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I am asking for what, if any, the preferred study skills of approaching classes are like in higher math education are like in general, and I am not asking necessarily for personal anecdotes for this question (though they are welcome if it's all you have to share).

My question is what are the better studying methods for learning higher mathematics(i.e., subjects involving more proofs instead of calculations like Analysis, Topology, Axiomatic Set Theory, Abstract Algebra, etc.)?

Let me explain what I mean by giving you an example. As an undergrad at first, courses like the Calculus sequence, Differential Equations could be approached by learning how to solve problems even if you weren't able to understand the proofs(which were usually skipped over in the textbook by the professor).

But now, that I have taken Linear Algebra, and I am about to take Analysis I usually take a much different approach:

$ $$\cdot$$ $ I start by rewriting all the definitions and theorems and proofs of a chapter and memorizing them. Then I will work through the examples, and ideally get to the exercises and finish off the chapter. And also spend time reflecting on the topics to get a more intuitive understanding of the concepts involved. I skip very little, anything if at all from the books I'm working with even if the topics are skipped in class.

This approach is very effective, especially when I am studying a topic for my own interests, and I am really able to understand things at a level that my peers usually have trouble following. The drawback is that I move at a much slower pace than my peers and I end up struggling in a class towards the end of a semester because I am behind on the syllabus. I end up risking a low grade though all but once I have managed an A- or above. Also am able to work out Jech's Set Theory which others have told me is out of my league but I actually find it the right amount of challenging using this approach.

The method followed by the math department at my school feels very superficial and not as the effective long-term. Usually, professors do not focus on the reasoning or intuition behind concepts. Tests are oriented so that we memorize by rote the proofs of the major theorems and regurgitate them on exams.

Let me give more concrete questions:

  1. I asked a math professor once, and he said that real mathematics is usually done where you understand 2 or 3 pages a day of a text on your first reading of it. Is this true for graduate level work for the average student?

  2. Do students who follow my immersive way of studying tend to have an advantage over those who don't when we get to grad school?

  3. In terms of learning theory and doing exercises, how much importance is recommended to place on each?

4. Are there study techniques used in more advanced math courses (like engaging in discourse with peers, focusing more on memorization before attempting to do problem sets, taking notes in a particular way) that are more fruitful than others?

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    $\begingroup$ I find that it can be hard to keep up with fast-paced graduate courses when taking such a thorough approach. I think it can be useful to learn in a "big picture first", coarse-to-fine manner. There is a (slowly expanding) core of material which you have grokked, and beyond that are large amounts of material for which you have a big picture understanding and can learn details of as needed. I think some people even learn in an extreme "backtracking" manner -- dive immediately into the topic you are interested in, and backtrack as needed to fill in missing knowledge. $\endgroup$
    – littleO
    Aug 20, 2017 at 0:29
  • $\begingroup$ I'm not even close to a math degree ( I talk to mathematical people online and barely know set theory stuff). I think the problem is there's a lot of interconnect and generalizations in math: Fermat's little theorem just a specific case of the more general Euler's theorem. 2-tuples just a specific case of n-tuples, functions just a specific type of morphism, ordering relations just a specific type of relation, total orders just a subset of partial orders. etc. $\endgroup$
    – user451844
    Aug 20, 2017 at 0:41
  • $\begingroup$ @littleO I think you're "big picture first" approach seems to be a good way to ensure that you get an education you can later expand on your own likeyou said. Unfortunately, there's a huge motivational problem because the reason a lot of people get into math like me is because they want to learn about some special gem of a problem and the "big picture" approach doesn't motivate your particular interests. Now I could of never understood the Undecidability problem or infinity when I first started math ofcourse but discussions on it helped keep me motivated to learn logic and calculus thoroughly. $\endgroup$
    – Red
    Aug 20, 2017 at 16:59
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    $\begingroup$ I just recently read "A Mind for Numbers" by Oakley. It seems to contain a lot of good advice about how to study mathematics. You might check it out. (I understand the book is the text for a Coursera course, "Learning How to Learn",which I have not taken.) $\endgroup$
    – awkward
    Aug 20, 2017 at 18:59
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    $\begingroup$ I would recommend to you for example the so-called Feynman technique, and techniques to solve problems and visualization techniques in mathematics, for example from a spanish author, Miguel de Guzmán. Honestly I was never a good student and did not use these techniques. My spirit some days on some issues is pessimistic. Thus I also want to be in solidarity with you, because this question has a great importance in the university world. I believe that some professor should be to write a book, or notes, about how to study mathematics at university level. Good luck. $\endgroup$
    – user243301
    Aug 24, 2017 at 12:49

2 Answers 2

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You question is waay too many questions in one for this website. Just FYI. Anyway...

"... real mathematics is usually done where you understand 2 or 3 pages a day of a text on your first reading of it. Is this true for graduate-level work for the average student??"

  • This completely depends on what the reading is and the person's background in it. Learning something completely new? Then yes, its probably true.

"Do students who follow my immersive way of studying tend to have an advantage over those who don't when we get to grad school?"

  • Sure, students who know more math upon entering graduate school have an advantage over the students who were content with just the details given in class. It's probably the self-motivation of the student more so than the actual knowledge that puts the student at an advantage to do well.

"In terms of learning theory and doing exercises, how much importance is recommended I place on each?"

  • Exercises confirm that theory was actually learned and understood. If you are doing exercises and you find them "easy", well then you probably have a very solid grasp of the theory. Do what feels right. Learn some theory, go back and see if you understand the theory.

"Are there study techniques used in more advanced math courses (like engaging in discourse with peers, focusing more on memorization before attempting to do problem sets, taking notes in a particular way) that are more fruitful than others?"

  • I've found that reading the book and taking notes (well) before class and then paying close attention in class is almost always sufficient to understand the material. I've also found that graduate students many times do not keep up with this regime of reading before the class, whether due to workload or general dislike for the material.

In summary, do what feels right, you'll learn a lot if you stay motivated, and enjoy.

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I found benefit in reading two books by Lara Alcock:

How to Study as a Mathematics Major

and

How to Think About Analysis

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