Why do we appreciate mathematics?
(Appeared in the Notices of the American Mathematical Society, 43, 533-534, May1996)
Though I agreed with almost everything while reading the article ``Why are we learning this?'' by Roman Kossak (Notices of the AMS, vol 42, no 12), still at the end I felt some dissatisfaction. The feeling was similar to the one I felt when (several times) I taught calculus using one or other of the standard (e.g. American) calculus texts. In most cases, these texts were written very carefully, the authors clearly considered the needs of a large and heterogeneous student population, so the result was usually attractive and convincing. Still I felt that something was missing. Should we put more emphasis on showing students that mathematics is attractive? Certainly we should. In the age of TV, videoclips, videogames, etc. when there are so many tools that can distract our young people from reading, thinking and studying we should certainly find new means to revitalize the teaching methods of our several thousand year old profession. Among other things (like calculus reform), it is certainly a good idea what Dr. Kossak suggests: to offer special, carefully planned mathematics appreciation courses that emphasize the extremely wide and powerful range of mathematics as it exists in our age, together with its rich history, and its role as an indispensable tool in many applications. However, a better use of existing (e.g. calculus) courses, taken by hundreds of thousands of students each year, is even more important to promote the same goal.
To find out what improvement can be achieved in existing courses I think first we should ask ourselves, who have chosen mathematics as a profession: ``Why do we appreciate mathematics?''. I hope that many of you share my answer: the first and foremost reason is that mathematics has an extremely solid logical structure, unparalleled by any other science or by any other result of human creativity. From ancient times, many generations of mathematicians have worked on extending its range, filling in its gaps, establishing mathematical truth beyond doubt. Though this program clearly can never be completed, still the mathematics of our time has a very complex and solid logical structure that joins essentially the whole into one great entity. This characteristic of mathematics is inseparable from its applications: mathematics is applied because it is able to supply extremely reliable models and methods, nothing else can offer.
Here I would like to convince the reader that an introductory calculus course (reformed or not) is especially suitable to reveal the logical structure of mathematics for average students. To clarify my point, I am not advocating endless lecturing on theorems of advanced calculus in an introductory course. I fully realize that this would be illusory in case of the typical, heterogeneous classes. But anyhow, there is a beautiful logical ``backbone'' that leads from the properties of natural numbers to multivariable calculus, and which is only partly shown by most standard calculus textbooks and courses. If you show only fragments of a ``backbone'', your students may miss the point. I believe that the ``solution'' that many textbooks gives in case of very important but theoretically involved theorems like the max-min theorem, or the Riemann integrability of continuous functions, namely, that ``the proof can be found in advanced calculus texts'', is not helping students to understand and appreciate the logical structure and the reliability of mathematics. So what could be done? As often in case of real life problems, there are conflicting facts here, and it is not easy to find a nearly optimal compromise. My proposals are as follows.
(a) A good calculus textbook should carry all of the ``backbone'' theory (and not only fragments of it) that leads from the properties of natural numbers to the theorems of analysis. The most difficult notions, definitions, theorems and proofs can be placed into optional sections, but the order of presentation should follow a path along which notions and theorems are built on each other. All these should be discussed in ``the easiest'' possible way. Proofs of repetitive nature can be omitted, difficult proofs could be replaced by the proof of a typical special case, but in my opinion no essential component should be missing. Since the size of a typical calculus textbook is around one thousand pages, my estimate is that all these would result only a few percents increase in size. It seems good to me if every student may see and the strongest students may even read the whole story. The strong ones can grasp its essence, and this way they may serve as sort of ``witnesses''. Also, such a first encounter with the essence of calculus may encourage our best students for further study of mathematics and better prepare them for advanced calculus.
(b) A good calculus teacher should emphasize the essential features of mathematics, how theorems are logical, necessary consequences of the axioms. He or she should mention that the specific order in which the material is presented is not arbitrary, new things are solidly based on old ones all the time. She or he could also reveal the logical structure without giving the difficult details in class, referring the students to the text.
(c) A good calculus exam should ask about the student's understanding of the logical structure of the material too, how concepts and theorems are built on each other, without asking the difficult details.
Certainly there exist textbooks and professors that follow similar methods in case of introductory calculus courses. But there are many that do not and I believe that the proposed changes could bring at least a small improvement in students' appreciation of mathematics.