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I find myself constantly being told I should not intentionally jot down example problems but understand the core logic of a topic. But some of my high level math classes tend to have very few examples and mainly definitions, i.e. words. Is it sound to keep a notebook of example problems related to the topic so you have a more practical understanding of the subject?

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I find myself constantly being told I should not intentionally jog down example problems but to understand the core logic of a topic.

You're offering a false dichotomy. If you can't generate concrete working examples, you don't really understand the definitions; conversely, if you can't prove things from the definitions, you don't really understand the examples. You should both work through examples and understand the definitions; the two approaches strongly reinforce each other. In particular:

  • If you are given only examples, you should figure out appropriate formal definitions and core logic yourself. Aim for both simplicity and generality. Look for weird corner cases. Work out new examples that stress-test your definitions.

  • If you are given only formal definitions and few or no examples, you should work out several new examples yourself. Aim for examples that both illustrate and stress-test the definitions. Together your examples should exercise every case, every word, and every symbol in the definitions. Failure to cover the definitions completely may mean that the definitions can be simplified; success may mean that the definitions can be generalized.

Write everything down. Fail, revise, repeat.

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    "If you can't generate concrete working examples, you don't really understand the definitions; conversely, if you can't prove things from the definitions, you don't really understand the examples." Both of these things seem true to me, but they can be harsh: e.g. for certain mathematical topics, quantifying over this implication one finds standard definitions and examples that no one really understands. Jun 12 '15 at 15:47
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    one finds standard definitions and examples that no one really understands — Yep.
    – JeffE
    Jun 12 '15 at 21:54
  • @PeteL.Clark As a fellow mathematician, I am hard pressed to come up with examples and definitions no one really understands unless you mean explicitly open problems, as in "here are the first 10^9 non-trivial zeros of the zeta-function; no one really understands why they are on the line Re s=1/2". Can you offer any better examples?
    – Boris Bukh
    Jul 26 '15 at 17:29
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To answer your question: Yes you should.

As has been noted already by fuesika you should think about the problems, but ultimately you become better at something by practicing it. Being a physics student I found it very helpful for the understanding of some of the graduate courses in math to have a set of problems/examples. A set of exmples or practical problems (here: real-world problems) allowed me to see the core use of the mathematical statements. Sometimes math is an art stating something easily understandable in a way such that only those initiated in math are able to understand it.

Knowing all the definitions might not hurt you, but it is not a necessary condition for the understanding of a topic. You can learn the definitions etc. in all their beauty after you understand the structure of the subject/course/problem and to this end I think that having a set of problems is indispensable.

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Yes, I believe it's a good idea.

I find myself constantly being told I should not intentionally jog down example problems but to understand the core logic of a topic.

It depends on how it's done. If the speaker or instructor puts an example up that you can later retrieve and he/she is focusing on telling you about the underlying logic, then lay down the pen and listen actively first. Most of the times this kind of comments were given when instructors want to give a fuller picture but everyone is just scribbling madly.

Is it sound to keep a notebook of example problems related to the topic so you have a more practical understanding of the subject?

Yes, but be flexible about the linkage between question and the underlying theory. Do not silo a certain question into a theory or vice versa. Instead, frequently revisit them and evaluate how different questions and theories can be interchanged, compared, combined, contrasted, etc. Some software function like hyperlinks and tags on EverNote would be an ideal tool to do that.

An added advantage is that if you later become a TA or a faculty you would have a good collection of applications. So, do yourself a favor and note the source of your examples as well so that you don't have to frantically look for them later.

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I think it's important to know questions and how they might be asked so you won't be blown away during tests. I normally find myself better prepared if I know HOW questions are going to be asked since each test and teacher/professor might ask it in a different way.

But if you are recording the question to memorize the answer, then yes that would fall into the pattern of not understanding the fundamentals.

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