as to the generality of conditions under which transfer may be expected. In accordance with these considerations the committee has not used the factor of "formal discipline" in determining the content of the mathematical courses to be recommended in this report. 5. The needs of the several groups. With these several principles and factors before us, we are now ready to consider more fully the needs of the several groups of users as distinguished above. We are particularly concerned to ask whether or not their respective group needs are compatible with one introductory course to be taken in common; and if yes, when the differentiation from such a common course should begin.

(1) The “general readers."-This group will need to use in "practical" fashion but little of mathematics other than ordinary arithmetic. As general readers, however, they will still require a certain acquaintance with mathematical language and concepts. Just what terms, symbols, and concepts would meet the requirements of this group will have to be determined by extensive inductive studies. Assuming, however, ordinary arithmetic and mensuration, some items can be at once named as fairly certain to be included: How to interpret and evaluate a simple literal formula; the meaning and use of an algebraic equation of one unknown; the notion and use of negative numbers in such simple cases as temperature, latitude, and stock fluctuations; the simpler conception of space. relations (inductively obtained); the notion of function (the dependence of one quantity upon another); the graph as a means of interpreting statistical information, with such terms as average and median.

(2) The group preparing for certain trades.-Under this head the committee would group those whose use of "practical" mathematics is, while generally quite definite, still relatively small-such, for example, as machinists, plumbers, sheet-metal workers, and the like. The general run of the need here contemplated can be gathered from the requirements laid down for machinists in one of the more recent vocational surveys simple equations, use of formulas, measurement of angles, measurements of areas and volumes, square root, making and reading of graphs, solution of right triangles, geometry of the circle. Much practice would of course be necessary to make even this small amount of mathematics function adequately. It is at once evident that if no more algebra is needed than formulas, simple equations and the graph, and no more geometry than is here suggested, then the ordinary high-school courses in these subjects are but ill-adapted to the needs of such pupils. It would seem to follow that this group of pupils has no need to follow courses in mathematics other than (i) arithmetic, (ii) the "interpretative

(introductory) mathematics discussed above, and (iii) the special applications of these to the specific subject matter of their several specializations.

This group might then well study in common with the preceding until the completion of the work there laid out. The presentation of this common course along the lines previously laid down (p. 11) would well harmonize the somewhat diverse interests of the two groups. What little additional content and whatever practice in specialized application this second group might need could then be given either in a parallel or in a succeeding course (or courses) especially devised for that purpose.

(3) The group preparing for engineering.-This group will consist mostly of boys intending to study in engineering schools. In contrast with the two preceding groups, appreciably more mathematics is here needed. In contrast with the following group, there are here specific aims external to mathematics itself which define and limit the mathematical knowledge and skill needed. Although recognizing that the individual teacher will require a certain leeway as regards content in getting his class effectively to work at any topic, we may still profitably ask as to the minimum content fixed. for this group by its peculiar needs.

The minimum mathematical content suitable for the use of this group can probably best be secured by working simultaneously along two lines: First, to ascertain inductively what mathematics the engineer needs (including experiment to find out what part of this can best be taught in the secondary school); second, to criticize the existing courses to see what they lack and what they include that is useless for this group. It is much to be hoped that necessary inductive studies and experiments along the first line of procedure may be vigorously pushed. The second in important respects waits for the first, but it is possible from certain inherent considerations at once to exclude some matter now customarily taught.

Taking the customary high-school mathematics as a basis for comparison, we find at least three principles of criteria for exclusion from the present offerings: (a) Exclude all those items which are not themselves to be directly used in practical situations or which are not reasonably necessary to the intelligent mastery or use of such "practical" items; (b) exclude all involved and complicated instances of otherwise useful topics or applications which do not serve to clarify the main point under consideration; (c) exclude all such proofs and discussions as do not in fact help the pupil to an intelligent use of the topic. It is probably correct to say that these exclusions relate to material introduced from considerations of theory rather than of intelligent practical mastery; from considerations of

the pleasure that theorizers (teachers mostly) get from the study of mathematics rather than from a conscious purpose to give that familiarity and grasp which the future practical man will need.

Under the head of (a) topics excluded as not needed in this group the committee would mention such as the H. C. F. and the L. C. M.; operations with literal coefficients (except for a few formulas); radical equations; the theory of exponents, except the simplest operations with fractional and negative exponents (these to be retained to give meaning to logarithms and the slide rule); operations with imaginaries; cube root; proportion as a separate topic (the simple equation suffices for the progressions).

Among (b) excluded complex applications might be mentioned the following: All lengthy exercises in multiplication and division; factoring beyond the simplest instances of the four forms (i) ax+ay, (ii) a2—b2, (iii) a2+2ab+b2, (iv) x2+(a+b) x+ab; all but the simplest fractional forms (the more complicated are in fact given to illustrate factoring); all radicals beyond √ab and Va÷b; simultaneous equations of more than two unknowns; simultaneous quadratics (except possibly a quadratic and a linear); the clock, hare and hounds, and courier problems and the like; the extended formal demonstrative geometry of our ordinary schools; most trigonometry beyond the use of sine, cosine, and tangent in triangle work.

(e) Proofs excluded or deferred are mostly cared for in (a) and (b). The chief instances in the past (too often yet remaining for the "specializers ") have been the distinction between negative quantities and negative numbers, the (supposedly) rigorous generalizing of ama", the proof of too evident propositions in geometry, the incommensurable cases in geometry, the general proof of sin (x+y).

It may be mentioned in this connection that teachers of mathematics from arithmetic onward only too frequently deceive themselves as to the place that the presentation of a rigorously logical proof plays in bringing conviction. The worth of a sense of logical cogency can hardly be overestimated; but we who teach not infrequently overreach ourselves in our zeal for it. The teacher of introductory mathematics can well take lessons from the laboratory, where careful measurement repeated under many different conditions will bring a conviction often otherwise unknown to the pupil who is not gifted in abstract thinking. Probably in most instances an inductively reached conviction is the best provocative of an appetite for a yet more thoroughgoing proof.

Everything so far points to one common introductory course. With this group as with the preceding, the point of differentiation would seem to come at the end of the interpretative course first discussed for the "general readers." Whether this third (engineering)

group should proceed further in common with the fourth group (of specializers), we later consider further in common with the fourth group.

(4) The group of specializers.-This group will include those pupils, both boys and girls, who "like" mathematics. While these best of all could continue to work with the present offerings, the considerations urged under the discussion on presentation suffice, in the committee's opinion, to demand even for this group a far-reaching reorganization of practically all of secondary mathematics.

Since we are here planning for those who specialize in mathematics, we are not called upon-after meeting the interpretative need-to consider any external demands upon mathematics, but only such a selection and arrangement within the subject itself as best furthers the mathematical activity. Hitherto the arrangement within the course has been made, as we saw in the discussion on presentation, in answer to considerations rather of "logical" organization than of psychological experiencing and growth. The results have not been satisfactory. Algebra, geometry, and, to a lesser degree, trigonometry have been treated as separate logical entities, with consequent loss to the pupil of both interest and power. The committee thinks that the selection and organization should be made in the light of experiment as to which conceptions do in fact prove successively most strategic in the pupils' continued approach to mathematical power. The result would probably take a form somewhat analogous to the "general science" course which is now being worked out in that field.

That this group should take its introductory work in common with the others has perhaps been sufficiently implied. The intelligent choice of a specialty could hardly precede the actual experiencing of taste and aptitude. How far beyond the common introductory course this group should go in company with the third (the preliminary engineering) is not easy to say. In all but the largest schools administrative considerations will probably keep the two together in whatever work is offered. Where numbers and funds suffice, differentiation may well begin immediately upon the completion of the common introductory course, according to considerations already laid down. In that case the preliminary engineering group would get their mathematics more in terms of engineering content and situations; those specializing in mathematics would get theirs more directly in terms of "pure" mathematics. The contents of such courses could well differ considerably.

6. Selecting mathematical ability.--From the point of view both of society and its needs and of the individual and his satisfactions, it is highly desirable that ability, or the lack thereof, be disclosed in order that intelligent choice may be made. Mathematical ability as

expressed in mathematical achievement and application is a most powerful agency in advancing civilization. In order that society may profit by its available stock of mathematical ability, there is urgent need of some process that shall disclose this ability. Analogous considerations demand that the individual learn by a less costly process than occupational trial what degree of probable success he may expect from an occupation in which mathematical ability is an important factor. We hope much from further psychological study in this field of disclosing specific abilities, but as matters now stand the opportunity in the high school for trial of mathematical success and liking is at least one important part in the disclosing of mathematical ability. This factor must be taken into account in arranging the introductory work in mathematics.

IV. SUGGESTIONS AS TO COURSES.

Each valid consideration in the foregoing discussion should have its effect in the resulting determination of the mathematics courses. Considerations of presentation demanded that we give up the "logical" arrangement of subject matter, especially for introductory work, and find instead an organization based upon the successful attack of projects and problems in connection with which the pupils already have both knowledge and potential interest. Four groups of pupils judged by probable destination showed four types of mathematical needs: (i) The "general readers," whose needs lie largely in the "interpretative" function of mathematics; (ii) those who, expecting to enter trades, would have a small but still definite need for "practical" mathematics; (iii) those who, as prospective engineers, would need a considerable body of content determined by the demands of engineering study and practice; (iv) those specializing in mathematics who would wish a content determined by the satisfactions inherent in the activity and by the demands of further study. From considerations of comparative values nothing should enter into the curriculum except as it can show probable value in relation to other topics and to time involved. "Formal discipline" was not considered by the committee in determining the content of courses to be recommended. Care should be given that at an early stage mathematical taste and ability may be disclosed so as to allow appropriate choice of school work and occupational preparation. It seemed clear that a new introductory course should be offered which all the students should, normally, take in common. College entrance considerations, except as inherently cared for above, are deliberately disregarded.

With these demands before us, can an appropriate school procedure be devised and feasibly operated?