How to Study as a Mathematics Major 的笔记

图书信息可参见这里

Preface

a student who expects mathematics to come in the form of procedures to copy will not know how to interact with material presented via definitions, theorems, and proofs. Indeed, research shows that such a student will likely ignore much of the explanatory text and focus disproportionately on the obviously symbolic parts and the exercises.

Introduction

That's why I'm writing this book: to give you a leg-up so that your academic life is easier and more enjoyable than it would otherwise be.

However, this book is not about some magical easy way to complete a mathematics major without really trying. On the contrary, a lot of hard work will be required. But this is something to embrace. A mathematics major should be challenging-if it were easy, everyone would have one. And, if you've got this far in your studies, you must have experienced the satisfaction of mastering something that you initially found difficult. The book is, however, about how to make sure that you're paying attention to the right things, so that you can avoid unnecessary confusion and so thatyour hard work will pay off.

Chapter 1 Calculation and Procedures

  • P4.

For now, however, note that knowing how to apply procedures is extremely important because, without fluency in calculations, it is hard to focus your attention on higher-level concepts.

  • P10.

In high school, your teacher probably took responsibility for deciding how much practice you should do, but a professor is more likely to set just one exercise and leave it to you to judge whether you'd benefit from inventing similar ones.

  • P15

If you are tempted to answer any of these questions by saying "that's just how it is;' be aware that there are reasons for all of these things, even if you don't yet know them.

  • P16

Indeed, plenty of mathematicians will tell you that learning procedures mechanically is bad—that you should always be striving for a deep understanding. This is a well-intentioned claim, but it is a bit unrealistic. For a start, there are plenty of situations in which good understanding is not really accessible to you at a given stage.

Sometimes we don't learn about the details of something for pragmatic reasons — there just isn't time. Sometimes we don't learn about them because they are fiddly and because studying them would distract us from understanding what a procedure achieves. There will still be situations like this during your major, but you will start to see more and more of the theory underlying mathematical knowledge.

Chapter 3 Definitions

  • P37

An axiom is an assumption that we agree to make, usually because everyone agrees that it is sensible....But there can be cases in which two people inadvertently assume that different, more subtle axioms hold, and then get confused by each other's claims. To avoid this, it's good mathematical practice to state all our axioms clearly up
front.

  • P51

You have to be willing to get your hands dirty when learning new mathematics; to try out lots of things that might help. As you build up a bank of example objects of various types, you'll get better at using them to help you understand new definitions.

  • P56

Mathematicians aren't usually trying to decide on the precise meaning of a word just for the sake of it. They're usually trying because they want to use it in formulating a theorem and an associated proof. Thus the formulation of the definition might depend on what is convenient for being able to state a nice, elegant, true theorem.

You need to be aware that you will occasionally come across a mathematical definition that does not quite correspond with your intuitive understanding. It is perfectly okay to find this a bit weird, but you have to deal with it. Mathematicians have made their collective decisions for good, sensible reasons. Those reasons might not be immediately apparent to you but, if the mathematical definition of a term does not match your intuition, it's your intuition that needs fixing up.


Chapter 4 Theorems

  • P75

sometimes rewriting in a different way can give us different ideas about sensible things to try.

  • P77

This brings us back to the idea that both logical form and example objects can contribute to mathematical understanding, though focusing
on each has different advantages and disadvantages. If you look mainly at examples, you might feel that you understand, but you might fail to appreciate the full generality of a statement or find it difficult to see the logical structure of a whole course. If you look mainly at formal arguments, you might be able to see how everything fits together logically, but you might find yourself complaining that it is very abstract and that you don't really understand what is going on.

Looking at lots of examples does not necessarily give any insight into why the conclusion of a theorem must always hold, which is what mathematicians
are really interested in.


Chapter 5 Proof

  • P93

Some people will explain this by saying that professors want you to practice writing rigorous proofs of simple things, so that you can use your proving skills confidently when you go on to work with more complicated things....
A better reason is that mathematicians value not just knowing that a theorem is true, but also understanding how it fits into a broader network of connected results that forms a coherent theory.

  • P125

When you see a proof that involves a clever insight, you should not worry that you wouldn't have thought of it; you should think about why it works,how it might be adapted, and how it relates to other ideas you have seen.


Chapter 7 Reading mathematics

  • P144

... successful mathematicians are smart and strategic. They don't want to spend time memorizing things when they could reconstruct them using a bit of effort and common sense.
There is one thing I do want to emphasize, though, about remembering things by reconstructing them: you have to give yourself a bit of time.

不要一想不起来就翻书/笔记,自己不妨花点时间回忆一下,从记得的知识开始构造所需要的结果。

  • P148

please try to exhaust all approaches that might lead to understanding before you resort to rote memorization-there are many things you could do that would be much more satisfying.


Chapter 8 Writing mathematics

  • P153

good presentation can make your thinking clearer to someone else, and can help you to avoid ambiguity.

Chapter 13 (Not) Being the best

  • P232

The fact that you don't understand everything in lectures doesn't mean that you are not good at mathematics any more, it just means that the mathematics is harder and the pace is faster. To keep up with it, you will have to do more (or better) work. But so will everyone else. And that's as it should be (if studying for a mathematics major was easy, everyone would do it).

  • P235

People who try to be fast often miss out on opportunities to consolidate their understanding, because racing on to the next problem as soon as they finish one means that they do not reflect on what they have learned, do not think about how it relates to things they already know, and do not tighten their grip on it so that they'll remember it and be able to apply it in new situations. Reflecting might only take a minute, and it might dramatically increase your understanding of what you have done.

In the worst case, trying to be fast can lead people to memorize procedures without understanding what they are doing at all.

不少中学老师和课外辅导机构就在给学生做这种事。后患无穷!

  • P236

Mathematicians don't want students to end up with encyclopedic knowledge but dodgy underlying understanding. They're more impressed by deep understanding, and, at least in upper-level courses, they tend to test for this by making at least some of the questions on their exams require original thinking or novel applications of the mathematical ideas.

posted on 2016-01-06 00:44  星空暗流  阅读(313)  评论(0编辑  收藏  举报

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