How do/should you study math? (graduate level)

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In summary, the conversation discusses the approach to studying textbooks, specifically in terms of working out every detail and balancing efficiency and honesty. The speaker shares their experience with different textbooks and how they handle typos and errata in some texts. They also ask for advice on whether to skip details or grind them out and whether skipping details means not being able to use results hinging on them. The speaker mentions their idealistic approach of working out every proof and filling in all gaps, but also acknowledges the issue of efficiency.
  • #1
greetings,

when i read textbooks, i always make sure to work out every single detail of every single statement and proof.

i'm in my first year of graduate school, and i find that sometimes i can't do this, either because the book I'm working on isn't very good, or because i don't have enough time.

for example, I'm currently up to page 117 of rotman's algebraic topology text. (rotman, as some of you might know, is notorious for typos and errata in his algebra texts. I'm actually compiling a list for him for errors I've found in this text.) I'm starting to not care about some of the combinatorial details of barycentric subdivision. i want to just get to the applications to euclidean space, and frankly, the qualifying exam probably won't ask about said details.

some textbooks i can just breeze through (e.g. munkres' book on manifolds, and his book on topology) while doing most of the problems and reading everything thoroughly. other books aren't as well written, though. (i've been unfortunate to have come across some of them in my first year, wasting tons of my time. this is a whole other issue.)

my question is: how do you and how should you study? should you skip over details, or do should you grind out each and every single detail or most details? i used to be idealistic and think if i don't see every detail, then i shouldn't be permitted to use results hinging on them. for instance, if I'm skipping a couple of details on barycentric subdivision, then should i be able to use invariance of domain?

it seems to me a question of balancing efficiency and honesty.
 
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  • #2
I take nothing for granted. For every theorem, I work out every proof and fill in all gaps. Depending on the textbook, this means writing up to 5 pages of detail for every page I read.
 
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1. How many hours a day should I study math at the graduate level?

The number of hours you should study math at the graduate level can vary depending on your individual learning style, the difficulty of the material, and your other commitments. Generally, it is recommended to spend at least 3-4 hours per day studying in order to fully grasp the concepts and practice problem-solving. However, some students may need more or less time depending on their background knowledge and understanding of the subject.

2. What is the most effective way to study math at the graduate level?

The most effective way to study math at the graduate level is to actively engage with the material. This can include attending lectures, taking notes, completing practice problems, and seeking help from professors or peers when needed. It is important to also review and reinforce concepts regularly, rather than cramming all the material at once. Additionally, finding a study group or study partner can be helpful in discussing and understanding difficult concepts.

3. Is it necessary to have a strong background in math before pursuing a graduate degree?

Having a strong background in math before pursuing a graduate degree can be beneficial, but it is not always necessary. Some graduate programs may require a certain level of math knowledge, while others may offer remedial courses for students who may need extra support. It is important to have a basic understanding of mathematical concepts and the ability to think critically and problem-solve, but with dedication and hard work, it is possible to succeed in a graduate math program with a less extensive background.

4. How can I stay motivated while studying math at the graduate level?

Staying motivated while studying math at the graduate level can be challenging, but there are some strategies that can help. Setting achievable goals and rewarding yourself for meeting them can provide a sense of accomplishment and motivation. It can also be helpful to connect with other students or professors who share a similar interest in math and engage in discussions or attend seminars or conferences. Additionally, taking breaks and engaging in activities outside of studying can help prevent burnout and maintain motivation.

5. What resources are available to help me study math at the graduate level?

There are many resources available to help students study math at the graduate level. Your university or department may offer tutoring services or study groups specifically for math courses. There are also online resources such as video lectures, practice problems, and forums where you can ask questions and receive help from other students or instructors. It is important to take advantage of these resources and seek help when needed in order to succeed in your graduate math studies.

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