Spens (1938)

Notes on the text
Preliminary pages Membership, Analysis, Preface, Introduction
Chapter 1 Development of the secondary curriculum
Chapter 2 The present position
Chapter 3 Physical and mental development of 11-16 year olds
Chapter 4 The curriculum of the grammar school
Chapter 5 Scripture
Chapter 6 English, classics, mathematics, general science
Chapter 7 The School Certificate Examination
Chapter 8 Technical schools
Chapter 9 Administrative problems
Chapter 10 Welsh problems
Chapter 11 Conclusions and recommendations
Appendix I List of witnesses
Appendix II Liberal education (Young)
Appendix III Secondary curriculum (Kandel)
Appendix IV Faculty psychology (Burt)
Appendix V Transfer of training (Hamley)
Appendix VI Curricula in the Dominions (Clarke)
Index

London: HM Stationery Office

Appendix V Memorandum on the cognitive aspects of transfer of training
by Professor HR Hamley, Professor of Education and Acting Director of the Institute of Education, University of London
[pages 439 - 452]

Dr Ballard in his book, The Changing School (1925) says: 'It sometimes happens that one section of the educational world is torn by eager controversy over some apparently vital problem, while other sections know nothing about it. The formal training controversy is a case in point'. I do not see any sign that the educational world is torn by controversy over this subject today, but, if I may judge from the letters that I am constantly receiving from teachers and administrators, there seems to be a widespread demand for authoritative information on this still unsettled question. My usual reply is to refer the enquirer to an excellent little report on the subject prepared for the British Association in 1930(1), and, if further details are asked for, to direct him to one of the summaries of experimental researches that have recently appeared. (2)

The problem of 'transfer of training', which is linked up with that of 'formal training' or 'mental discipline' has been stated in various forms: Are the effects of mental exercise general or specific? Can we, by exercising a mental capacity on a particular subject, strengthen that capacity as a whole? If so, is the strength thus gained available for other subjects or in other situations? Can the effects of training be transferred from one mental function to another? (3) The older or traditional answer to the last of these questions was emphatically in the affirmative. It was maintained that mental discipline was as essential to the development of the mind as physical discipline was to the development of the character. This belief had its effect on the curriculum; indeed, in large measure it determined it. Latin stood the test because, among other things, it improved the memory and cultivated the faculty of verbal accuracy; mathematics because it developed reasoning ability and science because it stimulated and developed the power of observation. In spite of all that has been asserted, and even proved to the contrary, such beliefs are still voiced today. In the introduction to the Report on Natural Science in Education (1918) (4), we are told that 'As an intellectual exercise it (science) disciplines our powers of mind and quickens and cultivates directly the faculty of observation'. And so, indeed, it may. The trouble is, however, that there is no guarantee that it will.

The main tenets of the doctrine of formal discipline have been well stated by Thorndike: 'The common view is that the words accuracy, quickness, discrimination, memory, observation, attention, concentration, judgement, reasoning etc, stand for some real and elementary abilities which are the same no matter what material they work upon; that these elemental abilities are altered by special discipline to a large extent; that they retain those alterations when turned to other fields; that thus in a more or less mysterious way learning to do one thing well will make one do better things that in concrete appearance have absolutely no community with it. (5)

The first murmurs against the faculty doctrine seem to have come from Ward and Herbart, but it was William James who made the first attempt to tackle the problem experimentally. James' conclusion, which was only mildly disturbing, had the effect of stimulating others to like endeavour. It was not until 1901, however, that Thorndike and Woodworth announced the results of the first serious attempt to deal with the problem empirically. Their conclusions have been described by Sandiford as 'a veritable bombshell in the educational camp', and indeed they were, not only because they seemed to shatter, at a blow, an age-long and cherished tradition, but also because they led the way to a revolution in curriculum construction. The latter consequence of Thorndike's experimental results is seldom realised. The general inference drawn by these workers may be given in their own words: 'Improvement in any single mental function need not improve the ability in functions commonly called by the same name. It may injure it. Improvement in any single mental function rarely brings about equal improvement in any other function, no matter how similar, for the working of every mental function-group is conditioned by the nature of the data in each particular case.' (6) Others had disposed of the 'faculty psychology' by denying that faculties exist; Thorndike achieved the same result by taking from faculties their real meaning. Henceforth all abilities became individual and specific. That being so, a new interpretation had to be given to the term 'a liberal education'. A liberal education must provide opportunities for a 'liberal' number of subjects. This theory, accepted in the United States, with less criticism than the importance of the subject merited, was largely responsible for the introduction of the 'unit' or 'credit' system, which has done so much to lower American educational standards. On the basis of these experimental findings Thorndike formulated his well-known theory of 'identical elements', which has persisted, with important modifications of interpretation, until the present day. According to this doctrine 'a change in one mental function alters any other only in so far as the two functions have as factors identical elements. The change in the second function is in amount that due to the change in the elements common to it and the first.' (7) Such a conclusion seems to be demanded as a logical necessity, for, if there is a correspondence or correlation between any two functions, there must be a common factor subsisting between them.

Admitting the theory of 'identical elements' as an explanation of the facts, our next question, and an important one, is: what is the nature of this common factor or identical element? Is it a mental or subjective factor, or is it a factor inherent in the objective situation? In other words, is it an element of mind or of material? Again, in modern mathematical language, we have to ask ourselves whether a condition which is 'necessary' is also 'sufficient'? There is abundant evidence that in this case it is not. We shall return to these points later, for it is in the neglect of distinction between necessary and sufficient conditions that so many errors of interpretation are to be found.

The long series of researches which followed Thorndike and Woodworth's pioneering effort have been summarised by Whipple in the Twenty-Seventh Yearbook of the National Society for the Study of Education (8), by Orata in his Theory of Identical Elements (9), and by Rugg and Vevia Blair in a monograph, prepared for the National Committee on Mathematical Requirements (USA), entitled The Present Status of 'Disciplinary Values' in Education (10) The latter, which is a detailed summary of all the important researches on this subject up to 1921, was compiled by Miss Blair on the basis of an earlier summary by Rugg. This appendix to the report of the National Committee is very interesting, for, in addition to the summary of researches, it gives the answers of a number of American psychologists to a questionnaire on 'disciplinary values in education'. The members of the Committee responsible for the report, among whom were some of the most eminent mathematicians in the United States, were driven to examine the disciplinary values of their subject as part of their enquiry into the aims of mathematical education. This admirable report, which is recognised in America as 'the charter of the mathematics teacher', has laid the foundation for mathematical education in that country which will stand the test of the mathematician and of the psychologist for many years to come. Not the least valuable chapter in tills report is that dealing with 'disciplinary values in education'. The balanced statement of the results of research is, I believe, largely responsible for the fact that the leading American teachers of mathematics have strenuously resisted the suggestion of the 'general educator' that the main aim of school mathematics should be utilitarian. As Bagley is sometimes quoted as an opponent of the idea of 'transfer' some of his remarks may be of interest: 'In my opinion, the possibilities of transfer are increased by the kind of teaching that makes the student conscious of the procedure as such, and keenly appreciative of its value as a general procedure. The theory of "identical elements" I regard as sound - but it is not rich in pedagogical suggestiveness. The theory of transfer through "concepts of method" and "ideals of procedure" furnishes a definite suggestion for teaching. The two theories are not inconsistent with one another; a person who has gained an understanding and an appreciation of a procedure will not be limited to the "identical elements" which come by accident; he will search for identities - for places at which the procedure may be applied.' It is strange that Orata, in his otherwise thorough analysis of the literature on this subject, makes no reference to this report of the National Committee.

As will be seen from an examination of these summaries, investigations into the subject of formal discipline fall roughly into three classes: those that deal with sensory and perceptual experiences, those that deal with such mental activities as memory, observation and reasoning, and those that have to do with methods of teaching and the relative disciplinary values of various school subjects. Thorndike's earlier investigations came within the first of these categories, but his conclusions were assumed to be of far wider application than the results warranted, and wider, I suspect, than Thorndike himself intended. Later investigations into the more complex mental functions do not, by any means, support the negative conclusions that were drawn from his earlier work. Orata has estimated that in over eighty per cent of the researches published between 1901 and 1927 there was evidence of 'definite' or 'appreciable' transfer (11); in nearly half of these it could be asserted that there was 'considerable transfer', Some of these experiments, it is true, may be discounted because of their faulty methods of procedure, and others because of their questionable statistical techniques, but, when these less satisfactory studies have been eliminated, there still remains a substantial balance in favour of formal training; there is certainly very little to support the assertion so frequently heard today that the doctrine of formal training is an 'exploded myth'.

The general procedure followed in all these researches was roughly the same, namely that of comparing the improvement shown by individuals or groups of individuals who had been set certain educational tasks. The statistical analysis of the results was not difficult because it involved little more than the computation of sums and differences. There are, however, snares even in simple statistics and these, as I have already indicated, many investigators were unable to avoid. The main weakness of these researches has been the lack of careful planning.

In most of the earlier experiments, and in some of the later ones, two groups of subjects were used for purposes of comparison: one, the 'training' or 'experimental' group, and the other, the 'control' group. It was not until it was realised that the real problem was not whether transfer of training actually occurs, but under what conditions it does occur, that the number of groups was increased to three or more. Let us suppose that we wish to compare the effects of a 'special' type of training with those of the accepted or 'conventional' type. Our problem is not only to determine whether the 'special' type of training is more efficacious than the 'conventional', but also to determine whether it is more effective than no training at all. Therefore, to compare any two methods of training, we have to employ three groups of subjects: the first being the 'training group' which is given 'special' training; the second, the 'practice' group which is given the 'conventional' training or ordinary practice; and the third, the 'control' group which is given no training at all. The omission of this third group was a serious defect in the early researches.

Woodrow's experiment (12) on the memorisation of poetry may be taken as an illustration of the procedure. The 'training' group were exercised in a definite technique of procedure (learning by wholes, attention to meaning, use of rhythm and grouping, use of secondary associations etc); the 'practice' group were given memory drill unenlightened by any special method of procedure. The 'control' group were denied both enlightenment and drill. The issue was, then, not between memory practice and no memory practice, but between practice supported by rational principles, and practice not so supported. This emphasis on consciousness of methods and procedures is the keynote of all modern research on this subject.

The results of Woodrow's experiment may be stated in his own words: 'In the case of all the end-tests except one, however, the difference in improvement between the practice and control group is small and statistically unreliable. The training group, on the other hand, shows for every end-test a decidedly greater improvement than either the practice or control group.' We are justified in concluding from this experiment that little or no 'transfer' follows unintelligent memory drill, but that considerable 'transfer' follows the conscious application of principles and methods of procedure. It may be objected that Woodrow's results do not show transfer of training from one material to another so much as the direct application of the same principles to material of more than one kind. There is point in this objection, of course, but it should be noted that no special effort was made during the course of the experiment to generalise the procedure by relating it directly to materials other than those which formed the subject of the experiment.

The conscious application of a special method to the study of a school subject is well illustrated by Miss EP Johnston's investigation into the value of geometry as a training in logical thinking. This experiment, the results of which were published in 1924, is important, for the reason that it was the first attempt, so far as I know, to test the difference between the conscious use of a set of principles and a rule-of-thumb or routine application of the same principles. The problem set was this: Can pupils of geometry be taught more effectively when trained to use consciously a technique of logical thinking, and furthermore, does such training, more than the usual method, increase the pupil's ability to analyse and see relationships in other non-geometrical situations? Miss Johnston's conclusions were set forth in her articles with commendable restraint: 'The data would seem to offer conclusive evidence, in so far as one experiment can be considered to do so, that when pupils are taught to use, consciously, a technique of logical thinking, they try more varied methods of attack, reject erroneous suggestions more readily, and without becoming discouraged maintain an attitude of suspended judgement until the method has been shown to be correct. Furthermore, the results in tests of reasoning other than geometrical would seem to indicate that such training in logical thinking with the materials of geometry tends to carry over these methods of attack and these attitudes to other problem situations not concerned with geometry.' (13) In other words, a training in demonstrative geometry has disciplinary value in so far as its logical principles are consciously recognised and applied. This is a conclusion with which those who admit a mathematical bias will readily concur.

I have quoted this passage at length because it is an admirable summary of the results that may be expected from teaching by an enlightened method. It is also an excellent example of cautious inference from data which would stand the test of more positive claims.

It is worth noting that Miss Johnston carried out this investigation as part of her ordinary class work and not in preparation for a degree. She had one advantage in that she was able to equate her groups for intelligence and thus eliminate one disturbing factor. I understand, from a private source, that occasionally she used non-geometrical illustrations. For example, when dealing with 'axioms', part of the Declaration of Independence was quoted. If it is axiomatic that 'all men are created equal' and 'are endowed by the Creator with certain inalienable rights', then certain conclusions regarding the Constitution must follow. Such references to non-geometrical illustrations are good teaching method, but to some extent depreciate the value of the experiment. In spite of this defect the main conclusions were, I think, unassailable.

On reviewing the literature on this subject one is struck by the dearth of well-planned and carefully executed research. We have, of course, Thorndike's elaborate statistical investigations into the mental discipline of various school subjects leading to the conclusion that 'the intellectual values of studies should be determined largely by the special information, habits, interests, attitudes, and ideals which they demonstrably produce. The expectation of any large difference in general improvement of the mind from one study rather than another seems doomed to disappointment. (14) The problem that we are at present discussing, namely, the conditions under which general mental training may be expected from the study of any subject was not the real object of Thorndike's investigation. (15)

Among the experiments that have followed the general lines of those carried out by Woodrow and Miss Johnston the following should be mentioned: Meredith, (16) on the transfer from the definition of scientific terms to the definition of terms in ordinary daily use; Judd (17), on the effect of consciously formulating guiding principles of generalisation; Winch (18) on improvement in logical reasoning following practice in problem arithmetic; Coxe (19), Haskell (20), and Hamblen (21), on transfer of training from Latin to English, following a conscious adherence to certain methods of instruction; Ruger (22), on the effect of developing formulae or general rules for the solution of puzzles. All of these investigators stress the importance of 'conscious formulation and application of general principles', or 'organisation of the material of study', or 'the development of meanings, concepts and generalisations'.

Space does not admit a detailed discussion of these researches, but since the classics have often been credited with peculiar value as mental discipline, Hamblen's study, which was in its main essentials similar to that of Haskell, may be outlined in a little more detail. The problem he set himself was to determine the extent to which the effect of the study of Latin upon English derivatives could be increased by the conscious adaptation of content and method to the attainment of this objective. In the strict sense some of Hamblen's results do not measure 'transfer', but rather the direct results of special teaching. He found, as one would expect, that exercise in 'derivative work' led to a great improvement in the students' command of English words. He also found that marked improvement in this respect followed exercise in logical selection and the organisation of subject matter even when derivative work was not given. Coxe's study related to the spelling of English words of Latin derivation. He reached the interesting conclusion that merely pointing out the similarities in spelling between Latin and their English derivatives produced a marked gain in the accuracy of English spelling.

One other study not mentioned in the above list may be cited. This is the investigation made by Whelden on the effect of Latin upon the quality of academic work at Yale. This research is worthy of notice because every precaution was taken to control selective factors by the use of modern statistical analysis. The author reports that no support was given to Latin 'as an intellectual discipline serving to extend the scope of intellectual capacity in whatever field it might be applied'. (23) I do not think the classicist need be unduly disturbed by this pronouncement, for the author goes on to add: 'These conclusions should be understood, of course, in terms of "Latin as taught" and "Other subjects as taught", with the possibility remaining that some change in the type or method of instruction in Latin, or some change in the manner of presentation of other subjects, might affect the relationship between the extent of a man's background in Latin and the quality of his work in other academic fields.' These results are, I am afraid, of no great help to the teacher of classics.

This account of researches into the subject of formal training would hardly be complete without some reference to ideals, which some writers (eg Godfrey Thomson and Fox) consider to be the key to the problem of transfer. According to Thomson, as ideals are progressively clarified they become more potent in transfer value: 'There comes first an unconscious employment of certain principles or ideals. These gradually become clearer and more definitely outlined. They are recognised by their owner and named, and thereby gain tremendously in effectiveness and in transfer value.' (24) Of the experimental investigations into the importance of ideals, that made by Ruediger, following an earlier study by Bagley and Squire, on the effect of inculcating neatness as an ideal, is probably the most reliable yet presented. Ruediger's conclusion may be quoted because there is little doubt that, paucity of evidence notwithstanding, it is capable of fairly wide application. 'Neatness made conscious as an ideal or aim in connection with only one school subject does function in other subjects.' (25)

It is not easy to extract from these researches and the interpretations that have been placed upon the results any central concept or general principle. The conclusion seems to be that transfer of training depends upon the conscious acceptance by the learner of 'methods', 'procedures', 'principles', 'sentiments', 'ideals', and schemata or patterns of thought. The only word which emerges from these concepts as common or generally applicable to them all is the word conscious. Orata, in his summary of the studies that have appeared since 1927, estimates that 70 per cent of the studies support the proposition that the effect of training is general and that, therefore, transfer takes place most effectively through conscious generalisation (to use the words of Judd), whereas about 30 per cent may be classified as supporting the theory that practice is specific, and that transfer therefore takes place through identical elements. (26) If these conclusions are to be reconciled, Thorndike's Theory of Identical Elements must be given a much wider interpretation than that commonly accepted. As Judd has consistently emphasised, common elements are common mental elements; they are not necessarily common elements in the objective situations. They are methods, procedures and ideals that are, in the Gestalt sense, 'functionally similar'. Burt has very clearly expressed this thought in his Report to the British Association: 'Transfer of improvement occurs only when there are common usable elements, shared both by the activity used for the training and also by the activity in which the results of the training reappear. The "common elements" may be elements of (i), material, (ii) method, (iii) ideal; they are most "usable" when they are conscious. A common element is more likely to be usable if the learner becomes clearly conscious of its nature and of its general applicability: active or deliberate transfer is far more effective and frequent than passive, automatic or unintentional transfer This seems especially true when the common element is an element of method rather than of material, an ideal rather than a piece of information.' (25) I regard this statement as the best summary of our modern views to be found in the literature on the subject.

The question may be asked: Can we not explain the facts by stating that transfer, when it does occur, is a single function of intelligence? The more intelligent the learner, the greater his powers of generalisation and the more varied his interests and enthusiasms. In other words: 'Transfer of training depends upon the degree to which we apply our minds to the task in hand.' This would seem to be obvious. More than one investigator has shown that the degree of transfer effected depends upon the intelligence of the learner, but this does not explain why the less intelligent of two groups may, under certain conditions, surpass the more intelligent in the critical tests. Several of my students have shown this to be true, when they have been forced to work by a rotation method with groups of unequal intelligence. If transfer is a function of intelligence, it is a function of all intelligence, low as well as high. My own view, supported by very little evidence, I admit, is that the degree of transfer is not a linear function of intelligence, but that it is a function of intelligence and other variables which are themselves related to intelligence.

It may be, however, that there is more in the statement quoted above than is evident at first sight. Transfer is made possible when we apply our minds, but what is it that impels us to apply our minds to any task? It is interest or enthusiasm which gives to intelligence its power. None of the investigators to whom I have referred has made an adequate estimate of the interest displayed by his learners; only two of them have mentioned the subject of interest at all. Again, no one seems to have made any estimate of the effort put into learning by various methods. Yet it should be evident that the problem of formal training is not a purely cognitive problem - no educational problem ever is. My main objection to the theory of identical elements is that it is expressed in cognitive terms and that it does not, even in its more modern form, give due consideration to the conative and affective concomitants of the cognitive elements involved. In our next advance in this subject we shall have to take much more serious account of these conative and affective aspects of the learning process.

In all transfer there is communication. Transfer of training is not the communication of 'elements' but of life; it is not a process but a living process. Methods, procedures, ideals, principles and patterns of thought are not inert entities; they are mental experiences impregnated with life. As Nunn has expressed it, in another connection: 'The prime contribution of the heroes of science to the world's cultural wealth is not the scientific method but the scientific life. Our business, then, is to teach the realisation of the life, not the mastery of the method.' The study of science, or mathematics, or Latin may be justified to a greater or less extent on utilitarian or disciplinary grounds but its full justification can only be sought in terms of a richer and more deeply significant mode of life. Method is not enough. This was in our minds when the distinction was drawn between necessary and sufficient conditions.

Statements such as those made above do not, however, indicate ways in which interest may be aroused, nor do they suggest the means by which teaching may be made more vital. Perhaps we shall gain some light on this problem after a short psychological digression.

Spearman has shown, in his Principles of Cognition that when two 'fundaments' (A and B) are presented to our minds we tend to educe a 'relation' (R) between them. For example, if I were to write two letters O and o, or S and s, a child would say: 'One is bigger than the other.' Again two numbers 7 and 11 would evoke the response: 'Their sum is 18' or 'Their difference is 4.' This process of eduction may be represented by a formula A---->R<----B, where the arrows express the fact that the relation is 'educed' or drawn from the fundaments. Eduction is not merely a result; it is also a process. The arrows suggest more than mere eduction, however; they suggest a tendency, a tendency to think in the direction of the relation, whenever the same fundaments, or similar ones, are presented. For example, having once seen that 7 and 11 lead to the number 4 by a process of subtraction, the child will the more readily educe that relation when the same numbers are again presented. The arrows may, therefore, be said to indicate a certain intentional stress, or, if we prefer it, a certain purposive integrity, which makes further thought in that direction natural and, therefore, easier. Thus, the eduction of a relation secures, at the time of eduction, one of the conditions of its own repetition. The strength of this condition depends partly on the intellectual ability of the person and partly on his interest in the subject. It has been said that the problem of education is the problem of interest. This is, of course, not the whole truth, but no one would doubt that learning is facilitated by interest. Interest gives the bonds that are symbolised by the arrows in the formula A---->R<----B a purposive set in the direction of eduction. Interest, like magnetic influence in a metallic field, gives strength to mental patterns and largely determines their mode of functioning.

When a relation is educed from two fundaments various degrees of eduction intensity are possible. A relation may be so closely held to the fundaments that it is inseparably bound to them; on the other hand, the relation may be free, that is so fully abstracted that it may be 'intended' apart from its context. Lorimer calls the capacity of the person to deal with relations apart from their fundaments as 'free intelligence'. As an illustration of free relations we may take the inculcation of habits. One teacher of mathematics insists on neatness in the mathematics lesson, on the ground that neatness tends to reduce the liability to error; another teacher of mathematics achieves the same end by presenting neatness as a desirable ideal. In the former of these cases, neatness will in all probability be bound to its mathematical associations; in the latter it is likely to be free, in the sense that it will be observed in non-mathematical situations. It is not an uncommon experience to find a child neat in the mathematics classroom and slovenly everywhere else.

Other illustrations could be cited from elementary problems of learning. Most teachers who appreciate the importance of concrete aids in early number teaching, also realise the danger of keeping the child too long in the concrete. Branford gives a very simple illustration of the inability of a child to use relations apart from their perceived fundaments. This child was invited by him to follow a demonstration of congruity with three triangles cut simultaneously from three sheets of paper in contact. The child put triangle A on triangle B and agreed that they were alike. She then put triangle C on triangle B and agreed that they were alike. He was on the point of saying, 'You see, then, that all three are alike', when the child put A on C. She was unable to reach this conclusion, nor would she agree to it, until A and C had been fitted together. (26)

Rignano gives some amusing illustrations of this lack of ability to deal with abstract relations in his Psychology of Reasoning. A neighbour owed an old lady 12 lira but was owed 7 lira by her. The neighbour proposed to settle the account by making a single payment of 5 lira, but the old lady was not satisfied until the whole transaction had been carried out in the concrete in accordance with the equation 12 - 7 = 5.

The bearing of this discussion upon the problem of transfer will be evident, for much depends upon whether an educed relation is 'free' or 'bound', whether the 'intentional gradient', if we may call it so, is in the direction of the relation or in the direction of the fundaments.

Miss Johnston's investigation, which I have cited as an example of good experimental procedure, was designed to test whether pupils could be taught geometry more effectively when trained to use consciously a technique of logical thinking. The issue lay, then, between a logical system bound to its geometrical fundaments and a logical system as far as possible free from its geometrical origins. As I have already indicated, the verdict was in favour of intellectual freedom. This favourable result, it may be noted, was not due to differences in intelligence but to differences in teaching method. The conclusion could have been stated as follows: geometry should be taught in such a way that the logical relations developed from it are free and accessible, available for use in non-geometrical situations. That is to say, relations and relation systems are of general value in so far as they are free and available for use.

There is one notable omission from Miss Johnston's account of her own work. Only passing reference was made to the interest displayed by her pupils in the work given them, but I understand from those who have been present during her lessons that the 'experimental' class found their work very interesting.

Some years ago I made a comparative study of the heuristic and didactic methods of teaching elementary school mathematics. The results showed that those who had been taught by the heuristic method were not only better able to apply their knowledge to unusual problems, but were also much more eager to do so. At the end of the year both classes were asked whether they had found their mathematical work interesting, and, if so, why? Those who had been taught by the heuristic method admitted two interests, one in the subject itself, and the other in the method by which the subject had been taught. One of the boys, better psychologist than he knew, wrote: 'The new method cleans and expands the mind. It gives us the desire to think.' When I asked the boy why he had used the word 'cleans', he replied that he had in mind 'a house kept in order'. I doubt whether any of the authorities I have quoted have set forth the results of true intellectual training with greater insight - a well-ordered mind, ready and eager to expand.

It is unfortunate, especially in view of the extravagant claims that are sometimes made for the disciplinary value of the classics, that no psychological study of classical subjects has been made comparable with those that have been carried out in other school subjects. That the study of Latin or Greek can be made a valuable intellectual discipline few would be ready to dispute. That these subjects are, by their very nature, superior instruments for this purpose is certainly open to question. (27) The study of Latin may be as useless as a mental training as that meaningless juggling with symbols which sometimes goes by the name of 'mathematics'. The question is not, whether the classics are valuable instruments of culture or of mental training, but under what conditions they can be made so. It is unfortunate that the answer to this question cannot be given on the basis of facts, for facts, as distinct from opinions, are distressingly few. Perhaps clues to the solution of this problem may be found in some of the conceptions that have been developed in this paper.

The main aim of classical studies is the communication of thought (28), through the agency of languages, which though structurally different, have certain conceptual correspondences. If that is so, translation is the communication not of word, but of thought, which in the process of communication becomes free of its linguistic origins. But thought cannot be liberated until words have been discriminated and language forms have been analysed. On the cultural side, classical studies are of value in so far as they refine and enrich the mind; on the disciplinary side they are of value in so far as they provide occasion for what may be called 'functional thinking'. The latter statement will need a word of explanation. The word 'function' may be used in two different senses, one scientific or physical and the other mathematical. When we speak of 'the function of the teacher' or 'the function of the liver', we use the word 'function' in the physical sense as synonymous with a duty, or a service, that the teacher or the liver is expected to perform, without imputing to either any necessary association with mathematics. But when we assert that 'the temper of the teacher is a function of the condition of his liver', we imply a correspondence between the state of the one and the condition of the other which could be expressed with equal precision of thought by a mathematical equation. The variables in this case are 'the temper of the teacher' and 'the condition of the liver' and the statement is to the effect that these two variables are related. A functional relationship may be defined as a determinate correspondence between two sets of things, a correspondence which, for mental economy, is sometimes expressed as a law or in a formula. 'Functional thinking' (29) is thinking in terms of numbers, or points, or even words, which exhibit some relationship or correspondence.

It is the mathematical meaning of the word 'function' which is to be read into the statement that the classics provide occasion for 'functional thinking', although we frequently use the word in the scientific sense, as when we speak of 'the functions of words' or 'functional grammar', meaning that words have certain duties or offices to perform. In our present discussion the variables are 'Latin' and 'English' - rich enough variables surely - and the relationship between them is that of correspondence of thought. And yet how often it is assumed that in the study of Latin or Greek all that is to be sought is a one-to-one correspondence between word and word. If we examine certain Latin primers, still popular among teachers of the old persuasion, we find that all the earlier exercises consist in the translation of isolated words, usually verbs. Could the subject possibly be made more barren and uninspiring? If the truth be told, as they stand, isolated words may not mean anything. The proper unit of language teaching is not the word but the whole story. It is the old error repeated, the subordination of the pupil's true intellectual needs to the supposed necessities of logical training. So it was with school mathematics until about twenty-five years ago, when Sir Percy Nunn, in a work of real genius, pointed the way out. Nunn's Algebra (30), which is a treatise on 'functional thinking', has rescued school mathematics from the crippling effects of its own relentless logic. I have no doubt that mathematics taught in his way is excellent mental training. Again, I have no doubt that Latin or Greek taught in an analogous way would have the same desirable virtue.

Before Nunn's treatise was published, the subject of algebra was either extremely mechanical or extremely abstract and was almost wholly unrelated to the child's natural interests or future needs. The study of algebra began with arithmetical substitutions, often of a meaningless kind, and continued through the fundamental operations, factors and the like, to quadratic equations. Since the publication of that work these formal exercises have given way definitely and finally to 'formulae', 'graphical representation', 'variables and functions', concepts that are both fundamental and dynamic. School mathematics is now based on the fundamental concepts of the subject rather than upon the acquisition of mechanical skills in logical order.

The comparative study of the heuristic and didactic methods of teaching school mathematics to which I have referred showed that algebra and geometry taught from the 'functional' point of view gave much better results with non-mathematical problem material than the more formal teaching methods. (31) In other words, 'functional' mathematics proved to be better mental training. Whether this superiority was due to the method followed or to the interest engendered by that method need not at present be decided. The one seemed to involve the other.

So far as I am aware, there has not appeared on the teaching of any other school subject, a work comparable with Nunn's Teaching of AIgebra. The distinctive features of that work are that it is based upon firm philosophical foundations, that it takes full account of the pupil's interest and needs, that it looks backward into the past (historical) and forward to the future (the practical needs of modern life), and that it is dynamic rather than static in its treatment. It is a treatise, indeed, not on mathematics but on mathematical thinking. I am venturing to suggest, with considerable trepidation, that the teacher of the classics would learn much from an examination of the general principles outlined in the Teaching of AIgebra. He would find, without an undue straining of analogy, that the two subjects have certain common features. Functional mathematics is based upon five main logical concepts: 'the class', 'order in the class', 'the variable', 'correspondence' and 'functional relationship'. These concepts have their counterparts in the study of the classics. For, corresponding to the 'variable', which is the type or symbol of a set of numbers or points arranged in a particular order, we have the 'word', which is the type or symbol of certain perceptual or conceptual meanings. With words as our materials translation from one language to another becomes a functional process. Considerations such as these lead one to the conviction that the full mental value of classical education could only be obtained by treating the subject 'functionally'.

Footnotes

(1) British Association: Report on Formal Training. Bristol (1930).

(2) See references below.

(3) In this memorandum 'transfer of emotion' from one set of conditions to another is not considered. Such 'conditioning' of emotional expression is a normal accompaniment of all education.

(4) Natural science in Education. Report of the Committee appointed by the Prime Minister; Chairman, Sir JJ Thomson. HM Stationery Office (1918).

(5) Thorndike, EL Educational Psychology, (1903), p. 84.

(6) Thorndike, EL and Woodworth, RS Influence of Improvement in one Mental Function upon the Efficiency in other Functions, Psychological Review, (1901), p. 250.

(7) Thorndike, EL Educational Psychology, (1903), p. 80.

(8) Whipple, GM The Transfer of Training, 27th Year Book of the National Society for the Study of Education, Pt. II. Ch. XIII. See also Sandiford, Peter. Educational Psychology, pp. 279 - 289, which gives a shorter summary of significant researches up to 1928.

(9) Orata, PT The Theory of Identical Elements. Ohio (1928). Also Transfer of Training and Educational Pseudo-Science, Mathematics Teacher, May 1935, p. 265. In the first of these works Orata gives reference to most of the researches up to 1927, and in the second he brings the list of references up to 1935.

(10) The Present Status of 'Disciplinary Values' in Education, V Blair - an Appendix to the Report of the National Committee on Mathematical Requirements entitled: The Reorganisation of Mathematics in Secondary Education. Mathematical Association of America, (1923).

(11) It should be noted that in these researches the word 'transfer' is used in relation to many different phenomena.

(12) Woodrow, H The Effect of Type of Training upon Transference, Journal of Educational Psychology, March 1927.

(13) Johnston, Elsie P Teaching Pupils the Conscious Use of Technique of Thinking. Mathematics Teacher, April 1924.

(14) Thorndike, EL Mental Discipline in High School Studies. Journal of Educational Psychology, Jan. and Feb. 1924.

Broyler, Thorndike and Woodward. A Second Study of Discipline in High School Studies. Journal of Educational Psychology, September 1927.

(15) Investigations of this type have their value, but so many factors are uncontrolled that a satisfactory interpretation of results is rendered almost impossible. The relative strengths of conscious and unconscious ideas and of interests and sentiments aroused were incapable of assessment.

(16) Meredith, GP Consciousness of Method as a Means of Transfer of Training. The Forum of Education, Feb. 1927. I regard this as the most significant work on the subject that has been carried out in this country.

(17) Judd, CH The Relation of Special Training and General Intelligence. Educational Review, Vol. XXXVI., 1908.

(18) Winch, WH The Transfer of Improvement in Reasoning in School Children. British Journal of Psychology, Vol. XIII., 1923.

(19) Coxe, WW The Influence of Latin on the Spelling of English Words. Journal of Educational Research, March 1923; March 1924. Monograph No. 7, 1925.

(20) Haskell, RI Doctor's Thesis, University of Pennsylvania, 1923.

(21) Hamblen, AA Doctor's Thesis, University of Pennsylvania, 1925. Both theses deal with the effects of the study of Latin upon the knowledge of English derivatives.

(22) Ruger, HA The Psychology of Efficiency, Archives of Psychology, 1910.

(23) Whelden, CH Training in Latin and the Quality of Other Academic Work. Journal of Educational Psychology, October 1933, p.497.

(24) Thomson, GH Instinct, Intelligence and Character (1936), pp. 144-5.

(25) Ruediger, WC Indirect Improvement of Mental Function through Ideals, Educational Review, No. 1919, p. 364. See also Mental Discipline, School and Society, 13 Jan. 1917.

(26) Orata, PT Mathematics Teacher, May 1935, p. 267.

(25) Burt, C Formal Training. Report of a committee appointed by the British Association and presented at Bristol in 1930, pp. 3, 4.

(26) Branford, B A Study of Mathematical Education. Oxford, (1921), p. 305.

(27) For a comprehensive study of the place of Latin in Education see: Valentine, CW Latin: Its Place and Value in Education, London, (1935).

(28) The word 'thought' is here used generally and includes the expression of feelings as well as ideas.

(29) The expression 'functional thinking' was first used by Felix Klein, the German mathematician. Klein maintained that training in functional thinking given in the mathematics classroom would carry over to similar non-mathematical thought processes.

(30) Nunn, TP The Teaching of Algebra, (1914); Exercises in Algebra, including Trigonometry, (1913-1914).

(31) Hamley, HR Relational and Functional Thinking in School Mathematics, being the Ninth Yearbook of the National Council of Teachers of Mathematics, America, 1934. This book gives an outline of the course followed by the experimental group.