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
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.
This is also applicable in physics.
No matter what long story problems they give, you can always extract only the relevant variables for the equations you will be using.
One example is in A level maths, APGP, H2 Maths 2008 P1 Q10. The question is a long question on money. compound interest. One can simply extract the first term, and common ratio, and from there, do merely algebraic manipulations to get to the answer.
It is the thought process and analytical skills that matter. Equations are used at the end of your thought processes and analysis to come out with the final solution. :D
thanks for sharing.