Is doing exercises with time-restrictions important? Seeking advice

Question

I'm currently on the verge of my undergraduate maths studies, and have taken some time to understand where my weaknesses lie and some more in understanding whether these should determine my future career. In particular, I have realised that I am quite a slow-working maths student: my absolute favourite part of learning maths is going through the exercises and even struggling with them; however, I am notoriously slow at solving problems, especially during written exams, which brings me to my question.

Should one be concerned about how slow they are at tackling problems? For example, I'm currently going through Atiyah-Macdonald's text on Commutative Algebra, my main goal is solving most of the exercises from it; this is taking me ages. Personally, I really don't find time a "problem"... even if I spend three hours trying to solve an exercise and making no progress whatsoever, I do find myself "discovering" little things in the process and making rather useless humble conclusions which satisfy me nonetheless. One of the reasons I ask this question is that I'm not really sure how I'm supposed to interpret my failure, when I get a grade I'm not satisfied with at an exam because I feel I didn't have enough time to finish all the exercises; I'm interested in knowing if this is an issue I should worry about and maybe think to improve since I often think about maybe doing research as a grown-up soon.

Any general advice is super welcomed. Thanks in advance!

Answer 1

This is all based on my experience as an undergrad in maths (graduating this semester), and someone who took a few grad-level courses as well.

In theory, time is not a factor in doing problems. If you are struggling with a research question or difficult proof in Analysis or Number Theory or PDEs, usually it takes some miraculous insight or new application of a theorem. Some people will come to that insight for that particular problem faster than others. For grad school, this is a really good place to be in practice problems, where your prime goal should be to understand, as much as possible, the content: in my experience, the more that I struggled the first time, the less that I struggle when applying the techniques and theorems to a similar problem.

In practice, in grad school, there are things like exams for class, or, even more, comprehensive exams for maintaining in grad school. The primary purpose of this is twofold.

1. You should eventually be able to see a problem and recognize the associated terms and theorem relatively quickly, especially if you continue with research since that's what you will want to be able to do to answer questions.

2. There is only a limited amount of time in the semester/trimester/quarter, and so you will have to test the students somehow.

For me, by no means would I consider myself a "fast learner," especially as I got to higher level real and complex analysis. BUT, the important thing is that once I finally got through a problem, picturing it in my head/on paper, and struggling and working through the associated theorems/lemmas, I came out much more prepared for the exam in the end. Even though I wasn't the number 1 student in my class, I was able to achieve relatively well by struggling in this way. Speed naturally will come from more practice, and seeing patterns in how to apply the theorems, similar to speed in taking derivatives.

If you feel that you are slow because of other factors, such as getting easily distracted, getting panicked, etc. you can contact your school's accessibility services, and they should be able to accommodate some issues (with the professor involved in the process) in a fair way. I personally did not use these services but a few of a my friends in graduate analysis did use them and it helped them a lot with their anxiety, especially because some of them struggled with English and so having translated tests made them much more comfortable.

Answer 2

I think when you do research, you are pretty often in the position you describe with your exercises. So if you find that satisfying, it actually makes you fit to do research. A research problem is seldom something you can quickly figure out, you will often spend two thirds of the time taken to solve a problem banging your head at a wall asking why nothing you know is working, and then, little by little, a realisation after realisation, going back and forth from one approach to another, find the right path, the right technology fit to solve the problem. Especially when working with things that are new or have few precedents.

However, with undergrad exercises, you must try to improve your timing, because if you don t get a bit quicker with those you grades are not gonna satisfies you, and also, when doing research, you are gonna spend to much time on things that are somewhat trivial (namely, the kind of things you do exercises about as an undergrad). But then, how to improve timing? Well, it is experience. As a relatively slow guy myself, what I can say is that you may find useful do a little bit less exercises for now, but try to understand, each time you waste a lot of time on one, what was the cause. Essentially, you need to figure out what got you confused in the first place and made you go for non-productive approaches instead of the right one. Clearly if you think carefully about it, next time you probably won’t do it. Also it helps choosing the right exercises. Make sure you follow some sort of progression, usually the one the lecturers set for the course. Don t jump to exercises that are too hard too early, as you ll actually waste more time. If a book has exercises that are too hard, pick an easier one and than after that go back to the good stuff. After all time is finite, and while it is good to spend a lot of time exercising, one must make sure it is being done productively. Sometimes, after having spent a good amount of effort on one, maybe it s good to just humbly ask around and learn something new rather then figuring it out on your own. Your brain will also absorb new thinking patterns from others, as well as developing them on its own. As an undergrad I’d say both things need to happen and a lot to quickly build up experience.

In conclusion, do not be concerned, as long as you enjoy the process, but do address the issue: exercises (the ones in exams at least) that are supposed to be solved quickly must, if you do not manage to solve them quickly, teach you what was wrong in the way you were looking at them, so that next time you won’t make the same misjudgement. Little by little your experience will build up so much so solving exercises will be a lot quicker because you ll have a good method. Finally as a rule if thumb, don t obsess on one single exercise that does not come easy. It is good to take time off it and look at it with new eyes time later, maybe after studying other stuff. And as always, have fun.