in visualizing spatial relations and figures, in representing such figures on paper, and in solving problems in mensuration. For many of the practical applications of mathematics it is of fundamental importance to have accurate space perceptions. Hence it would seem wise to have at least some of the work in solid geometry come as early as possible in the mathematical courses, preferably not later than the beginning of the eleventh school year. Some schools will find it possible and desirable to introduce the more elementary notions of solid geometry in connection with related ideas of plane geometry. The work in solid geometry should include numerous exercises in computation based on the formulas established. This will serve to correlate the work with arithmetic and algebra and to furnish practice in computation. The following provisional outline of subject matter is submitted: (a) Propositions relating to lines and planes, and to dihedral and trihedral angles. (b) Mensuration of the prism, pyramid, and frustum; the (right circular) cylinder, cone and frustum, based on an informal treatment of limits; the sphere, and the spherical triangle. (c) Spherical geometry. (d) Similar solids. Such theorems as are necessary as a basis for the topics here outlined should be studied in immediate connection with them. Desirable simplification and generalization may be introduced into the treatment of mensuration theorems by employing such theorems as Cavalieri's and Simpson's, and the Prismoid Formula; but rigorous proofs or derivations of these need not be included. Beyond the range of the mensuration topics indicated above, it seems preferable to employ the methods of the elementary calculus. (see Section 6, below). It should be possible to complete a minimum course covering the topics outlined above in not more than one-third of a year. The list of propositions in solid geometry given in Chapter VI should be considered in connection with the general principles stated at the beginning of this section. By requiring formal proofs to a more limited extent than has been customary, time will be gained to attain the aims indicated and to extend the range of geometric information of the pupil. Care must be exercised to make sure that the pupil is thoroughly familiar with the facts, with the associated terminology, with all the necessary formulas, and that he secures the necessary practice in working with and applying the information acquired to concrete problems. 4. Trigonometry.-The work in elementary trigonometry begun in the earlier years should be completed by including the logarithmic solution of right and oblique triangles, radian measure, graphs of trigonometric functions, the derivation of the fundamental relations between the functions and their use in proving identities and in solving easy trigonometric equations. The use of the transit in connection with the simpler operations of surveying and of the sextant for some of the simpler astronomical observations, such as those involved in finding local time, is of value; but when no transit or sextant is available, simple apparatus for measuring angles roughly may and should be improvised. Drawings to scale should form an essential part of the numerical work in trigonometry. The use of the slide-rule in computations requiring only three-place accuracy and in checking other computations is also recommended. 5. Elementary statistics.-Continuation of the earlier work to include the meaning and use of fundamental concepts and simple frequency distributions with graphic representations of various kinds and measures of central tendency (average, mode, and median). 6. Elementary calculus.-The work should include: (a) The general notion of a derivative as a limit indispensable for the accurate expression of such fundamental quantities as velocity of a moving body or slope of a curve. (b) Applications of derivatives to easy problems in rates and in maxima and minima. (c) Simple cases of inverse problems; e.g., finding distance from velocity, etc. (d) Approximate methods of summation leading up to integration as a powerful method of summation. (e) Applications to simple cases of motion, area, volume, and pressure. Work in the calculus should be largely graphic and may be closely related to that in physics; the necessary technique should be reduced to a minimum by basing it wholly or mainly on algebraic polynomials. No formal study of analytic geometry need be presupposed beyond the plotting of simple graphs. It is important to bear in mind that, while the elementary calculus is sufficiently easy, interesting, and valuable to justify its introduction, special pains should be taken to guard against any lack of thoroughness in the fundamentals of algebra and geometry; no possible gain could compensate for a real sacrifice of such thoroughness. It should also be borne in mind that the suggestion of including elementary calculus is not intended for all schools nor for all teachers or all pupils in any school. It is not intended to connect in any direct way with college entrance requirements. The future college student will have ample opportunity for calculus later. The capable boy or girl who is not to have the college work ought not on that account to be prevented from learning something of the use of this powerful tool. The applications of elementary calculus to simple concrete problems are far more abundant and more interesting than those of algebra. The necessary technique is extremely simple. The subject is commonly taught in secondary schools in England, France, and Germany, and appropriate English texts are available.' 7. History and biography.-Historical and biographical material should be used throughout to make the work more interesting and significant. 8. Additional electives.-Additional electives such as mathematics of investment, shop mathematics, surveying and navigation, descriptive or projective geometry will appropriately be offered by schools which have special needs or conditions, but it seems unwise for the National Committee to attempt to define them pending the results of further experience on the part of these schools. III. Plans for Arrangement of the Material In the majority of high schools at the present time the topics suggested can probably be given most advantageously as separate units of a three-year program. However, the National Committee is of the opinion that methods of organization are being experimentally perfected whereby teachers will be enabled to present much of this ma Quotations and typical problems from one of these texts will be found in a supplementary note appended to this chapter. terial more effectively in combined courses unified by one or more of such central ideas as functionality and graphic representation. As to the arrangement of the material the committee gives below four plans which may be suggestive and helpful to teachers in arranging their courses. No one of them is, however, recommended as superior to the others. PLAN A. Tenth year: Plane demonstrative geometry, algebra. PLAN B. Tenth year: Plane demonstrative geometry, solid geometry. Twelfth year: The calculus, other elective. PLAN C. Tenth year: Plane demonstrative geometry, trigonometry. PLAN D. Tenth year: Algebra, statistics, trigonometry. Additional information on ways of organizing this material will be found in Chapter XII. SUPPLEMENTARY NOTE ON THE CALCULUS AS A HIGH SCHOOL SUBJECT. In connection with the recommendations concerning the calculus, such questions as the following may arise: Why should a college subject like this be added to a high school program? How can it be expected that high school teachers will have the necessary training and attainments for teaching it? Will not the attempt to teach such a subject result in loss of thoroughness in earlier work? Will anything be gained beyond a mere smattering of the theory? Will the boy or girl ever use the information or training secured? The subsequent remarks are intended to answer such objections as these and to develop more fully the point of view of the committee in recommending the inclusion of elementary work in the calculus in the high school program. By the calculus we mean for the present purpose a study of rates of change. In nature all things change. How much do they change in a given time? How fast do they change? Do they increase or decrease? When does a changing quantity become largest or smallest? How can rates of changing quantities be compared? These are some of the questions which lead us to study the elementary calculus. Without its essential principles these questions cannot be answered with definiteness. The following are a few of the specific replies that might be given in answer to the questions listed at the beginning of this note: The difficulties of the college calculus lie mainly outside the boundaries of the proposed work. The elements of the subject present less difficulty than many topics now offered in advanced algebra. It is not implied that in the near future many secondary school teachers will have any occasion to teach the elementary calculus. It is the culminating subject in a series which only relatively strong schools will complete and only then for a selected group of students. In such schools there should always be teachers competent to teach the elementary calculus here intended. No superficial study of calculus should be regarded as justifying any substantial sacrifice of thoroughness. In the judgment of the committee the introduction of elementary calculus necessarily includes sufficient algebra and geometry to compensate for whatever diversion of time from these subjects would be implied. The calculus of the algebraic polynominal is so simple that a boy or girl who is capable of grasping the idea of limit, of slope, and of velocity, may in a brief time gain an outlook upon the field of mechanics and other exact sciences, and acquire a fair degree of facility in using one of the most powerful tools of mathematics, together with the capacity for solving a number of interesting problems. Moreover, the fundamental ideas involved, quite aside from their technical applications, will provide valuable training in understanding and analyzing quantitative relations-and such training is of value to everyone. The following typical extracts from an English text intended for use in secondary schools may be quoted: "It has been said that the calculus is that branch of mathematics which schoolboys understand and senior wranglers fail to comprehend. ✶ ✶ ✶ So long as the graphic treatment and practical applications of the calculus are kept in view, the subject is an extremely easy and attractive one. Boys can be taught the subject early in their mathematical career, and there is no part of their mathematical training that they enjoy better or which opens up to them wider fields of useful exploration. The phenomena must first be known practically and then studied philosophically. To reverse the order of these processes is impossible." The text in question, after an interesting historical sketch, deals with such problems as the following: A train is going at the rate of 40 miles on hour. Represent this graphically. At what rate is the length of the daylight increasing or decreasing on December 31, March 26, etc.? (From tabular data.) A cart going at the rate of 5 miles per hour passes a milestone, and 14 ininutes afterwards a bicycle, going in the same direction at 12 miles an hour, passes the same milestone. Find when and where the bicycle will overtake the cart. A man has 4 miles of fencing wire and wishes to fence in a rectangular piece of prairie land through which a straight river flows, the bank of the stream |