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SOME HINTS ON TEACHING MATHEMATICS TO
BY FLORIAN CAJORI, Professor of Mathematics and Dean of the School of Engineering,
The tendency of the present time is to arithmetize mathematics. The earlier explanation of irrational numbers, like that of fractions, involved the idea of measurement. Formerly an irrational number was defined as the expression of the incommensurable ratio of two geometrical quantities — that is, as the ratio between the quantities which have no common measure. But this mode of treatment involves certain logical difficulties which G. Cantor, K. Weierstrass and others have endeavored to remove by treating irrational number in a manner free of ratio and measurement, and of all geometrical considerations. The pupil is warned to take no notice of geometric figures or diagrams. Among them, if geometry is studied, "geometry without diagrams is the order of the day.”
The question naturally arises whether in the training of engineers the teacher should endeavor to follow in the footsteps of the modern logicians of mathematics. Should he aim at extreme mathematical rigor? Probably the majority of teachers agree that this should not be the case, that it is well, as a rule, to lead beginners along the same road that the race has taken in the acquirement of knowledge. In the historical development of mathematics the naïve treatment has invariably preceded the more strictly logical. Geometrical intuition has antedated strict logic. The geometrization of mathematics has come earlier than the arithmetization of mathematics.
The introduction to a science should be concrete in form. Too many refinements should not be attempted. No teacher need worry over the fact that his pupils believe all curves to have tangents. Not only is the use of visible figures desirable, but students should be required to draw a great many diagrams. The teacher should insist upon the maintenance of intimate relations between geometry, graphics and analysis.
If this be the correct attitude, then the methods of exposition of the fundamental principles of the calculus, advocated by the noted German mathematician, Alfred Pringsheim, and by others, can hardly be adopted for elementary instruction. Says Pringsheim: “That the older method, based upon geometric evidence, of introducing the irrational, as well as of the closely related concepts of limits and continuity, do not appear valid, and that the arithmetical theories of the irrational mark in the direction indicated a substantial advancement and indeed are well adapted, according to modern views, to supply a good foundation for the logical development of analysis, is recognized by the large majority of scientific mathematicians.” Pringsheim's article, from which this quotation is taken, contains a plea for the introduction of the modern theories of the irrational into elementary lectures on the calculus. This plan appears to us of doubtful expediency in teaching any class of beginners and especially one of engineering students. The ideas are too difficult and arid. But recent theories of the irrational are not the only modern devices to be avoided. Embodied in some text-books which are used in a few of our engineering schools are the e-definition of a limit and a considerable use of e-proofs. These should be avoided in introductory courses as too abstract and hazy. To beginners of the calculus perceptual intuition should be made a great help in grasping fundamental ideas. Instead of making microscopic examinations of fine points in logic, the pupil should be given much exercise in working problems in geometry and mechanics. He should become infused by the consciousness of a new power for working problems which previously were beyond his reach. If possible, he should acquire something of the enthusiasm which the seventeenth and eighteenth century mathematicians displayed in solving problems which were published as challenges to feel the pulse of their rivals.
One important result of intuition is the infinitesimal right triangle which was introduced by Isaac Barrow, the teacher of Isaac Newton. This triangle not only gives the pupil at once the fundamental relation ds=dx2 + dyż, but gives him also dy/dx as the slope of the tangent at the point x, y of the curve. In the treatment of tangents there appears to be room for reform in our courses of study. Tangent lines are usually found in the calculus by the differential method and in analytical geometry by the secant method. This use of two methods is time-killing. Nor is the secant method always readily understood or easily applied. Why not use the differential method throughout? Let the pupil learn the differentiation of algebraic integral polynomials while he is studying algebra; let him apply the process in the discussion of equal roots of equations; then let him use differentiation in finding the slope of tangent lines not only in the calculus, but also in analytical geometry. This mode of procedure will bring analytical geometry and calculus closer together. It will save much time and will remove a source of perplexity to many students.
In the computations that the engineer carries out, absolute accuracy is not needed, nor is it possible. The data from which he starts are experimental and, therefore, are only approximations to the truth. Yet the engineer should know about how close or how remote he is from the actual facts. If the measurements of the length of a base line are 505.09,505.35,505.15,505.42, an untrained student, after obtaining the mean value 505.2525, may imagine that his accuracy reaches to the fourth decimal. The student may compute accurately, but that is not sufficient. He should be made to effect his solution, if possible, within certain prescribed degrees of approximation; he should be required to ascertain what the precision of the final results will be, when he knows how precise the component measurements are.
DISCUSSION. PROFESSOR GEORGE F. Swain: I have nothing in particular to say in regard to the points which have been mentioned in this paper, but there is one other matter which has occurred to me, and in regard to which I would like to have an expression of opinion from those present, and that is in regard to the desirability of teaching differential calculus and integral calculus together. In most schools differential calculus is taught first and before the integral calculus is taken up, while in a few at least the two subjects are taught together. It would seem to me it would be better to teach them together so that the students could see the relation between differentiation and integration and be prepared to use integration earlier than if taught by the other method. I would like some expression of opinion from some of the members in regard to this point.
DEAN FREDERICK E. TURNEAURE: Engineering mathematics at the University of Wisconsin is in charge of a professor who has a thorough appreciation of engineering problems and he has come to the conclusion it is desirable to teach the two subjects at the same time; and, further, he finds it desirable to bring in a good deal of analytical geometry.
PROFESSOR ARTHUR N. TALBOT: Is the time given to the subject the same as ordinarily given to the differential calculus?
DEAN TURNEAURE: The same amount as formerly is given to it.
PRESIDENT CHARLES S. Howe: While the subjects are taught together we also separate them. I think the students have had a better idea of the subjects where we have used both methods together, and I think that the student likes the calculus better by that method, or at least has a better idea of calculus. He devotes himself entirely to the differential calculus for a certain length of time and then takes up the integral calculus, using what he has done in the differential. In the end he does have a thorough knowledge, but he is