LB .145 Copy 1026 Class L'B I 6 2, Co Book . V\i5 Gopyi ight N" COPyRIGHT DEPOSrn INDUCTIVE vs. DEDUCTIVE METHODS OF TEACHING !E&ufatt0ttal ^Bgrliologg Monograyl|g No. 11 Inductive versus Deductive Methods of Teacliing: An Experimental Researcli W: H. WINCH External Member of the Psychological Board of Studies for the University of London Chairman of the Committee of the Teachers' Guild of Great Britain and Ireland on Psychological Research in Schools ; Lecturer for the London County Council on Pedagogical Methods in Schools. Author of "Problems in Education,'^ ''German Schools, ^^ "When Should a Child Begin School, ' ' etc. laltimorp. 1. &. A. WARWICK & YORK, Inc. 1913 .V\i5 Copyriehl, 1913 WARWICK & YORK. Inc. EDITOR'S PREFACE. It affords me great pleasure to call editorial atten- tion to this interesting and instructive contribution to experimental pedagogy. Mr. Winch writes with the authority of long experience born of his profes- sional duties as one of the official inspectors of Eng- lish schools. He is, indeed, well known as the first Englishman to bring the technique of experimental and statistical methods to bear upon the actual prac- tical problems of the school. Those who have followed with any care the modern developments of educational theory know how sig- nificant is that trend of investigation which seeks to study the concrete problems of education at first hand in the classroom and with all the exactness of experimental control. The movement for experi- mental pedagogy is yet in its infancy, but it has already shown the possibilities that lie before it. In the Journal of Educational Psychology, with which this series of Educational Psychology Monographs is affiliated, there has appeared of late an important series of articles which show for various school sub- jects what important problems offer hope of solution by experimental investigation. This monograph presents what is at the very least a first approxima- Z INDUCTIVE VS. DEDUCTIVE METHODS. tion toward the solution of one of these vexed ques- tions of educational practice : Is it better to follow deductive or inductive methods in the teaching of various types of subject-matter 1 The presentation has the special merit of being sufficiently detailed that any teacher who desires to do so may of himself repeat the experiments and verify the conclusions. G. M. W. AUTHOR'S PREFACE. This is, I believe, the first attempt to decide be- tween the conflicting claims of 'inductive' and 'de- ductive ' methods by experimental procedure. In the 'world of science' it is not usual for results to be accepted unless the methods by which they have been obtained are described in such detail as enables other workers to repeat, corroborate, or modify them. Nor are they regarded as valid unless they are obtainable under widely differing external circumstances. To produce similar evidence for educational science will be the aim of all serious workers in education during the next two or three decades, and I am therefore offering this research as a contribution to the scien- tific knowledge of the results of inductive and de- ductive methods in actual application under school conditions. I am quite well aware that much valuable know- ledge is collected by school administrators and school inspectors during the ordinary course of their work. They know much about the results of the application of different methods in different schools. But to dis- entangle all the contributory factors — even to realize them — is very difficult, and inspectors are likely to be misled ; for the teacher is, naturally, mainly desirous of showing that his school is a good one, and not of 4 INDUCTIVE VS. DEDUCTIVE METHODS. settling, by experimental tests, tlie value of a par- ticular method. The work reported in this mono- graph is not subject to this source of error, since the teachers, in this case, were working luitJi the experi- menter, and not against him. It is my firm and ever- growing conviction that without that kind of co-op- eration on the part of teachers there can never be, in an applicable sense, a 'Science' of Education. W. H. W. London, September, 1912. CONTENTS. statistical Note 7 I. Introduction 11 II. The Problem of the Experiments 17 III. The General Plan of the Experiments 20 IV. First Series of Experiments : 1. General Characteristics of the Children Who Worked the Exercises 2.3 2. The Preliminary Tests 23 3. The Method of Marking the Preliminary Tests . 25 4. The Teaching of the TSvo Groups 31 5. The Immediate Testing of the Two Groups ... 35 6. The Marking of the Tests 30 7. The Subsequent Testing of the Two Groups on the Same Subject-matter 37 8. The Testing of the Two Groups on New Material . 38 9. The Marking of the New Material 40 10. Chronology of the Experiment 44 11. Results : (a) The Marks for the Preliminary Tests ... 45 (b) The Marks for the Test Immediately After the Definitions Had Been Taught and Learnt 4fl (c) The Marks for the Tests of Deferred Repro- duction 47 (d) Correlation Between Immediate and De- ferred Reproduction 49 (e) Results When the Two Groups Are Tested on New Material 50 12. Pedagogical Conclusions 53 V. Second Series of Experiments : 1. General Plan 55 2. The Preliminary Tests and the Method of Marking. 50 3. Chronology of the Experiment 59 4. The Final Tests and the Method of Marking . . 60 5. Results of the Experiment : (a) Results of the Preliminary Tests .... 61 (b) Results of the Tests in Immediate and De- ferred Re]iroduction 63 5 (c) Correlation Between Immediate and De- ferred Reproduction 65 (d) Results of the Test on New Material ... 67 VI, Third Series of Experiments : 1. General Plan 69 2. The Preliminary Tests and the Method of Marking. 70 3. Chronology of the Experiment 75 4. The Final Tests and the Method of Marking . . 76 5. Results of the Experiments : (a) Results of the Preliminary Tests .... 90 (b) Results of the Tests in Immediate and De- ferred Reproduction 92 (c) Correlation Between the Results of Immedi- ate and Deferred Reproduction .... 95 (d) Results of the Test on New Material ... 96 VII. Fourth Series of Experiments: 1. General Plan 100 2. The Preliminary Tests and the Method of Marking. 101 3. Chronology of the Experiment 104 4. The Tests of Immediate and Deferred Reproduc- tion 107 5. The Test of Application to New Material .... 107 6. Results 114 VIII. Fifth Series of Experiments: 1. General Plan 119 2. The Preliminary Tests and the Method of Marking. 120 3. Chronology of the Experiment 122 4. The Tests of Reproduction 124 5. The Test of Application to New Material .... 129 6. Results : (a) Of the Preliminary Tests 133 (b) Of Immediate Reproduction 134 (c) Correspondence Between Immediate and De- ferred Reproduction 135 (d) Results of the Test on New Material . . .138 IX. General Summary 140 STATISTICAL NOTE. Suppose we have two measurements of any mental function for a number of children, that the second measurement gives higher results than the first in most cases, and that the average mark for the second measurement is a little higher than for the first. May we, therefore, conclude that some general tend- ency is at work, or must we regard the higher aver- age of the second measurement as the result of 'chance' or mere variability? To answer this ques- tion I propose to illustrate the usual statistical check on results of this kind by means of one or two exam- ples. Suppose the children are measured for their power of spontaneous definition ; that, a week later, they are measured again, and that the marks are as shown in the following table : First Second Name. measurement. measurement A. B 9 10 CD 8 9 E. F 7 8 G. H 6 7 I. J 5 6 K. L 4 5 M. N 3 4 O. P 2 3 R. S 1 2 Average, 5 Average, 6 Common sense has no difficulty in deciding that there is a 'general tendency' to imj^rovement from one exercise to another. Let us now calculate the 8 INDtrCTIVE vs. DEDUCTIVE METHODS. 'probable errors.' The 'probable error' of an aver- age is i , where 'a' is the standard deviation, Vn and *n' is the number of cases measured.* Worked out on this formula, the 'probable error' of the aver- age 5 is approximately .6, and of the average 6 is also approximately .6. The 'probable error' of the dif- ference hetiveen two averages is .67449 V^^-^ ' n where '^i' is the standard deviation of the first aver- age, 'o-o' is the standard deviation of the second aver- age, and 'n' is the number of cases measured. Ap- plying this formula to the present example, we have the 'probable error' of the difference between the + arj^Ao /(2.6)^4- (2.6)^ ,., . two averages — .6/449 V j — -— —, which is n approximately .8. It is required statistically that the difference be- tween two means shall be twice (or more) the 'prob- able error' of that difference before the difference is supposed to be 'significant,' that is, indicative of a general tendency. But the difference between the means in this case is only 1 and its 'probable error' is .8, so that, apparently, we have no 'significant' difference at all. But let us consider one more illustration in which the averages are the same, but in which common sense would not find a general tendency to improve- ment: ♦Simple illustrations, in which V is found from easy examples, are given in the statistical note attached to my monograph, When Should a Child Begin School? STATISTICAL NOTE. First Second Name. measurement. measurement. A. B 9 2 C. D 8 3 E. F 7 4 G. H 6 5 I. J 5 6 K. L 4 7 M. N 3 8 O. P 2 9 R. S 1 10 Average, 5 Average, 6 The difference between the means is again equal to 1, and the 'probable error' of the difference, cal- culated just as before, is .8. Statistically, therefore, we are precisely in the same position as in the pre- vious example, and there is no 'general tendency' to improvement. But quite obviously the two cases are by no means similar and their ' probable errors ' are not the same, for we have overlooked the positive correspondence between the first and second meas- urements of A. B., C. D., and the rest in the first illustrative case and the negative correspondence in the second illustrative case. The theory of statistics takes account of this correspondence, or lack of cor- respondence, in the following formula for the 'prob- able error ' of the difference between two averages : .67449 V "'- + "^^~^^"^^% n where ' (Ti ' is the standard deviation of the first aver- age, 'o-o' is the standard deviation of the second aver- age, 'r' is the coefficient of correlation between the two series of measurements, and 'n' is the number of cases measured. I suggest now that the correlation 10 INDUCTIVE VS. DEDUCTIVE METHODS. coefficient be worked out for both cases. In the first case 'r' will be found to be + 1 and in the second case — 1.* Let us then apply the formula, corrected for correlation, to the two illustrative cases. The 'prob- able error' in the first case is .67449 y^ (2.6)''+(2.6)'--2(+l)2.6~X2:6 n It will be seen at once that the expression disap- pears, for 2 (2.6) (2.6) = (2.6)^ + (2.6) = : that is to say, the difference between the averages is perfectly representative of the two series of measurements, as common sense would suppose. In the second case the 'probable error' is .67449 / (^-6)- + (2.6)- - 2 (- 1) 2.6'xT6 ^ n that is, double what it was when the negative corre- lation was neglected. It now reaches 1.6, and is greater than the difference between the averages, which is only 1. Hence the conclusion is against any general tendency, again in accordance with common sense. These illustrations will probably be sufficient to show that the use of probable error formulae without regard to correlation may be very misleading, and also that mere averages, without some indication of the nature and extent of the variability of the meas- urements, may be even more so. *Easy illustrations will be found in the statistical note previously referred to. INDUCTIVE VERSUS DEDUCTIVE METHODS OF TEACHING: AN EXPERIMENTAL RESEARCH. I. INTRODUCTION. In England — it is for Americans to speak for their own country — there is a widely-spread opinion that the theory and the practice of teaching are two very different things. The young student leaves the nor- mal school or training college, and, doubtless crudely enough, begins to put, or to try to put, into practice some of the pedagogical methods which he has been taught as theoretically sound. Not infrequently — I had almost said invariably — his confidence in his theoretical instruction receives a violent shock. His superiors in the school assure him that he will do no good if he goes on like that. "What is worse and much more disconcerting (for, after all, principals and head masters must find fault; it is their metier), his confreres look on with amused tolerance and 'chaff' him about his 'new- fangled' ways. Then, dropping into friendly confi- dence, they explain to him that it was all very well for him to 'get up' and describe methods of that sort in examination papers; it was expected of him, and, naturally enough, he wished to get his certificate of 11 12 INDUCTIVE VS. DEDUCTIVE METHODS. proficiency and to do credit to liis college. Exam- iners required these things; they were unpractical persons like professors, but, of course, a wise student humored them; besides, how else could he pass his examinations ? Let these fellows take off their coats and come and do a day's teaching in the schools, and they would very soon change their opinions. Their stuff is all theory, and in actual school life is simply no good. Now you have become a man, you must put away from you childish things ; and so on. Thus the experienced and disillusioned confreres to the neo- phyte. It is not clear that taking off their coats would assist much in such professional conversions as are here anticipated, but the suggestion is a protest against what the teachers regard as a rather vision- ary and unpractical existence. If this rude shock resulted in complete divorce, there would be some hopes of other and happier mar- riages between theory and practice later on ; but, in England at least, what happens is rather of the na- ture of a judicial separation. The theoretical methods are not absolutely dis- carded; they are laid by and put in evidence only on special occasions ; the practical methods do duty day by day. For it is dangerous, from the standpoint of professional advancement, for the teacher boldly to renounce the methods he has been taught ; he is pur- sued all his life by unpractical, theoretical persons, to wit, inspectors, and he will often deem it to his advantage to profess a method he does not believe in. Head masters, too, mindful of the repute of their INTRODUCTION. 13 schools, will say, "Yes, that's all right, but don't do that when Mr. I r comes ; he does not like it ; he thinks 5^011 ought to teach that this way." Well, yes, no doubt, but what is there in all this but the usual difficulty which besets the young in- structed person when he takes an actual place in the working world : it is the old difficulty of science ver- sus practice. In a few years the teacher, like other people, will have allowed his theory and his practice to interpenetrate, and both will have been improved. In such wise may we suppose an experienced admin- istrator pooh-poohing my criticism. If I could admit this, my complaint would indeed lose much of its poigancy. But I do not admit it. On the contrary, I believe that with the great majority of teachers there remains permanently an irrecon- cilable breach between dominant theory and current practice. It is true that experienced teachers^ — some of them — will attend lectures about educational topics. Two men speaking somewhat loudly after leaving a lecture hall — modesty forbids me to name the lecturer — said one to the other: "I didn't get much from him ; he 's like all the rest of 'em. " " Oh ! I don't know," said the other, with a give-the-devil- his-due air, "one gets ideas." "Yes," was the prompt reply, "but they don't work." And this, let us quite clearly understand that, is not merely an expression of a private grumble ; it is a strongly held and often a clearly reasoned view. There are always "new methods" in education, of course, and I hope that those of us who hold the very newest of them are more or less prepared to see our darlings cold-shouldered for a newer birth. Still, behind all temporary fluctuations, there is a line of 14 INDUCTIVE VS. DEDUCTIVE METHODS. steady meaning in such phrases as 'new method,' 'inductive procedure,' 'developmental method,' 'psy- chological method,' et id omne genus. And behind all temporaiy oscillations there is a steady trend of opinion amongst experienced teachers that these methods have certain serious disadvantages; that though they may be valuable for show j^urposes in teaching, they are too slow, and the information thus acquired is not really available when it is wanted. An experienced head master in London wrote to a lecturer on pedagogy in the following terms : "I (recently) asked a question on the difficulty of covering a present average course (by using the new methods) in the time given to it on the school time- table, and I should like to press the point and illus- trate its importance from my knowledge of the views held by others, and especially by the class teachers in my own school, for at almost every conference with my staff this question arises strongly. "It seems impossible to train children by much individual work in class by inductive methods, much questioning and the consequent necessary waiting for the child's expression to be formulated in a suffi- ciently acceptable form, and at the same time to get through the course set in a given time, and especially properly to prepare the children also for examina- tion purposes. ' ' For instance, in an illustration given of obtaining from the class the definition of the diameter of a circle, the time taken, if similarly applied to other parts of the course, would not permit of a present average syllabus being more than about half com- pleted, nor would the information got be available INTRODUCTION. 15 throughout the class for reproduction at a more or less distant examination. "Another illustration: Two years ago an inspector {fons et origo malorum W. H. W.), examining Standard V (approximately American Grade VI), asked for a definition of a proper noun, and, not get- ting a satisfactory answer, tried to obtain it from the boys with the aid of many questions and illustrations. He took up twenty minutes of the lesson, and failed in the end to get what was wanted. "Of course, all the time the children were being educated on the best lines; they showed eagerness, interest and active thought. (This, I fear, is a con- cession to the lecturer as a theoretical person. W. H. W.) But, the time taken, in view of the rest of the syllabus, was excessive, and the result at the end was not satisfactory." Then follows a paragraph in which the writer ex- presses his willingness to carry out the new methods provided the educational authority will dispense with tangible results. This is a strong letter. It expresses views which are very common, and which, moreover, are held by some of the most successful schoolmasters I know. And they will have to be met by educational science. I, myself, believe that, until these questions are dealt with in such a way as to be acceptable to teachers, the breach between theory and practice will remain. Professors, teachers of method and insx^ectors may continue as now to receive lip-homage for the meth- ods they advocate, but their directions will be hon- ored rather in the breach than in the observance. In actual practice there will be little, if any, change. 16 INDUCTIVE VS. DEDUCTIVE METHODS. What, then, do I suggest I I propose the experi- mental determination of these disputed questions in the schools themselves. There is an increasing num- ber of teachers willing — nay, anxious — to carry out such experiments if adequate guidance be given to them. To the description of one attempt at an expe- rimental solution of some of these disputed points I. now proceed. II. THE PROBLEM OF THE EXPERIMENT. No one can liope to solve all the questions raised in the never-ending controversy about inductive and deductive methods by means of a single experiment or by means of a, single series of experiments. Yet, if one attempts to deal with the matter experimen- tally, one must deal with some definite subject-mat- ter. There is danger in this, since we may find out afterwards that our conclusions are true only for subject-matter of that particular kind; but it is a danger which must be faced. A good plan, if one is conscious of bias, is to select subject-matter which favors the chances of the method in which personally one does not believe. So I, believing in inductive rather than in deductive methods, chose geometrical definition as the subject- matter for my experiment. There is much good opinion in favor of deductive treatment of definitions, especially when, as in demonstrative geometry, a sort of system of reasoned conclusions is to be built up upon those definitions. It is argued that a jDupil ought to know exactly what the definition means, that the exact wording of it is very important for that purpose, and that at some stage in the procedure the defin- itions should be memorized. There is no question here about the introduction to demonstrative geom- etry. It is supposed by both parties to the dispute 17 18 INDUCTIVE VS. DEDUCTIVE METHODS. that manual work and geometrical construction are necessary propaedeutics to any rational system of geometry. But if we are ever to have a system of demonstrative geometry we must, it is said, have exact meanings for our terms or we shall never be able to reason in words at all. This is by no means a weak case, and to it addi- tional importance is given at the present juncture, when so much dissatisfaction is being expressed as to the 'chaos' into which geometrical teaching is falling through what is alleged to be an excess of inductive method. Those who advocate purely inductive methods urge that the memorizing of the definition and the study of its application to examples is not, in the truest sense, knowing the definition ; it is urged that it can be known better if it is built up from the ex- amples. It is asserted that the memorizing of defini- tions leads to bad errors, 'howlers,' of which the following, once given to me, is a choice example : "A circle is a figure bounded by a straight line, which is such that every point within it is at equal distance from every other point." There is a tendency to concede that inductive methods are slow; there is some tendency also to concede the point that induc- tive methods will not prepare a pupil so well for examination purposes. But it is argued that what he does learn he will remember longer, and that he will be made more intelligent. The use of the word 'intelligent' in educational disputes amounts almost to a public scandal, and I do not propose to use it without giving an opponent some way of checking the assertion. By intelligence, in this case, I am going to mean THE PROBLEM OF THE EXPERIMENT. 19 the power gained to deal with new material in con- sequence of the mental processes which the pupil has passed through in acquiring the old. I have found this interpretation of the word acceptable to both parties in the dispute. Let me now endeavor to disentangle the threads and see how far the assertions made, first on one side and then on the other, are susceptible of experimen- tal determination. First of all, we can quite easily find out whether pupils taught inductively or deductively know the required definitions better immediately after they have acquired them. We shall demand exact mean- ings, but not a stereotyped form of words. A child taught inductively would not, of course, know a par- ticular form of words for a definition, but no devia- tion from accuracy of meaning must be allowed. Secondly, we can find out, by repeating the exer- cises later on, whether the pupil forgets more or less after one method than after the other. We are thus testing the durability of his knowledge. Thirdly, we can find out how many 'had errors' are made by pupils who are taught by inductive and deductive methods, respectively. Fourthly, we can find out which of the two methods gives the pupil the greater power of attacking new material successfully. In so far as we can determine these issues, we are in possession of the clues which will lead us to reasonable conclusions on most, if not all, of the questions raised in this section and in the section entitled Introduction. III. THE GENEEAL PLAN OF THE EXPERI- MENT. One difficulty in all work of this kind is to find some unsophisticated material with which to experi- ment. I wanted to work, if I could, with some school children who had never learnt or even heard of a geometrical definition throughout the whole of their school lives. I think I succeeded in getting this con- dition fulfilled with some Standard V girls in the southwest of London and some Standard III boys in the southeast. In the course of the experiment one of the boys' fathers told him it was Euclid, which it wasn't, and gave him one or two 'tips' which spoiled some of his papers ; but, with that exception, nothing transpired to indicate that we were not working on virgin soil. I worked also with a Standard VI and VII girls' class in a poor school. These girls, though not au courant with geometrical definition, had nevertheless done much constructive work and were accustomed to express themselves freely and exactlj^ I worked also with a Standard VII boys' class in a poor school. There was a little difficulty here with one or two of the definitions, owing to the boys' attendance at the Manual Training Center, where they had learnt something about them. And, finally, the work was done with an ex- VII boys' class, the highest class of a higher grade school. The teacher of this class was 20 GENERAL PLAN OF THE EXPERIMENT. 21 a man who had for j^ears attended lectures on psy- chology, and was accustomed to teach very largely by inductive methods. I think it will be conceded that we have here a good variety of material; and, since in every case we worked with all the pupils of the class, and not merely with selected pupils, it will also be conceded that if any tendencies show themselves throughout the whole range of our material, the probability that they are chance results is very small indeed. So much for a general survey of the material; it re- mains to be seen how it was utilized. Briefly, though there were local variations in procedure, the same general plan was followed throughout. A whole class, under one teacher and working un- der the same syllabus of instruction, was divided into two equal groups. The children were required to attempt the definition of some geometrical shapes which were placed before them in the form of large drawings, and on the results of these attempts the class was divided. One of the two groups subse- quently acquired the definitions inductively. The other group learnt them, but the children were in- structed that the exact words given them in the defi- nition were not required as long as they gave all the meaning. The two groups were tested immediately, and after shorter and longer intervals. And, after some interval of time, new material of a somewhat analogous kind was given to the children to define with a view to discovering which of the two groups could better apply their old knowledge, as we say, though it is, in some aspects, rather an application of a method than an application of knowledge. Two of the classes were taken by me in the indue- 22 INDUCTIVE VS. DEDUCTIVE METHODS. tive acquisition of the definitions ; in the three other cases they were taken by their own teachers. In one of the schools in which I took the exercises myself one of the other teachers inquired of the teacher of the class whether I wished the inductive method to succeed. ' ' I think he does, ' ' was the reply. ''Then," came the prompt rejoinder, ''he ought to have let you take it ; you would have got it into them much better than he would." With this unsolicited testimonial to my handling of the method, I proceed to a detailed description of the five series of experi- ments actually carried out in the schools above cited. IV. FIRST SERIES OF EXPERIMENTS. 1. General characteristics of the children who worked the exercises. This experiment was carried out during the months of September, October and November of the year 1911. The work was done with the whole of a Stand- ard V class of girls of an average age of 11 years 8 months. The children knew nothing about geomet- rical definitions and were not biased by practice or novelty towards either of the methods employed. It is, of course, necessary to know the customary lines of teaching in a school in order to prevent one from drawing misleading conclusions from the results of experiments of this kind. The infant school work which these girls had done some four, five or six years previously was very little affected by tend- encies to the kindergarten or sensory type of instruc- tion. . The school was situated in a neighborhood slightly above the average for municipal elementary schools, and the girls of the class were accustomed to give full attention to their school work. 2. The Preliminary Tests. Drawings of squares, triangles, oblongs and diam- eters of circles were placed upon large blackboards, with their names written above them, thus : 23 24 INDUCTIVE VS. DEDUCTIVE METHODS. ^ Wu,ayTj&a qV -r-C ^-^^fl/ f G.'i^C^C^ FIRST SERIES OF EXPERIMENTS. 25 The children had the drawings pointed out to them, with the accompanying words, ''These are squares," etc. In the case of the hist set of figures tlie straight line was pointed to with the words, "This is a diameter of a circle, and this is a diameter of a circle," etc. Then upon a blackboard the fol- lowing questions were written : 1. What is a square? 2. What is a triangle? 3. What is an oblong? 4. What is a diameter of a circle? All the children in the class were required to an- swer the questions. They were told to do so thought- fully and without hurry. No time limit was imposed in this or in any other of the subsequent exercises of this experiment. 3. The Method of Marking the Preliminary Tests. The importance of this question merits a short dis- cussion of the principles involved. I suppose one's first notion is something like this: Let us just take any currently accepted definition of a square, etc., Euclidean or other, and mark the children's exer- cises in accordance with their correspondence or non- correspondence with that. It will not matter very much which definition we take, provided that we keep to the same one all through the marking. A perusal of the answers shows us immediately, if we had not known it before, that this method will not do. The children will hardly be likely, for ex- ample, to write down that a square is a rectilinear quadrilateral with a right angle, as one good defini- tion gives it. They know the meaning of a straight 26 INDUCTIVE VS. DEDUCTIVE METHODS. line ; they know the meaning of four sides ; they do not know the meaning of right angle, and if they did they would tell you quite candidly, if they had been properly taught, that the definition just given was wrong. A square has four right angles, not oyie, they would urge. The teacher might quite authoritatively inform them that every four-equal-sided straight- lined figure, if it have one right angle, must have four ; therefore, why say four in the definition % and that a definition should not say more than it need. I hope the teacher won't, because the redundancy is no redundancy to the child at this stage ; indeed, is no redundancy at all until the child is in a mental condition to deduce some of the properties from the others. Then, and only then, can one strike out the derivable properties and be satisfied with the others as sufficiently defining the term. This is an exceed- ingly interesting exercise, but its time is not yet. ''Well," I can hear an impatient mathematician say, "this isn't mathematics; this is psychology'; the chil- dren are not going to be marked on real definitions at all." On the contrarj^ I assert that they are going to be marked on the very realest of definitions ; they will be marked according to the number of qualities and properties which they can themselves see to be common to all the specimens called by the same name. And I go further, and assert that the mathema- tician's definition, suitable and right for those who know the system of knowledge within which the defi- nition finds a place, is just mere arbitrariness to those who do not. And, moreover, to tell the child that he may mention some of the coimnon properties that he finds, and that he may not mention others, un- less he is in a position to see for himself that some FIEST SERIES OF EXPERIMENTS. Zl are derivatives or derivable, is to shut him up sliarp before a mystery. He won't do much spontaneous defining after that. Perhaps, some day, we may have a system of demonstrative geometry built up by psy- chological research. If so, the definitions will change as knowledge accumulates and reasoning becomes more penetrating, just as our definitions do of the common objects of daily life. Imagine a system of geometry for school cliildren with evolving defin- itions. It is doubtless too horrible, and I do not, at present, ask my reader to accept such a thing, but only to grant that if I want to mark fairly the efforts of untaught children in spontaneous definition I must be guided by what they do, and not mark on an a priori scheme, settled beforehand by Euclid or by another geometer, or by myself, who am no geometer. What, then, do the children say in answer to these questions, "What is a square?" etc. There are be- fore us between forty and fifty sets of answers, and, though it would be illuminating for anyone to read the whole of them, they cannot be reproduced here. The interest in them lies in the fact that they repre- sent untaught, spontaneous attempts; it is an inter- est which is at first psychological; the pedagogical interest comes later. Perhaps one of the best papers and one of the poorer ones may be found worthy of attention. I request the reader to look at the draw- ings whilst considering the children's definitions. The specimen papers follow exactly as they were written : Nellie W., aged 12 years 3 months, wrote : 1. A square is an object with four corners and four lines, two for the sides and one for the top and bottom. 2. A triangle is an object with three corners, and three lines, two slanted ones and one straight one at the bottom. 28 INDUCTIVE VS. DEDUCTIVE METHODS. 3. An oblong is an object with four corners and four lines, the lines at the top and bottom being longer than those of the sides. 4. A diameter of a circle is a line drawn right across the circle from one side to the other each line is called a diameter. Winnie T., aged 12 years 8 months, wrote : 1. A square is a kind of box with four lines all the same length. 2. A triangle is a thing with three sides not all the same length. 3. An oblong is a thing with two short sides and two long sides. 4. A diameter is a circle with a number of lines going from one side to the other. Even from these two papers one may get a hint as to the way the children are going to sum the figures up, and a careful search through all the papers re- veals that by one or another the following points of accurate definition are mentioned : Common qualities mentioned in children's defini- tions. 1. A square is a shape, figure, drawing. It has lines or sides. It has four lines or sides. It has equal lines or sides. It has straight lines or sides. It has corners. It has four corners. It has equal corners. (A total of eight points.) 2. A triangle is a shape, figure, drawing. It has lines or sides. It has three lines or sides. It has corners. It has three corners. (A total of five points.) FIRST SERIES OF EXPERIMENTS. 29 3. An oblong is a shape, figure, drawing. It has lines or sides. It has four lines or sides. It has straight lines or sides. It has two long sides. It has two short sides. The two long sides are the same length. The two short sides are the same length. The two long sides are opposite each other. The two short sides are opposite each other. It has corners. It has four corners. The corners are all the same size.) (A total of thirteen points.) 4. A diameter of a circle is a line. It is a straight line. It goes through the middle or center of a circle. It goes from one side or edge of the circle to the other. (A total of four points.) The papers were marked thus : One mark was al- lowed for each common attribute correctly given. It was decided not to allow thing or object as equivalent to shape, diagram, etc., for it was thought that 'thing' was so wide a term as to be hardly available for the purpose of these definitions, and that by the word 'object' children usually meant a material thing in three dimensions of space. It is not contended that all the above units of marking are exactly equal in value ; it is contended only that marking on these units is easy, practically serviceable, and yields results from which one can draw reliable conclusions for practical purposes. 30 INDUCTIVE VS. DEDUCTIVE METHODS. On the results of this marking all the children of the class were divided into two equal groups, one of the best children being placed at the head of the first group — Group A — an equivalent child being placed at the head of the second group — Group B — then the children next in order were placed, one in each group, and so on down the list, carefully preserving the bal- ance all the way down, till all the children were di- vided between the two groups. There is a weakness here which needs attention. It is not usually satisfactory to divide a class on the basis of one test only. It is probably much more sat- isfactory where the higher mental functions are con- cerned than it is where simple sensory functions are measured, but it is risky even in the former case. It adds enormously to the probability of a valid result if several tests of the same kind are taken and the results of these correlated. One then feels confi- dence, if the results of the tests correlate highly with one another, that one is testing some function or group of functions which is operating steadily, and that each child is working at about its 'true form' as compared with the others. In some of the subse- quent experiments of this kind I adopted that plan, but as I wished to be present during the whole time, and on each occasion when exercises were done in this case, I reduced the number of Preliminary Tests to one, but I did so with a full consciousness that I should feel less reliability in the equality of my two groups. The marks obtained in the Preliminary Tests by the two groups, respectively, will be shown in the section headed 'Results.' FIRST SERIES OF EXPERIMENTS. 31 4. The Teaching of the Tivo Groups. About a week later the two groups were separately instructed in the subject-matter of the definitions. As each child had already made some attempts for herself under test conditions to' define the terms, she was in a favorable condition for the reception of knowledge by any method. I had decided that one of the two groups should have the definitions written out for them, with illus- trative drawings underneath, and that the children of this group should be instructed to study and learn the definitions. They were told that they were going to be examined afterwards, that they might then write down the exact words if they liked, but that as long as they got down all the meaning they would lose no marks because they had failed to remember the exact words. This resembled the way in wiiich I remember myself to have learnt the Euclidean defi- nitions. I studied the words and let my attention pass from words to figures to see the illustrations of the general statements in the definitions — it may fairly be called a deductive method. There is, how- ever, one important difference. The definitions given to the children are such as they themselves are capa- ble of making up. That does not mean that every child, nor even any child, in the class could say pre- cisely all these things by itself; it does mean that they are on the lines of the child's own evolutionary track. The Definitions as Learnt Deductively. They were written down with illustrative examples drawn underneath the definitions. The drawings were the same as those given before in the Prelimi- dZ INDUCTIVE VS. DEDUCTIVE METHODS. nary Tests ; they will not be reproduced here. The definitions were worded thus : 1. A square is a shape with four sides and four corners. The sides are straight and they are all of the same length ; the corners are all of the same size. 2. A triangle is a shape with three sides and three corners.* 3. An oblong is a shape with four straight sides and four corners. Two sides are long and two are short. The two long ones are opposite to each other and are of the same length, and the two short ones are opposite to each other and are of the same length. All the corners are of the same size. 4. A diameter of a circle is a straight line going through the middle of a circle from one side of the circle to the other. The Definitions as Taught Inductively. The teacher, myself in this case, having the points of each definition in mind, taught up to them, but no instruction was given by the teacher otherwise than by questioning. In another school, in which also I did the necessary teaching myself, a discussion arose afterwards be- tween the two groups of girls because one group had done rather better than the other in the subsequent testing. ''You ought to do best," said a girl of the deductive group to the inductive group ; ' ' the gentle- man taught you the definitions ; we had to learn 'em for ourselves. ' ' ''That's just where you're wrong," she was an- *It will be remembered that one of the triangles drawn had curvilinear sides. FIRST SERIES OF EXPERIMENTS. 66 swered; ''the gentleman never told us a thing; we told him all about it. ' ' But perhaps a little more explicitness may be use- ful to those who may wish to repeat the experiment. Let me illustrate by means of the last example : Pointing to all the diameters drawn, the teacher says: ''What can we say about all these?" The an- swer 'lines' will be received. He can then ask the question: "What is a diameter of a circle!" He will be answered, if he chooses his questionee well : "A diameter of a circle is a line." One feature of the method is that the teacher accepts all the answers given to him and translates them into concrete form. He draws a curved line on the blackboard, but not within one of the circles, and asks: "Is that a diam- eter of a circle?" He is answered: "No, because it is not a straight line. ' ' He draws a straight line, still outside the circle, and asks : "Is that right ? ' ' The answer comes: "No, because it's not in a circle." "Very well," the teacher says, "let us try again. What is a diameter of a circle f'i If he chooses a child's answer, as he should, from among the least proficient of the class, he will be answered: "A di- ameter of a circle is a straight line inside a circle. ' ' He accepts the answer and draws a straight line in a circle which neither passes through the center nor touches the circumference on either extremity. He again asks : " Is that right ? " He will be told : ' ' No, because the line does not go to the edge on both sides." He corrects his drawing, producing the line each way to the circumference. He will now be told his line is wrong, because it must pass through the middle or center of the circle. He then draws a fresh line passing through the center, but cutting the cir- 64: INDUCTIVE VS. DEDUCTIVE METHODS. cumference, and lie is now told the Hue must reach the edges, but not pass over them. At this stage he can rely upon receiving a correct definition from the great majority of his pupils ; but it is essential, if we are to keep this method distinct from the other one, that he does not ask a number of children to give the correct definition. One or two may do so, and the teacher then passes on; otherwise the mnemonic repetitive factor comes in here as in the other method, and for the purposes of this experiment it is usually desirable to avoid this. I do not propose to go in detail over the method employed for teaching the remaining definitions. Any experienced teacher — and this paper is not writ- ten for other than experienced teachers — will be able to ask analogous questions and get the answers cor- rected in analogous ways. With children taught fre- quently on this method it is quite possible to get the necessary drawings and corrections, or most of them, done by members of the class, so that the machinery of correction and amplification is mainly in the hands of the class, with the teacher there to see fair play, and direct the discussion to profitable issues. But I do not press this latter point ; the work of concrete exemplification of error was, in the case of all the experiments about to be described, solely in the hands of the teacher. Teachers who would not kgree with the method of mutual correction may quite well agree with this. There are one or two points of detail, however, which may cause difficulty. It is of little use for the teacher to point to the squares drawn and ask,'' What are all these!" for he will naturally be answered, "Squares." Indeed, the word squares is written FIRST SEEIES OP EXPERIMENTS. 35 above the drawings. But if he points to the squares, and the triangles, and oblongs as well, and asks the same question, he will get answers like ''drawings," "shapes," "figures," "diagrams." He can then start his detailed questioning to bring out the defini- tion of a square. Among his answers will very likely be, "A square is a shape with four lines and four cor- ners." It is obvious that many figures which are not squares can be drawn to comply with this definition, and the correction will proceed as before. The size of a corner is a difficulty for young children; they confuse corner with edge. It helps to ask, (pointing to angles of different sizes) "If these were the cor- ners of a room, how much sand or how many blocks could I put in that corner, and in that one ? " In some such way the size of the comers becomes thinkable to the young child. The question has been raised whether items of considerable difficulty, like this one, should not carry more than one mark. The theoret- ical justification is conceded, but it is argued that in practice a mark for each unit gives sufficiently steady and reliable results. Children will often give a quality which is true of only one or two of the squares or triangles. It is only necessary to point to the other ones in these cases. I need, perhaps, hardly say that these children, like the others, knew they were going to be examined im- mediately afterwards on the work they were then doing. 5. The Immediate Testing of the Two Groups, As soon as the teaching of the Inductive Group was completed, the group which had been learning the definitions in another room also stopped their work ; 36 INDUCTIVE VS. DEDUCTIVE METHODS. and in a third room, so tliat the environment of both the groups should be changed, both sets of pupils answered the following questions : 1. "What is a square!" 2. "What is a triangle?" 3. "What is an oblong?" 4. "What is a diameter of a circle?" 6. The Marking of the Tests. The papers were marked exactly as in the case of the Preliminary Tests, so far as the positive units were concerned, but a new feature was added to the marking. It is well known that bad teaching and bad learning produce errors, and errors of a noxious kind. But some statements that we sometimes call errors in the work of school children are rather irrelevances and redundancies than errors. For instance, when a child, in defining a square, after mentioning the prop- erties of a square quite correctly, says: "and the corners are opposite each other," the statement is worth no positive mark, but neither is it worth any negative mark. Or when a child, in defining a tri- angle, says : ' * Some of the lines are curved and some are straight," though this, strictly speaking, is no part of the definition (which includes only the quali- ties common to all the triangles given), yet it can hardly be called a bad error. But it is a bad error for a child to say: "A triangle is a shape with three equal lines and three corners." Such an answer re- ceives five positive marks — one each for shape, lines, three (lines), corners, three (corners). But 'equal' receives a negative mark as a *bad error.' Again, when a child, in defining a square, says, amidst much FIRST SERIES OF EXPERIMENTS. 37 wliicli is correct: ''The corners all come under eacli other," the statement is marked as a bad error. Or when a child, speaking of a diameter, says : "It must stand upright," the statement is regarded as a bad error. The first diameter in the drawings was up- right, hence the error. The definition was being- elaborated from a memory of the one drawing with- out comparison with the memories of the others. In the case of the experiment in this school, there- fore, besides giving the positive marks obtained by each group, I shall also give the negative marks, and, in addition, the positive marks with the negative marks subtracted from them. It is interesting, before turning to the section show- ing the results, to try to guess from our general knowledge of children's minds of the given ages (roughly from eleven to twelve), and our opinions as to the methods of teaching and learning, which of the two groups gained the more positive marks and which group made the more bad errors. 7. Subsequent Testing of the Two Groups on the Same Subject-Matter. In discussions among teachers the question is fre- quently raised as to the relation between the quick- ness and the permanence of knowledge. Teachers are prone in theory to back the tortoise rather than the hare, though in practice they repeatedly prod the tortoise up. How far does the present experi- ment throw light on the matter ? Are we justified in supposing that a test given to two groups of cliildren, immediately after certain knowledge has been ac- quired, supplies us with comparative results which 38 INDUCTIVE VS. DEDUCTIVE METHODS. will be true, say, a week later, a month later, and so on? To test tills point tlie exercise above described was repeated a week later. The children were not aware that they would ever have to do this work again. Then, once again, more than a month after the first test (the exact chronology of the experiment will be given later on), the test was repeated a second time. The papers were marked in both these cases exactly as in the first test, positive marks being given for the points remembered and negative marks for the bad errors. The results will enable us to see how far the comparisons between the groups based upon the im- mediate results are corroborated when the results for deferred reproduction are taken into considera- tion. Again, it is worth while to try to think the answer out on general principles before turning to the actual results. 8. The Testing of the Two Groups on New Material. It will be remembered that the children of the class were divided on the results of a test in which they were required to find definitions for themselves of a square, etc. How far has the teaching or learning by different methods affected their power to attack new material of an analogous kind? This is one of the most important questions that can be asked of any method of teaching or learning. In ordinary pedagogical discussions it would be implied by assertions that children would be made more intelligent by one method than by another. To investigate this point experimentally the following tests were made : Drawings were shown of rhombuses, etc., and their names written above them, thus : FIRST SERIES OF EXPERIMENTS. 39 yf'fiff'wy^^-u. Name. First Second Third Total (Iidtials onlj'.) test. test. test. marks. D. A 17 22 22 61 W. E 19 20 19 58 n. L 18 17 20 55 A. R 10 10 11 31 L. L 11 7 13 31 Averages 14.5 15.5 16.3 46.3 M. V.'s 2.6 2.8 2.2 Care was taken also that the children should be so arranged in the grouping that the ages of the one group should very closely approximate to those of the other. The average age of Group A worked out to 13 years 1 month (mean variation 7.2 months), and of Group B to 13 years 0.5 months (mean varia- tion 5.6 months). (b) Results of the Tests in Immediate and Deferred Reproduction. It now remains to be shown what marks were ob- tained after the one group had been taught the defi- nitions and the other group had learnt them. The total marks will be shown for the three Preliminary Tests, with the marks for immediate reproduction and for the two tests of deferred reproduction — the one given a few days later and the other about a month later than the test of immediate reproduction : 64 INDUCTIVE VS. DEDUCTIVE METHODS. Table IX, shoiiing the work of the Inductive and Deductive Groups compared, section by section, in the Preliminary Tests and in the Tests of Reproduction (positive marks only). Group A (Deductive). Marks for , Average Marks — three No. Pre- First Second preliminary of liiuinary repro- repro- tests. cliildreu. tests. duction. duetion. Over 50 18.4 20.8 2«.5 40 to .50 fi 15.6 27.7 26.5 30 to 40 5 11.7 25.0 22.6 Third repro- duction. 26.0 25.0 23.0 Group B (Inductive). Marks for , Average Marks — three No. Pre- First Second preliminary of liminary repro- repro- tests. children, tests. duction. duction. Over 50 6 18.4 28.8 28.3 40 to 50 6 15.4 26.5 26.2 30 to 40 5 11.7 25.8 24.8 Third repro- duction. 26.8 25.2 25.2 It seems clear that in this case the children taught inductiv^ely were just as successful as those taught deductively, even in immediate reproduction, and that after a. month's interval they were rather more so; they had lost less of what they had been taught. This will, perhaps, he shown more clearly in the fol- lowing tables: Table X, shoicing the icork of the two groups compared, in the Preliminary Tests, and in the Tests of Immediate and Deferred Reproduction (positive marks only). Average mark for three pre- liminary tests. Inductive group 15.4 M. V.'s 2.4 Deductive group 15.4 M. V.'s 2.4 1 ■Average Marks. -^ First Second Third repro- duction, repro- duction. repro- duction. 27.1 1.8 26.6 1.9 26.5 1.9 25.4 2.2 25.8 2.2 24.8 2.4 , Av erage Marks. \ First Second Third repro- duction. repro- duction. repro- duction. 26.8 1.9 26.3 1.8 26.2 1.8 25.0 2.3 25.2 2.3 23.9 2.0 SECOND SERIES OF EXPERIMENTS. 65 Table XI, showing the icork of the tivo groups compared in the PreUminary Tests and in the Tests of Immediate and Deferred Reproduction (ichen the negative marks have been subtracted fioin the positive 7nar}:s). Average marlc for three pre- liminary tests. Inductive group 15.4 M. V.'s 2.4 Deductive Group 15.4 M. V.'s 2.4 The balance of advantage seems even more clearly on the side of the group taught inductively.* (c) Correlation Between Immediate and Deferred Reproduction. It seems likely from Table IX, already given, that there is considerable' positive correlation between the results of inmiediate reproduction and those of deferred reproduction. That is to say, the girls who are best immediately after teaching and learning are Filso the best after an interval, and those who are worst immediately after teaching and learning re- main the worst after some time has elapsed. But in Table IX the children are classified on the basis of their marks for the preliminary tests, and this classi- fication tends to obscure much of the correlation ^vhich undoubtedly exists. In the following tables the classification is based on the marks obtained in the test of immediate reproduction : *In Table.s X and XI the difference between the means of the ;\'ork of the two groups in deferred reproduction is about twice the probable error' in each case, even on the assumption that the series are not positively correlated. 66 INDUCTIVE VS. DEDUCTIVE METHODS. Table XII, showing the results for Immediate and Deferred Re- production compared, of the Inductive and Deductive Groups {positive marks only). Deductive Group. Marks in No. immediate of Average Marks in Reproduction. retH'oduction. girls. First. Second. Third. Over 28 3 29.3 27.3 26.0 28 5 28.0 25.8 24.8 27, 26 5 26.2 26.0 24.6 Below 26 4 23.3 22.5 24.0 Inductive Group. Marks in No. immediate of Average Marks in Reproduction, reproduction, girls. First. Second. Third. Over 28 5 29.4 28.8 27.0 28 4 28.0 27.8 27.3 27, 26 5 26.6 25.0 25.2 Below 26 3 23.0 23.7 22.7 It is quite obvious, from the foregoing table, that considerable positive correlation exists between im- mediate and deferred reproduction, but such a table gives us no numerical equivalent for correlation. The correlation coefficients have, however, been worked out, and for the Deductive Group the coeffi- cient for the first and second reproduction is + .62 (probable error .10), and for the second and third is + .58 (probable error .11). In the Inductive Group the correlation coefficient between the first and second reproductions is + .76 (probable error .07), and between the second and third is -\- .76 (probable error .07). All the figures indicate high reliability for the results, and a com- parison of the correlation coefficients for the Induct- ive and Deductive Groups shows the work of the former to be the more consistent. SECOND SERIES OF EXPERIMENTS. 67 (d) Results of the Test on New Material. In the case of the previous school we found that with younger children of a lower standard the de- ductive method seemed the better for purely repro- ductive purposes. In this school the inductive method seems better, even for purposes of reproduction. We have now to see whether, when application is made to new material, the results for these children agree with or differ from those of the preceding school. First let me show the results for the two groups as wholes : Table XIII, sJiowing the work of the two groups compared in the Preliminary Tests and in the Tests of Application to Ncio Material. Average INIarks for New Material. When Average mark negative for throe Positive marks preliminary marks have been tests. only. subtracted, Inductive group. . . 15.4 25.5 24.2 M. V.'s 2.4 2.7 3.4 Deductive gi'oup. . . 15.4 23.3 21.9 M. V.'s 2.4 2.9 2.9 We have a clear advantage, in both cases, on the side of the Inductive Group. The difference between the averages amounts to about three times its 'prob- able error,' even on the assumption that the series are not positively correlated. Once again, then, we liiid the inductive method triumphant when applica- tion is made to new material. Let me now show how far this is a difference which is to be found all along the line, i. e., for the weaker as well as for the abler pupils : 68 INDUCTIVE VS. DEDUCTIVE METHODS. TaJjle XIV, showing the icork of the tioo groups compared, section hy section, in the Preliminary Tests and in the Test of Appli- cation to New Material (positive marJcJ, and positive marks after deduction of the negative marks). Group A (Deductively Group B (Inductively Taught). Taught). Marlvs for New Marks for New Marlis for Material. Material. three No. (Posi- (After No. (Posi- (After preliminary of tive de- of tive de- tests. girls. onlv). duetion). girls. ouly). duetion). Over 50 (J 25.8 24.5 27.3 26.0 40 to 50 6 23.7 21.7 6 24.2 22.7 30 to 40 . . . . 5 19.8 19.2 5 25.0 24.0 There seems little doubt that the group inductively taught shows a superiority which is general — a supe- riority which, somewhat unexpectedly to me, how- ever, seems most clearly marked in the weakest (ini- tially considered) of the three sections into which each group is divided. VI. THIRD SERIES OF EXPERIMENTS. 1. General Plan. As in the previous experiments, a whole class of children, working under the same teacher, with the same curriculum, and according to the same time- table of work, was divided into two equal groups on the basis of several tests in geometrical defmition, which the children attempted without instruction and without help. Then, subsequently, one of the two groups was taught inductively and the other group learnt the definitions. There were tests of immedi- ate reproduction immediately after the lesson, and another test, which might also be called a test of im- mediate reproduction, on the following day. About a week later there was a test of application to new material, and, three weeks after this, two further tests were given, which will be referred to as tests of deferred reproduction. The work was done with fifty children, whose av- erage age was 9 years 3 months. They were graded as Standard III of a municipal elementary school for boys, situated in a very good suburban neighborhood in the southeast of London. The inductive teaching was done in this case not by me, but by the teacher of the class; whilst the group which studied the written definitions was taken, during that particular lesson, by the head master of the school. All the tests were G9 70 INDUCTIVE VS. DEDUCTIVE METHODS. administered by the class teacher, who had had some experience of research work in biology as well as in experimental pedagogy. One of the boys' fathers told him he was doing Euclid (which he wasn't), and gave him a 'tip' or two which affected some of his papers adversely; but, with that exception, the suc- cess of the experiment was not hindered by any pre- vious knowledge on the part of the children. Whereas, with the Standard V class of girls, in the experiment just described, the teacher's methods were instructional rather than either definitely in- ductive, deductive, or memoriter, and with the Stand- ards VI and VII class of girls, in the experiment which has just been described, the teacher's methods were both inductive and memoriter, according to the subject-matter dealt with; in this third case the re- action against unintelligent teaching had gone so far that, whilst the inductive teaching was extremely good, the memoriter work was decidedly novel to the children. Novelty has a stimulating influence, we all know, but it is unlikely that its influence is more ef- fective in result than that of habitual practices. In any case it is essential to try the experiment with classes differently taught. 2. The Preliminary Tests and the Method of Marking. Just as before, drawings of squares, triangles, ob- longs and diameters of circles, with their names writ- ten against the drawings, were shown to the children ; the questions, "What is a square!" etc., were written on the blackboard; the children were told to look at THIRD SERIES OF EXPERIMENTS. 71 the squares, triangles, etc., and to answer tlie ques- tions in writing as well as they could. The units of marking, as before, were obtained after a careful review of the answers actually given, and it was found that the units previously adopted were quite suitable. A few instances of the children 's attempts at spontaneous definition may be worth quoting. It must be remembered that these children were considerably younger than either of the classes of girls whose work has previously been described, and that they were graded as Standard III as com- pared with Standards V, VI and VII. On the other hand, the school was much more favorably situated socially than either of the schools for girls. More- over, it was a boys' school; and boys, whether through greater natural ability or more training, are more proficient, geometrically, than girls. R. D., aged 9 years 1 month, wrote : 1. A square is a four sided figure with four points and the sides are all equal. 2. A triangle is a three sided figure with three points and the sides equal. 3. An oblong is a four sided figure with two sides long and two sides short. 4. A diameter is a strait line that goes anything like a circle and will go across any way. If we mark this paper — R. D.'s first preliminary test — on the system of marking adopted in the pre- vious experiments,* we see that for his definition of a square he receives one mark for 'figure,' one for the adjective 'sided,' one for the numerical adjective ' four, ' and one for the equality of the sides. ' Points ' are taken as equivalent to angles or corners, and *The reader is recommended to turn to page 28 for the list of units. i'l INDUCTIVE VS. DEDUCTIVE METHODS. therefore receives a mark, whilst the numerical ad- jective 'four' also scores. This gives a total of six marks for the definition of the square. The definition of triangle receives one mark for 'figure,' one mark for 'sided,' one mark for 'three,' one for 'points,' and another for 'three' (points). 'The sides equal' receives a mark as a 'bad error,' but there were so few of these in the preliminary tests that they were not tabulated. The boy's definition of oblong receives a mark for 'figure,' one for 'sided,' one for 'four,' one for 'two sides long,' and another for 'two sides short.' His last definition is rather weak. He obtains a mark for 'hue' and one for 'strait,' and that is all. When one remembers that these are untaught, spontaneous definitions given by a boy 9 years of age, we shall, I am sure, regard them as affording evidence of considerable ability. Four times the boys answered these questions without help and without criticism, and advanced a little each time. This is what K. D. wrote on his fourth attempt — the fourth preliminary test — three days after the first: 1. A sqnai'e is a four sided figure ^yitll four equal sides and four sliai'p points. 2. A triangle is a three sided figure with three equal sides to it and it has three sharp points, 3. An oblong is a four sided figure with four points but the sides are not all the same two sides one length and the other sides another length. 4. The diameter of a oirele is a line that is going from one side to the other side of the circle and that is called the diameter of a circle and the line is quite strait. Let US see how far this fourth paper is in advance of the first. The definition of a square receives the same mark as before; it is slightly more concise in THIRD SERIES OF EXPERIMENTS. 73 expression, but the units of correct description are the same in number in both cases. The definition of triangle remains unaltered. It is interesting and important to notice that even a clever boy may go on perpetrating a 'bad error' unless his attention is drawn to it, which, of course, the conditions of the experiment did not permit us to do in these tests. That E. D. is clever for a nine-year-old boy is clearer from his next two definitions than from those of the square and triangle. He nearly doubles his previous mark for his definition of an oblong. He now receives marks for 'points,' for 'four' (points), for 'two sides long' and for 'two sides short,' and also for 'two long sides equal' and 'two short sides equal.' His definition of diameter has also improved. He has now included the condition that it must go from one side of the circle to the other. These papers of R. D. are extremely good ones, and do not represent the average mark of the class, which ranges from 11 to 13 units, rather than from 18 to 23, which R. D. obtains for his first and fourth papers, respectively. Let me now give examples of some papers below the average. J. C, aged 9 years 2 months, answered his first pre- liminary test as follows : 1. A square is four put into one shape with equal sides. 2. A triangle is a thing that has no equal sides, two are equal and one is not, and it has three sides. 3. An oblong is not a square, but it is a long one. 4. The diameter is a line drawn through the midle of a circle. Side by side with this — J. C.'s first preliminary 74 INDUCTIVE VS. DEDUCTIVE METHODS. test — let us compare the paper worked by him three days later — his fourth preliminary test : 1. A square is a shape of a block with four equal sides. 2. A triangle is a long square with only three sides, the two side ones are both the same and the top one is not. 3. An oblong is a square that is long, with two equal sides and two ends whicli are not the same size. 4. The diameter of a circle is a line drawn down the midle. The marks for the definition of a square are in both cases the same: 'shape' receives a mark, 'sides' receives one, 'four' gets one, and 'equal' (sides) gets one. The two definitions of a triangle receive the same mark: there is a mark for 'sides' and one for 'three' (sides), and that is all. The first definition of an oblong receives no marks at all, whilst the one given later receives a mark for 'sides,' a mark for 'two equal' (sides), and one for ' ' two ends which are not the same size as the others. ' ' His definition of a diameter remains unchanged throughout the preliminary tests ; in each case it re- ceives two marks only, one for 'line' and one for ' drawn through the middle. ' J. C. advances from a mark of 8 in the first test to 11 in the fourth.* Having given some indications of the work done in the Preliminary Tests, on the results of which the class was divided into two equal groups, let me set out in detail the chronological progress of the experi- ment. *The average improvement from test to test is shown on page 91. THIRD SERIES OF EXPERIMENTS. 75 3. Chronology of the Experiment. A first Preliminary Test was given at 9.40 A. M. on Monday, October 23, 1911, immediately after Scripture lesson; a second on Tuesday, October 24, a third on Wednesday, October 25, and a fourth on Tlmrsda)^, October 26, at the same hour and after the same lesson on each occasion. On the results of these four tests the class was divided into two equal groups. On Thursday, November 9, at 9.40 A. M., one of the two groups was taught the definitions inductively by the methods already described, whilst the other studied them as written out, with reference to the drawings appended to the verbal descriptions. Twenty-three minutes were taken by the teacher to teach the definitions inductively ; the same time was allowed to the group which was studying the defini- tions with a view to remembering them. Both groups of children were aware that they were to be tested on their work at the close of the lesson. Ac- cordingly, at 10.15 A. M., a test was given in immedi- ate reproduction. In this school, since the children were young and the exercises very novel, we thought it best to take another test, identical with the test of immediate reproduction, at the same hour on the next day, Friday, November 10, to see how far the first day's test was reliable. These two tests will be referred to as the two Tests of Immediate Eepro- duction. At 9.40 A. M. on Thursday, November 16, a test was given on the application of what had been learnt to new analogous material with the object of discov- 76 INDUCTIVE VS. DEDUCTIVE METHODS. ering which of the two groups attacked the new mate- rial the more successfully. Finally, two tests of deferred reproduction were given at 9.40 A. M. on Thursday, December 7, and Friday, December 8. The children were quite una- ware that they would be required to take any of these tests, with the exception of the one immediately after the teaching and learning on Thursday, November 9. 4. The Final Tests and the Method of Marking. The two tests of Iimnediate Reproduction were repetitions of the Preliminary Tests, as were also the two tests of Deferred Reproduction. The units of marking previously used in the Preliminary Tests were found quite satisfactory. The tests of deferred reproduction received negative as well as positive marks. One or two specimens of the worked papers may be of interest. L. 0., aged 9 years, who scored 13, 16, 18, 18 in his preliminary tests, and was taught inductively, for his first test of Immediate Reproduction on November 9, wrote as follows : 1. A square is a shape with four lines all the same size and for corners all the same size. 2. A triangle is a shape with three corners and three lines. 3. A oblong is a shape with two long lines the same size, and two shorter lines the same size. 4. A diameter of a circle is a line which goes from one part to the opposite part touching the middle of the circle and keeps inside the circle. This is a good set of answers for a boy only nine years of age. Marked on the system of units previ- ously used, the definition of a square receives seven marks, the definition of a triangle receives five THIRD SERIES OF EXPERIMENTS. 77 marks, that of the oblong receives seven marks, and that of the diameter of a circle receives three marks. It will be seen that, compared with the standard defi- nitions, there is a loss of one mark in the definition of a square, since the description 'straight' is not applied to the 'lines' or 'sides.' The definition of triangle is correct. Six marks are lost on the oblong. ' Four equal cor- ners ' are omitted, carrying three marks. 'Straight' is omitted in describing the lines or sides, and the two long lines and the two short lines are not described as opposite. One mark only is lost on the definition of 'diam- eter;' the line is not described as 'straight.' The marks, totaled, amount to 22. On the next day's test L. 0. goes down one mark. His definitions of square and triangle remain un- changed. In his definition of oblong he omits the two points previously inserted, namely, that the two long lines are of the same length, and that the two short lines are of the same length. But in the definition of the diameter of a circle he inserts the description 'straight' which he had omitted the day before alto- gether. He thus scores 21 marks for his second test. Let us now see what happens a month later when the same test is applied a third time. I give his paper in full. L. 0., aged 9 years 1 month, on December 7, 1911, in his first test of Deferred Eeproduction, wrote as follows : 1. A square is a shape with four lines all the same length, and four corners all the same size. 2. A triangle is a shape with three lines and three corners. 3. An oblong is a shape with four lines two long lines both the same size, and two shorter lines both the same length. 78 INDUCTIVE VS. DEDUCTIVE METHODS. 4. A diameter of a circle is a line inside which goes from one part of the circle and touches the middle of the circle goes on to the opposite part of the circle to where it started and it must be a straight line. This is a very good paper, and scores a total of 23 marks, an advance on the work of the raonth before. On the day following, on which was given the second test of Deferred Reproduction, L. O. scored 24 marks, for the description 'straight' of the sides of the square, omitted on December 7, was included on December 8. His average mark for his two tests of Immediate Reproduction, those, namely, of the 9th and 10th of November, was 21.5; his average mark for his two tests of Deferred Reproduction was 23.5. It must not be thought that every boy obtains more marks a month after the lesson than he does for his immediate tests, but many of them do; and the aver- age result shows only a slight decline, rather more marked in the group taught deductively than in the group taught inductively. This is explained by the fact that both the teaching and the learning were well within the comprehension of the boys. When this is the case, and they work in consequence with interest and enthusiasm, they forget surprisingly little. It may now be of some value if I give the corre- sponding papers of a boy in the Deductive Group. R. S., aged 9 years 2 months, who scored 14, 19, 19, 18 marks in his four preliminary tests, in his first test of Immediate Reproduction wrote : 1. A square is a shape with four sides and four corners. The sides are straight and all the same length. The corners are all the same size. 2. A triangle is a shai)e with three sides and three corners. 3. An oblong is a shape with four sides and four corners. The sides are straight and there are two long sides and two short sides. THIRD SERIES OF EXPERIMENTS. 79 The long sides are opposite oue another and are the same length, and the two short sides are opposite and are the same length. The diameter of a circle is a straight line which goes through the centre of the circle. Only two units of definition are omitted : the equal- ity of the angles is left out in the definition of the oblong, and the delimitation of the diameter by the opposite parts of the circumference of the circle is omitted in the last definition. It is an excellent pa- per, appearing on the face of it, if one compares it with the verbal expression of the definitions which were given to be studied, to owe a great deal to a highly developed rote memory. If that is so, it is memory for statements that are really understood, since they persist unchanged without the subsequent intrusion of stupid errors, and an unusually high mark is obtained by this boy for his power of appli- cation to new material. I propose to defer consid- eration of the latter issue, since just now we are con- cerned only with the tests of Immediate and De- ferred Reproduction. The next day R. S. obtained 29 marks, as compared with 28 of the previous day. There were slight changes of verbal expression. For instance, the tri- angle became "a three cornered figure with three sides." The equality of the angles was omitted in the square, but on this occasion, though not in the previous test, was included in the definition of the oblong. In the definition of the circle an improve- ment was shown ; the point was included which the day before had been omitted; it was now men- tioned that the line went from ''one side to the oppo- site side of the circle." One month later R. S. scored 28 marks. He 80 INDUCTIVE VS. DEDUCTIVE METHODS. omitted the description 'straight' in his definition of a diameter of a circle which he had before included. On the day following he made the same omission. Otherwise his definitions are just as good as those which he had given a month before. His average mark for Immediate Reproduction is 28.5, and for Deferred Reproduction is 28.0. Let me give one more illustration, the work of a boy who obtained 6, 6, 7 and 7 marks in his four pre- liminary tests, and who also was taught in the De- ductive Group. H. W., aged 10 years 1 month, in his first test of Immediate Reproduction wrote : 1. A square is a shape with four corners aud four sides the same size. 2. A triangle is a shape with three corners and three sides. 3. An oblong is a shape with two small sides, and two big sides opposite one another. 4. The diameter of a circle is a line passing through the middle of the circle. Marked on the same units as before, the definition of a square obtains six marks; the definition of tri- angle obtains five marks ; the definition of oblong re- ceives six marks, for it is called a 'shape,' its 'four' 'sides' are implied, its 'two long sides' and its 'two short sides' are noted, and the fact that its 'two long sides are opposite each other.' The definition of diameter receives two marks. This is not a strong paper; it scores 19 marks only as a total, but it im- plies a very great advance on this boy's preliminary tests. One point of interest lies in this. Whereas, in the papers of R. S., recently given, there was an appearance of rote learning in the answers, there is, in the case of this boy, no direct indication of that. H. W.'s next test of Immediate Reproduction, THIRD SERIES OF EXPERIMENTS. 81 worked on the following day, receives the same num- ber of positive marks, namely, 19. A 'bad error' has crept in, for the corners of the triangle are described as all the same size. The 'two small sides' of the oblong are now called "two small tops," but this and the 'bad error' are the only changes. One month later, for his first test of Deferred Reproduction, H. W. wrote : 1. A square is a shape with four corners and four sides opposite one another and they are all of the same length. 2. A triangle is a shape with three corners and three sides they are not opposite one another. 3. An oblong is a shape with three corners and three sides, it is a zig-zag shape not all the same length. 4. A diameter of a circle is a line passing through the middle of it. Considerable changes are evident in this paper. There are, as before, six positive units of correct description in the definition of the square; but the statement "four sides opposite one another" has been adjudged a 'bad error.' It is, of course, the confused application of some phrase remembered, but not understood. Let it not be supposed, however, that no child inductively taught makes similar errors. The definition of triangle receives five marks as before. The memory of the oblong has largely gone. It is still remembered that it is a 'shape' and has 'corners' and 'sides,' and thus three positive marks are obtained. But to give an oblong ' three ' corners and 'three' sides and to call it 'zig-zag' shape is held to involve three bad errors. The definition of diam- eter remains unchanged, and scores two marks. The paper as a whole receives 16 positive marks, and there are four marks for bad errors. 82 INDUCTIVE VS. DEDUCTIVE METHODS. But on the next clay, in his second test of Deferred Eeproduction, H. W. made a decided recovery. He then wrote : 1. A square is a shape with four corners and four sides they are opposite one another, with all the sides and corners an equal size. 2. A triangle is a shape with three sides and corners it is a zig-zag shape. 3. An oblong is a shape with four sides, two long sides and two short tops they are opposite one another. 4. A diameter of a circle is a line passing through the middle of it. This is undoubtedly H. W. 's best paper. He scores the highest marks he has yet scored for the definition of the square, namely, seven positive marks, since, for the first time, he has mentioned the equality of the corners, but he retains liis 'bad errors.' The defi- nition of a triangle remains unchanged in correct units ; it is held inadmissible to call the triangle a zig- zag shape. The definition of oblong has returned to its first condition; indeed, it is rather better, for it is easier now to regard H. W. as implying that the 'two long' and 'two short' sides are pairs of equals. The mark for the double equality is, however, not given, as the meaning is somewhat doubtful. The definition of diameter remains unchanged. H. W. scores 20 positive marks for his paper and one nega- tive mark for a 'bad error.' Again we find the marks for Deferred Reproduction not much inferior to those of Immediate Reproduction in this case; in- deed, the last paper is the best the boy did through- out the series. I trust that the inclusion of these papers will be of service in giving life and body to the rather bloodless array of figures, which I give subsequently, dealing THIRD SERIES OP EXPERIMENTS. 83 with the results of the tests in Immediate and De- ferred Reproduction. The Test of Application to New Material was iden- tical witli that used in the experiment j^reviously de- scribed. Drawings of rhombuses, trapeziums, rhom- boids and diagonals of squares, with their names appended, were shown to the children, and they were required to answer in writing the questions: "What is a rhombus?" etc. It may add to the facility with which the progress of this experiment is understood if I give verbatim one or two of the worked papers. In the test of application to new material negative marks were assigned as well as positive marks. L. 0., aged 9 years, a boy who worked in the In- ductive Group, whose work in Immediate and De- ferred Reproduction has already been quoted, wrote the following paper in this test : 1. A rhombus is a shape contaiiiiug four lines all the same length, so that if you loolved at it one \A'ay it seems to bend back- ward, and if you look at it again it looks to bend forward. 2. A rhomboid is a shape also containing four lines, two long lines both the same length, and two shorter lines both the same lengtli. 9. A trapezium is a shape with four lines three long ones, and one short one. 4. A diagonal of a square is a streight line going from one cor- ner to its opposite one. -^ L. 0. receives four marks for his definition of rhombus — one for 'shape,' one for 'lines,' one for 'four,' and one for all the same 'length.' He receives three marks for his definition of trapezium, one for 'shape,' one for 'lines,' and one for 'four.' His statement that there are three long lines and one short one was not held to be equivalent to the state- ment that the sides were unequal, but it was not con- sidered a ' bad error. ' For his definition of rhomboid 84 INDUCTIVE VS. DEDUCTIVE METHODS. he obtains seven marks — one for 'sliape,' one for 'lines,' one for 'four,' one for 'two long- lines,' one for 'two shorter lines,' and two for the pair of equal- ities in the length of the lines. The definition of diagonal receives four marks, one for 'line,' one for "streight," and two for "from one corner to its opposite one. ' ' The paper scores a total of 18 positive marks, and there are no 'bad errors ;' the average mark obtained by the boys of the Inductive Group is rather lower than this. R. S., aged 8 years 5 months, who also was taught in the Inductive Group, wrote : 1. A rhombus is a figure with four straight sides and four equal corners. 2. A trapezium is a figure with four corners which are equal with four sides. 3. A rhomboid is a figure with two small sides which are hori- zontal and two bigger parlerlell lines equal. 4. A diagonal of a square is a straight line from one corner to another corner. The first definition receives a mark for 'figure,' a mark for 'sides,' one for 'four,' one for 'straight,' one for 'corners,' and one for 'four;' a total of six positive marks; but there is one 'bad error' — the corners are not equal: boys taught inductively can obviously make the same sort of blatant error as boys taught deductively. The second definition receives a mark for ' figure, ' one for ' corners, ' one for ' four, ' one for ' sides, ' and one for 'four' (sides) ; a total of five positive marks ; but, again, there is a 'bad error' — the angles of the trapezium are not equal. The definition of rhomboid obtains seven positive marks — one for 'figure,' one for 'sides,' one for 'two small' (sides), one for 'two THIRD SERIES OF EXPERIMENTS. 85 bigger' lines, a mark for saying the two bigger are equal, and one for saying the two bigger lines are parallel. There is one 'bad error;' the two small sides were in one case only drawn horizontally. The definition of diagonal receives three positive marks — one for 'line,' one for 'straight,' and one for "from one corner to another : ' ' the further specification of 'opposite' corner is omitted. The paper, as a whole, gains 21 positive marks, with three negative marks for ' bad errors. ' Let me quote one more illustration from among the boys who were taught inductively. H. B., aged 9 years 2 months, wrote : 1. A rhombus is a fugare which is like a square and has fore corners. 2. A trapezium is something like a triangle only it has fore corners. 3. A rhomboid is a sought of fugare which is something like an oblong. 4. A diagonal of a square is a square with a line across the midal. This is a very weak paper ; it was worked by a boy who was almost at the bottom of the Inductive Group in the preliminary tests, and he seemed to jus- tify his position. It is psychologically interesting that he apprehended the similarity between the work now required and the work he had been taught, but was unable to specify the points of similarity and ditference between the figures of the first set and the figures of the second set. He had but little know- ledge and could not apply much of that. His marks are: three for his definition of rhombus, two for his definition of trapezium, one for his definition of rhomboid, and one for the definition of the diagonal of a square. "Across the midal" is not held to be 86 INDUCTIVE VS. DEDUCTIVE METHODS. wrong, though it might be ; in any case it is not re- garded as sufficiently definite to obtain a mark. It is regarded as a 'bad error' to say that the diagonal of a square is a square. H. B. receives a total of seven positive marks, with one negative mark for bad errors. It is, perhaps, worthy of note that this boy falls from 17 in his test of Immediate Reproduction to 9 in his test of Deferred Reproduction. He can- not apply his old knowledge and he cannot remember it for more than a day or two. Let us now turn to some illustrative examples of the work of the group taught deductively. G. M., aged 8 years 1 month, wrote : 1. A rbombus is a figure with two slanting sides and two straight ones arranged so that two of the sides are facing each other and the other two opposite each otlier and also four corners. 2. A trapezium is a figure with four slanting sides arranged so that there are two sides nearly the same length, these two are generally touching each other. Then there is a smaller one and yet a smaller one still, so that there are four sides and two equal ones the others ofcourse are not. 3. A rhomboid is a figure with two slanting sides and two straight ones and also four corners two of the sides are longer than the other two and also are opposite one another and so are the two shorter sides. There can be ones upright and lying down and also slanting ones. 4. A diagonal of a square is a line drawn from one corner to the other it need not have to be drawn from a corner for it could be from the middle of the top to the middle of the bottom, but you can't have it so that it is from the middle of the one side to the middle of the bottom or to the middle of the top. For the diameter is the greatest and longest line you can have across it or down it and that wouldn't be the longest, not nearly. This is an excellent paper for a boy of eight years of age. He was taught in the Deductive Group, but evidently he is quite capable of applying what he has learnt. It would be a serious error to suppose that because a boy has learnt a set of definitions therefore THIED SERIES OF EXPERIMENTS. 87 he cannot apply tliem. In a very large number of cases he certainly can. The contention raised in this monograph is that inductive teaching produces a higher transfer to new material than deductive, not that deductive teaching involves no transfer at all. This first-rate paper may do something to prevent an exaggerated conclusion which the subsequent fig- ures may not succeed in adequately moderating. Let us mark the paper on the usual system of units. G. M. is evidently using the word straight to mean, as it often does with boys, horizontal and vertical ; he does not mean that only two of the lines are ' straight ' in the proper sense. And he is wrong on his own mean- ing, for one of the rhombuses drawn had neither ver- tical nor horizontal lines, but two of them had, and to these he has apparently confined his attention. He receives a mark for * figure,' a mark for 'sides,' and one for 'four' (sides), which is involved in his pair of twos, and one for 'corners' and one for 'four.' He gets two marks for seeing that the opposite sides are paired. This marking yields a total of seven pos- itive marks, whilst he receives a negative mark for being wrong on his own meaning of 'straight.' In his definition of trapezium he receives a mark for 'figure,' one for 'sides,' and one for 'four.' His first description of the sides is held to be equivalent to saying they are unequal, so he receives a mark for that. Later he is marked for a 'bad error' in saying that two of the sides are equal. They are so in one of the trapeziums only. For the definition of trape- zium, then, he gets four positive marks, with one neg- ative mark for a 'bad error.' Again, in his definition of a rhomboid we find a misuse of the word straight, and again he is wrong, even on his own meaning. 88 INDUCTIVE VS. DEDUCTIVE METHODS. But be obtains positive marks for 'figure,' for 'sides,' for 'four' (sides), for 'two long' (sides), for 'two shorter' (sides), and two marks for noting tbe pairs of opposites. He also notes tbe 'four corners.' He tbus receives nine positive marks and one mark for a 'bad error.' His definition of diagonal is extremely interesting. He receives two positive marks only — one for 'line' and one for "from one corner to anotber." After tbat, alas! tbe transfer from diameter (tbe corre- sponding definition wbicb was learnt) bas been too tborougb. No diameters were drawn in tbe squares wbicb were before tbe boy's eyes, and it is not unfair to call tbe lapse into diameter a 'bad error.' Tbis definition receives therefore two positive marks and one negative mark. Tbe paper, as a wbole, receives a total of 22 positive marks, and tliere are four bad errors; it is considerably above tbe average of tbe papers worked by tbe Deductive Group generally. H. W., aged 10 years 1 month, whose work in Im- mediate and Deferred Reproduction has already been quoted, wrote tbe following in bis test of appli- cation to new material : 1. A rhombus is a shape something like the shape of a diamond. 2. A trapezium is a shape with four corners not opposite one another their are different shapes of trapeziums they are a zig- zag shape some corners longer than others, they are not squares. 3. A rhomboid is a shape with two small tops both opposite one another, and with two long sides with the corners exactly opposite one another. 4. A diagonal of a square there is a square and a line passes right through. Sometimes they pass from side to side Qther times from corner to corner. H. W.'s definition of rhombus receives one mark only — a mark for tbe description 'shape.' For tbe definition of traiDezium three positive marks are THIRD SERIES OF EXPERIMENTS. 89 gained — one for 'shape,' one for 'corners,' and one for 'four.' There are no 'bad errors.' It was not thought admissible to regard the expression "some corners longer than others" as involving the ine- quality of the angles. His definition of a rhomboid receives a mark for 'shape,' one for 'sides,' one for 'corners,' and one for 'four' sides, for the number of sides is involved in the rest of his answer. He also receives a mark for "two small tops," one for "two long sides," and one for noting that the two small sides are 'opposite' each other. The opposition of the angles has not been allowed for in the system of marking. This definition therefore receives a total of seven positive marks. The definition of diagonal receives two positive marks only — one for 'line' and one for 'from corner to corner.' It was regarded as a bad error to say that "sometimes they pass from side to side. ' ' The total marks for this paper amount to 13 positive marks, from which one has to be de- ducted for ' bad errors. ' Let me now pass to the work of a boy who scored 13, 12, 12 and 11 in his four preliminary tests. It seems likely from these figures that we are dealing with a boy of little educability, and this suggestion is confirmed by his later work.. In his two tests of Im- mediate Reproduction he scores an average of 18.5 marks; in both tests of Deferred Reproduction he scores 13 marks, so that a month afterwards he is back again to the position he occupied before he learnt the definitions, and he completely fails in ap- plying what he has learnt. A. R., aged 8 years 6 months, the boy whose work has just been described generally, wrote : 1. A rbouabiis is a square which is not strate up. 90 INDUCTIVE VS. DEDUCTIVE METHODS. 2. A trapezium is a four-sided thing wliicli sides are not all strate. 3. A rboiuboid is lilve an oblong but its lines are not strate up. 4. A diagonal of a squai'e is a diameter of a circle only it is a squear. A. R. has seen some general resemblance between the 'figures' of his first set of definitions and those of his second set, bnt the resemblances have hindered rather than helped him, for a rhombus is not a square, and a diagonal of a square is 7iot a diameter of a circle. The meaning of the word ''strate" is misconceived ; his reference to the sides of the trape- zium is not incorrect on the basis of his own mean- ing. Of positive marks, on the system of marking adopted, he can obviously obtain very few. He scores no marks for his definition of rhombus, two for his definition of trapezium, one for his definition of rhomboid, and none for his definition of diagonal. His three positive marks are subject to a deduction of two for the 'bad errors' previously indicated. Boys of this kind are the despair of the teacher, but the evidences ^delded by his work do not point so much to stupidity as to ineducability. Possibly the reader may already have gathered from a perusal of the papers which I have used as illustrations some opinions of his own as to the rela- tive applicability of the two methods of teaching and learning. But all such opinions need to be confirmed or modified by a consideration of the tables of results which are set out in the next section. 5. Results of the Experiments, (a) Results of the Preliminary Tests. The marks for the four preliminary tests were fairly steady, decidedly so, when the age of the chil- THIED SERIES OF EXPERIMENTS. 91 dren was taken into consideration, A'^ery few of the boys made any violent jumps, and there was a gen- eral improvement from exercise to exercise. In the first test the average mark was 11.1, in the second 12.3, in the third 12.9, and in the fourth 13.1. The correspondences between the results of the first, second, third, and fourth Preliminary Tests are shown compendiously in the following table : Table XV, shoicing the correlation beticecn the results of the four Preliminary Tests. Marks in the four No. ,— Average Marks in Preliminarj' Tests.— ^ preliminary of First Second Tliird Fourth tests. boys. test. test. test. test. 70 and over 4 16.5 19.8 18.3 18.8 60 to 70 8 14.8 14.4 16.5 17.3 50 to 60 11 12.5 13.4 13.9 14.3 40 to 50 15 10.4 11.3 12.1 11.9 Below 40 12 6.7 7.9 8.9 9.0 There is obviously high positive correlation be- tween the results of the successive preliminary tests. The mental functions we are testing appear to be working very steadily. Exact numerical values for the coefficients of correlation have been worked out from the 50 individual cases on the Pearson formula. Between the results of Tests 1 and 2 the correlation coefficient is + ,76 (probable error ,04), between Tests 2 and 3 is + .79 (probable error ,03), and be- tween Tests 3 and 4 is -j- ,80 (probable error ,03). These high correlations between the results of the successive tests give us reasonable expectations that a division into two equal groups may be satisfac- torily effected. The boys were divided into two equal groups containing 25 children each. The fol- lowing table will indicate how the division was made : 92 INDUCTIVE VS. DEDUCTIVE METHODS. Table XVI, showing the Division into Tiro Equal Groups. Group A. Name (Initials , Marks for Preliminary Tests. , only). First. Second. Third. Fourth. Total. R. D 18 22 18 22 80 A. C 17 18 19 16 70 L. 13 16 18 18 Go W. G i 9 6 12 31 G. k .3 5 7 10 25 Averages 11.1 12.4 12.9 1.3.3 49.7 M. V.'s 3.5 2.9 2.5 2.4 Group B. Name (Initials , Marks for Preliminary Tests. n only). First. Second. Third. Fourth. Total. H. B 17 20 17 17 71 R. S 14 19 19 18 70 C. L 16 17 16 18 67 S. B. . 9 8 il 5 33 A. W 4 8 6 6 24 Averages 11.1 12.3 13.0 12.8 49.3 M. V.'s 2.7 2.8 2.7 3.3 The average mark per boy per test for Group A was 12.4 (mean variation 2.6), and for Group B was 12.3 (mean variation 2.6). The average age of Group A was 9 years 3 months, and of Group B was also 9 years 3 months. (b) Results of the Tests in Immediate and Deferred Reproduction. It now remains to be shown which of the two groups was the more successful when tested on pre- cisely what they had been taught or learnt. THIRD SERIES OF EXPERIMENTS. 93 First, let me give the marks of the two groups as wholes, together with their variability: Tabic XVII, slioicing the icork of the Inductive and Deductive Groups compared, in the Preliminary Tests and in the Tests of Immediate and Deferred Reproduction (positive marks only). 1 First Second IKS. ^ For all imme- imme- First Second four diate diate deferred deferred preliminary repro- repro- repro- repro- tests. duction. duction. duction. duction. Inductive group. . . 12.4 18.8 18.6 18.0 18.1 M. V.'s 2.6 2.6 2.5 3.0 3.2 Deductive group. . . 12.3 20.5 20.6 18.8 19.4 M. V.'s 2.6 3.4 4.1 3.4 3.5 In the tests for deferred reproduction, it will be remembered, negative marks were given as well as positive marks. The marks for the two groups are given below after the negative marks have been sub- tracted from the positive marks : TaWe XVIII, showing the marks (after deduction) for the Induct- ive and Deductive Groups compared, in the Preliminary Tests, and in the Tests of Deferred Reproduction. Average mark 1 Average First Marks. ^ Second for four pre- deferred deferred Inductive gi'oup. . M. V.'s liminary tests. 12.4 2.6 reproduction. 17.7 3.2 reproduction. 17.8 3.4 Deductive group.. M. V.'s 12.3 2.6 18.6 3.6 19.1 3.7 There seems no doubt that, when the tests are given on precisely the subject-matter which has been learnt or taught, the group which learnt the definitions did better work than that which was taught inductively, and this is true both in immediate and deferred re- production, and for both positive and negative 94 INDUCTIVE VS. DEDUCTIVE METHODS. marks. This conclusion must, of course, be drawn subject to the age and mental proficiency of the pu- pils. It now remains to be seen whether the differ- ence between the groups is one which is common to the more proficient as well as to the less proficient pupils : Tabic XIX, showing the marks of the two groups compared, section by section, in the Preliminary Tests and in the Tests of Imme- diate and Deferred Reproduction (positive marks only). I Inductive Group. , , Deductive Group. , Average Average mark of Average mark of Average two tests mark of two tests mark of Marks imme- two tests imme- two tests in four No. diate deferred No. diate deferred preliminary of repro- rejiro- of repro- repro- tests. boys, duction. duction. boys, duction. duction. 70 and over. .. 2 19.0 19.2 2 24.0 21.0 60 to 70 4 21.2 19.5 4 22.5 20.9 50 to 60 5 19.7 10.2 6 22.9 21.0 40 to 50 8 18.0 18.5 7 18.8 18.3 Below 40 6 17.0 14.8 6 17.8 16.0 It seems clear that there is a balance of advantage all along the line in favor of the group which learnt the definitions, so far, at least, as the positive marks are concerned. It now remains to be shown whether this is also true when the negative marks are de- ducted from the positive marks : Table XX, showing the marks of the two groups compared, section by section, in the Prelim inanj Tests and in the Tests of De- ferred Reproduction (after the negative marks have been de- ducted). r-Inductive Group.— ^ ^—Deductive Group.— ^ Marks Average Average in four No. mark in two No. mark in two preliminary of tests deferred of tests deferred tests. boys. reproduction. boys. reproduction. 70 and over 2 19.3 2 21.5 60 to 70 4 19.1 4 20.5 50 to 60 5 19.1 6 20.9 40 to 50 8 18.4 7 18.4 Below 40 6 14.2 6 15.4 THIRD SERIES OF EXPERIMENTS. 95 Again, there seems a decided balance of advantage on the side of the group which learnt the definitions deductively. (c) Correlation Between the Results of Immediate and Deferred Reproduction. It would seem likely from the tables given above that the tests given immediately after the teaching and learning may be regarded as fairly significant of the relative position of the two groups even after considerable time has elapsed — in this case after a month. As this is a very important issue for experi- mental pedagogy, it may be well to subject the hy- pothesis to further determination. The following tables will show in a general way how far the sug- gestion may be taken as valid : Table XXI, showing the correlation heticeen the marks obtained in the various Tests of Reproduction {positive marks only). Inductive Group. Marks for the first test of No. Average Marks per Boy in the Repro- inimediate of ductive Tests. reproduction. boys. First. Second. Third. Fourth. Over 25 2 27.0 26.5 23.5 25.5 20 to 25 3 22.3 19.7 20.7 20.7 18 to 20 7 19.6 19.7 19.1 19.7 16 to 18 6 17.7 18.1 16.0 15.7 15 to 16 4 16.0 15.0 16.0 16.0 15 and under 3 14.3 15.3 16.0 14.3 Deductive Group. Marks for the first test of No. Average Marks per Boy in the Repro- immediate of ductive Tests, reproduction. boys. First. Second. Third. Fourth. Over 28 2 29.0 27.0 25.5 26.0 25 to 28., 3 26.7 26.0 22.3 22.7 21 to 25 4 23.5 23.2 22.2 23.2 17 to 21 8 19.7 20.4 18.5 19.7 16 to 17 4 17.0 18.7 16.2 15.5 16 and under 4 13.7 13.2 12.5 12.7 96 INDUCTIVE VS. DEDUCTIVE METHODS. It is obvious that considerable positive correlation exists between the results of the successive exercises. A more precise determination may, of course, be made by means of a correlation coefficient. Worked out by the standard formula from the individual cases, the following are the coefficients : For the In- ductive Group the results of the first Test of Eepro- duction correlate with those of the second to the extent of + .78, the second with the third to the ex- tent of + .57, and the third with the fourth to the extent of + .85. For the Deductive Group the corre- lation coefficients are : first and second tests, + .86 ; second and third tests, + .68 ; third and fourth tests, + .94. It is obvious that tests given immediately may be fairly regarded as indicative of what will happen later on, at least in such exercises as these, when we are making comparison of one group with another. (d) Results of the Test on New Material. We have seen already, when the tests required an exact reproduction of what had been learnt or taught, that the children in this school who learnt deduc- tively, like the Standard V children of the girls' school in the experiment first described, did better work than the group taught inductively. But in both the experiments previously described it was found, when the test given was on new material, that the children taught inductively did better work than those who learnt their definitions. Is this advantage also to be found on the side of the inductive group in this school? These children are younger and are less proficient mentally, according to school grading, THIRD SERIES OF EXPERIMENTS. 97 than either of the girls' classes whose work has been described. Moreover, they are boys, not girls. Are these variations in conditions such as to produce a difference in the results? It will further be remem- bered that the inductive group, in this case, was taught by its own teacher, and not by me, so that any intensity of impression due to personal novelty was thereby eliminated. Perhaps I may be pardoned for a sentence of ap- parent digression. I hold it extremely important for the science of experimental pedagogy that no result should be taken as valid for general application un- less the use of it is justified by its success in the hands of the usual teachers of the school. Its success in the hands of the specialist or other exceptional per- son is quite insufficient to recommend it for general adoption. Let us, then, see what the results were when the whole experiment was conducted by the teachers themselves. I shall show the work of the two groups compared both in the Preliminary Tests and in the Test of Application to New Material. First, let me give the results of the two groups as wholes : TaT)le XXII, showing the toorJc of the tiro groups compared, in the Preliminary Tests and in the Test on Neto Material. Average mark for four preliminary tests. Inductive gi*oup 12.4 M. V.'s 2.6 Deductive group 12.3 M. v.'s 2.0 Again we find, notwithstanding the decidedly su- perior acquisition of the material studied (see Table Average Mark for New Material. Positive Marks after marlis. deduction. 16.3 15.6 3.5 4.0 15.7 14.9 5.8 5.2 98 INDUCTIVE VS. DEDUCTIVE METHODS. XX) on the part of the deductive group, that they are inferior to the other in their power to attack new material of an analogous nature. Four boys in the deductive group completely failed to make a reason- able application of their old knowledge, obtaining only 6, 7, 2 and 4 marks, respectively, whilst only one boy in the inductive group failed to do so, and he obtained 8 marks. Let us now see, as we have in previous cases, how far this difference is to be found for the weaker as well as for the abler children of each group : Table XXIII, shoiciiig the irork of the tico groups compared, sec- tion by section, in the Prcliminarii Tests and in the Test of Application to Neiv Matrrial (positive marks, and positive inarks after the deduction of the negative marks). Group Inductively Taught. Marks for Application to Marks New Material. in four No. , -> preliminary of Positive After tests. boys. only. deduction. 70 and over 2 18.5 18.0 60 to 70 4 18.3 17.0 50 to 60 5 17.8 17.6 40 to 50 8 15.5 15.0 Below 40 6 14.0 13.2 Group Deductively Taught, Marks for Application to Marks New Material. in four No. r ^ preliminary of Positive After tests. boys. only. deduction. 70 and over 2 18.0 10.5 60 to 70 4 15.0 14.2 50 to 00 17.8 17.2 40 to 50 7 14.0 13.3 Below 40 6 15.3 14.6 Except in the case of the least proficient section of boys at the bottom of each group, there seems to be THIRD SERIES OF EXPERIMENTS. 99 an advantage all along the line in favor of the in- ductive group. When, therefore, the tests are tests of the application of knowledge rather than an exact reproduction of it, we are, perhaps, entitled, on the whole, to conclude that inductive methods are the better. It may be noted that whilst the marks for the inductive group proceed regularly downwards from the highest section to the lowest, those for the corre- sponding sections of the deductive group do not. The variability for this group is disproportionally high, due, doubtless, to the psychological fact that for some children of this age the step from knowledge to the application of it is a very considerable one; whereas, of course, the children of the other group had been through a process of applicable method when they had received their inductive lesson. The variability of the work is, however, decidedly high, and the difference between the means of the work of the two groups is very small; and did this experi- ment stand alone, I should hesitate before putting much confidence in the conclusion which I have indi- cated above. But its consilience with the previous results lends strength to the conclusion, especially when the differences in the conditions under which it was obtained are taken into account. VII. FOURTH SERIES OF EXPERIMENTS. 1. General Plan. In the experiment now to be described, just as in those previously recounted, the work was done with all the children of one class, under one teacher, with the same curriculum of study, and working according to the same time-table of instruction. The experi- ment was carried out in a municipal higher grade school for boys, an elementary school situated in a somewhat mixed neighborhood. The class chosen for the experiment was the First Class in the school, containing 35 boys, graded as Ex. VII on the English standard system of school grading, of an average age of approximately 131/2 years.* The teacher of the class had a theoretical acquaint- ance with psychological work, and had already car- ried out some observations in educational psychol- ogy. He was, especially, capable of temporary dis- sociation between the pedagogic and psychologic attitudes — a necessary capacity in an experimenter. Beyond this, he was a first-rate teacher who varied his methods according to the subject-matter with which he was dealing. As in previous cases, the class was divided into two equal groups on the results of tests in spontane- *In America this would constitute Grade VIII, or rather, per- haps, the First Year of High School. 100 FOURTH SERIES OF EXPERIMENTS. 101 ous definition, but the test on which the division was effected was not the same as that used for the pur- pose in the former tests. But, as before, there were tests of immediate and tests of deferred reproduc- tion, and a test of application to new material of an analogous kind. Further relevant conditions will be given in the details which follow. >s' 2. The Preliminary Tests and the Method of Marking. The first test in this series was the spontaneous definition of squares, triangles, oblongs, and diam- eters of circles, which were drawn in the way already indicated, and the questions (with which by now the reader will be quite familiar) : ^'What is a square?" etc., were set for written answers. The papers were marked on the system of units which has already been described, and an average mark was gained of 19.1 out of a possible maximum of 30. This, as might have been expected, was by far the highest mark that had been obtained by any class doing this test. It was not proposed to use this test on squares, tri- angles, etc., for the purpose of dividing the class, but it served a useful purpose as a preparatory exercise. Next week the teacher taught all the children of the class how to arrive at the definitions inductively in the way that I have described in the first series of experiments (p. 33). This lesson also rendered val- uable service. It gave full opportunity to all to un- derstand quite clearly what they had to do when they were set to attack the preliminary test on which the class ivas to be divided. The questions used for the preliminary test were 102 INDUCTIVE VS. DEDUCTIVE METHODS. the same as those which, in the previous schools, had been used as a test of the power of application to new analogous material. In one sense it is, of course, in this case, also a test of application to new material, for one inductive lesson on the square, etc., had already been given. We may, indeed, look upon our division into two ' equal groups ' in this case as being effected during the course of a series of lessons in- stead of at the very beginning of it. The questions were: ''What is a rhombus?" "What is a trapezium?" "What is a rhomboid?" and "What is a diagonal of a square?" The an- swers were marked on the system of units which has already been described.* One or two of the papers worked may be of service in enabling an experienced teacher to gauge the men- tal level of the boys taking these tests. Edward S , aged 14 years 8 months, wrote : 1. A rhombus is a figure, it has 4 equal straight lines, it has angles, there are 4, 2 equal large ones and 2 equal small ones. 2. A trapezium is a figure, it has 4 straight lines of different lengths, it has angles, there are 4, all of different sizes, any shape. 3. A rhomboid is a figure, it is enclosed by 4 straight lines, 2 equal long ones, two equal short ones. It has 4 angles, 2 equal small ones and two equal large ones. 4. A diagonal is a straight line going from one corner to the other of a square terminating at both ends dividing the square into 2 triangles. This is not the best paper ; there are four boys in the class who get higher marks, but it is obvious that we are here dealing with a very different mental level, geometrically, from those at which we have previously worked. With the table of units at hand, *The reader is recommended to turn to page 41 for the list of correct units of description. FOURTH SERIES OF EXPERIMENTS. 103 it is quite easy to mark this paper. The only diffi- culty occurs in the case of the last definition, in which the qualification opposite is omitted when the corners of the square are mentioned. It is held, however, that the statement '^ dividing the square into 2 tri- angles" is equivalent to the limitation of from one corner to the 'opposite' corner. A total of 37 posi- tive marks was gained — 11 for the definition of rhombus, 8 for the definition of trapezium, 14 for the definition of rhomboid, and 4 for the definition of a diagonal of a square. There are no 'bad errors.' Charles B , aged 13 years 9 months, wrote : 1. A rhombus is a figure or drawing consisting of 4 straight lines. All the lines are the same length and the two lines opposite one another are parrallel to one another. It has 4 corners and four equal angles. 2. A trapezium is a drawing. It has 4 straight lines, 4 angles, all the lines are of different lengths. It has two long and two short sides. The angles are all different. 3. A rhomboid is a drawing consisting of 4 lines which are straight. It has 2 long and 2 short. The 2 short are parrallel to one another, and the 2 long are parrallel to one another. It has 4 corners and four angles. The 2 long sides are the same length and the 2 short are the same length. All the angles are not equal. 4. A diagonal is a line, must be straight. It is drawni from one corner to the one opposite. It passes through the centre of the figure. It does not go outside the figure. It must touch the corners. This also is a good paper, gaining one mark more than the average for the whole class. The marks are quite easily given. The definition of rhombus gains 9 positive marks. There is one 'bad error' — the four angles are not equal; but throughout this experiment we worked with positive marks only. The definition of trapezium gains all the positive marks possible on our scale of marking, namely, 8. It is interesting to note that the term 'corners' is not in C. B.'s mind synonymous with ' angles. ' In every one of his defi- 104 INDUCTIVE VS. DEDUCTIVE METHODS. nitions in which the confusion can occur lie makes the same duplication, but these duplications are not 'bad errors' according to our system of marking. The definition of rhomboid receives 12 positive marks, and the definition of a diagonal of a square receives full marks, namely, 4. The total marks for this paper amount to 33. It is scarcely necessary to multiply examples; sufficient have been given to show how much more competent, geometrically speaking, these boys are than those with whom we worked in the previous boys' school. I turn now to the chronology of the whole of the experiment. 3. Chronology of the Experiment. The first test in this series was given on Friday at 9.30 A. M., September 29, immediately after Scrip- ture lesson. In this test the boys were asked, un- taught and unaided, to define square, triangle, etc. Exactly one week later all the boys in the class were taught inductively how to arrive at the definitions of square, triangle, etc. On Tuesday, October 17, at 11 o'clock, immediately after recreation, the test used as a Preliminary Test in this school, ' ' What is a rhombus 1 ' ' etc., was given, on the results of which the class was divided into two equal groups. In this experiment one test only was given for purposes of division. It was hoped that the preparatory work with the squares, triangles, etc., together with the greater age and proficiency of the children, would result in the necessary steadiness, and that the boys' variability would be so small that one test would suffice. On Thursday afternoon, October 19, from 2.30 to 2.50, one of the groups was taught inductively how to FOURTH SERIES OF EXPERIMENTS. 105 arrive at the definitions of rhombus, trapezium, rhomboid, and diagonal of a square. Exactly the same method was followed as that used by me in the first and second experiments. The teacher of the class had heard me 'teach' the definitions, so there was no danger that he would vary the method essen- tially ; but a minor variant was employed. He jotted down on the blackboard (which I did not), in an ab- breviated form, the 'units' of description as the boys supplied them. His argument for the variation was that the boys who were going to study the definitions deductively would have visual memories of verbal descriptions to help them, and that the inductively taught group ought also to have some visual verbal memories to assist them. Whilst the boys of one group — Group A — were being taught the definitions, the other group — Group B — went into the school hall and had a reading lesson under a student-teacher. From 2.55 to 3.15 the boys of Group A went into the school hall and took the reading lesson, whilst Group B came back to their own teacher and studied the definitions of rhombus, etc., which had been con- structed from the spontaneous descriptions of the Preliminary Test and had been already written in preparation upon a blackboard, with the appropriate drawings. Definitions of Rhombus, Trapezium, Rhomboid and Diagonal of a Square in the Form in Which They Were Given to the 'Deductive' Group to Study. A rhombus is a figure enclosed by 4 equal straight lines. Two sides opposite are parallel, and the other two sides opposite are parallel. It has 4 angles, 2 106 INDUCTIVE VS. DEDUCTIVE METHODS. large and 2 small. The 2 large angles are equal and opposite, and the 2 small angles are also equal and oj^posite. A trapezium is a figure enclosed by 4 unequal straight lines. It contains 4 unequal angles. A rliomhoid is a figure enclosed by 4 straight linos, 2 long and 2 short. Two long sides are equal, oppo- site and parallel, and the two short sides are equal, opposite and parallel. It has 4 angles, 2 large and 2 small. The 2 large angles are equal and opposite, and the 2 small angles are equal and opposite. A diagonal of a square is a straight line which starts at an angle and passes across the square to the opposite angle. The boys were told to study the definitions, and they, as well as the boys of the inductive group, were made aware that they would be required to answer questions on them. The time from 3.15 to 3.30 was spent by the boys of both groups in the playground. At 3.30 all the boys returned to their classroom ; the questions, "What is a rhombus?" etc., which had been written on the blackboard, were exposed to view, and the boys wrote the answers. There was one other variant from the method which I had used myself, for the drawings of the figures were placed before the boys whilst they were answering the ques- tions in their tests of reproduction. Exactly one week later, on Thursday, October 26, at 3.30 P. M., the boys of both groups worked a test in deferred reproduction, following immediately upon the recreation interval, as in the test of immediate reproduction. Two weeks after this test, at 10 o'clock in the morning, on Wednesday, November 8, following two FOUKTH SERIES OF EXPERIMENTS. 107 short lessons on Scripture and French reading, the test of Application to New Material was given. Perhaps a summarized note showing the main chronological issues involving differences from other experiments may be of service. First, both groups had inductive teaching, as well as inductive practice, before the Preliminary Test. There was one pre- liminary test, and only one. The test of deferred re- production was given one week after the test of im- mediate reproduction. The test of application to new material was given three weeks after the teach- ing and learning which we were relying on to differ- entiate the groups, and two weeks after the test of deferred reproduction. 4. The Tests of Immediate and Deferred Repro- duction. In these tests all the boys in the class answered in writing the following questions: ''What is a rhom- bus?" etc. The questions were written on the black- board, and the drawings of the rhombus and other figures were again shown to the boys. I have already pointed out that this was a variation on the method previously adopted. 5. The Test of Application to Neiv Material. Drawings of hexagons, pentagons, tangents and quadrilaterals (similar figures), with the names ap- pended, were shown to the boys thus : 108 INDUCTIVE VS. DEDUCTIVE METHODS. HEXAGONS. PENTAGONS. TANGENTS TO CIRCLES. (The tangents are drawn in dots.) The sides of LMNO were each IV2 times the corresponding sides of ABCD, so that no easily recognizable ratio should appear. The figures were drawn so that CD and NO were not quite in the same straight line. In the diagrams actually used the tangents were continuous Imes drawn in red. FOURTH SERIES OF EXPERIMENTS. 109 Then the following questions were written on a blackboard and the boys required to answer them in writing : 1. ''What is a hexagonl" 2. "What is a pentagon!" 3. "What is a tangent to a cicle?" 4. "In how manv ways is ABCD like LMNO!" The boys were allowed, nay encouraged, to give thought and time to their answers. It will, doubtless, be remembered that no time limits were imposed in any of the tests and exercises in these experiments. The system of marking the papers could, no doubt, be inferred by analogy from the units of correct de- scription which the boys and girls have given in other cases and which we have adopted. But it is unneces- sary for us to infer what our units ought to be; they emerge quite clearly from a consideration of the pa- pers actually worked. Let me give one or two by way of illustration be- fore listing the units on which the boys' papers were marked. Frederic R , aged 13 years 11 months, who worked in the deductive group, wrote : 1. A hexagon is a figure enclosed by six equal straight lines. It has six angles, all equal. The two opposite sides are parallel in the three cases. 2. A pentagon is a figure enclosed by five equal straight lines. It has five angles all equal. None of the sides are parallel to each other. 3. A tangent to a circle is a straight line any length, which must touch the side of the circle anywhere, but must not cut it. 4. The first thing why ABCD differs from LMNO is its size. The 4 angles are the same in both figures. The 4 straight lines are the same only in proportion, LMNO is about half the size again as ABCD, M.N.O. angles are the same as B.D.C. only the sides are 110 INDUCTIVE VS. DEDUCTIVE METHODS. different lengths. A. angle is exactly the same as L. angle. Both the figures are exactly the same shape. The only thing why one is different from the other is in size. Even without a list of units of correct description it is not difficult to assess this paper. The definition of hexagon receives a mark for 'figure,' four marks for ''six equal straight lines," three marks for "six angles equal," and six marks for noting that there were three pairs of opposite sides, and that three pairs were parallel. F. R. thus obtains a total of 14 marks for his definition of hexagon. The definition of pentagon receives eight marks — one for 'figure,' four for "five equal straight lines," and three for "five angles equal." The definition of tangent re- ceives three marks — one for 'line,' one for 'straight,' and one for "touching the side of the circle." A boy's conception of touching would be satisfied if the line impinged upon the circumference of the circle in such a way that, if produced, it would cut the cir- cumference. Consequently it is necessary to add the limitation 'if produced, will not cut the circle.' The fourth answer is a good one, but it is unfortunate that the boy is bothered by the notion that he has to find differences, which every now and again intervene among the similarities. He calls ABCT) and LMNO both 'figures,' for which he receives a mark; for '4 angles' he receives two more ; for noting that the four angles are equal, each to each, in the two figures, he receives four marks; for "4 straight sides" three more, and for tlie similar proportionality of the sides he obtains four more. Finally, he notes that the fig- ures are alike in shape, for which he receives a mark. F. R.'s total mark for this answer is 15, and his mark for his whole paper 40. His marks were 38 for his FOUETH SERIES OF EXPERIMENTS. Ill preliminary test, 49 for immediate reproduction after teaching, 49 for deferred reproduction, and 40 — the present mark — for application to new mate- rial. If these are compared with the average marks given later, it will be seen that he is five or six marks ahead of the average throughout the entire series. I will give one or more worked papers before setting out the units of correct definition which were ac- cepted as the basis of marking. Robert S , aged 14 years, who worked in the inductive group, wrote : 1. A hexagon is a figure enclosed by six, straight, equal, sides. The opposite sides are equal and parallel. One side is exactly balanced by the opposite one. It has six angles, which are all equal. Three are on one side and three on the other. 2. A pentagon is a figure enclosed by five straight sides. They may be equal or unequal. No sides are opposite and no sides are parallel. It have five angles. They may be equal or unequal. 3. A tangent to a circle is a straight line. It may be drawn at any angle. It must touch the circle but not cut it. 4. Both have five sides. The base in each case is horizontal. They have five angles each. The angles are the same number of degress in each case. There are two large ones and two small ones. The two large ones are formed by the base and sides and the two small ones from the top and sides. The smallest angle is A in 1 and correspondonds with L in 2. The largest C in 1 and N in 2. The definition of hexagon obtains 14 marks — one for ' figure, ' four for ' ' six straight equal sides ; " six for noting that there are three pairs of opposite sides, and that they are parallel, each to each, and three marks for ''six equal angles." The definition of pentagon obtains six positive marks — one for "fig- ure," three for "five straight sides," and two for "five angles." The statement 'no sides are opposite 112 INDUCTIVE VS. DEDUCTIVE METHODS. and no sides are parallel' is held to be of too negative a nature for inclusion within the definition. To say that the sides and the angles may be equal or unequal would be accounted 'bad errors,' though, as I have said before, we did not tabulate the 'bad errors' in this fourth experiment. The definition of tangent receives three marks — one for 'straight,' one for 'line,' and one for 'touch the circle.' "It may be drawn at any angle" is too vague to be regarded as either positive or negative. The point is missed that the tangent, if produced^ will not cut the circle. In the fourth answer there are two curious errors. Tlie figures have 4 sides and 4 angles, and not 5, as R. S. says. He obtains marks for mentioning 'sides' and 'angles' as pertaining to both. Nearly all the rest of the answer is occupied with the equality of the angles each to each, for which 4 marks are obtained. One mark is gained for noting that the base lines in each case are horizontal ; that is regarded as equiva- lent to noting that their inclination is the same. This marking yields a total of 30 positive marks, with 4 'bad errors.' I give this paper because I wish to make it quite clear that boys inductively taught could quite well make 'howlers' as well as boys de- ductively taught, though these boys, in both groups, make extremely few. R. S.'s other marks were 26, 42 and 43 ; in all cases, except that of the Deferred Reproduction Test, well below the average. Prob- ably the perusal of the papers given above may make clearer the usefulness of the units of correct defini- tion which are now appended. FOUETH SERIES OF EXPERIMENTS. 113 Units of Correct Description or Definition of Hexagon, etc. 1. A hexagon is a figure. It has sides or lines. It has straight sides or Hnes. It has equal sides. It has six equal sides. Two sides are opposite. Two other sides are opposite. And the two other sides are opposite. Two opposite sides are parallel. Other two opposite sides are parallel. And the other two opposite sides are parallel. It has angles. Its angles are six in number. And they are equal. Its angles are greater than right angles. (A total of 15 points.) 2. A pentagon is a figure. It has sides or lines. Its sides are straight. The sides are equal. There are five sides. It has angles. Its angles are five in number. The angles are equal. And they are greater than right angles. (A total of 9 points.) 3. A. tangent to a circle is a line. It is a straight line. The line touches the circle. And, if produced, does not cut it. (A total of 4 points.) 114 INDUCTIVE VS. DEDUCTIVE METHODS. 4. ABCD and LMNO are both figures. They both have sides. They both have straight sides. Their sides are in both cases unequal. And they are 4 in number in both figures. They both have angles. Their angles are 4 in number. And are in both cases unequal angles. BA is the same fraction of LM. As BC is of MN. As CD is of NO. As AD is of LO. BA has the same slant or is parallel to LM. BC has the same slant or is parallel to MN. CD is parallel to NO. And AD is parallel to LO. The angle at A equals the angle at L. The angle at B equals the angle at M. The angle at C equals the angle at N. The angle at D equals the angle at 0. The figures have the same shape. (A total of 21 points.) It is, of course, not urged that the common proper- ties of the figures have been exhaustively enumer- ated, but only that the units of correct description are such as are actually used by boys and are service- able for marking papers in such experiments as these. 6. Results. First, let me give the coefficients of correlation be- tween the results for the various exercises in so far as they may be of value. The marks for the Prelimi- FOURTH SERIES OF EXPERIMENTS. 115 nary Test in the A Group were very closely corre- lated with those of the B Group ; the boys were most successfully paired in the two groups, from the best downwards to the worst. Worked out on the regular formula, the coefficient of correlation amounted to + .98. The results of the test in immediate repro- duction correlated with that of deferred reproduction to the extent of -f .752 in the inductive group and -f .777 in the deductive group. There was a falling off on the average of about one unit in the marks. There were 7 cases out of 34 in which the mark for deferred rej)roduction was higher than for immediate reproduction, 10 cases in which it was the same, and 17 cases in which there was a decline. The decline of the whole class was from an average mark per boy of 44.94 to 43.50, with mean variations approximat- ing to 4 in both cases, and a correlation coefficient between the results of immediate and deferred re- production of 4- .78. Though the difference is small, we are entitled statistically to say that there is a general tendency to decline, since the difference be- tween the means is from three to four times the probable error of that difference. A general slight decline seems, therefore, clear. The inductive group falls from 45.2 to 43.5; the deductive from 44.7 to 43.5. But the fall is too irregular to enable us to con- clude that there is any greater tendency to loss on the part of the children inductively taught than of those deductively taught. Let us now consider the results of the test on new material. It is clear that the difference between the results of the two groups is very small in this school, though it favors the inductive group, as, indeed, is the case in all the experiments. But the variability 116 INDUCTIVE VS. DEDUCTIVE METHODS. is such that without very high positive correlation between the two series the probable error of the dif- ference between the means will be considerable. Now let me give the average results in gross, treat- ing the groups as wholes. There were 17 boys in each group : Table XXIV, shoioing the tvork of the Inductive and Deductive groups compared, in the Preliminary Test, in the Tests of Im- mediate and Deferred Reproduction, and in the Test of Appli- cation to Netv Material. Test of Test of Pre- immediate deferred Test on liminary repro- repro- new Inductive Group : test. duction. duction. material, Average mark . . . 32.06 45.18 43.53 35.65 M. V . 3.13 3.67 3.97 3.24 Deductive Group: Average mark . . . 32.12 44.71 43.47 35.00 M. V , . 2.98 4.69 4.80 3.53 The boys of the Inductive Group appear to hold the advantage throughout, though they were slightly weaker in the Preliminary Test. A closer analysis of the results is given in the next table : TaWe XXV, shoxcing the work of the Inductive and Deductive Oroups compared, section by section, in the Preliminary Test, the Tests of Immediate and- Deferred Reproduction, and the Test of Application to New Material. Inductive Group. Marks in No. preliminary of test. boys. Over 35 4 30 to 35 6 25 to 30 7 Marks in No. preliminary of test. boys. Over 35 4 30 to 35 6 25 to 30 7 Pre- Immediate Deferred > liminary repro- repro- New test. duction. duction. material. 37.50 47.00 45.50 37.50 32.50 47.00 45.83 37.66 28.57 Deductiv 42.57 e Group. 40.42 32.85 1 -Average Mark per Boy. , Pre- Immediate Deferred liminary repro- repro- New test. duction. duction. material. 37.50 47.25 47.75 37.00 32.33 44.16 43.16 36.60 28.85 43.71 41.28 32.42 FOURTH SERIES OF EXPERIMENTS. 117 Only in one test — that of application to new mate- rial — does there appear to be a regular sectional ad- vantage on the side of the inductive group, both for the weaker as well as for the stronger boys. In both reproductive tests the balance of advantage shifts from side to side. We may justifiably conclude that the results of this experiment, having regard to the greater age and mental ability of the children, are consilient with those of the former researches. The inductive method has shown itself the better when application to new analogous material is the test employed. We are unable to say with any confidence which of the two groups has been the more successful in immedi- ate and deferred reproduction. The average results are slightly in favor of the inductive group, but the balance of advantage fluctuates from side to side, and is decidedly uncertain. But this is the fourth case in which the inductive method has shown itself supe- rior in application to new material, and the second case in which the inductive method has equaled the other, even for purjDOses of reproduction. In both these classes there had been much previous inductive teaching. But it must be remembered as well that the class of much younger hoys, in which the deduct- ive group scored heavily in reproductive work, were also accustomed to much inductive work. Age ap- pears to be a factor ; perhaps it is the younger chil- dren who reproduce better on a deductive and memo- riter method. This hypothesis will be put to the test in the last of this series of experiments. VIII. FIFTH SERIES OF EXPERIMENTS. 1. General Plan. Just as in the previous experiment, a whole class, working under one teacher, and according to the same syllabus of instruction, with the same time- table of work, was divided into two equal groups on the results of a test in the definition of geometrical forms, which the boys attempted, untaught and un- aided. Then one group worked inductively and the other deductively. Tests were given immediately after the teaching and learning, also in deferred re- production a week later, and in reproduction, still further deferred, about seven weeks after the first test of deferred reproduction. About two weeks after the teaching and learning a test of application to new material was given. The boys who did the work were graded as Standards VI, a, and VII ; their average age was 12 years Qi'o months, and their teaching generally had been clear and efficient, but had tended rather towards deductive than inductive methods. The school was situated in a poor neigh- borhood in the southeast of London, and the average mental ability of its pupils was low ;* but the boys of the highest class, with whom the experiment was made, were by no means without ability ; in fact, in ♦The natural mental ability of the pupils of this school was well known to me from the results of a number of mental tests which had been applied to every child over eight years of age. 119 120 INDUCTIVE VS. DEDUCTIVE METHODS. consequence of certain exigencies of organization, the class contained more children of ability than would ordinarily be found in a top class of such a size in a school of this social type. 2. The Preliminary Tests and the Method of Marking. The Preliminary Test, on the results of which the boys were divided into two equal groups, was the same as that used in the experiment just described, which took place in the higher grade school. The teacher had already used the questions: "What is a square ? " " What is a triangle ? " " What is an ob- long?" and "What is a diameter of a cicleT' (with the consideration of the appropriate drawings) as a kind of general propaedeutic to the experimental series, and the boys had already been shown in- ductively how to work out the definitions of square, triangle, oblong and diameter just as they had in the higher grade school. The Preliminary Test, given two or three weeks later, consisted in the questions: "What is a rhom- bus?" "What is a trapezium?" "What is a rhom- boid?" and "What is a diagonal of a square?" The appropriate drawings were shown and the boys at- tempted to answer the questions. I give below one or two of their worked papers. William L , aged 13 years 8 months, wrote : 1. A Rhombus is a figure with all side equal two sides slope at 60° and the other two run parallel. 2. A Trapezium is a figure with four unequal sides, and it as a right angle in it. 3. A Rhomboid is a figure with two long sides equal and two short sides equal, but none of the corners have right angles. 4. A Diagonal of a Square is the distance across from corner to another corner which slopes at 45°. FIFTH SEEIES OF EXPERIMENTS. 121 W. L., in his first definition, gains a total of six marks. His second definition receives four marks. "It as a right angle in it" is not true as applied to all the trapeziums; it is a 'bad error,' but there are so few of these that they are not tabulated. The definition of rhomboid receives eight marks. The negative statement that there are no right angles, though correct, receives no mark, as we could hardly have made allowance for all the negative statements which may be truly made about the figures. The defi- nition of diagonal receives two marks only — one for 'distance' and one for "from corner to another corner." W. L.'s paper receives a total of 20 posi- tive marks. It is one of the best papers worked in the class, and is assessed considerably above the av- erage mark, which is 12.25 for this preliminary test. Let me now give the paper of a boy who is among those toward the bottom of the lists. Frank B , aged 12 years 4 months, wrote : 1. A Rhombus is a square turned in shape. 2. A Trapezium is a figure all sides unequal. 3. A Rhomboid is an oblong with the two smallest perpendicular lines slanting. 4. Diagonal of a square is a line drawn from top to bottom of the corners. As we have seen in former cases of 'unintelligent' children, the similarities between these figures and those which they have previously dealt with are ap- prehended, even to the extent of error, for a rhombus is not a square. That a square is one shape and a rhombus is another shape is probably dimly under- stood by the boy ; he is giving, perhaps, what he con- ceives to be a genetic definition of a rhombus, but he receives no marks for it on our system of marking. 122 INDUCTIVE VS. DEDUCTIVE METHODS. For his definition of trapezium he obtains three marks. The rhomboid he defines genetically; his definition is worth, perhaps, two marks — one for 'lines' and one for ''two smallest lines." His defini- tion of diagonal is worth two marks ; he describes it as a 'line' and notes that it goes from one corner to another. F. B. thus receives a total of seven marks, which is a little above half the average mark for the class. The two examples of worked papers, given above, will enable teachers to see the limits of the mental level with which we are dealing. These boys are very obviously much below the first-class boys of the higher grade school whose work we considered in the experiment previously described. 3. Chronology of the Experiment. First of all came the inductive work with the squares, triangles, oblongs and diameters of circles. This was done with all the class. A week or so later, on Wednesday, October 11, at 10 A. M., follomng immediately upon Scripture lesson, the Preliminary Test was given, on the results of which the boys were divided into two equal groups. Most of the boys had finished their work in 20 minutes, though no one was hurried, and one or two took a few minutes longer. On Thursday, October 12, immediately after registra- tion, the teacher of the class taught one of the groups how to arrive inductively at the definitions of the geometrical figures which thej^ had attempted in the Preliminary Test. The teacher had heard me teach similar definitions and was well acquainted with the method as I employed it. The teaching took 22 min- FIFTH SERIES OF EXPERIMENTS. 123 utes, from 2.14 to 2.36 P. M. Wliilst the one group was being taught by their own teacher, the other group, under another master, studied the definitions with reference to the drawings of the figures which were appended. They knew that the exact words of the definitions were not to be required, but that they might use them if they chose. Both groups of boys were aware that they were to be tested immediately afterwards on what they had been taught or learnt. The definitions given to the 'deductive' group ran as follows : Definitions of Rhombus, Trapezium, Rhomboid and Diagonal of Square to Which Appropriate Drawings Were Appended* 1. A Rhombust is a figure with four straight equal sides; the opposite sides are parallel. It has four corners, two big ones opposite and equal, and two smaller ones opposite and equal. 2. A Trapezium is a figure or shape with four straight unequal sides and four unequal corners. 3. A Rhomboid is a figTire with four straight sides. The two long sides are opposite, equal and parallel. The two short sides are opposite, equal and parallel. It has four corners, two big and two small. The two big ones are equal and opposite, and the two small ones are equal and opposite. *The drawings may be seen on page 39. fPerhaps a slight amendment miglit usefully have been made in this definition of the rhombus ; it is not clear on this wording that there are tico pairs of opposite sides which are parallel ; the form of words used in the previous experiment seems more satisfactory. 124 INDUCTIVE VS. DEDUCTIVE METHODS. 4. A Diagonal of a Square is a straight line drawn from one corner to the opposite corner. At 2.40 P. M., a few minutes after the teaching and learning, both groups answered in writing the ques- tions: "What is a rhombus!" etc. No time restric- tions were laid down ; each boy was permitted to go on until he could do no more, but the superiority of the pace of the boys who had learnt the definitions was evident in this and in all succeeding exercises. After a lapse of one week, at the same hour in the afternoon, namely, 2.40, and on the same day of the week, Thursday, October 19, both groups answered the questions : ' ' What is a rhombus I ' ' etc. This will be referred to as the first test of deferred reproduc- tion. None of the boys were aware that they were ever again to be required to answer these questions ; it was only the test of immediate reproduction of which they had been forewarned. One week later, again on Thursday at 2.40 P. M. (October 26), the boys worked a further test — a test of application to new material — and on Thursday, December 7, at 2.40 P. M., two months after the test of immediate reproduction, a second test of deferred reproduction was given, in which the old questions, "What is a rhombus?" etc., were repeated; and, as before, the boys answered them in writing. 4. The Tests of Reproduction. These were in all cases the same. They consisted, as previously stated, of answers in writing to the questions : ' ' What is a rhombus ? ' ' etc. One or two papers to indicate what these boys could do after teaching and learning may be of interest. FIFTH SERIES OF EXPERIMENTS. 125 Thomas G , aged 13 years 6 months, one of the best boys who worked in the deductive group, in his exercise in immediate reproduction, wrote : 1. A Rhombus is a figure with four equal straight liues. The opposite lines are parallel. It has four corners, one pair of oppo- site corners being equal and the other pair of opposite corners being equal. 2. A Trapezium is a figure with four unequal sides, and four unequal corners. 3. A Khomlwid is a figure with four straight sides, two long sidesi and two short ones. The two long ones are equal and opf>o- site each other, and the two short ones are equal and opposite each other. The figure has four corners, two big ones and two little ones, The two big ones are equal and opposite, and the two little ones are equal and opposite. 4. A diagonal of a square is a straight line drawn from one corner to the opposite corner. This is an excellent pajjer ; the definition of rhom- boid, for example, where it differs from the wording of the definition studied, is better than the definition we provided. The boy rightly says ''two long and two short" sides, before speaking of "The two long" sides. Our own definition is faulty in that respect. The word 'The' is not only distinguishing, but rela- tive, and, indeed, very often distinguishing because relative. Let us mark the paper in accordance with the list of units of correct description given on page — : The definition of rhombus receives single marks for 'figure,' 'four,' 'equal,' 'straight,' 'lines,' 'four,' 'corners,' and eight marks for noting two pairs of opposite, parallel sides and two pairs of opposite equal corners — a total of 15 marks. For his defini- tion of trapezium he receives obviously every mark but one. He has omitted 'straight' in his descrip- tion of the sides, thus receiving seven marks out of eight. Every possible point on our system of mark- 126 INDUCTIVE VS. DEDUCTIVE METHODS. ing is scored by his definition of rhomboid, with the exception of two ; he omits the two pairs of parallels, thus receiving 18 marks. The definition of a diag- onal of a square receives full marks, namely, four. F. G. 's total mark for his immediate reproduction is 44, which is much above the average of his group. In Ms next test, one week later, he loses four marks on his first definition, for he now omits to note the two pairs of opposite parallel sides. His mark for trapezium remains unchanged. In his definition of rhomboid he now notes the two pairs of parallels, which he omitted to do in his test of immediate re- production, and on this occasion receives full marks, namely, 20. The definition of a diagonal of a square remains unchanged. F. G., therefore, has gone down two marks in one week. Let us see how many he has lost seven weeks after this. The definition of rhom- bus suffers most; the parallelism of the opposite sides does not reappear, and it is doubtful whether F. G. remembers that there are tivo pairs of opposite equal angles, for his expression is dubious. He has now lost four of the marks he originally obtained for this definition. The definition of trapezium remains unchanged. In the definition of rhomboid two marks are lost, for he now omits to note that there are two obtuse angles and two acute angles. The total mark for this definition is 18. The definition of a diagonal of a square remains unchanged, and receives four marks as before. Two months after learning the definition F. G. receives 40 marks for his reproduc- tive test, against 44 marks in his test of immediate reproduction, and 42 marks in his first test of de- ferred reproduction, which took place one week after he learnt the definitions. He loses very little ; he had FIFTH SERIES OF EXPERIMENTS. 127 evidently understood the definitions as well as learnt them. Indeed, his understanding is shown by his power of 'transfer,' for he receives a very good mark for his application to new material. Bearing in mind that this pupil worked in the de- ductive group, let us compare his work with the cor- responding papers of one of the best boys in the in- ductive group. George H , aged 13 years 9 months, in his test of immediate reproduction, wrote : 1. A rhombus is a 'figure,' sides, four sides, all equal, four angles, two opposite sides are parallel, other sides are parrallel, all sides are straight. 2. A trapezium is a figure, sides, of four, all unequal, four angles, angles unequal, all sides are straight. 3. A rhomboid is a figure, of sides, four sides, two opposite sides are equal, parrallel, and has four angles, two opposite angles are equal, sides straight, two opposite sides straight, two sides are longer than the other pair of sides. 4. A diagonal of a square is a line from one corner to the oppo- site corner, it is also straight. G. H.'s paper is, like F. G.'s, an excellent one. There is a certain staccato utterance which is a little irritating, but it is a peculiarity of his own and is not shared by the members of the inductive group gen- erally. For the definition of rhombus he receives single marks for 'figure,' 'four,' 'sides,' 'equal,' 'straight,' 'four,' 'angle;' two marks for noting a pair of parallel sides, and that they are opposite sides ; and one mark for noting the other pair of par- allel sides; but he fails to note that the other pair of parallel sides are opposite also. He also receives two marks for noting that one pair of angles are equal and opposite. His total mark, therefore, for this definition is 12. His definition of trapezium re- ceives full marks. The definition of rhomboid is not 128 INDUCTIVE VS. DEDUCTIVE METHODS. SO good. He receives single marks for 'figure,' 'four,' 'sides,' 'straight,' 'four,' 'angles;' three marks for noting that one pair of opposite sides are 'equal,' 'opposite' and parallel; two marks for not- ing that one pair of angles are equal and opposite, and two marks for stating that two sides are longer than the other two — a total of 13 marks. The defini- tion of diagonal scores full marks. G. H. thus re- ceives a total of 37 marks for his exercise in imme- diate reproduction. In a week's time, when he takes his first test in deferred reproduction, he obtains one mark less. In his definition of rhombus he omits the parallelism of the 'other sides,' losing a mark which he had gained the week previous. His definition of trapezium re- mains unchanged. In the definition of rhomboid, though it is expressed with some slight differences, he obtains all the marks which he received before, namely, 13. The definition of diagonal remains un- changed; for this he obtains four marks, as before, making a total of 36 marks. Seven weeks later there is a somewhat more seri- ous loss. He still receives 11 marks for the definition of rhombus, which has remained unchanged. His definition of trapezium has improved, for, though it contains no further units of correct description ac- cording to our scale, he notes that the smallest angle is opposite the smallest side and the biggest angle is opposite the biggest side. These statements are not quite clear, but indicate the commencement of a fresh idea about the trapezium. Two marks on his pre- vious record are lost in his definition of rhomboid; he omits now that there are two long and two short sides. The last definition remains unchanged. For FIFTH SERIES OF EXPERIMENTS. 129 the second paper in deferred reproduction, there- fore, Gr. H. receives 34 marks. These papers written by F. G, and G. H,, though much superior to the average work, are typical in the slowness with which points like these of definitional description are forgotten when they have been duly understood, and expressed in a way which is really a result of work on the part of the pupil himself. 5. The Test of Application to New Material. This, after all, is the supreme test of what teachers call 'intelligence.' We have seen in the two papers given above that the boy who learnt deductively knew more of what he had actually studied than the boy taught inductively, not only immediately after the work, but after two months had elapsed ; and with the boys of this class we shall find this difference to be true generally between the boys of the deductive and the boys of the inductive groups. The older children hitherto — girls and boys graded as Standard VII and upwards — have not shown this difference, though the younger and less proficient children have. I incline to attribute this to the relative i^redominance of de- ductive work in the usual teaching of this class, whereas in the two preceding classes of elder chil- dren, both boys and girls, the teaching was predomi- nantly inductive. Are we about to find that these boys give us results which differ from those of the older children previously experimented with, and, indeed, from all the children previously experi- mented with, when test is made of their power to apply their knowledge to new material? The test of application was the same as that used 130 INDUCTIVE VS. DEDUCTIVE METHODS. with tlie Ex- VII class in the Higher Grade Boys' School. Drawings of hexagons, tangents to circles, pentagons and quadrilateral similar figures were shown. The questions: ''What is a hexagon?" "What is a tangent to a circle?" "What is a penta- gon?" and "In how many ways does ABCD resem- ble EFGH?" were written on the blackboard and the children answered them in writing.* I will illustrate what the boys did by means of two papers, both above the average, one from the 'de- ductive' and one from the 'inductive' group. Harry W., aged 13 years 6 months, who worked in the 'deductive' group, wrote: 1. A hexagon is a figure with six straight sides all of which are equal, it has also six equal corners or angles. 2. A tangent to a circle is a straight line, drawn so that it touches the circumference of the cii'cle. 3. A pentagon is a figure with five straight sides and five angles, all sides being equal and all angles being equal. 4. A. b. c. d. is the same as E. f. g. h. They vary by the sides, and the angles, if you look at them closely and then measure the angles they will all be different on one and all the same as the first on the other. They look different by the size. With the exception of the last definition, this is an easy paper to mark. The definition of hexagon re- ceives a total of 8 marks. The definition of tangent receives 3 marks. The definition of pentagon re- ceives 8 marks. In the last answer about the simi- larity of the quadrilateral figures, it is clear that H. W. wishes to express the inequality of the angles in both figures and the equality of the angles, each to each, of one figure with those of the other, for which he receives 5 marks. Thus H. W., taught deductively, ♦The drawings may be seen ou page 108. One of the two similar quadrilaterals was lettered EFGH on this occasion. FIFTH SERIES OF EXPERIMENTS. 131 scores 24 marks for his test of application to new material. Frank C , aged 13 years 2 months, who was taught inductively, wrote : 1. A Hexagon is a six straight sided figure, liaving all sides equal, it has six angles equal, larger than right Angles. 2. A Tangent to a circle is a line which is straight and is just touching the boundary of a circle. 3. A Pentagon is a five, equal, straight sided figure, it has five equal angles larger than right angles. 4. Both have four sides. Both have four angles. Both have four angles which are larger than right angles. A angle equals E angle. B " " F angle. C " " G angle. D " " II angle. Both have sides with the same slope. Both are placed on the same side. The definition of hexagon receives a total of nine marks; the definition of tangent three marks; and that of pentagon nine marks. The last answer is more difficult to mark. Both figures have ' sides ;' this car- ries one mark. There are four sides in both figures ; this carries another mark. Similarly, ''Both have four angles ' ' carries two marks. The next statement is wrong; it is not true that both have four angles which are larger than right angles. Then there are four marks for noting the equality of the angles, each to each, and four marks for noting that the sides of the figures are parallel. One further mark is gained by F. G.'s statement that both the figures are on the same side (of the base). This answer, therefore, re- ceives a total of 13 marks. The paper is an excellent one, and carries a total of 34 marks ; it is, in fact, one of the best papers worked in either group in the test of application to new material. 132 INDUCTIVE VS. DEDUCTIVE METHODS. Lest the reader should carry away a quite exag- gerated notion of the power of application of these pupils (I am using the expression 'application' in the strictest sense), I propose to give one further paper by a boy who worked in the deductive group and made very little application of his knowledge. He obtained 37 marks in his test of immediate repro- duction and 34 marks a week after. But he obtained a very poor mark when he worked on new material, and seven weeks later he sank to 23 marks when tested on his old knowledge. There are evidently some boys who learn quickly and forget quickly. The pedagogical error, now happily being rectified by psychologists, has been to regard these boys as the rule rather than the exception. This boy, George L., aged 12 years 4 months, wrote : 1. A Hexagon is a six sided figure. Each of the six sides are straight equal and opposite and Paralled. 2. A tangent of a circle is a straight line dra\vn which is slant- ing and the circle stands on it. 3. A Pentagon is a figiu'e with five sides, they are all straight. The Three small ones are equal and opposite, and the two long ones are equal and opposite. 4. a. b. c. d's. has two straight long sides equal and the other tAvo sides unequal E. f. g. h's has two long straight sides equal, and the other two unequal only a. b. c. d is smaller than E. f. g. II. "Equal and opposite" has, unfortunately, trans- ferred itself too successfully. For his definition of hexagon he receives 5 marks. * * Each of the sides are opposite and parallel" is considered to be too con- fused to gain positive marks, but is not regarded as involving 'bad errors.' The definition of tangent gains 2 marks only; the latter part of his definition was drawn from one figure only. The statements that the tangent is slanting and that the circle stands FIFTH SERIES OF EXPERIMENTS. 133 on it were not trne of all the tangents drawn, and are considered 'bad errors.' In Ins definition of penta- gon he receives 4 marks only. It is considered a 'bad error' to say that "three small ones are opposite." No positive marks are allowed for saying that "three are equal" and "two are equal," and it is counted an error to say there are "two long" and "three small" sides. In his last answer G. L. receives 2 marks; both the figures have 'sides,' and in each case two are longer than the remaining two. But none of them were equal; though, as two of them were not very different in length, the statement was not accounted a 'bad error.' The statement as to the size of the two figures is irrelevant; the boys were asked for 'resemblances,' not for differences. This is one of the worst papers in the class. The boy had acquired the knowledge of the definitions of rhombus, etc., but he could not apply it, and he speedily forgot it. Possibly, with these examples before him, the reader may find greater interest in the tabulated results, which I now give. 6. Results, (a) Of the Preliminary Tests. In the Preliminary Test the highest mark gained by any boy was 19, the lowest was 6, and the average mark was 12.25. There were 16 boys in the group deductively taught and 16 boys in the group induct- ively taught. The average mark of the boys of the first group was 12.25 (mean variation 3.0), and of those in the second group was 12.25 (mean variation 3.0) . The correlation between the total results of the corresponding boys in the two groups was practi- 134 INDUCTIVE VS. DEDUCTIVE METHODS. cally perfect, amounting to + .97 on the product- moment formula. In so far as one test can in any way be satisfactory as a basis of the division of a class into equal groups, it seems fair to suppose that an adequate division has been made. These boys, it will be remembered, had had some special inductive teaching concerning the square, triangle, etc., though I should not describe the general methods of their teacher as predominantly inductive. I incline to think this special inductive propaedeutic may have been an advantage to us in making the division, but it may, I fear, serve to throw some bias on the in- ductive side and unduly favor the inductive group. We may, however, remember that we have three ex- periments already described in which no such propae- deutic was given. (h) Of Immediate Reproduction. What marks did the two groups obtain immedi- ately after the teaching and learning? In two pre- vious experiments with older children, girls as well as boys, the group taught inductively appeared to advantage from the first. Is that also the case with these Standard VII boys I We can say quite defin- itely that it is not. The average mark obtained by the boys of the deductive group was 34.2 (mean vari- ation 6.0), and of the inductive group 31.4 (mean variation 4.5). This difference between the means and its prob- able error justify us statistically in asserting the FIFTH SERIES OF EXPERIMENTS. 135 existence of a general tendency in favor of the 'de- ductive' group. The superiority of the work of the deductive group in immediate reproduction may also be shown compendiously in the following table : Table XXVI, sJioivhig the tcork of the Deductive and Inductive Groups compared, section by section, in the Preliininaru Test and the Test of Immediate Reproduction. , Deductive Group. , , Inductive G roup. \ Av. mark Av. mark Average in imme- Average in imme- mark in diate mark in diate Marks in No. prelimi- repro- No. prelimi- repro- preliminary of nary duction of nary duction test. boys. test. test. boys. test. test. Over 15 .. 4 17.0 38.0 4 17.0 34.5 10 to 15.... . . 7 12.4 32.7 7 12.5 29.6 5 to 10.... .. 5 8.2 33.2 5 8.0 31.4 (c) Correspondence Between Immediate and De- ferred Reproduction. But, after all, the important question in education is not so much what can be done by pupils immedi- ately after they have just been taught, but what they can do some time afterwards. Do they remember what they once knew, and how far can they apply their knowledge? To the second of these questions I hope to give an answer when dealing with the re- sults of the test on new material. Let me turn for a while to the first, and let me break it up into a number of constituent questions. The boys gain certain marks immediately after teaching and learning. What do they gain a week later, and, more important still, what do they gain two months later ? 136 INDUCTIVE VS. DEDUCTIVE METHODS. In a rough way we can find the answers to our questions in the following table : Table XXVII, slion-ing the work of the Inductive and Deductive Groups compared, section by section, in the Tests of Immediate and Deferred Reproduction. Deductive Group. Imme- diate Maries for No. repro- immediate of duetion reproduction. boys. test. 40 and over 5 41.4 35 to 40 3 37.7 30 to 35 4 30.8 25 to SO 3 26.7 Below 25 1 22.0 Inductive Group. -Average Marks. n Deferred Deferred Marks for No. immediate of reproduction. boys. 40 and over 35 to 40 6 30 to 35 3 25 to 30 G Below 25 1 37.0 32.0 27.3 20.0 repro- duction, first test. 38.6 38.0 28.3 22.3 20.0 repro- duction, second test. 38.4 32.7 25.8 23.0 19.0 Imme- diate repro- duction test. -Average Marks.- Deferred repro- duction. first test. 33!7 29.3 22.8 17.0 Deferred repro- duction, second test. 3i!7 27.0 24.5 24.0 The conclusions seem clear. The Inductive Group contains no boys at all equal to the highest section of the Deductive Group. The best boys in the In- ductive Group correspond to the second section of the Deductive Group, but even then they are inferior to that section, both in the immediate and deferred tests. The work done in immediate reproduction may be very well taken as representative of what the work will be later on in exercises of this kind, for the various sections into which the groups are divided FIFTH SERIES OF EXPERIMENTS. 137 retain their relative positions throughout the whole experiment. Calculated exactly, the correlation co- efficients between the results of Immediate Repro- duction and those of the first Deferred Reproduction Test in the Deductive Group is + .804, and between Immediate Reproduction and the second Deferred Reproduction Test (two months later) is + .859. The corresponding figures for the Inductive Group are + .616 and + .619. Summarizing the results and treating the groups as wholes, the averages and variabilities are as fol- low: Table XXVIII, showing the icorJc of the Inductive and Deductive Groups compared in the Tests of Immediate and Deferred Reproduction. Imme- First Second diate deferred deferred repro- repro- repro- Deductive Group : duction. duction. duction. Average mark 34.2 32.5 30.1 M. V 6.0 6.3 7.0 Inductive Group : Average mark 31.4 27.8 27.6 M. V 4.5 5.1 4.1 The Deductive Group has outdistanced the In- ductive Group quite clearly, both in immediate and deferred reproduction, not only in positive marks, for, perhaps, I ought to add, it has also made fewer 'bad errors.' It is the third result in which this has been found to be the case. We shall, therefore, again have to admit the contention urged against induct- ive methods in the earlier chapters of this mono- graph. We must certainly conclude that, in exami- nations on precisely what has been taught or learnt, children taught by what we have called deductive methods may be more successful than children taught 138 INDUCTIVE VS. DEDUCTIVE METHODS. inductively. Also we see that children need not be young to be taught successfully by deductive meth- ods. Let us now turn, however, to the Test of Ap- plication to New Material and see whether the same relation between the two groups holds there. (d) Results of the Test on New Material. We have seen that for purposes of immediate, and even of deferred, reproduction the more mechanical method has shown itself superior to the less mechan- ical. Is the same relationship retained between the two groups when the test is no longer one of simple reproduction, but requires a transfer of knowledge or method to analogous material? We can say at once that the same relation is not maintained. The inductive group now comes to the front, but the dif- ference between the means of the two groups is a small one and the variability of the averages is high. The deductive group scores an average mark of 20.5 (mean variation 5.9), and the inductive group an average mark of 21.1 (mean variation 4.4). But let us look a little more closely into the composition of these averages: Table XXIX, showing the worlc of the Inductive and Deductive Groups compared in Immediate Re product ion and in the Test on Neiv Material. , — Deductive Group. — n , Inductive Group. n Marks Average Maries for Average MarliS for in imme- Imme- Imme- diate No. diate No. diate repro- of repro- New of repro- New duction. boys, duction. material, bovs. duction. material. Over 35 8 40.0 2.S.0 37.0 24.8 30 to 35 4 30.8 17.8 3 32.0 19.3 25 to 30 3 20.7 19.7 (5 27.3 17.5 Below 25 1 22.0 12.0 1 20.0 15.0 FIFTH SERIES OF EXPERIMENTS. 139 The figures certainly suggest a superiority on the side of the inductive group in three of the corre- sponding sections into which the groups are divided ; and the regiihir decline of the figures in both groups (with the exception of the average of 19.7 in the third section of the Deductive Group) would appear to in- dicate that there is a general tendency in favor of correlated transfer in the Inductive rather than in the Deductive Group. The coefficient of correlation between the results of the Inductive and Deductive Groups, when tested on new material, is, however, not very high. With high variability as well, this involves a high probable error. So that we may con- clude in this case merely that the Inductive Group does better work on the whole than the Deductive Group, but we have not the usual statistical justifi- cation that there is a strong general tendency in that direction. We shall, however, hardly feel disposed to attribute the superiority of the Inductive Group to chance, since in every one of the five experiments, with different teachers, with children of different ages, of different abilities and of different sexes, we have found the inductively taught group the more competent when tested on the power of application to new material. IX. GENERAL SUMMARY. In five different schools in different parts of Lon- don, attended by cliildren varying in social class, ex- periments have been made to test the relative values of 'inductive' and 'deductive' methods of teaching as applied to geometrical definition. Both girls and boys, of ages ranging from 8 to 15 years, were set to do the work. The main problems were two in num- ber. In the first place, an attempt was made to dis- cover which of the two methods gave the better re- sults when the children were tested on precisely what they had been taught or had learnt. In the second place, an endeavor was made to find out which of the two methods gave the better results when the chil- dren were tested on new material. The answer to the first of these two questions was not the same in all of the five schools tested. In three of them, two of the three boys' schools and one of the two girls' schools, the conclusion was unambiguously in favor of the 'deductive and memoriter' method. This was the case with the younger and less profi- cient boys and girls, and at first sight it looked as if age were an important factor in the ]:>roduction of this result, but the same result was obtained with a class of boys who were much older, so that age was certainly not the only factor of differentiation. In two classes, the oldest class of boys and the oldest class of girls who did the work, the inductive method 140 GENERAL SUMMAEY. 141 was just as successful as the 'deductive,' even for purjDoses of exact reproduction, immediately after- wards, of what had been taught or learnt. There were some indications that the children inductively taught lost rather less of what they had known than those deductively taught when they were tested some time afterwards; but, on the whole, the tests of de- ferred reproduction gave the same comparative re- sults as those of immediate reproduction. The im- portance of this consideration in testing school methods where exact reproduction is required is obvious.. The answer to the second of the two main issues was the same in all of the five schools tested. The children who were taught 'inductively' did better work than those taught 'deductively' in every case when they were required to apply themselves to new material. This research, therefore, offers an experimental justification of what are known, among teachers, as ' intelligent ' methods of teaching, and of the superior 'transfer' effect of certain methods. Many pedagogical corollaries may be drawn from the experiments, but it will be sufficient in this place to emphasize a consideration already alluded to in the body of the text. Examinations, whether internal, that is, conducted from within by the school authorities, or external, that is, conducted by external educational authori- ties, should always include questions on subject-mat- ter which is not identical with that set down in the syllabuses of instruction if the examination is to test good method in teaching. But if the tests are to serve any useful pedagogical purpose, the new mate- 142 INDUCTIVE VS. DEDUCTIVE METHODS. rial, though it should not be identical, ought to be analogous to that which has been dealt with in the school curriculum. Questions on new analogous ma- terial are probably the best questions of all (if the same set of questions be required to serve a double purpose), for they test, with fair adequacy, whether the work set down in the syllabuses has been effi- ciently done, and they also test, with admirable ade- quacy, whether the methods by which the school work has been done were such as to give the pupil power to apply his knowledge. INDEX 'Bad' errors, meaning of, 36. and mechanical method, 51. Chance or Variability, 7. Children's Definitions, spontaneous, 27, 28, 57, 71, 72, 73, 74, 102, 103, 120, 121. after teaching and learning, 76, 77, 78, 80, 81, 82, 125, 127. of new analagous material, 40, 83, 84, 85, 86, 88, 90, 109, 111, 130, 131, 132. Circle, definition of diameter of, 29, 32. drawing of diameter of, 24. Classes taking the experiment, 20, 23, 55, 69, 100, 119. Co-operation of Teachers, 4. Correlation coefficients, 9, 10, 50, 62, 66, 91, 96, 115, 137. value of, 30. Deductive Method, method of learning by, 37. "*J^ method of testing, 19. shown to be the better, 46, 93, 134, 136, 137. Deferred Reproduction, 44, 47, 63, 65, 92, 107, 124, 135. Definitions, 'real,' 26, 27. arguments in favor of deductive treatment of, 17. arguments in favor of inductive treatment of, 18. units of marking of, 28, 29, 41, 42, 43. as learnt deductively, 31.:. \ as learnt inductively, 32. children's spontaneous, after teaching and learning, of new analagous material, see Children's Defini- tions. of diameter of circle, 29, 32. of hexagon, 113. of oblong, 29, 32. 143 144 INDUCTIVE VS. DEDUCTIVE METHODS. of pentagon, 113. of rhomboid, 42, 106, 123. of rhombus, 41, 105, 123. of square, 25, 28, 32, 35. of diagonal of square, 39, 106, 124. of tangent to circle, 113. of trapezium, 42, 106, 123. of triangle, 28, 32. Demonstrative Geometry, introduction to, 17, 27. Diagonals of Squares, 39, 106, 124. Diameter of Circle, definition of, 29, 32. drawing of, 24. Durability of knowledge, 19, 37, 48, 49, 64, 66, 95. Education, method of settling disputed questions in, 16. Educational Science, 3. Equal groups, how formed, 30, 45, 56, 61, 92, 133. use of, 52. Errors, 'bad,' meaning of, 36. and mechanical method, 51. ^ Errors, method of correcting inductively, 34. 'X 'probable errors,' method of determining, 8-10. in spelling not counted, 43. Experiment, use of, 16. Experimental Pedagogy, 1. Geometrical Definitions, see Definitions, Geometrical Teaching, alleged cause of 'chaos in,' 18. Geometry, Demonstrative, introduction to, 17, 27. r^. Groups, equal, how formed, 30, 45, 56, 61, 92, 133. use of, 52. Hexagon, definition of, 113. drawings of, 108. Immediate Keproduction, 19, 44, 47, 63, 65, 92, 107, 124, 135. Inductive Method, an objection to, 14. "y" ; INDUCTIVE VS. DEDUCTIVE METHODS. 145 arguments for, 18. ^ method of learning by, 32. ^ method of correcting by, 34. method of testing, 19. shown to be the better, 51, 52, 61, G(3, 67, 98, 116, 117, 138. Intelligence, meaning of, 18. test of, 38, 129. training of, 53. Knowledge, durability of, 19. relation between quickness and permanence, 37, 48, 49, 64, 66, 95. Marks, positive and negative, 36, 38. Material, 'new,' see 'New Material.' Negative marks, 36, 38. 'New material,' meaning of, 53, 54. application to, 19, 38, 50, 67, 96, 107, 116, 129, 138. 'New methods,' general tendency of, 13, 14. alleged disadvantages of, 14. Novelty, influence of, 70. Oblong, definition of, 29, 32. drawings of, 24, Pedagogy, Experimental, 1. Pentagon, definition of, 113. drawings of, 108. Positive Marks, 36, 38. Practice versus Theory, 11, 13, 15. 'Probable Errors,' method of determining, 8-10. Reproduction, deferred, 44, 47, 63, 65, 92, 107, 124, 135. immediate, 19, 44, 47, 63, 65, 92, 107, 124, 135. Rhomboid, definition of, 42, 106, 123. drawings of, 39. Rhombus, definition of, 41, 105, 123. drawings of, 39. 146 INDUCTIVE VS. DEDUCTIVE METHODS. School Classes taking the experiment, 20, 23, 55, 69, 100, 119. Science, Educational, 3. 'Science' of Education, 4. Spelling errors not counted, 43. Spontaneous definitions, 27, 28, 57, 71, 72, 73, 74, 102, 103, 120, 121. Square, definition of, 25, 28, 32, 35. diagonals of, 39, 106, 124. drawings of, 24. Tangent to Circle, definition of, 113. drawings of, 108. Teaching, the divergence of Theory and Practice, 11. breach between Theory and Practice, 13, 15. unintelligent, reaction against, 70. Teachers, co-operation of, 4. Theory versus Practice, 11, 13, 15. Time taken for the exercises, 44, 59, 75, 104, 106, 122. Trapezium, definition of, 42, 106, 123. drawings of, 139. Unintelligent teaching, reaction against, 70. Unsophisticated material, 20. Variability or Chance, 7. lEhufaltonal }^Hyrl?oIogg j^ottograyl)a ^hitth by (Suy MonUoBe mijipplp WARWICK & YORK, Inc. laltittwrp, ai. &. A. Moto- Sensory Develop- ment Observations on the First Three Years of a Child. By GEORGE V. N. SEARBOBN Price: 215 + vi pages, frontispiece. $1.50. Few subjects are of greater interest to the parents of young children or to school teachers with the truly scientific spirit of their profession than the evolu- tion of a child's mechanism of efficiency. To the psychologist, and to a less extent to the physiologist, acquaintance with the average course of this human unrolling is clearly a technical necesstiy. All these surely should welcome every competent new account of the first three years of human life. This book, as its name implies, dis- cusses both the motor and the sensory development of an average child. It con- sists of careful observations of the steps in individual evolution with the addition of numerous notes and brief theoretic discussions of the observations. The chief emphasis has been put on the be- ginnings of voluntary movement and on the forerunning phenomena. These are considered from both the physiologic and psychologic points of view. The affective side of child-development is more fully treated than are the purely intellectual processes, although the moto- sensory evolution of ideation as exhibited in learning to talk is as amply considered as circumstances allowed and as was ex- pedient. A feature of the book is a careful chronologic epitome of the observed de- velopment, perhaps more detailed than in any work since the pioneer treatise of Preyer. This is given in two tables of considerable length, one of them ar- ranged alphabetically and the other by weeks. For purposes of reference these tables will be found of value. Throughout the book there is continual reference to the temporal and other re- lationships of mental development as noted in similar accounts by Preyer, Dar- win, Shinn, Moore, Major and others. These notes facilitate the use of the book for pedagogical purposes, and they also enable parents to judge more accurately of the natures of their children in com- parison with the average. WARWICK & YORK, Inc., BALTIMORE. M3>. Spelling Efficiency in Relation to Age, Grade and Sex, and the Question of Transfer An Experi- mental and Critical Study of the Function of Metliod in tlie Teacliing of Spelling. By J. E. WAIiIiACE Price: Vimo, cloth, vi, 91 pages. $L26. There are few elementary school sub- jects in whicli inefBciency is more surely detected and reprobated in later life, and in the teaching of which the elementary schools are charged with more extrava- gant waste of time, than spelling. 7.22 per cent, of the time of the child in the elementary schools in ten of our largest cities is devoted to the study of spelling, and yet the complaint continues to be almost universally voiced that the ele- mentary and secondary school graduates have not learned how to spell. School superintendents and teachers have felt the justice and sting of these criticisms, and have attempted to pro- vide a remedy either by increasing the time devoted to spelling or by changing the methods of teaching. The results, however, have not in all cases proved satisfactory. Dr. Wallin, who has been offering courses in educational psychology and the principles of teaching in schools of edu- cation for a number of years, points out briefly in this monograph some of the fallacies involved in the exclusive use of the incidental method of teaching spell- ing, based upon the psychological prin- ciples which condition the reduction of mechanical subject-matter to the plane of automatism (spelling is of an instru- mental nature). By means of the re- sults of the very researches made in the past to demonstrate the adequacy of the incidental method, it is shown that its use has not justified the claims made in its behalf. On the other hand, the su- periority of a spelling drill technique, based upon the laws of habit formation, is shown, partly by the author's own in- vestigation and partly by the results of a thoroughgoing application of the meth- od under control conditions during four years in a large school system. The booli also discusses the relation of spelling efficiency to age, grade and sex ; the facts derived from the tests are sup- ported by numerous tables, a number of practical conclusions are offered, and a bil^liography is appended. When Should a Child Begin School ? An Inquiry Into the Relation Between the Age of Entry and School Progress. By vr. H. WINCH Few educational questions have excited more general interest in recent years than that of the age at which children suould commence their attendance at school. On the one side we have the rule-of-three conclusion, felt rather than expressed as an inference, that the more teaching the child gets and the sooner he begins school the more progress he is sure to make. On the other we have had a strong feeling, now grov/ing in inten- sity and range, that attendance in school, particularly in England, begins too early and that there is an educational disad- vantage in commencing as soon as the children of Great Britain do. While this investigation by Mr. Winch has special reference to England, where the school life begins at a much earlier period than in either America or Germany, the re- sults set forth by the author are of vital interest to all who have to do with the education of children. Tlie effect of age of entry is considered from several points of view : 1. Does early entry at school enable the pupil to make more rapid advancement in school standing than entry at a later age? In other words, in a given grade are those pupils who entered school earlier found to constitute the younger portion of the class? 2. In the same grade some pupils may be doing work of a high degree of efficiency, others work of an inferior qualitv. To what extent does early entry correlate with high efficiency when tested by examinations? 3. How far does early entry depend upon social circumstances? 4. What is the influence of early entry upon the subsequent behavior of pupils and upon their attentiveness to school work ? The results of Mr. Winch's inquiry are now published for the first time. Some Vrti'f °^ them have been privately circulated, '^'^"^''- and a few of the tables, together with 12mo cloih, the methods employed, were discussed 1AC '«/.«». some years ago at a meeting of the In- iu» pages. gpectors of the Education Committee for J1.25. I^ndon. ^ WABWZOX & TOBK, Zno., BAIiTXBIORE, MD. Mental Fatigue "Die Gcistige ErmUdung.' By Translated from the German by GUT BIONTBOSZ; WHZFFI.B Price: ^mo, cloth, via, 133 pages. $1.25. This noteworthy monograph ia a com- prehensive exposition of the nature of mental fatigue, of the methods proposed for measuring it, and of the results that have thus been obtained, with special reference to their application to class- room problems. The text is an amplification of a lecture delivered Ix^fore the Munich association of gymnaslal teachers, and its primary purpose is not to contribute to the ex- perimental Investigation of fatigue, but to inform and to interest teachers. The following are amoog the topics dis- cussed : The nature and forms of fatigue, the symptoms of fatigue, the measure- ment of fatigue by physiological and by psychological methods, the factors other than fatigue that affect efficiency of men- tal work— practice, adaptation, warming- up, spurts, enthusiasm, etc. — and the laws of fatigue. In considering the application of these laws to school-room problems, attention is given to the dependence of fatigue upon individual differences, upon age, puberty, the length of lesson periods, the number of lessons per day, the day of the week, the introduction of various rest pauses (recesses, holidays, vacations, etc.), change of occupation, the fatigue coefficient of the different studies, also to hygienic arrangement of the school pro- gram and other practical problems. A selected bibliography closes the mono- graph.. The translation is offered with the con- viction that it will meet a very general demand on the part of the teacher of educational psychology and of the hy- giene of instruction for a clear and sys- tematic presentation of the problem of mental fatigue and its relation to school work. WABWZOK & YOBK, Inc., BALTIUOBE, MD. Relative Efficiency of Phonetic Alpha- bets An Experi- mental Inves- tigation of the Compara- tive Merits of tlie Webster Key Alphabet and the Proposed Key Alphabet Submitted to the National Education Association. By GUY MONTBOSZ: WHIFPIiE, Price: Svo, 60 pages 35c. paper binding. WARWICK This monograph will exert a two-fold appeal to those who aim to keep abreast of present-day movements in education. First, in that it offers an excellent ex- ample of the application of the experi- mental method to a pedagogical problem, and in this respect will take its place as a contribution to experimental pedagogy ; secondly, in that it deals with an im- portant topic just now a matter of gen- eral discussion in educational circles. The National Education Association has under consideration the adoption of a new key-alphabet for phonetic nota- tion. The merits of the proposed alpha- bet have been the subject of extensive and lively debate, but no one has hither- to done the obvious thing and tried out the new alphabet under experimental conditions. This Dr. Whipple has ac- complished, and the results will interest every teacher who uses a phonetic alpha- bet in his class work as well as every educator who believes with the author that, in the school as well as in other realms of life, "you can tell by trying." In view of the fact that the subject of phonetic alphabets will be given much attention by educators during the next year, this work is offered at a price which will place it easily in reach of teachers in city and rural schools, and also the members of clubs and reading circles. & VOSX, Inc., BAI.TXMOBZ:, MD. Back- ward and Feeble- Minded Children A Series of Studies 1b Clinical Psychology. By EDMUNB B. SXTBT Price: 12m 0, 200 pages, illut. $1.40. Bach of the more populous States has several thousand mental defectives, large numbers of whom are attending the put>- lic schools. They usually make little progress and are distressingly disturbing factors In the regular classes. In Ger- many, and recently in Prance, and in some of our own cities, these children are being placed in special classes or in special schools, according to the degree of defect. Teachers and school experi- ence immediate relief, and the children themselves are the greatest beneficiaries. All the schools have these defectives, and the problem of recognizing and caring for them is an immediately pressing one In all our cities, towns and rural dis- tricts. Following a yenx in the clinics of Paris, Dr. Huey's posit on at Lincoln for nearly a year and a half Involved making a mental examination of each new ad- mission to this, one of the largest state institutions for the feeble-minded. As research psychologist to the insti- tution Dr. Huey made careful psychologi- cal study of 35 selected cases which rep- resent the transition zone between feeble- mindedness and non-feeble-mindednesa. These are Just the border cases that puz- zle the school principal or the clinician. In this volume he presents case after case representing various types and groups of backward and feeble-minded children. The mental and physical char- acteristics of each child and the salient features of different groups are clearly stated, with charts which graphically present the results of various measure- ments and tests. The methods of making examinations and tests and of making observations and gathering data needed for the interpre- tation of any given case are illustrated in detail. The concreteness of the ma- terial and the abundance of illustrative examples will be appreciated by all, and make the studies intelligible even to those unfamiliar with psychological technique. WARWICK & YORK, Xnc, BAI^TIlllIORE, MD. Experi- mental Studies of Mental Defectives ▲ Critique of the Binet- Slmon Tests and a Contri- bution to the Psycliology of Epilepsy. By J. E. WAI^ULCE WAI.I.XN, FhJ>. About 160 pages. $1.25. The Binet-Simon tests liave be«D bailed by popular writers and even by some scientific workers as a wonderful mental X-ray machine, which will enable us to dissect the mental and moral mechan- isms of any normal or abnormal indi- vidual. But those who have had ex- tensive experience with these tests linow that, despite their very great practical value, they have numerous imperfections and definite limitations. These imperfec- tions and limitations can be made known only by thoroughgoing trial on large groups of individuals by expert investi- gators. Dr. Wallin is well qualified by training and experience to undertake this work, and he has presented in this, the seventh of the series of Educational Psychology Monographs, a systematic critical study of the results of the Binet Scale when applied to a colony of epi- leptic children, and has included a guide for the conduct of the tests. In the course of his study certain facts have been revealed concerning the men- tal status of the epileptic which should interest the schoolman as well as the alienist and the physician, for epileptic children constitute a numerous class which grades nearer the public school laggard than do feeble-minded children, and which cannot be reached by the cut- and-dried methods of the schools, but re- quires a special educational regime. Moreover, epilepsy, despite the investiga- tions of many alienists, still remains a little understood pathological condition with marked disturbance of mentality. We commend this contribution to the attention of physicians, alienists and all schoolmen who are interested in the scientific examination of mental de- ficiency. WARWICK & YOBK, Inc., BAIiTmOBE, MD. Varia- tions in the Grades of High- School Pupils By CIiABENCZ: TBUMAN GBAV. 12mo, Cloth ca, 120 pages. $1.25. Ten years ago no serious attempt had been made to study scientifically the relative merits of various systems of grading students, despite the fact that statistical methods for undertaking such studies were fully available and that grading plays so large a rOle in the school career of hundreds of thousands of school children. In the last five years, however, this inviting field has been the scene of numerous important investigations, so that we have at least arrived at a better understanding of the nature of the problem and of the general line along which progress must be made. In the present monograph Mr. Gray reports the methods and results of his investigation of one phase of the general problem, viz., the nature, degree and causes of the variations occurring in the grades of high-school pupils. The gen- eral aim of his study is to base an edu- cational investigation upon school grades. It is usually argued that such marks are inaccurate, that they are complex, that they are not scientific, and, above all, that it is impossible to measure mental traits by such cold statistics as grades afford. In direct contrast to these arguments stands the fact that all promotions from the kindergarten through the university are based upon this so-called inaccurate, complex, unsci- entific and cold estimates of progress and achievement. One of the most vital and fundamental principles of any school system is its plan of promotions, and because of the close relation between promotions and grades there is the most urgent need that schoolmen become in- terested in the problems of grading. Variations in the Grades of Hiffh-School Pupils should interest all teachers, and more particularly all school administra- tors, because the author not only shows clearly how unreliable are the grades commonly given by teachers, and makes evident the need of Instruction and train- ing in grading, but also presents a rela- tivelv simple method by means of which any "high-school principal can study the condition of the grading in his own school and take due steps to remedy the faults that he may find. WARWICK & YOBK, Inc., BAIiTIMGBZ;, MD. How I Kept My Baby Well By ANNA a. NOVES. i2moy Cloth, Illustrated, ca, 180 pages. $1.25. WABWICl The fact that the Journal of Educa- tional Psychology has defined its scope to include the consideration of child psy- chology and hygiene justifies the inclu- sion in the allied series of Educational Psychology Monographs of the material set forth in the present volume. Mrs. Noyes has made a contribution of real interest to physicians and nurses, to mothers and fathers, and to students of childhood generally. The value of her work is twofold. On the one hand, it points the way to a method and type of observation tliat any intelligent mother can undertake with profit to herself and to others, and in so far disproves the contention of some critics of the child- study movement that observations of young children by their own mothers can never yield data of real value ; on the other hand, it furnishes generalizations in the shape of principles or rules gov- erning feeding, clothing and the general control of infant development that will be of direct utility to those who, like the author, face that vital problem- how to keep the baby well. Mrs. Noyes has displayed commendable caution in drawing these generalizations. It is not asserted that what applied to her own baliy will apply invariably to any other baby, but only that it undoubtedly will apply to many babies, and that her method of attacking the problem is, at any rate, a method that other mothers may follow to advantage when confront- ed with the same situation. The conservation of human life by the reduction of infant mortality is a noble undertaking, and it is hoped that this little contribution may in some measure further that undertaking. The volume is profusely illustrated. The author and Mr. Noyes followed the life of the child through his first two years with a camera just as faithfully as the mother followed him with her charts and memorandum pad. As a con- sequence there appear as illustrations more than sixty pictures of the baby, most of them full-page cuts. The book also contains some forty or fifty full- page charts. Both photographs and charts greatly enhance the value of the book. &i YOBX, Inc., BAI^TZMOBi:, MD. Al^G 5 1913 LIBRARY OF CONGRESS 021 344 895 A