Abbildungen der Seite
PDF
EPUB

trol of subject matter. This reversal of method, coupled with a distrust of the theory of discipline, has thus not only reduced grammar to a small fraction of its former self, but has, besides, greatly rearranged and rewritten the study.

Keeping before us the demand for economy in learning and effective control of subject matter, what can we say about method? How does learning in fact take place? (1) Repetition is a factor in learning known to all. (2) An inclusive "set" which shall predispose the attention, focus available inner resources, and secure repetition is a necessary condition less commonly considered. (3) The effect of accompanying satisfaction to foster habit formation is a third factor to be noted. These three factors are necessary, then, to adequate consideration of the problem of method. It accords with these considerations and with undisputed observation that, other things being equal, any item is more readily learned if its bearing and need are definitely recognized. The felt need predisposes attention, calls into play accessory mental resources, and in proportion to its strength secures the necessary repetition. As the need is met, satisfaction ensues. All factors thus cooperate to fix in place the new item of knowledge. The element of felt need thus secures not only the learning of the new item, but it has at the same time called into play the allied intellectual resources so that new and old are welded together in effective organization with reference to the need which originally motivated the process.

Lest some should fear that by need is here meant a mere "bread and butter demand," the committee hastens to say that it is psychologic and not economic need which acts as the factor in learning. Economic need may indeed be felt; and, if so, may then serve to influence learning; but there is nothing in the foregoing argument to deny that a purely "theoretic " interest might not be as potent as any other to bring about the learning and organization of subject

matter.

To speak of the bearing and need of any new material is to imply the presence and functioning of already existent purposes and interests. From this consideration thus related to the foregoing the committee believes that, speaking generally, introductory mathematics-ordinarily conceived as separate courses in algebra, geometry, and trigonometry-should be given in connection with the solving of problems and the executing of projects in fields where the pupils already have both knowledge and interest. This would make the study of mathematics more nearly approximate a laboratory course, in which individual differences could be considered and the effective devices of supervised study be utilized. The minimum of

1 The behaviorist psychologist by definition rejects the subjective connotation of "satisfaction." If we had access to the actual psychology involved, possibly the difference of statement would in effect disappear.

the course might well in this way be cared for in the recitation period, reserving the outside work rather for allied projects and problems in which individual interests and capacities were prominent factors.

The significant element in this conception is the utilization of ideas and interests already present with the pupils as a milieu within which the mathematical conception or process to be taught finds a natural setting, and from which a need to use the conception or process can as a consequence be easily developed. Where this state of affairs exists, the bearing and felt need utilize the laws of learning as was discussed above, and the mathematical knowledge or skill is fixed in a manner distinctly economical as regards both present effort and future applicability.

As was stated at the outset, this suggested procedure reaches beyond the questions of economy of learning and application-controlling though these here are--to the question of content. The procedure here contemplated makes definite demand for an appropriate introductory content. To work along this line there must be made a selection of conceptions and processes which can serve the pupils as instruments to the attainment of the ends set before them in the projects or problems upon which they are at work. This instrumental character becomes then the essential factor in any introductory course. It is these instrumental needs and not "logical" interconnectedness which must give unity to such a course. A content thus instrumentally selected will, on the one hand, be free of the old formal puzzles, the complex instances, the verbal problems which in the past have wasted so much time and destroyed so much potential interest; and will, on the other, run across the divisions heretofore separating algebra, geometry, and trigonometry.

A distinct advantage in the procedure here suggested is the better promise it holds out of meeting in one introductory course the needs of both those who will go on to advanced study in mathematical lines and those who will not. Where the basis of selection and procedure is instrumental, all can begin together. The future specializers in mathematics will as the course proceeds take increasing interest in the mathematical relationships involved and will stress this aspect in their individual problems and projects. Those whose tastes and aptitudes lead them elsewhere will in the meanwhile have had the opportunity to learn in practical situations some of the mathematical concepts and processes which they will later use in their own chosen fields. Their individual projects in the course can serve well as connecting links between the mathematics taught and their later field of vocational application.

After the introductory course has been completed, and differenti.ation has begun, the same principles still hold, though in the different

fields. Those who have chosen to continue the study of mathematics as such will find their problems or projects within the field of mathematics itself, quite likely examining anew in the light of wider acquaintance assumptions freely made in the earlier period. Euclid's system of axioms and postulates might here receive its first careful consideration. Those who had elected to prepare for engineering and the like might continue to find their mathematics in connection with problems or projects devoted now particularly to a preliminary engineering content. Conceptions usually reserved for college analytics and calculus-if not indeed already used in the introductory course-can well have a place here. Their rich instrumental character will justify their presence, even if they lack somewhat in relationship to a fully developed logical system.

2. The several needs for mathematics-Among the multiplicity of specific occasions for using mathematics and among the various types of subject matter, there are certain possible groupings which promise aid in the determination of the mathematical courses.

Without implying the possibility always of sharp differentiation, we may distinguish in the realm of mathematical knowledge (i) those items the immediate use of which involve a minimum of thinking, as, for example, adding a column of figures, and (ii) those items which are primarily used as notions or concepts in the furtherance of thinking. It is clear that the distinction here is of the way in which the knowledge is used and not of the knowledge itself; for any item of knowledge might at one time serve one function and at another time the other. It would still remain true, however, that certain groups of people might have characteristically different needs along the two lines. Under the first head we should include the mechanic's use of a formula, the surveyor's use of his tables, the statistician's finding of the quartile. The man in the street would call this the "practical" use of mathematics. Under the other head we should include the intelligent reader's use of mathematical language by which he would understand an account of Kepler's three famous laws. Some may wish to call this the "cultural" use of mathematics. The term "interpretative" might, however, more exactly express the differentiating idea.

We may next ask whether there are differentiable groups among high-school pupils whose probable destinations or activities determine within reasonable limits the extent and type of their future. mathematical needs. In a democracy like ours, questions of probable destination are of course very difficult. There must be no caste-like perpetuation of economic and cultural differences: and definite effort must be made to keep wide open the door of further study for those who may later change their minds. But differentiating choices are in fact made; and in view of the wealth of

offerings on the one hand and of individual differences on the other, such choices must be made. Properly safeguarded by an intelligent effort to adopt social demands to individual taste and aptitude, these choices should work to the advantage both of the individual and of the group. The committee considers that four groups of users of mathematics may be distinguished:

(a) The "general readers," who will find their use of mathematics beyond arithmetic confined largely to the interpretative function described above.

(b) Those whose work in certain trades will make limited, but still specific, demand for the "practical" use of mathematics.

(c) Those whose practical work as engineers or as students of / certain sciences requires considerable knowledge of mathematics.

(d) Those who specialize in the study of mathematics with a view either to research or to teaching or to the mere satisfaction of extended study in the subject.

It is at once evident that these groups are not sharply marked off from each other; and that the needs of the first group are shared v by the others. It is, moreover, true that the "general readers' represent a wide range of interest. The committee has taken all these things into account, and still believes that the division here made will prove of substantial utility in arranging the offerings of high-school mathematics.

3. Comparative values.-Out of the conflict of topics for a place in the program there emerges one general principle, already suggested in these pages, which is being increasingly accepted for guidance by students of education. In briefest negative terms: No item shall be retained for any specific group of pupils unless, in relation to other items and to time involved, its (probable) value can be shown. So stated the principle seems a truism, but properly applied it proves a grim pruning hook to the dead limbs of tradition. A final method of ascertaining such comparative values remains to be worked out; but the feasibility of a reasonable application of the principle will hardly be denied. In accordance with this, many topics once common have been dropped from the curriculum and more are marked to go. Thus our better practice has ceased to include the Euclidian method of finding the H. C. F.. because the knowledge of this method is nowhere serviceable in life; and in secondary algebra itself little if anything else depends on it. Indeed, the H. C. F. itself might well go, as it is used almost exclusively in simplifying fractions made for the purpose.

In a full discussion, many terms of the statement would need consideration. What constitutes value is probably the point where most questioning would arise. The committec takes this term in its broadest sense, specifically denying restriction to a "bread and but

ter" basis or other mere material utility, though affirming that remunerative employment is normally a worthy part of the worthy life. What the statement then in fact demands is (i) that the value of the topic be not a mere assumption—a positive case must be made out; and (ii) that the value of the topic so shown be sufficiently great in relation to other topics and to the element of cost (as regards time, labor, money outlay, etc.) to warrant its inclusion in the curriculum.

This principle of exclusion seems especially applicable to those items which now remain merely as a heritage from the past and to those which have been introduced mainly to round out the subject or where the unity of the subject matter has been found in the content itself and not in the relation of the content to the needs of the pupil.

In offering such a principle for guidance, the committee considers that it is merely stating explicitly what has been implicitly assumed in all such controversies. The committee none the less believes that conscious insistence on the point is necessary in order to disclose whatever indefensible elements may be in our present program of studies.

4. "Formal discipline."-A full discussion of this topic, of course, is impossible within the limits of this paper. Such a discussion is, moreover, for our purpose unnecessary, because we shall wish to use only the most general conclusion, in which there is substantial concurrence. We can thus avoid the niceties of elaboration, about which agreement has not yet been reached. The older doctrine assumed uncritically a very high degree of what we now call "general transfer" of training. Modern investigation, to speak generally, restricts very considerably the amount of transfer which may reasonably be expected, and inquires strictly into the conditions of transfer." Under the older doctrine it was a sufficient justification for the requiring of any subject that pupils should gain through it increased ability in the use of any important "faculty," because the increase in ability was naively assumed to mean an increase in the equally naively assumed faculty itself and would accordingly be effective wherever the faculty was used. As pupils show such an increase of ability in one or more " faculties" by the simple fact of learning any new subject, the convenience of this older doctrine for curriculum defense is evident. When this old psychological doctrine was first called in question by scientific measurement, the idea gained popular currency that all transfer was denied. No such claim has serious support. The psychologists, however, have so far found it difficult to agree upon any final situation as to the amount of transfer which in any particular situation may be a priori expected. All agree, none the less, in greatly reducing the old claim both as to the amount and

« ZurückWeiter »