Peter Alfeld, --- Department of Mathematics, --- College of Science --- University of Utah

Understanding Mathematics

a study guide by Peter Alfeld.


I wrote this page for students at the University of Utah. You may find it useful whoever you are, and you are welcome to use it, but I'm going to assume that you are such a student (probably an undergraduate), and I'll sometimes pretend I'm talking to you while you are taking a class from me.

Let's start by me asking you some questions. If you are interested in some suggestions, comments, and elaborations, click on the Comments. Do so in particular if you answered "Yes!". (In making the comments I assume you did say "Yes!", so don't be offended if you didn't and are just curious.)

If your answer to all of these questions is a resounding "No!" then you should read no further and return to the study of mathematics. I'd also like to meet you, please drop me an e-mail! Otherwise I'm hoping that this page has something to offer you (and of course you may send me an e-mail anyway). I believe that many students struggle with mathematics only because they don't know what it means to understand Mathematics and how to acquire that understanding.

The purpose of this page is to help you learn how to approach mathematics in a more effective way.


Understanding Mathematics

You understand a piece of mathematics if you can do all of the following:

By contrast, understanding mathematics does not mean to memorize Recipes, Formulas, Definitions, or Theorems.

Clearly there must be some starting point for explaining concepts in terms of simpler concepts. That observation leads to deep and arcane mathematical and philosophical questions and some people make it their life's work to think about these matters. For our purposes it suffices to think of elementary school math as the starting point. It is sufficiently rich and intuitive.

All of this is neatly summarized in a letter that Isaac Newton wrote to Nathaniel Hawes on 25 May 1694.

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).

Here's an

Example: Powers

for complicated mathematics building on simpler mathematics.

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

Example: Conversion of logarithms.

Example: Solving a quadratic equation.

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

Frequently Asked Questions

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.

Mathematical Gems


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

Examples of simple but unsolved problems

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:


Books

I frequently get inquiries about which book to use for self study of mathematics. There is of course a vast amount of literature and online information, but two books stand out:
  1. My favorite of all books I have ever read, or otherwise examined, is "What is Mathematics" by Courant and Robbins. This book first appeared in 1943, and it is still in print! It is available as an inexpensive paperback: What Is Mathematics? An Elementary Approach to Ideas and Methods by Richard Courant and Herbert Robbins, and updated in 1996 by Ian Stewart. Don't be mislead by the term "elementary". The book does start at the beginning, but it covers a huge swath of mathematics, and is suitable for many years of reading and careful study. It is intended to describe the spirit and contents of mathematics to the serious and curious, but perhaps uninitiated, and it is as close to being perfect as a book can be. Oxford University Press, USA; 2 edition (July 18, 1996), ISBN-10: 0195105192 and ISBN-13: 978-0195105193.
  2. A much more recent book is Math Overboard! (Basic Math for Adults) Part 1 by Colin W. Clark. This is an earnest, and largely successful, attempt to explain the contents of basic mathematics comprehensively and in detail to adults. More so than any other book of this kind that I have seen, including many textbooks, it is true to the principles outlined in these web pages. The first part covers Arithmetic, Algebra, Geometry, Functions and Graphs. A second part covering trigonometry, exponential and logarithmic functions, statistics and probability and "advanced topics" is forthcoming. I sometimes get inquiries from people asking for a book to help them review or prepare for the study of College Algebra or Calculus, and this is the book I will henceforth recommend. Dog Ear Publishing (September 26, 2012), ISBN-10: 1457514818 and ISBN-13: 978-1457514814.