Last revised March 31, 1996

This is how most of the world, and most mathematics students, see mathematics. In this view, mathematics consists of techniques for getting answers to quantitative problems.

This is in fact a very important part of mathematics. Calculation is at the very root of mathematics. It is perhaps appropriate for graduate students to have a distain for mere calculation, as so many do. But I don't think it's appropriate for a mathematician to have such a distain. My experience is that one obtains a powerful new insight into a mathematical theory when one sees how it relates to concrete calculations. And to some extent, I think that a mathematical theory which can't prove its value in terms of concrete calculations is not likely to stand the test of time.

- In solving concrete problems, an investigator enriches himself, and in this way he discovers new methods and extends his mental outlook. He who searches for new methods without any specific problem in mind just wastes time for nothing.
- --David Hilbert

We faculty in mathematics tend to think of courses as either theory, or "cookbook" courses, and it be rather distainful of the later. And yet there's a vast realm in between the theoretical development of mathematics and the mere rote learning of formulas. This is the realm of mathematical problem solving, and this is what I mostly saw in many of my courses when I was an undergraduate. This is a sort of mathematical thinking which doesn't seem to exist very much any more in our courses, the sort of thinking which involves taking a non-routine problem and covering a blackboard with calculations until one has found a way to make it tractable with the standard set of tools.

This kind of mathematical problem solving is probably every bit as creative as proving theorems. And I don't think that the computer has made it obsolete, since the essence of it consists of setting the problem up, rather than cranking the formulas.

Despite the common belief, doing calculations and finding answers is not the way mathematics is most commonly used in applications. I was very struck by this when I first got my B.S. and worked in the aerospace industry. Many of the engineers I worked with were not, in fact, very good at doing the sorts of calculations commonly taught in calculus and other undergraduate courses, with the exception of certain specific techniques relevant to their particular work. (Many of the EEs, for instance, were pretty proficient at LaPlace Transforms.)

Mathematics was nonetheless quite pervasive in the engineering work I observed, and engineers commonly told me that while most of the things they had been taught in college were irrelevant to the work they actually did, mathematics was the exception. Mathematics was indispensible to their work.

This apparent contradiction puzzled me. As I observed the engineers and thought about this, I started noticing that much of the mathematics they used was in fact extremely simple. And more often than not, what they needed from mathematics was not a numerical answer. When they did want quantitative results, they usually put their problem on the computer.

In part, mathematics gave the engineers a theoretical framework which enabled them to formulate problems for the computer. But more important than anything else, it gave them a language to express relationships that would be difficult to express otherwise.

For the engineers I encountered in the aerospace industry, it was not often very important to be able to compute an integral. But the concept of an integral was extremely important in order to describe many situations.

Mathematics as a language is something we seldom teach explicitly. Students pick it up without even thinking about it much. But this seems to be the really important aspect of mathematics.

When lack of mathematical education stands in someone's way, whether they want to learn more physics, or electronics, or economics, or basic statistics, it is almost never the ability to do calculations that is the stumbling block. When someone picks up a book on, say, thermodynamics, and realizes that they don't know enough mathematics to read it, it's not because reading the book involves doing a lot of calculations. What one needs is the fundamental concepts and the ability to follow mathematical reasoning.

Now it's important to realize here that the ability to follow mathematical reasoning is not all that far removed from the ability to calculate. Simply knowing what all the symbols in an equation mean for some reason just isn't enough to understand a book with lots of equations in it, just as simply knowing the vocabulary of a foreign language isn't enough to be able to read books in that language. To some extent, one has to be able to follow the derivations given in a mathematically oriented book in order to be able to understand what is being said.

However it is not clear that the sort of very routine "plug and chug" type calculating taught in most university mathematics courses today is much help in learning to follow mathematical derivations.

It is a very difficult idea for most students to understand that mathematics does not consist simply of techniques, but that there is a subject matter to mathematics, just as much as there is to physics or astronomy. Mathematicians are not people who devote their lives to doing calculations.

When graduate students in other disciplines, especially not in the physical sciences, learn that I have a Ph.D. in mathematics, they sometimes ask me in a puzzled way, "What do you have to do to get a Ph.D. in mathematics? Do you write a... thesis?"

It doesn't compute for them. They don't understand how anyone could write a thesis in mathematics. "Does it have to be... original?" they ask. It's clear that they can't imagine how mathematics could ever be original. "Do you solve some equation?"

Mathematicians are, in fact, people who devote their lives not to solving equations, but to trying to find the answers to unanswered questions. These questions are just as legitimate as the ones that physicists or biologists do research on.

For many students, this aspect of mathematics is rather unwelcome. It is what they call "theory," and a common question students ask about a course is, "Does it have a lot of theory in it?" It is usually clear that most students do not hope for a positive answer to this question.

Since those of us who teach mathematics at research universities devote our lives to proving theorems, I believe that our courses often tend to slight other aspects of mathematics. We tend to give slight attention to those aspects of problem solving or mathematical thinking which are not relevant to the proof of theorems.

It is questionable whether this attitude serves our students well.

The process of learning mathematics necessarily involves learning certain ways of thinking, and most people find it plausible that learning the kind of thinking one does in mathematics has value outside the realm of mathematics. In any case, this kind of thinking is certainly indispensible in order to be able to use mathematics as a useful tool, and is closely related to the types of thinking involved in the other physical sciences.

When I used to teach trig, for instance,
after teaching the addition formula for *sin(x+y)*,
I would then ask students if they could predict
the double angle formula for *sin(2x)*.
Very few of my students would respond to a question like this
with anything except a blank stare.

Somewhat analogously, when I was a graduate student I once eavesdropped on a pair of students doing their topology (or maybe analysis) homework together. At the moment, they were trying very unsuccessfully to prove that every finite set is closed. They were attempting some rather elaborate proofs, taking various sequences and looking at the limits. I heard them explicitly mention the fact that single points are closed and that the union of a finite number of closed sets is closed, but they pretty quickly passed over these facts, which didn't seem very useful -- probably because they were too simple. They didn't see the connection.

Seeing connections is a big part of mathematical problem solving and theorem proving. When I'm trying to do a problem or prove a theorem, it's as if I see every formula, fact, or whatever, surrounded by lots of links to other facts.

Is it reasonable to make a judgement that trigonometry students
are fairly stupid for not being able to see that the formula
for *sin(2x)* is obvious from the formula for
*sin(x+y)* simply by letting y=x,
or that my fellow graduate students were rather stupid
for not immediately thinking of a finite set
as being a finite union of singletons?
I have to admit that to me it does seem rather stupid
not to see these things.
And yet at the same time
I seem to remember that I was fairly stupid this way myself
when I first started learning mathematics.

Learning mathematics, in this way, is like learning to play any other game. When you first start learning to play any game, whether it's chess or go or backgammon, at first you make some incredibly stupid moves. But as you play longer, you learn to become sensitized to the logical connections inherent in the game and start to learn the standard combinations. In chess, for instance, the first time your opponent puts your king in check and then grabs your queen with his rook or bishop after the king moves out of the way, it comes as a big shock. Later on, you become sensitized to the dangers of having a piece pinned or vulnerable to an X-ray attack.

The thing is, though, that after you learn to play a lot of different games, you start to learn to pick up standard patterns a lot faster. In the process of learning to play many games, you learn something that is on a higher logical level than the patterns inherent in any one game. You have become sensitized not just to the patterns in particular games, but to the idea that in any games there will also exist patterns to be exploited.

The same sort of thing seems to happen in mathematics.

One thing that is certainly important in mathematical thinking is the ability to look beyond content and see the underlying logical structure, and to realize when two situations which seem, on the basis of their surface content, very dissimilar, are actually identical when one looks at the logical structure.

I think, in fact, that this may be the most essential aspect of mathematical thinking. But it is certainly not something that is easy, even for experienced mathematicians.