Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

Are you taking mathematics only because it's required?

That's too bad because there are better reasons to take mathematics and if you are sitting through a class only because you have to you are making your life miserable.

Perhaps it might help if I tell you how I got interested in mathematics. Mathematics and Physics were the two subjects in which I excelled in High School. I was fascinated by how you could build these logical systems that were so powerful, creative, and subtle. The whole way of thinking appealed to me.

But there was another more practical aspect: I was always bad at languages in School (I failed French and barely passed English) because they seemed so arbitrary and useless. (I got to use English for my survival only long after I left High School.) It was such a pain to have to memorize things. Now this may surprise you and so I'll put it in large letters:

Understanding mathematics requires no memorization whatsoever

Instead of memorizing, as described in detail elsewhere in these notes, you understand the way in which everything in mathematics depends on everything else. In the process, of course, you'll encounter some things over and over again, and so eventually you can't help remembering something, like certain definitions or even approximations of some numbers like pi and e. But it's a totally useless activity in understanding mathematics to try to cram facts into your memory. If you find yourself doing that you are on the wrong track and you should work through the home page of these notes.

There was a particular and truly marvelous book that helped turn me on to mathematics. It was written in the forties, but it's now coming out as an inexpensive updated paperback, and it has lost none of its luster. It's by Courant and Robbins and called "What is Mathematics".

There's another thing that motivated me to pursue mathematics, and that's the fact that there are a large number of simple sounding but unsolved mathematical problems that make sense to a high school student.

Fine print, your comments, more links, Peter Alfeld, PA1UM