Understanding Mathematics

Understanding Mathematics 1

Understanding Mathematics 2
Understanding Mathematics

Understanding Mathematics 3

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

a study guide by Peter Alfeld.

Understanding Mathematics 4

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.)

  • Do you feel
    • That being lost in mathematics is the natural state of things? Comments.
    • That lectures and textbooks are incomprehensible? Comments.
    • That the amount of material in any math course is so overwhelming that you (or anybody else) could not possibly absorb it? Comments.
    • That mathematics is just a collection of formulas and theorems that one somehow has to cram into one’s head? Comments.
    • That the solution of problems requires a collection of tricks whose conception was based on a generous allowance of magic? Comments.
    • That math courses are just hurdles one has to cross as an undergraduate student? Comments.
    • That mathematics is irrelevant? Comments.
  • Are you taking mathematics only because it’s required? Comments.
  • Are you frustrated working through heaps of meaningless problems that are all alike? Comments.
  • Does the thought “I can look it up if I have to ” occur to you frequently? Comments.
  • When solving assigned problems, do you often think “What does he want us to do?”? Comments.
  • Do you find yourself frequently searching through literature in vague hopes of finding a helpful formula or theorem? Comments.
  • Do the proofs you find in your textbook or in the literature look contrived to you? Comments.
  • Do you often wonder why your teachers make you study a piece of mathematics that could not possibly ever be useful? Comments.
  • Are you upset that the teacher insists on doing proofs rather than telling you how to solve problems? Comments. >
  • Do you often wish your teacher would do more examples? Comments.
  • When you are trying to solve a problem, do you find yourself frequently spinning your wheels, hoping for an idea that never comes? Comments.
  • Do you ask questions in class? Comments.
  • Do you sell your textbook after you take a math class? Comments.
  • Do you find it difficult to prepare for an exam? Comments.
  • Do you worry about your grade a lot? Comments.
  • Do you skip class a lot? Comments.
  • Do you often leave class early? Comments.
  • Do you often repeat courses? Comments.
  • Do you cheat on exams? Comments.

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:

  • Explain mathematical concepts and facts in terms of simpler concepts and facts.
  • Easily make logical connections between different facts and concepts.
  • Recognize the connection when you encounter something new (inside or outside of mathematics) that’s close to the mathematics you understand.
  • Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

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.

Understanding Mathematics 5

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:

  • Always strive for understanding as opposed to memorization.
  • If this means you have to go back, do it! Don’t postpone clarifying a point you miss because everything new will build on it.
  • It may be intimidating to be faced with a 1,000 page book and having to spend a day understanding a single page. But that does not mean that you’ll have to spend a thousand days understanding the whole book. In understanding that one page you’ll gain experience that makes the next page easier, and that process feeds on itself.
  • Read the sections covered in class before you come to class. That’s one of the most useful ways in which you can spend your time, because it will dramatically increase the effectiveness of the lecture.
  • Do exercises. The teacher may suggest some, put you can pick them on your own from the textbook or make up your own. Select them by the amount of interest they hold for you and the degree of curiosity they stimulate in you. Avoid getting into a mode where you do a large number of exercises that are distinguished only by the numerical values assigned to some parameters.
  • Always check your answers for plausibility.
  • Whenever you do a problem or follow a new mathematical thread explicitly formulate expectations. Your expectations may be met, which causes a nice warm feeling (and you should probably also look for a new and different problem). But otherwise there are two possibilities: you made a mistake from which you can recover, now that you are aware of it, or there is something genuinely new that you can figure out and which will teach you something. If you don’t formulate and check expectations you may miss these opportunities.
  • Find a class mate who will work with you in a team. Have one of you explain the material to the other, on a regular basis, or switch periodically. Explaining math to others is one of the best ways of learning it.
  • Be open and alert to the use of new technology. (I know you are because you are reading this web page.) You can go from here directly to computing help. But don’t neglect thinking about the problem and understanding it, its solution, and its ramifications. The purposes of technology are not to relieve you of the need to think but:
    • To check your answers.
    • To take care of routine tasks efficiently.
    • To do things that can’t possibly be done by hand (like the visualization of large data sets).

    Keep in mind R.W. Hamming’s famous maxim: The purpose of computing is insight, not numbers.

  • Once you are done with a course Keep Your Textbook and refer back to it when you need to. You have spent so much time with that book that you know it intimately and know how to use it and where to find the information you need. The small amount of money you might get by selling it does not come close to offsetting the loss in time and energy you waste being thwarted by a lack of understanding a particular piece of mathematics that you easily refamiliarize yourself with by consulting your old friend, the textbook. Here’s a more passionate elaboration on this theme.


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.

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