Understanding Mathematics
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 never ask questions in class?
Comments.

Do you sell your textbook after you take a math class?
Comments.

Do you find it diffcult 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 email!
Otherwise I'm hoping that this page has something to offer
you (and of course you may send me an email 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.
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
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
0691080976. 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:

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.
Fine print, your comments, more links, Peter Alfeld,
PA1UM
[30Dec2004]