THE IMPROVEMENT OF THE FRESHMAN YEAR OF MATHEMATICAL INSTRUCTION IN TECHNICAL SCHOOLS.

BY CHAS. S. SLICHTER,

Professor of Applied Mathematics, University of Wisconsin, and Consulting Engineer, U. S. Reclamation Service.

I shall consider this subject very briefly in two parts: first, the course of study; second, the instructional force.

I. Reform of the Freshman Course in Mathematics. -It is no mere accident that the movement for reform in elementary instruction in mathematics was coincident with the recent period of phenomenal growth in technical education in this country. The enormous expansion of the technical schools has exerted its influence in more than one way upon the development of pure science. It is no secret that the pick of our bright youth in this country has gone into technical courses of study during recent years to the detriment of other lines of intellectual work. Many of these men would have taken courses in pure science under conditions existing twenty years ago. The prosperous years for applied science have therefore not been contemporaneous with years of equal prosperity in pure science. Not only has the number of students of pure sciences remained nearly stationary, but the complaint is very general that the caliber of the men has materially fallen off. The brightest and most ambitious of our youth have been attracted to technical courses, and, to a less degree, to courses in history and economics, but not to the natural sciences. These facts, I am confident, account for the very gen

eral conversion of the teachers of mathematics in our higher institutions to the belief that the elementary instruction should be thoroughly improved and enriched. The great vigor with which the Perry movement has been pushed in England during these same years has made it easy to get aid and comfort for the American campaign, and to profit in many other directions of advance. It is significant that those in charge of the mathematical work in our technical schools have, as a class, been somewhat conservative in their attitude toward the new movement. They are too vitally interested in a strong and scientifically developed course of preparatory mathematics to desire to risk much by revolutionary changes, and they no doubt fear that the new movement has some danger of running wild as an educational fad, to the real detriment of proper foundational work in secondary mathematics. This country has little to learn, except by way of bad example, from technical instruction in Great Britain. It is probably a fact that real reform in secondary instruction in mathematics in this country had more to avoid than to simulate in the Perry movement.

Fortunately there is every indication that the changes that are being made in mathematical instruction in the American secondary schools are being planned with care and good judgment. The introduction of graphical methods and of mathematical work of a practical sort is becoming very general and in most instances is not being carried to extremes.

The reform movement has not been confined to the secondary schools. Some technical schools have made radical changes in the mathematical work of the college course.

Those who have worked out the matter

independently have arrived at very similar results. The changes consist for the most part of modifications that naturally follow from the movement in the secondary schools. The tendency is to blot out entirely the present lines of distinction between algebra, trigonometry and analytic geometry. These changes have involved, not only the early and constant use of graphical methods, and abundant use of numerical data, but have also included the early consideration and application of vector analysis, and a study of the theory of nomographs and similar subjects.

My own experience with the modified courses has been very encouraging. I have been delighted with the success of the students and of the instructors with every change that has been attempted. In fact the encouragement has been so great that I am convinced that the time is ripe for very extensive and radical modification of our first year's work in technical mathematics. I have had an opportunity to discuss this matter with the heads of mathematical departments in several of our technical schools and I have found general accord with the reform and little difference of opinion concerning the general character of the changes that should be made. There will be a great advantage if the changes in the courses can take place in a large number of institutions at the same time. For a number of years I have introduced a large amount of geometrical work in the course in higher algebra. In doing this I have tried to avoid what seems to me to be the greatest danger in this work, namely, the danger of falling into merely routine graphical work rather than work that is thoroughly scientific and that calls out at every point all that is in the student. I hand you herewith some

samples of cross section paper which we had printed in order to aid in bringing out the possibilities of such work. The examples given on the cover page are merely suggestions of the work that the students do. We find it very easy to give the student a very thorough insight into the properties of the common functions by the use of this paper. The various plates are drawn to scale to fit an ordinary 10-inch C. E. rule, and the bond paper upon which the plates are printed permits many interesting combinations by superposing sheets before blueprinting.

I have been slow to break down completely the barriers between algebra, trigonometry, and analytic geometry, largely because of the difficulties that arise in adjusting the courses of the migrants that come to us from small colleges and other institutions. This difficulty would be partially met if the reform were taken up in a number of institutions at about the same time. Independent of the action of other institutions, we have reached the point at the University of Wisconsin where we shall entirely break down the barriers between the courses in higher algebra, trigonometry and analytic geometry. We shall convert these into a single course in analysis, with the various topics introduced at the proper and logical time and independent of the traditional order in which they now appear in the separate texts. Our most treasonable act is the deposing of the conic sections, which have been the reigning family for so many years, from their exalted place in our course of study. This gives an opportunity for the more abundant enrichment of the course and a better comprehension of those things which the student of applied science needs. Such a course not only conforms more nearly to the actual

needs of the student, but it even has the advantage of being more logical and scientific than our old courses. As a matter of fact all that the engineering student learns in his usual course in mathematics is a simple comprehension of the properties of the algebraic function and of the circular and exponential functions of the real variable. Indeed we might define the work of the engineering student, during his two years of mathematics, as the study of the exponential functions of the real variable. Indeed we might define the work of the engineering student, during his two years of mathematics, as the study of the exponential and circular functions: in his first year he considers these functions in ordinary calculus; in his second year he considers them in infinitesimal calculus. He never gets beyond these elementary functions. The modification of the freshman course that we propose recognizes the logical unity of the material of study and is in fact more scientific than the system at present in vogue.

The danger in the new program lies in the possibility that the new work will be less strenuous and less scientifically complete than the usual courses in analysis. This can only be avoided by watchfulness on the part of those who plan the courses and by assistance and criticism of results on the part of the engineering departments that depend upon good mathematical preparation for their students.

II. Instructional Force.-When we come to consider the make-up of the instructional force we find the very greatest difference of opinion among those that are interested in the subject. Personally I hold to the opinion, which has been so well expressed in recent editorials in the ENGINEERING NEWS, that the instruc