Strategies and Tips: What to do when stuck on math?

Since math students will be stuck on some math at some point, what strategies or tips can help
(to assuage this recurrent reality of maths)?

Certainly, this wonderful website helps; here: one can ask questions, interact with different people, be referred to other information. Still, to what else can be one resort?

I thought to start with some quotes from the following two articles:

[Source:] As research mathematicians, Littlewood and Poincaré knew the value of letting a problem go. They both structured into their lives time for allowing the attention to disengage from the level of actively thinking about a problem. Littlewood went for walks in the country. Poincaré went on a geology excursion. And occasionally, with good luck, the sudden flash of deep inspiration occurred.

[Source:] ■ The first theme has to do with sleep and the role it plays in doing mathematics and AHA! experiences. Many of the participants commented at one time or another about the phenomenon of waking up to a solution, either in part or even fully formed.

■…he discusses how “a bridge gets built between two apparently unrelated fields”.

■ Certain gaps in knowledge needed to be filled and my main role was to feel that these gaps could be filled. – Dick Askey

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This, probably, is a more appropriate answer to: “Strategies – What to do when stuck on some math problem“, but I thought it was still worth mentioning.

I’ve always thought that section 2.1 of this book is a rather nice and concise list of concrete advice regarding how to solve problems in analysis, particularly useful–I think– for students who might not have a large experience in tackling proofs. If I’m stuck on a problem, having a read through it, sometimes, gives me an idea of what else I can try.

I know this is an old post, but stuck at math, or stuck on a problem, this is a timeless issue. I’m trying to personally close the gaps that keep me from getting better marks and I think that while Polya, mentioned above, is a great book, and a wonderful resource, it was written first in 1945 and then a second edition was published in 1957. And while the techniques used for old problems is almost canonical (geometry = Euclid), the way we speak in about math has changed. So we need a younger, more contemporary voice. I started with Polya. But it is hard for me to relate to. There had to be another resource.

So, instead, I can recommend to anyone reading this and looking for help on this topic, take a look at Terrence Tao’s book “Solving Mathematical Problems” and work through the problems. And really try to absorb the process as you do it. I use spaced repetition to remember definitions and theorems, and an index card with a succinct list of steps (there are 8 or 10, if I recall) one can commit to memory to prompt oneself when struggling on a proof or exam question. Each of the worked examples considers the value of all the data you’ve collected on the issue, trying to get the reader to internalize the questions to ask at each step.

I hope this helps any new viewer of this thread.