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 geometrical 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. to 5. Elementary statistics.-Continuation of the earlier work 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: 1 (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 thorough ness. 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.3 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 material more effectively in combined courses unified by one or more of such central ideas, functionality and graphic representation. Quotations and typical problems from one of these texts will be found in a supplementary note appended to this chapter. 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. 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 on Mathematics in Experimental Schools. 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 highschool 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 can not 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. Morever, 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 an 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 minutes 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 being utilized as one side of the inclosure. How can he do this so as to inclose as much land as possible? A circular tin canister closed at both ends has a surface area of 100 square centimeters. Find the greatest volume it can contain. Post-office regulations prescribe that the combined length and girth of a parcel must not exceed 6 feet. Find the maximum volume of a parcel whose shape is a prism with the ends square. A pulley is fixed 15 feet above the ground, over which passes a rope 30 feet long with one end attached to a weight which can hang freely, and the other end is held by a man at a height of 3 feet from the ground. The man walks horizontally away from beneath the pulley at the rate of 3 feet per second. Find the rate at which the weight rises when it is 10 feet above the ground. The pressure on the surface of a lake due to the atmosphere is known to be 14 pounds per square inch. The pressure in the liquid x inches below the surface is known to be given by the law dp/dx=0.036. Find the pressure in the liquid at a depth of 10 feet. The arch of a bridge is parabolic in form. It is 5 feet wide at the base and 5 feet high. Find the volume of water that passes through per second in a flood when the water is rushing at the rate of 10 feet per second. A force of 20 tons compresses the spring buffer of a railway stop through 1 inch, and the force is always proportional to the compression produced. Find the work done by a train which compresses a pair of such stops through 6 inches. These may illustrate the aims and point of view of the proposed work. It will be noted that not all of them involve calculus, but those that do not lead up to it. Chapter V. COLLEGE ENTRANCE REQUIREMENTS. The present chapter is concerned with a study of topics and training in elementary mathematics that will have most value as preparátion for college work, and with recommendations of definitions of college-entrance requirements in elementary algebra and plane geometry. General considerations.-The primary purpose of college-entrance requirements is to test the candidate's ability to benefit by college instruction. This ability depends, so far as our present inquiry is concerned, upon (1) general intelligence, intellectual maturity and mental power; (2) specific knowledge and training required as preparation for the various courses of the college curriculum. 1 Mathematical ability appears to be a sufficient but not a necessary condition for general intelligence. For this, as well as for other reasons, it would appear that college-entrance requirements in mathematics should be formulated primarily on the basis of the special knowledge and training required for the successful study of courses which the student will take in college. The separation of prospective college students from the others in the early years of the secondary school is neither feasible nor desirable. It is therefore obvious that secondary-school courses in mathematics can not be planned with specific reference to college-entrance requirements. Fortunately there appears to be no real conflict of interest between those students who ultimately go to college and those who do not, so far as mathematics is concerned. It will be made clear in what follows that a course in this subject, covering from two to two and one-half years in a standard four-year high school, and so planned as to give the most valuable mathematical training which the student is capable of receiving, will provide adequate preparation for college work. Topics to be included in high-school courses.In the selection of material of instruction for high-school courses in mathematics, its value as preparation for college courses in mathematics need not be specifically considered. Not all college students study mathematics; it is therefore reasonable to expect college departments in this sub 1 A recent investigation made by the department of psychology at Dartmouth College showed that all students of high rank in mathematics had a high rating on general intelligence; the converse was not true, however. 36 |
