# Understanding Mathematics

People wrote differently in those days, obviously the vulgar mechanick may be a man and he that is able to reason nimbly and judiciously may be a woman, (and either or both may be children).

**for complicated mathematics building on simpler mathematics.**

The following examples illustrate the difference between the two approaches to understanding mathematics described above.

You won’t be able to learn how to understand mathematics from abstract principles and a few examples. Instead you need to study the substance of mathematics. I’m hoping that the answers to the following

will illustrate how mathematics is meant to make sense and is built on a logical procession rather than a bunch of arbitrarily conceived rules.

One of the main things that turned me on to mathematics were certain concepts and arguments that I found particularly beautiful and intriguing. I’m listing some of these below even though they may not be frequently asked about. But I’ll hope you’ll enjoy them, and perhaps get more interested in mathematics for its own sake.

## Solving Mathematical Problems

The most important thing to realize when solving difficult mathematical problems is that one never solves such a problem on the first attempt. Rather one needs to build a sequence of problems that lead up to the problem of interest, and solve each of them. At each step experience is gained that’s necessary or useful for the solution of the next problem. Other only loosely related problems may have to be solved, to generate experience and insight.

Students (and scholars too) often neglect to check their answers. I suspect a major reason is that traditional and widely used teaching methods require the solution of many similar problems, each of which becomes a chore to be gotten over with rather than an exciting learning opportunity. In my opinion, each problem should be different and add a new insight and experience. However, it is amazing just how easy it is to make mistakes. So it is imperative that all answers be checked for plausibility. Just how to do that depends of course on the problem.

There is a famous book: G. Polya, How to Solve It , 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6. It was first published in 1945. This is a serious attempt by a master at transferring problem solving techniques. Click here to see an html version of Polya’s summary.

The main thing that keeps mathematics alive and interesting of course are **unsolved problems**. Many open problems that are important in the contemporary view are hard just to understand. But here are

for which you can form your own conjectures. The word simple in this context means that the problem is easy to state and the question is easy to understand. It does not mean that the problem is easy *to solve.* In fact all of these open problems are **difficult.** (That’s why they are unsolved, it’s not that nobody tried!)

## Acquiring Mathematical Understanding

Since this is directed to undergraduate students a more specific question is how does one acquire mathematical understanding by taking classes? But that does not mean that classes are the only way to learn something. In fact, they often are a bad way! You learn by doing. For example, it’s questionable that we should have programming classes at all, most people learn programming much more quickly and enjoyably by picking a programming problem they are interested in and care about, and solving it. In particular, when you are no longer a student you will have acquired the skills necessary to learn anything you like by reading and communicating with peers and experts. That’s a much more exciting way to learn than taking classes!

Here are some suggestions regarding class work: