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Authors: Ian Stewart
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relate to squares. Or you may be wondering why the uniqueness property of prime factorization is relevant. If so, you don’t need help to understand the proof that the square root of two is irrational; you need help rehearsing rational numbers, prime factors, or basic geometry.
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 tutors 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 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.
This process is very similar to a general method for solving a maze, which computer scientists call “depth-first search.” If possible, move deeper into the maze. If you get stuck, backtrack to the first point where there is an alternative path, and follow that. Never go over the exact same path twice. This algorithm will get you safely through any maze. Its learning analogue does not come with such a strong guarantee, but it’s still a very good tactic.
As a student I took this method to extremes. My usual method for reading a mathematics text was to thumb through it until I spotted something interesting, then work backward until I had tracked down everything I needed to read the interesting bit. I don’t really recommend this to everyone, but it does show that there are alternatives to starting at page 1 and continuing in sequence until you get to page 250.
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. Books are expensive, but universities 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.
The summer before I went off to Cambridge to study math, I read dozens of books in this easygoing way. One of them, I remember, was about “vectors,” which the author defined as “quantities that have both magnitude and direction.” At the time this made very little sense to me, but I liked the elegant formulas and simple diagrams with lots of arrows, and I skimmed through it more than once. I then forgot it. In the opening lecture on vectors,suddenly it all clicked into place. I understood exactly what the author had been trying to tell me, before the lecturer got that far. All those formulas seemed obvious: I knew why they were true.
I can only assume that my subconscious had been stirred up, just as Poincaré claimed, and during the intervening period, it had created some order out of my desultory wanderings through that book on vectors. It was just waiting for a few simple clues before it could form a coherent picture.
When I say “read around your subject,” I don’t mean just the technical material. Read Eric Temple Bell’s Men of Mathematics , still a cracking read even if some of the stories are invented and women are almost invisible. Sample the great works of the past; James Newman’s The World of Mathematics is a four-volume set of fascinating writings about math from ancient Egypt through to relativity. There has been a spate of popular math books in recent years, on the Riemann hypothesis, the four color theorem, π, infinity, mathematical crackpots, how the human brain thinks mathematical thoughts, fuzzy logic, Fibonacci numbers. There are even books on the applications of mathematics, such
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 tutors 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 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.
This process is very similar to a general method for solving a maze, which computer scientists call “depth-first search.” If possible, move deeper into the maze. If you get stuck, backtrack to the first point where there is an alternative path, and follow that. Never go over the exact same path twice. This algorithm will get you safely through any maze. Its learning analogue does not come with such a strong guarantee, but it’s still a very good tactic.
As a student I took this method to extremes. My usual method for reading a mathematics text was to thumb through it until I spotted something interesting, then work backward until I had tracked down everything I needed to read the interesting bit. I don’t really recommend this to everyone, but it does show that there are alternatives to starting at page 1 and continuing in sequence until you get to page 250.
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. Books are expensive, but universities 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.
The summer before I went off to Cambridge to study math, I read dozens of books in this easygoing way. One of them, I remember, was about “vectors,” which the author defined as “quantities that have both magnitude and direction.” At the time this made very little sense to me, but I liked the elegant formulas and simple diagrams with lots of arrows, and I skimmed through it more than once. I then forgot it. In the opening lecture on vectors,suddenly it all clicked into place. I understood exactly what the author had been trying to tell me, before the lecturer got that far. All those formulas seemed obvious: I knew why they were true.
I can only assume that my subconscious had been stirred up, just as Poincaré claimed, and during the intervening period, it had created some order out of my desultory wanderings through that book on vectors. It was just waiting for a few simple clues before it could form a coherent picture.
When I say “read around your subject,” I don’t mean just the technical material. Read Eric Temple Bell’s Men of Mathematics , still a cracking read even if some of the stories are invented and women are almost invisible. Sample the great works of the past; James Newman’s The World of Mathematics is a four-volume set of fascinating writings about math from ancient Egypt through to relativity. There has been a spate of popular math books in recent years, on the Riemann hypothesis, the four color theorem, π, infinity, mathematical crackpots, how the human brain thinks mathematical thoughts, fuzzy logic, Fibonacci numbers. There are even books on the applications of mathematics, such
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