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Intuiting Mathematical Objects using Kinetigrams - Hearing Morris Kline's Positive Advice

John Pais

Suppose we ignore the controversial and scathing indictment of U.S. mathematicians and mathematics teaching found in Kline (1977), and instead try to hear only his constructive suggestions for change. On p. 143 and p. 214, respectively, we find the following two passages:

The proper pedagogical approach to any new subject should always be intuitive. The strictly logical foundation is an artificial reconstruction of what the mind grasps through pictures, physical evidence, induction from special cases, and sheer trial and error.

...From a conceptual standpoint [addressed to beginners] the most difficult mathematical subject is calculus. The concepts can be far more readily understood intuitively, and this is how mathematicians grasped the subject until, after two hundred years of effort, they managed to erect the proper logical foundations...


Kline was a passionate advocate both for the intuitive approach to learning, and for the integration of mathematics and science, believing that the meaning of the mathematical concepts could most effectively be carried and transmitted through the integration of significant applications into the primary content of math courses. His book (Kline, 1967)provides a lovely demonstration of how to do this in calculus, and in (Kline, 1977, pp. 149-150), he discusses why this approach is so important for the student audiences that populate undergraduate mathematics courses in the U.S.:

When challenged that the values of mathematics proper do not mean much to potential users, many professors retort that students will learn the applications in other courses. But to ask students to take seriously theorems and techniques whose worth will be apparent one, two, or several years later is a grievous pedagogical error. Such an assurance does not stir up incentive and interest and does not supply meaning to subject matter. As Alfred North Whitehead has advised, whatever value attaches to a subject must be evoked here and now.

...Moreover, the teaching of mathematics is expedited by tying it in with applications. These provide motivation, which mathematics proper does not. Equally important, the only meaning the concepts had for the mathematicians who created them and the only meaning students will find in courses such as calculus derive from physical or, more generally, real situations.


The best current example I know of a text that integrates life science applications into mathematical content is Yeargers et al. (1996). In addition, I hope that in the present article I have been successful in persuading you that my online text (Pais, 1997-2001) can also make a useful contribution both to potential life science users of mathematics, and to undergraduates who may be merely trying to understand and survive their terminal math experience.

John Pais, "Intuiting Mathematical Objects using Kinetigrams - Hearing Morris Kline's Positive Advice," Convergence (October 2004)