Hi,
This short article is taken from "Letters to a Young
Mathematician" by Ian Stewart. It discusses some
practical methods to learn the subject.
The vast amounts that have been written about teaching
math might give the impression that all of the difficulties
encountered by math students are caused by teachers, and it is
always the teacher's responsibility to sort out the student's
problems. This is, of course, one of the things teachers are paid
to do, but there is some onus on the student as well. You
need to understand how to learn.
Like all teaching, math instruction is rather artificial.
The world is complicated and messy, with lots of loose ends, and
the teacher's job is to impose order on the confusion, to convert a
chaotic set of episodes into a coherent narrative. So your learning
will be divided into specific topics following a carefully
specified syllabus.
Because the lectures progress through set topics, one
step at a time, it is easy for students to think that this is how
to learn the material. It is not a bad idea to work systematically
through the book or set of notes, but there are other tactics you
can use if you get stuck.
Many students believe that if you get stuck, you should
stop. Go back, read the offending passage again; until light dawns
- either in your mind or outside the library window.
This is almost always fatal. I always tell my students
that the first thing to do is read on. Remember that you
encountered a difficulty, don't try to pretend that all is
sweetness and light, but continue. Often the next sentence, or the
next paragraph, will resolve your problem.
Whenever you get stuck on a piece of mathematics, it
usually happens because you do not properly understand some
other piece of mathematics, which is being used without
explicit mention on the assumption that you can handle it easily.
In that case, it is worthwhile to go through the other
piece of mathematics.
It takes a certain insight into your own thought
processes as well as a certain discipline to pinpoint exactly what
you don't understand and relate it to your immediate difficulty.
Your teachers know about such things and will be on the lookout for
them. It is, however, a very useful trick to master for
yourself, if you can.
To sum up: If you think you are stuck, begin by plowing
ahead regardless, in the hope of gaining enlightenment, but
remember where, why and how you got stuck, in case
this doesn't work. If it doesn't, return to the sticking point and
backtrack until you reach something you are confident you
understand. Then try moving ahead again.
Let me urge upon you another useful trick. It may sound
like a huge amount of extra work, but I assure you it will pay
dividends.
Read around your subject.
Do not read only the assigned text or set of notes. Books
are expensive, but schools have libraries. Find some books on the
same topic or similar topics. Read them, but in a fairly casual
way. Skip anything that looks too hard or too boring. Concentrate
on whatever catches your attention. It's amazing how often you will
read something that turns out to be helpful next week, or next
year.
There are also some specific techniques that will improve
your learning skills. The great American mathematical educator
George Polya put a lot of them into his classic How to Solve
It. He took the view that the only way to understand math
properly is the hands-on method: tackling problems and
solving them on your own. He was right.
Polya offers many tricks for boosting your problem
solving abilities. For example, if the problem seems baffling, try
to recast it in a simpler form. Look for a good example (say in the
text or notes) and try your ideas out on the example; later
you can generalize to the original setting.
To end, the rewards for self-sufficiency are great, if
one is willing to put in effort and discipline to learn math.
Thanks!
Cheers,
Wen Shih