# Advice on mathematical thinking and problem-solving

I have Autism. For me, this means that instead of seeing the whole picture, I see the details more than anything else. This means that I from time to time get stuck. Getting stuck in a certain path of thinking, for example at exams, is not good as it 'steals' valuable time and do not help you to solve the problems in the exam/test.

There exists different strategies for not getting stuck and move on to new perspectives if stuck in one perspective, as one example.

Have you experienced the same problems as me? How did you solve it? What was the best thing that you could do?

Thank you.

• well, it is hard to generalize but I'd say that lots of mathematicians struggle with some variant of the "getting lost in details" problem. As math is very much concerned with detail it can be very difficult to tell the difference between sensible attention to detail and obsession. Personally, if I find that I am going over the same reasoning repetitively, with little progress or change, I will force myself to try a different approach or, if that too fails, think about something else for a bit. But again, it is hard to generalize...what works for one person may not help another. good luck!
– lulu
Feb 4, 2017 at 22:01
• use headphones to block the outside world to focus. Feb 5, 2017 at 11:47
• Short story from me. I'm at university studying electrical engineering and I'd say I'm quite good overall. It was my first semester so I had Analysis I. Home work were so difficult we mostly couldn't solve them on our own (only with help from assistans) and the first test was 50/50 theory&calculating. I guess many people here will argue that, but in the end mathematics is something you have to learn by heart. Proofs, special integral, limes etc. you have to learn every little trick, for sure you must combine them in the end, but thats icing on the cake. Just my 2 cents. Feb 6, 2017 at 12:04
• About the "mentally exhausted " subject.And about more than half the things you wrote, I can realte. If I listen to psychedelic trance it helps my mind to recover faster. Check the "Psychedelic Universe" channel on youtube . Try this one "Sesto Sento - Tick Tock" start from the start, and listen to at least 2 minutes. Edit: do this when you when you have the feeling you described. Feb 6, 2017 at 12:47
• Due to your edits, I have reopened this question. One of the reasons why your post had remained closed before was because of edits including rants. But posts should focus on mathematical content only. Now that you have edited those out, I've reopened this post to give it another shot. I am not certain whether it not it belongs, but I think it might. Good luck to you, both on MSE and elsewhere. Feb 8, 2017 at 23:31

I don't know that this is an adequate answer for the question, but I certainly have sympathy for the trap of getting caught up in details ... that may be very subordinate. For myself, although I've mostly managed to avoid getting stuck in details, when I was much younger I would occasionally-and-unfortunately fixate on small things, since I'd been led to believe that every detail had the same significance in mathematics.

The latter is only "formally true", in the idealistic sense that if any link in a chain of logical reasoning fails, then the whole fails. However, live mathematics is not so "boolean" in its legitimacy or art. In particular, NOT all details are of equal significance in mathematical real life, despite various logical ideals.

I do also tell my PhD students and other grad students this, that one should be willing to let quite a few details be postponed, and try to discern the significant ones... all the more so because many of the small details become completely clear (only) with sufficient hindsight. Truly, in a strong sense, many details are genuinely unfathomable "in prospect", since the true explanation will only come later. That is, even if one does want to insist on careful explanations, the immediate formal seeming-explanations in typical sources are in fact not correct... and therefore all the more unpersuasive or baffling. Thus, duh, leading a serious person to be baffled, and think that it's their own internal "problem", rather than appreciating (since we are not often let in on the secret) that purely logical correctness is not at all a reliable explanation.

Especially if one is sensitive to such things, the disconnect can be nearly fatal, or at least severely impairing.

I hope there will be other answers about other aspects...

• Can you give a specific (hypothetical) example of how a student should let quite a few details be postponed and try to discern the significant ones? Also, can you give examples of details that are unfathomable in prospect since the true explanation will only come later? (I suspect I would learn something from hearing your examples.) Feb 5, 2017 at 9:49
• @littleO, appropriate examples obviously would depend on context, on the level of the student... but, still: worrying about the $\epsilon-\delta$ notion of limit, continuity, derivative, and integral before one sees the amazing utility of calculus is silly, because continuity has substantial intuitive content without any formal definition. Worrying about whether $0$ can be $1$ in a ring is a misguided concern, since it is a choice that rarely matters... although many sources declare their choice to be the unequivocal definition, and ask prankish questions about their convention. [cont'd] Feb 5, 2017 at 22:34
• [cont'd] Worrying whether $\int \lim_n f_n=\lim_n \int f_n$ in an unsophisticated setting is misguided, because we certainly hope so in practice, though, sure, counter-examples can be created (with pointwise limits). It is a bit silly to fret too much about whether various physically meaningful topological spaces are Hausdorff or not, because usually we want spaces to be locally compact, Hausdorff (and probably $\sigma$-compact) if we intend to do analysis on them... and it is not constructive to spend much energy wondering whether you're being pranked. Such things. Feb 5, 2017 at 22:39

Have you ever noticed that after the exam is over, on your way back home you quickly figure out that math problem without even trying all that hard?

During the exam, when your work goes around in a loop, your mind needs a break. You need to allow your mind to wander off, to give it a chance to end up somewhere outside of that loop.

Mathematics is also famous for different ways of attacking the same problem. Sometimes, some approaches are deliberately long and elaborated in order to keep the topic within the boundary of a subject (algebra, analysis, geometry etc.). Luckily (or when we are lucky), those approaches have alternatives which some people find more intuitive. Of course knowing different approaches to a problem tends to be very useful too.

Let me give you a few personal examples...

• I used to have troubles with understanding this proof of Fermat's Two Squares Theorem, while Minkowski's approach is so much easier.
• I only understood properly the $\varepsilon - \delta$ techniques in analysis (that was ~25 years ago) after my teacher plotted an alternative model on the $x$ axis and roughly defined what a vicinity is. A few years later, I discovered topology as a new subject in mathematics.

To conclude, try and seek for alternatives. Ask, if you can't find any (including on MSE). Maybe you are more drawn towards geometrical understanding, rather than "dry" $\varepsilon - \delta$ (as an example)? Once you develop a good intuition, you can attack long and elaborated methods.

I have challenges (not autism) too and have difficulties with tests, homework, organization, communication, etc.

I am registered with disability services at my university. I get a form that explains to my professors that I am in disability services. Then I schedule my tests with disability services and take them in a private room with extra time. This extra time and private room is enough to make it so that if I do get stuck I always have enough time and space to recover.

I recommend disability services if they are available to you. They have been helpful for me. I did not do too well in high school, so I went into the military. Unfortunately I did not do too well in the military. When I got out of the military I went into the industry and somehow managed to do well as a software tester (I even got credited in a video game!). Finally I got enough documentation together and landed at a university with disability services. With these services I sort of became a math prodigy at my local community college. Go figure -- I was never considered "skilled" at math growing up. Now I'm at the top of my class and have a really good shot at a good graduate school. I love math. I just didn't have the help I needed.

Registering for disability services ranks as one of the better decisions that I've ever made. It has boosted my confidence and helped me perform better academically. If I get stuck, I have the time to recover my thoughts and proceed.

I suggest to avoid algebraic computation for a while and try to see geometric proofs.

Geometry is a good medium for giving general insight instead of local calculation.

I recommend the book "Proof without words"

You can try to exercise learning the logic behind the formulas. I suppose it can help.

Best Wishes...

Well, mathematics is a subject which requires detailed analysis. And getting to the core of detailed analysis requires developing some strategies. Calculations are fun when you enjoy it. View your calculations in a different way, for example, if you’re good at arithmetic then leave the other part of calculus, geometry or differential equations and stay focused on arithmetic itself. Arithmetic was my favorite too and still is when I interact and teach my students with autism at Accel Centre. However, visualizing the problems and then solving it with music and videos can save you from a lot of stress. This way you’ll remember the easy-go procedure to solve it. To take my advice, it’s a better way to learn with visualization because it will help you remember the details when you’re stuck with something.