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Authors: Ian Stewart
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as D’Arcy Thompson’s classic On Growth and Form , about mathematical patterns in living creatures. It may be outmoded in biological terms—it was written long before the structure of DNA was found—but its overall message remains as valid as ever.
Such books will broaden your appreciation of what math is, what it can be used for, and how its sits in human culture. There will likely be no questions about any of these topics on your exams. But awareness of these issues will make you a better mathematician, able to grasp the essentials of any new topic more confidently.
There are also some specific techniques that will improve your learning skills. The great American mathematical educator George Pólya 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. He was right. But you can’t learn this way if you get stuck on every problem you try. So your teachers will set you a carefully chosen sequence of problems, starting with routine calculations and leading up to more challenging questions.
Pólya offers many tricks for boosting your problem-solving abilities. He describes them far better than I can, but here is a sample. If the problem seems baffling, try to recast it in a simpler form. Look for a good example and try your ideas out on the example; later, you can generalize to the original setting. For instance, if the problem is about prime numbers, try it on 7, 13, or 47. Try working backward from the conclusion: what steps must we take to get there? Try several examples and look for common patterns; if you find one, try to prove that it must always happen.
As you remarked in your letter, Meg, one of the main differences between high school and college is that in college the students are treated much more like adults.
This means that to a much greater extent, it’s sink or swim: pass, fail, or find another major. There is plenty of help available for the asking, but that too takes more initiative than it did in high school. No one is likely to take you by the hand and say, “It looks like you’re having trouble.”
On the other hand, the rewards for self-sufficiency are much greater. Your high school was mainly grateful if you were not a problem requiring some sort of extra attention, and unless you were extremely lucky, the most it could offer an exceptional student (beyond the grades certifying him or her to move on) was an extracurricular club and perhaps an award or two. In a university you will encounter real scholars who are on the lookout for young people capable of doing real mathematics, and they are just waiting for you to stand out, if you can.
8
Fear of Proofs
Dear Meg,
You’re quite correct: One of the biggest differences between school math and university math is proof. At school we learn how to solve equations or find the area of a triangle; at university we learn why those methods work, and prove that they do. Mathematicians are obsessed with the idea of “proof.” And, yes, it does put a lot of people off. I call them proofophobes. Mathematicians, in contrast, are proofophiles: no matter how much circumstantial evidence there may be in favor of some mathematical statement, the true mathematician is not satisfied until the statement is proved . In full logical rigor, with everything made precise and unambiguous.
There’s a good reason for this. A proof provides a cast-iron guarantee that some idea is correct. No amount of experimental evidence can substitute for that.
Let’s take a look at a proof and see how it differs from other forms of evidence. I don’t want to use anythingthat involves technical math, because that will obscure the underlying ideas. My favorite nontechnical proof is the SHIP–DOCK theorem, which is about those word games in which you have to change one word into another by a sequence of moves: CAT, COT, COG, DOG.
At each step, you are allowed
Such books will broaden your appreciation of what math is, what it can be used for, and how its sits in human culture. There will likely be no questions about any of these topics on your exams. But awareness of these issues will make you a better mathematician, able to grasp the essentials of any new topic more confidently.
There are also some specific techniques that will improve your learning skills. The great American mathematical educator George Pólya 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. He was right. But you can’t learn this way if you get stuck on every problem you try. So your teachers will set you a carefully chosen sequence of problems, starting with routine calculations and leading up to more challenging questions.
Pólya offers many tricks for boosting your problem-solving abilities. He describes them far better than I can, but here is a sample. If the problem seems baffling, try to recast it in a simpler form. Look for a good example and try your ideas out on the example; later, you can generalize to the original setting. For instance, if the problem is about prime numbers, try it on 7, 13, or 47. Try working backward from the conclusion: what steps must we take to get there? Try several examples and look for common patterns; if you find one, try to prove that it must always happen.
As you remarked in your letter, Meg, one of the main differences between high school and college is that in college the students are treated much more like adults.
This means that to a much greater extent, it’s sink or swim: pass, fail, or find another major. There is plenty of help available for the asking, but that too takes more initiative than it did in high school. No one is likely to take you by the hand and say, “It looks like you’re having trouble.”
On the other hand, the rewards for self-sufficiency are much greater. Your high school was mainly grateful if you were not a problem requiring some sort of extra attention, and unless you were extremely lucky, the most it could offer an exceptional student (beyond the grades certifying him or her to move on) was an extracurricular club and perhaps an award or two. In a university you will encounter real scholars who are on the lookout for young people capable of doing real mathematics, and they are just waiting for you to stand out, if you can.
8
Fear of Proofs
Dear Meg,
You’re quite correct: One of the biggest differences between school math and university math is proof. At school we learn how to solve equations or find the area of a triangle; at university we learn why those methods work, and prove that they do. Mathematicians are obsessed with the idea of “proof.” And, yes, it does put a lot of people off. I call them proofophobes. Mathematicians, in contrast, are proofophiles: no matter how much circumstantial evidence there may be in favor of some mathematical statement, the true mathematician is not satisfied until the statement is proved . In full logical rigor, with everything made precise and unambiguous.
There’s a good reason for this. A proof provides a cast-iron guarantee that some idea is correct. No amount of experimental evidence can substitute for that.
Let’s take a look at a proof and see how it differs from other forms of evidence. I don’t want to use anythingthat involves technical math, because that will obscure the underlying ideas. My favorite nontechnical proof is the SHIP–DOCK theorem, which is about those word games in which you have to change one word into another by a sequence of moves: CAT, COT, COG, DOG.
At each step, you are allowed
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