Math for a Future in the Sciences

Date: 02/08/2001 at 18:25:31
From: Ian
Subject: Studying and understanding math for a future in the Sciences

I've had a hard time with math for the last three years. The theorems, 
equations, patterns etc. all seem to pile up - it's impossible to 
memorize them all. Or sometimes I won't understand a concept.

Well, I've become obsessed with a field called Nanotechnology (or 
Molecular Engineering/material sciences). This field is intense in 
math and physics (I love physics; I can't get enough of it). My 
problem is, I want to be able to understand as much math as possible 
(geometry, Trig, Algebra, Calc, etc.) and keep this information with 
me for the rest of my life. But once I learn something, I forget it 
quickly - I can barely remember ANY algebra from last year!

Is there anything you can recommend for someone who wants to learn 
math on his own, and be able to retain information for the years to 
come?

Thanks a million.

Date: 02/09/2001 at 13:53:26
From: Doctor Ian
Subject: Re: Studying and understanding math for a future in the 
Sciences

Hi Ian,

I can sympathize with you, because I have a similar problem. I've 
_never_ been able to remember formulas, and the way that I've had to 
compensate is that whenever I learn a new formula, I have to learn how 
to derive it from first principles - by which I mean, the things that 
are so basic that I _can't_ forget them.

(One of the luckiest breaks I ever caught in my life was when I took
second-semester calculus in college.  That's the semester when you 
have to memorize about a zillion formulas that all look pretty much 
the same - one plus or minus the sine or cosine of plus or minus 
something, under or over the square root of something similar - so I 
fully expected to fail. But on the first day, the professor said: "I 
don't expect you to remember anything that I can't remember, and the 
only thing that I can remember is that cos^2 + sin^2 equals 1." So 
you're not alone!)

One trick you can use when trying to remember patterns or principles 
is to encode them as examples. For example, for the life of me I can 
never remember whether (a^b)^c is a^(b+c) or a^(bc).  So whenever I 
need to know, I drag out this example:

  (a * a) * (a * a) * (a * a) = a^6, so (a^2)^3 = a^6

which means that it must be (bc), and not (b+c).  Does this get 
tiresome? You bet. Is it preferable to guessing wrong? You bet!

Two other tricks I can recommend are:

1) Try teaching what you've learned to someone else. This is probably 
the single most effective way of learning anything, especially if the 
other person is having difficulty learning it. It forces you to think 
of new ways to understand the material, in order to avoid presenting 
it in the same old way. 

2) Try imagining yourself _using_ what you're learning at any given 
moment. The more vivid your imagination, the stronger the effect it 
will have on your memory. With math, the thing to do might be to try 
to find examples in physics where you would use any new concept or 
technique. 

I never used to be able to remember quotations from books, and then at 
some point I decided that I'd like to be able to do that. So now 
whenever I read a book, I highlight every interesting sentence or 
phrase that I want to remember... and I try to imagine myself in a 
situation where I would quote it. And it works!
  
In the end, you may find that your memory just isn't going to work as 
well as you want it to, in which case... there's always paper and ink!  
That's what they're for, really, so don't be ashamed to make use of 
them. If you have to choose between remembering things you don't 
understand, and understanding things that you can't remember, I 
strongly recommend the latter.  :^D

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/