i." "I'll try 2 just to see. No, I'll try i." "Why isn't the rule that on the even numbers I've got to start with 2?" "Yes, that's the rule." "Oh, I can't beat you on 9, and on 10 I've got to start with 2." "Oh! I mustn't take 2 the first time; I must take i." "I believe my hypothesis was right about there being first the i- draw and then the 2-draw. On 4 I had to draw — I forget just how I did draw on 4. On 10 I drew i and on 11 I took 2. 1 think I must take 2 on 11." 12b, o: "I can't win 12; it is a multiple of 3." 50b, I 7b, I " 2 3 8b, I <( 3 « 4 9b, I lob, 2 lib, 2 76 JOHN C. PETERSON 15b, o: "I can't get it." i6b, o: "Now I'll take i." 17b, I : "I'll draw 2." The subject was then asked how he would draw from 59 beads. "Why," he said, "it goes this way. I draw I first, then you draw 2 and I keep on with i. Then I beat you. The next timt^^ I start with 2 and you start with i, and I keep on with 2 and win. But I can't tell how a given nuiriber must be drawn on. I just know that it goes along i and 2. What is this number? (17) Well, on 17 I had to draw 2 didn't I? Well, I should guess that on 59 I should start with 2, and then you would draw i and I would continue with 2. (Why?) The only reason I say that is that it is just before 60, and 60 is a number that you beat me on just as you do on 18. Seventeen is just be- fore 18 and 59 is just before 60." In reply to a question as to how he would draw from 43 beads the subject said: "I would draw I on 43, because on 42 I couldn't beat you, and with the numbers following those on which I can't beat you I must al- ways start out on i." The random making of hypotheses upon the basis of momen- tary suggestions is here too evident to require further comment. Examples might be produced ahuost indefinitely from Series 1-2. A few examples from higher series and from the efforts of sub- jects to get a generalization of universal application for all series will perhaps be worth while. The following evidences of random hypothesis-making are taken from the record of Subject viii. In attacking Series 2-3 this subject fluctuated a good deal among a number of hypothe- ses, as is shown in his comments : 7b, 2 : "I think now that I will win on the basis of 7, i.e., I Avill try al- ways to draw so that there will be 7 left to draw from." 8b, 2 : "It appears that I can win on a basis of 5." 9b, 2 : "It appears that 7 doesn't work." " 3 : "It doesn't work. I forgot about the i." " 6: "It will work on a basis of 6 as well as of 5, i.e., on multiples of both 3 and 2, I believe. But I am not certain." 32 Note the possibility of confusion in the phrase, "next time." It may mean the next trial upon the same number or it may mean the next higher number. These two meanings fluctuate in the mind of the subject, owing perhaps to the ambiguous wording. The same ambiguous wording appears in the comments of this subject at 47b, i (p. 94), though here it is clear that the subject has the correct idea in mind. But it is the wording rather than the idea, that functions in the immediately following problems (47b, i to 7b, 3) in such a manner as to produce a number of errors. Several other instances were found of ambiguous wording which led other subjects into confusion and error. HIGHER MENTAL PROCESSES IN LEARNING 77 15b, 3 : "I am a little in doubt about the multiple of 3 and 2. It doesn't seem to work." 20b, 3 : "Twenty can't be worked. I didn't think of that multiple until I had worked for a long time. It is a multiple of 5." 2ib, I : "Sometimes the multiple of 3 works and sometimes it doesn't. Either a multiple of 2 or a multiple of 3." 2ib, s : "At ifirst I thought I could work on the basis of 2 or 3, but one of the multiples of 3 didn't come out right. I almost think now that I made a mistake and that I can't work on this basis." 26b, 2: "That hypothesis couldn't have been right because 12 is a multiple of both 2 and 3 (S won 12 repeatedly), so I have given that up." The subject was tormented by these and numerous other hy- potheses until he had solved all of the numbers of this series from 6 to 5 1 and then repeated the process up to 29 beads. Perhaps the most striking exhibition of this sort of random effort was found in the attempts of subjects to formulate a gen- eral solution for all series. Here it was usually necessary to give them a list of the critical numbers of all discontinuous series in order to refresh their memories. ^^ All of the subjects had made some discoveries, as for example, that all multiples of L + H are critical numbers. The generality of some of these earlier formulatonis was readily recognized if not already known. But some were not universal in their application, and here is where the effort began to be more sporadic. The subject would usu- ally begin by restating one of his old generalizations or by for- mulating a new one on the basis of one or two problems, and then go from series to series testing it out until a series was dis- covered to which it would not apply. Upon finding a refractory series he would stop and attempt to modify his view to suit the case in hand and then pass on to test it out upon other series. Often the subject would have to modify his generalization for every group of series which were in any marked degree different from the preceding ones, sometimes discarding the notion only to return to it later in his fumbling efforts to find a generaliza- tion which would be applicable to all series. Some subjects failed after working for hours, and most of the others succeeded in getting a general solution only by a tedious "fitting on" of ^3 Series in which only L and H may be drawn. For a list of these simi- lar to that presented to the subjects see page 7. 78 JOHN C. PETERSON numerous hypotheses suggested by individual series and often wholly at variance with conditions which had been met repeated- ly in other series. A better exhibition of random effort could not have been given by an animal in a problem box. Nor is it to be supposed that in the application of generaliza- tions to new situations we have an entire escape from trial and error procedure. Numerous errors and omissions of application point to the random character of efforts to apply generalizations. Certain hindrances to application and some sources of error are fairly apparent in the transition from the continuous to the dis- continuous series of problems and in passing from the first to the second series of the latter group. Three of the more important elements of their previous gen- eralizations which subjects brought over into Series i or 3 are (i) the critical-number concept, (2) the idea of the serial rela- tion of the critical numbers of the series, and (3) the knowl- edge of the fact that the common difference between successive terms of the various series of critical numbers is equal to L -)- H. No difficulty was experienced in applying (i) and (2) directly to Series i or 3. The trouble with the application of (3) arises from the fact that a new series of critical numbers is created by the restriction of the draws to L and H. The first term of this new series is 2 and the common difference is L -|- H as before. This new series of critical numbers should have given but little difficulty if the subjects had stuck to the old generalization for primary critical numbers and looked for an explanation for the secondary^^ critical numbers only. But there were too many pitfalls in the way. In the first place, the first critical number encountered was a secondary one. This refractory case imme- diately cast doubt upon the applicability of the old formula and lessened its chances of later consideration. Again, the secon- dary critical numbers fall into line with the primary ones in such a manner as to form a continuous series with a common dif- L-fH ference of . This is, of course, a coincidence peculiar to 2 2* See above, p. 8 for definition of these terms. HIGHER MENTAL PROCESSES IN LEARNING 79 this series and not common to all. Finally, the common differ- L + H , ence of is 2, in this particular case, making all even num- 2 bers critical. Thus the mind of the subject is led unawares away from the general principle built up in previous series and di- rectly into the rut of the old concepts of even and odd numbers. The power of these superficial relations and the familiar con- cepts aroused by them to divert the attention of subjects from the application of generalizations formed in previous series, is evident from the fact that twelve of the thirteen subjects who solved this series, stated their final formulae in terms of even numbers, though five of them had played with the old formula while working upon the early problems of the series. Some interesting errors of application were made by Subject viii in passing from Series i or 3 to Series i or 5. After having generalized correctly for Series i or 3 he began his attack upon the following series correctly enough by saying : "I would think right away that 6 is the lowest critical number and that all multi- ples of 6 are critical." But he quickly got off the track and began to believe that all multiples of 3 were critical. Later he sup- posed that the critical numbers were all multiples of 9, but finally changed his conjecture to multiples of 12. After obtaining a solution for this series he stated that the errors had arisen from his efforts to adapt the generalization of the preceding series — i.e., that "All multiples of 2 are critical" — to the present one. Examining the former series to find the basis of the distribution of critical numbers, he observed that 2 lies midway between i and 3, the L- and H-draws respectively, and straightway in- ferred that multiples of 3 were critical in Series i or 5 because 3 lies midway between the L- and the H-draw in this series. Find- ing this hypothesis to be incorrect he tried the sum of all the numbers between i and 5, i.e., 9, because it was observed that "2 constitutes all the numbers between i and 3. This failing, he "thought that a critical number had to be a multiple of all of the numbers between i and 5, i.e., a multiple of 12. One of these errors was repeated by this subject in passing from Series 2 or 6 to Series 2 or 10. In generalizing upon the 8o JOHN C. PETERSON former series he said : "Since about the beginning of the last series I have felt that it must go by multiples of 4, because 4 is intermediate between 2 and 6 and is a factor of both." Imme- diately afterward, when presented with Series 2 or 10, the sub- ject said without hesitation: "I should say the critical numbers are multiples of 6 and numbers which are i above those multi- ples; that is, I add 2 to 10 making 12 which I know is a critical number. Then I reduce to 6 — No, I didn't, did I? — as I had, in the other case, reduced 8 to 4." In generalizing upon Series 3 or 9 Subject ii said: "The criti- cal numbers are multiples of the difference between the high and the low draw." When presented with the next series, he said: "Immediately I take the difference between 4 and 12, which is 8. This formulation applies very well to all series in which H = 3L, but to no others. Although the subject did not explicitly say so, he undoubtedly carried this generalization over into Series i or 4 and Series i or 6 much to the detriment of his progress. While working upon 9 beads in Series i or 4 he decided that 3 was a critical number and said : "I feel that I am on the edge of some- thing here ... 3 I lose. It reduces to 3." Shortly afterwards he added : "I've been mistaken there. I lose 3 and 5. No, I win 3." After thinking this matter over for two minutes he continued : "I believe I can win 3, 6, and 9." A little later, after a short pause to determine how to draw from 9 beads, the subject said : "Now that's funny. I can win 9." The idea that multiples of 3 are critical numbers in this series seems finally to have been discarded at this point, but the generalization upon which it rested, — i.e., that the difference between successive criti- cal numbers is equal to H — L, — persisted into the next series and seems to have been the cause of considerable difficulty there. The idea, carried over from Series i or 3, i or 5, and i or y, that even numbers are critical, is also much in evidence in the records of Series i or 4 and i or 6. The ill effect of these gen- eralizations from earlier series upon the progress of the subject through Series i or 4 and i or 6, is evident in his records. Though in the speed of progress throughout the entire experi- ment this subject ranks second, he is tenth in rank in Series i HIGHER MENTAL PROCESSES IN LEARNING 8i or 4 and i or 6 combined. The other nine of the ten subjects who solved all of the series devoted only lo per cent of all their trials to these two series, whereas Subject ii devoted 32 per cent of all his trials in the entire experiment to the solution of them. It appears, therefore, that approximately two-thirds of the diffi- culty encountered by Subject ii in these two series, was due to the interference of certain generalizations carried over from pre- ceding series. ^° Many other examples of the difficulties attending the applica- tion of generalizations could be given but the principal types are perhaps sufficiently illustrated in the foregoing cases. The omis- sions and errors of application occurring in the records of all subjects are numerous enough to warrant the statement that, under the conditions of our experiment at least, the application of generalizations depends very much upon trial and error pro- cedure. It appears that any incompleteness of analysis, either in the old situation from which the generalization is evolved or in the new situation to which it is to be applied, is likely to render application difficult and perhaps erroneous. There is no apparent reason why difficulties of this nature should arise from the con- ditions of our experiment more readily than from the practical situations of life. 4. Summary Explicit generalization did not often occur until the number of beads presented was high enough to preclude the possibility of direct, perceptual foresight of the consequences of all possible draws, i.e., until the numbers presented were high enough to place a premium on the use of symbols in their solution. Num- 35 It is of course possible that other factors contributed to this retardation of progress. Fatigue may have affected the progress in Series i or 4. This series was completed shortly before noon when the experiment had been in progress approximately three hours. The subject stated that he was be- coming somewhat fatigued though not until after much difficulty had been encountered. However, the greater portion of the effort Vv'as expended upon Series i or 6 where fatigue conld not have been a factor. This series was begun after an intermission of one and a half hours, when the subject de- clared that he was thoroughly rested. 82 JOHN C. PETERSON bers lower than this were seldom taken into account even in later attempts at generalization. Several fairly distinct stages in the development of the criti- cal-number concept were found. Each advance to a higher level of meaning began usually with that critical number, above the range of perceptual solution, v^^hich had been reacted to most frequently, and affected the higher critical numbers largely in the order of the frequency of their repetition. This is also the order of their numerical and temporal nearness to the goal, i.e., the end of the trial. In the early stages of their development the meanings involved in the concept of the critical number v\rere found to be extremely unstable in character. The stability of these meanings gradually increased with continued reaction to the situations from which they were evolving. The selection of the essential elements of a series and their combination into adequate generalizations was effected mainly by means of trial inferences, or random hypotheses, which were made upon the basis of only a few cases — often only one — and tested out more or less persistently before final acceptance or re- jection by the subject. In the application of generalizations to situations which are in part new, much difficulty was experienced, and a consider- able amount of random modification often occurred before a successful adjustment to the requirements of the new situation could be made. False analogies arising from the observation of superficial relations often resulted in confusion and error. Sometimes the solution of new problems was much retarded by the attempted application of inadequate or irrelevant generaliza- tions formulated from the elements of earlier series. F. Transfer The term, transfer^ is used by Ruger broadly "to include the effect of any given experience on any subsequent one whether the effect results directly or by means of an idea, whether the transfer is one of method or of material, or of motor processes, and whether it is positive or negative."^" The term will be used 36 Op. Cit., p. 85. HIGHER MENTAL PROCESSES IN LEARNING 83 in the same broad sense in this discussion. This usage is becom- ing fairly common. Ruger's defense of it need not detain us. I. Degree of Transfer Time and other limitations have not permitted such experi- mental evaluation of the various problems as would be required for a highly accurate measurement of the transfer of the effects of learning from problem to problem. However, the facilitating and, sometimes, the inhibiting effects of earlier upon later efforts, are great enough to give a fair insight into the causes and con- ditions of transfer without a very precise determination of its amount. A rough estimate of the relative difficulty of various problems and series of problems can be made from some of the experimental data and other facts. Within a given series the higher problems are undoubtedly more difficult than the lower ones. If the subject's draws were determined by pure chance, two successive winnings would re- quire 8 trials upon 5 beads, 32 trials upon 8 beads, 128 upon II beads, and 512 upon 14 beads. This increasing difficulty is sufficient to conceal the effects of transfer in many instances, especially in the lower numbers of the series. Though no ex- perimental evaluation of the various problems of any series was attempted, the records of Group III when compared with those of Group II A show that the solution of 14 beads was consider- ably more difficult than that of lower numbers of the series. Group HA began with 4 beads and followed the same procedure as Group I. Group III followed the same procedure but began with 14 beads instead of 4. The number of trials required by each member of this group for the solution of 14 together with some data for comparison from Group IIA, are given in Table XII. The subjects of Group III required a greater number of trials for the solution of 14 beads than were required by those of Group IIA for the solution of the first seven problems of the series, and over half as many as were required by the latter group for the solution of the first ten problems. Assuming that the two groups were possessed of equal ability to solve problems 84 JOHN C. PETERSON Table XII Group IIA [otal No. i of trials Total No. of trials required \ tor the required for so- No. of trials required for solution of all lution of all solution of I. 4 problems from problems from beads 4 to 10 in- 4 to 13 in- inclusive inclusive Group IIA Group III Subj. A 32 48 Subj . I 8 " B 39 49 5 2 II " C 25 68 2 3 15 " D 56 88 4 4 27 " E 34 106 4 5 29 " F 59 84 4 6 35 " G 36 112 22 7 42 " H 74 97 25 8 50 " I 47 159 2 9 60 " J 39 92 32 10 II 12 63 180 Total 441 903 100 606 Average 44-1 90.3 lO.O 50.6 of this sort, the transfer from an average of 90.3 trials upon the first ten problems to the eleventh problem of the series was equal to 40.6 trials, or to 80.2 per cent of the average number of trials required for the solution of 14 beads when presented as the initial problem. Further profitable discussion of transfer from prob- lem to problem within a series must await a careful determina- tion of the difficulty of these problems. From logical analysis it would seem that all series of the same order are of approximately equal difficulty and that series of higher orders are somewhat more difficult than those of lower orders. ^^ Some experimental evidence of the substantial equality 2^ In Series 1-2 there are just four combinations of draws by which sub- jects can win non-critical numbers. Upon problems in which the initial number of beads is greater by i than a multiple of H -f- L the subject can win half of the trials by taking i bead at every draw. The other half he can win by taking i at his first draw and 2 at every draw thereafter through- out the trial. He can win one half of the trials upon problems in which the initial number is less by i than a multiple of H -f- L, by taking 2 beads at every draw. The other half of these problems he can win by taking 2 at his first draw and i thereafter throughout the trial. If L and H be substi- HIGHER MENTAL PROCESSES IN LEARNING 85 of difficulty of Series 1-2 and Series 1-3, is found in the records of Groups IIA and IIB, which are summarized in Table XIII. ^® The subjects are listed at the left in the first column. The num- ber of problems solved by each subject in each of the series and the number of trials and amount of time required for the solu- tion of the series, are given in the succeeding columns. tuted for i and 2, these four combinations of draws are effective for the solution of some problems in every series of the first order. But for all problems in which the initial number is removed from a multiple of H -]- L by an interval greater than i, a new combination of draws must be made by varying the first draw. In Series 1-3 a new combination is required for only those problems in which the initial number of beads is an odd multiple of 2; i.e., for one-third of the non-critical numbers of the series. In Series 1-4 it is required for half and in Series 1-5 for three-fifths of the non- critical numbers, etc. Moreover, the discovery of the successful combinations of draws is not favored at so early a point in the higher as in the lower series, owing to the magnitude of the H-draws which effect a reduction of small numbers to o before a sufficient number of repetitions has been made to impress the uniformity of response upon the mind of the subject. For the same reason the early discovery of the principle of drawing by opposites is less likely in the higher than in the lower series. Again, because of the longer inter- val between critical numbers in the higher series, a greater number of prob- lems must be solved in these series than in lower ones in order to reveal a given number of critical numbers upon which to base and with which to test generalizations. On the other hand, each critical number in the higher series is more firmly fixed in mind than those of the lower series by the more frequent reaction of subjects to it before the next higher critical number is presented. This would favor the more ready utilization of critical numbers in the higher series for purposes of generalization and so facilitate the solution of these series. This higher degree of learning is the only factor revealed by analysis which clearly favors the more rapid solution of higher than of lower series of the same order. How important these various factors are cannot be stated definitely without further experimentation. It does not seem un- reasonable, however, on the basis of this analysis, to assume that the higher series are at least as difficult as lower series of the same order. Critical numbers are relatively more numerous in higher than in lower orders of series, and their relations to one another are more complex. It is practically certain, therefore, that a series of a higher order is more diffi- cult than one of a lower order. 38 The subjects of these groups were members of a class in educational psychology, whose class work had been observed by the writer for from three to seven months. The division into groups was made upon the basis of reasoning ability as judged by the writer. 86 JOHN C. PETERSON Table XIII Group IIA Series 1-2 Subj. A B C D E F G H I 7 No. Problems 10 15 IS 16 21 74 27 76 81 162 Trials 48 62 85 112 220 300 332 389 617 823 Time (in seconds) 1833 1555 1840 4210 6450 19133 7440 13663 15988 18572 No. Problems 12 Series 1-3 Trials 37 18 13 24 15 31 o 16 17 524 Total Average 321 32.1 2196 219.6 81136 8113.6 71 7.1 158 15.8 Time 1130 230 248 685 130 1545 o 312 162 10260 Total 497 2988 90684 153 695 14702 Average 49-7 No. 298.8 Series i 9068.4 Group IIB -3 Time 15-3 No. 69.5 Series 1-2 1470.2 Subj. Problems Trials (in seconds) Problems Trials Time a 16 48 2545 9 14 300 b 16 50 2062 12 29 763 c 19 56 4545 9 IS 407 d 16 69 2640 4 10 135 e 17 95 2090 3 7 70 f 23 179 9585 I 2 120 g 38 210 8455 6 14 270 h 26 337 10131 12 31 532 i 52 376 18648 3 6 150 J 98 776 20435 12 30 1054 3801 380.1 The average number of trials per subject in Series 1-2 is 298.8, P.E., 51.9. In Series 1-3 the average number of trials per subject is 219.6, P.E., 46.3. The difference between these aver- ages is 79.2, P.E., 69.5. In so far, therefore, as the calculation of unreliability means anything when based upon so few and so variable data as those at our command, these data support the view that there is no substantial difference in the difficulty of dif- ferent series of the same order. HIGHER MENTAL PROCESSES IN LEARNING 87 In terms of the number of trials required to solve the series, the transfer from Series 1-2 to Series 1-3 is yG.y per cent. From Series 1-3 to Series 1-2 it is 92.8 per cent. The lower percentage of transfer in the former case is due almost wholly to the be- havior of Subject J. The insight of this subject into the first series at the time of her successful generalization was unusually superficial and, contrary to custom, five days were permitted to pass between her completion of the first series and her attack upon the second. If we leave her record out of account, the de- gree of transfer for the group becomes 92.1 per cent. The trans- fer from either series to the other is here so great as completely to overshadow such differences as may exist in the difficulty of the two series. The degree of transfer from series to series within the vari- ous orders of continuous series is shown in Table XIV. The percentages of transfer are based on the supposition that all series in any given order are equal in difficulty. The series are listed at the left of the table in the usual order. Individual sub- jects of the group are indicated by the Roman numerals at the top. Though not all of the series were specifically solved by all subjects the general solutions given for all series of an order were such as to indicate that all higher series of the order could be solved without manipulation of the beads. We. have there- fore figured the average number of trials per series just as if every series had been specifically solved by every subject. The degree of transfer may be further traced from lower to higher orders of series. As already stated, the higher orders are probably more difficult than the lower ones, but in the absence of experimental data upon this point we shall assume that all orders are of equal difficulty and calculate the percentages of transfer upon that basis. The transfer from lower to higher orders, as found in the records of the thirteen subjects who solved all of the continuous series, is shown in Table XV. 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