The Bearing of the New Psychology upon the Teaching of Mathematics
by Agnes L. Rogers - 1916
By the new psychology we indicate the comparatively modern psychology based upon experiment and measurement. The results obtained have come from investigations conducted under controlled conditions and which can be repeated therefore by other investigators and verified. The older analytical psychology, which used the method of introspection, made valuable contributions to our knowledge of the structure of mental processes, but from a practical point of view it is more important to know their dynamical character and functional relations. This is true particularly in teaching, since the fundamental problem for education is “the production and prevention of changes in human beings.”
By the new psychology we indicate the comparatively modern psychology based upon experiment and measurement. The results obtained have come from investigations conducted under controlled conditions and which can be repeated therefore by other investigators and verified. The older analytical psychology, which used the method of introspection, made valuable contributions to our knowledge of the structure of mental processes, but from a practical point of view it is more important to know their dynamical character and functional relations. This is true particularly in teaching, since the fundamental problem for education is the production and prevention of changes in human beings.
Psychologists, however, have made relatively few experimental investigations in the field of mathematics. If we disregard arithmetic, which of late has received considerable attention from the experimentalists, we find that most of the publications dealing with the psychology of the subject have used the older methods of introspection and observation. This is not surprising when we recall how little is known as yet of the psychology of thought and how indispensable systematic observation is as a preliminary to experimentation. The result is that we have at present various theories of a speculative character as to the nature of mathematical ability which still await confirmation.
Möbius has advanced the opinion that mathematical talent is a special capacity independent of other mental capacities and characterized by unusual ability in understanding relations of number, in judging relations of size, and in concrete imagery.
Hüther, on the other hand, maintains that mathematical genius involves no specific, fundamental capacity; it consists merely in an exceptional ease in carrying out certain thought operations and involves marked development of concrete imagery, synthetic imagination, and mathematical understanding. Betz, who agrees with Hüthers general theory, offers a different explanation of mathematical ability. He contends that the mathematical type of mind is characterized by a special clearness of certain minimal or highly abstract ideas and by the ability to manipulate and vary these ideas with ease and precision.
Henri Poincaré, the distinguished mathematician, on the basis of introspective analysis, gives it as his judgment that mathematical ability has nothing to do with a very sure memory or a special power of attention. It is a feeling for order and the concealed relations of numbers that distinguishes the mathematicians from other men. He divides mathematical reasoners into two distinct classes, the geometrical or intuitional and the analytical or logical types.
All such theories are interesting but singularly barren of fruitful, practical suggestions. At best they merely indicate to the experimental psychologists promising subjects of research. It should not be inferred, however, that experimentation alone can furnish information of value to the mathematics teacher. On the contrary, such a method of attack as that of Judd, furnishes an illuminating account of the mental activities that mathematics demands. This account presents an excellent survey of the simpler psychological processes underlying mathematics, which have been experimentally investigated, describing typical mental reactions involved in mathematical thinking and analyzing the psychological implications of the text books in use and of current class-room procedure. In comparison the new psychology has little to offer; but by its refinement of method it undoubtedly gives promise of richer results.
Early experimental effort naturally was directed to analysis of the mental processes involved in the simplest branches of the subject, namely, arithmetical operations. The work in this field has been extensive and significant for mathematics in general. For our purpose the most important results of the statistical studies by Rice, Thorndike, Stone, Bonser, Courtis, Winch and Starch i are the demonstration of the wide range of individual differences in capacity and the evidence in support of the specialization and independence of the different abilities involved in arithmetic. The extent of individual differences had already been shown by investigations upon other mental functions; its demonstration in the case of arithmetic, however, was exceptionally striking. Equally remarkable was the discovery that a high degree of excellence in the fundamental processes may be present along with a low degree of skill in arithmetical reasoning, and vice versa. Indeed, it was found that a similar variability prevailed among the fundamental processes themselves. These results led Fox and Thorndike to prophesy that the abilities testedaddition, multiplication, fractions, rational computation and problemsbear little resemblance to those of the mathematician.
The amount of experimental investigation accomplished in algebra and geometry falls far short of the work done in arithmetic. If we exclude the recent efforts to establish standards for algebra by Thorndike, Monroe, and Rugg, a new line of activity which cannot but have considerable effect upon the teaching of algebra, we find that in all the experimental investigations published, with two exceptions, the data have been school and college marks or class lists. Correlations have been calculated between mathematical ability as a whole and ability in other school subjectsii, and in brief the result has been that fairly high positive correlations have been obtained.
An interesting attempt to secure a more complete and detailed analysis of mathematical intelligence was made in 1910 by William Brown. He used the same statistical method of correlations, obtaining his data from a school examination in algebra, geometry, and arithmetic. He corrected the papers by two methods, first, according to ordinary school standards, and second, according to a differential system of marking based on an introspective analysis of the intellectual processes involved in answering. His principal results were that geometrical ability and algebraic ability are not related except through their connection with arithmetical ability, which is of some interest in connection with the present effort to correlate these subjects more closely, and that memory of preceding propositions is the central ability in geometry, being related most intimately to other forms of geometrical ability. This is in harmony with his opinion that school mathematics and higher mathematics relate to different forms of ability and it raises the whole question of the distinction between school mathematics as it is and as it might be. The fact that it is now advocated by some mathematicians that the concepts of higher mathematics should be introduced into the secondary school suggests that even if Browns conclusion is true of the present state of affairs, it need not be with different methods of teaching and other standards. In any event his experimental procedure should lead us to accept his statements with caution; for obviously the psychologizing of examination papers is an unsatisfactory manner of measurement, and further the number of individuals examined was relatively small.
Another statistical study, carried out in the Dartmouth pedagogical department under the direction of F. C. Lewis, deserves mention because of the new departure in method. Instead of using ordinary school marks as data, tests were given in originals in geometry and in practical reasoning and the scores made in these were correlated. It may be doubted whether these tests were adequate measures of the abilities in question; but the mode of procedure plainly marks a step in advance and the results are noteworthy. The students of each of twenty-four groups were arranged in two series, the first according to their ranking in mathematical reasoning and the second according to their ranking in practical reasoning. It was found that of the first five mathematical reasoners from each group, 63% were at the foot of the practical reasoning series, conspicuous for their inefficiency in practical reasoning; and of the pupils at the foot of the mathematical reasoning series, 47% were conspicuous for their positions at the head of the practical reasoning series.
These statistical inquiries, like the earlier accounts based on introspection and observation, are interesting rather than helpful. Their general outcome can be expressed briefly in the statement that a high correlation exists between efficiency in mathematics and efficiency in other subjects.
For further guidance we must turn to general psychology and here the crucial question is the transfer of training. Not only do experiments on transfer yield the most useful suggestions for methods of teaching, but upon them rests ultimately the defense of the place accorded to mathematics in the curriculum of the secondary school. For the mathematicians themselves have admitted that all the facts that a skilled mechanic or engineer would ever need could be taught in a few lessons. Consequently, very little of the mathematics given at present in the high schools could be retained on the ground of its practical usefulness. Nor does the conventional value of mathematics justify the time and effort it entails. Society does not regard an individual as grossly ignorant or ill-informed if his knowledge of mathematics is extremely meagre. Current opinion rather assumes that mathematical skill is highly specialized and unrelated to general intelligence or culture. The ultimate defence of the retention of mathematics in the curriculum rests therefore upon its general educational value.
What, then, are the established results as regards transfer of training? At the present time no psychologist of repute denies that transfer of training is possible. Experimentation has shown conclusively that practice in one function affects other functions. The points at issue are the extent to which transfer takes place and the methods by which it is secured. It has been established that the amount of transfer varies with the degree of difference between the functions in question. Change either in the content or in the method of study reduces the extent of the spread of improvement. Accordingly the indirect effect of practice invariably is less than the direct. Furthermore, transfer can be negative; the habits or mental acts developed by a particular kind of training may inhibit rather than facilitate other mental activities.
Investigators have found very different degrees of transfer effect in accordance with the different functions tested and the different experimental conditions, and frequently it has been extremely small in amount. On the whole, however, there is ground for the assertion that if transfer often is exceedingly small, it is probably always (to some extent) present, and a very small spread of training may be of great educational value provided it covers a wide enough field. As Thorndike points out, if a hundred hours of training in being scientific about chemistry produces only one-hundredth as much improvement in being scientific about all sorts of facts it would be a very remunerative educational force. Certain facts also must be kept in mind with regard to the smallness of the transfer obtained in experimental investigations. For the most part adults have been used as subjects in these investigations and with adults we would not expect to obtain as much transfer as with children. Whereas in the young and immature mental habits are still in process of formation, in adults they virtually are established and in consequence any improvement made in training is probably due to adaptation to special conditions and therefore not susceptible of generalization. The results of Dallenbachs experiments on visual apprehension in school children strongly contrast with those of Whipple and Foster on adults regarding practice effects and cast doubt upon the common assumption that conclusions derived from a limited number of selected adults necessarily are true of growing children. Again, certain features of the experiments made on transfer seem distinctly unfavorable to spread of improvement. Thus, the practice periods usually have been short and the training given can be described fairly as work at high pressure. On general educational grounds we do not believe that improvements so effected are likely to transfer.
We may say further that the processes tested in the laboratory investigations are comparatively simple. They differ in marked fashion from the complex processes involved in Latin, mathematics, or science. In short, the conditions of the experiments depart so radically from ordinary class-room conditions that it well may be questioned whether results so obtained can determine even approximately the amount of transfer possible in the case of the school subjects. Even where experiments have been carried out in the school room as by Winch and Sleight the methods of securing greatest transfer have not been utilized fully. It seems reasonable, therefore, to conclude that even if but slight transfer effect has been found in experimental work, judgment against a wide spread of improvement in general should be suspended, since the conditions favorable to generalization were absent.
At present we have only prophecy, not knowledge. The experimental results so far obtained are so paltry and limited compared with the mass of facts to be measured in the case of any of the secondary school subjects that an extreme view stands discredited. It is possible, to say the least, that the amount of general effect produced by the high school training in mathematics is greatly in excess of the typical transfer effects of experimental investigation. But measurement of the actual changes made by mathematics has still to be accomplished.
Psychology has a more positive contribution to make to the second point in dispute. There is, to be sure, a variety of opinions as to the ways and means by which transfer is facilitated. Identical elements, development of attention, will, mental imagery, ideals, divesting the essential process of inessential elements, improvement in the technique of learning, all these have been suggested as causes of transfer and probably all do function to some extent in the spread of improvement. There is, however, a growing consensus of opinion as to the factors operative in transfer as a result of the progress made by those investigators who have subjected their results to careful analysis. Prominent among such studies is that of Ruger. In this article his conclusions only can be summarized briefly.
Ruger found four general factors in transfer of training. They were ideals, attitudes, concepts of method, and high level of attention. The formation of ideals, such as a general idea of efficiency, was an important element in the spread of improvement from one activity to another. Similarly, the attitudes adopted facilitated or hindered transfer. Thus, a self-conscious attitude restricted progress and checked transfer while an attitude of self-confidence was very favorable. Again, concepts of method played a most important part in the study. The conscious control of assumptions, the active search for new hypotheses, the effort to distinguish between suggestions and to classify them appropriately, the deliberate testing of hypotheses, and the generalization of these methods themselves together with the realization of the value of such generalizations contributed greatly to transfer. Above all, a high level of attention was an indispensable precondition. Of the special factors in transfer as shown in Rugers study the most important were related ideas. Upon these, to a large extent, depends the possibility of generalization. We have been apt to believe that if we gave individuals a theory they would be able to apply it appropriately. But the fact is that children have to be taught as carefully to apply theory as to understand it and unless related ideas are pointed out to them they often will fail to perceive their connection. Ruger found in his experiments with wire puzzles that geometrical concepts played no part in hastening their solution and that the greatest transfer was from similar puzzles. Finally, as to the relation of habit to transfer, the fact that certain established habits could be utilized advantageously in new conditions did not lead necessarily to their use. The recognition of the identical elements in the two situations was often absent and the degree of transfer varied directly with the precision of analysis of the similarity of the new case to the old.
These results suggest the following practical injunctions: First, proper attitudes should be cultivated in the pupil. Secondly, attention should be focused on the art of learning and on methods of procedure in the solving of problems so that the pupils should be stimulated to analyze the situation, to formulate hypotheses, to criticize and evaluate each suggestion, to be systematic in selecting and rejecting these and in verifying them. Further, each step should be generalized as a method so that there should be deliberate control of assumptions. Thirdly, attention should be directed to related ideas in order that as many as possible may be recalled or discovered. Lastly, motivation should be secured and attention should be kept at a high level. By such means the experience gained in mathematics will tend to be generalized and made available in other fields.
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i See bibliography in H. B. Howells A Foundational Study in the Pedagogy of Arithmetic
ii See Columbia Contributions to Philosophy, Vol. XI, No. 2.