**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.

Tamás Szabados

Hungary