How many problems do we have to do as undergraduate mathematicians in order to learn a subject?

Question

I'm wondering how many problems are needed in order to learn a subject, let's say Calculus of Several Variables. We know that the professors often assign us a list of problems to solve as homework, but I find that this is often insufficient.

I have found that solving all problems on a book or two is enough to handle (and understanding) most of the problems on a given subject and but I'm wondering if there is some more clever method to learn something because it is not always practical to do all the exercises of all subjects that we learn as math undergraduate students.

Is there any method that has proven that works? Thanks foy your help.

Answer 1

I think your question is good, but a little vague. When you say "how many problems," would ten easy problems be counted the same as ten difficult problems? If exactly the same problem was posed twice, would it count as one problem or two problems? As you can imagine, the answer to your question would depend on the subject to be learned, the types of problems asked, and the background of the learner.

I personally subscribe to the educational philosophy called mastery learning. From Wikipedia:

Mastery Learning (or as it was initially called, “learning for mastery”) is an instructional strategy and educational philosophy, first formally proposed by Benjamin S. Bloom in 1968. Mastery Learning maintains that students must achieve a level of mastery (i.e. 90% on a knowledge test) in prerequisite knowledge before moving forward to learn subsequent information. If a student does not achieve mastery on the test, they are given additional support in learning and reviewing the information, then tested again. This cycle will continue until the learner accomplishes mastery, and may move on to the next stage.

So, if your question was changed to "How many problems do we have to do in order to master a subject?" then I would answer "as many as are necessary to show mastery in the subject," which often means a score of 90% or better.

In practice, this number is much larger than what is usually done in typical schools. For example, a typical school would perhaps ask a dozen questions on mixed number addition. But to achieve mastery might involve a hundred questions or more, depending on the learner.

Answer 2

This is a good question, but it is impossible to answer in general terms. It depends upon (at least) the subject, the person, and (perhaps most importantly) what you mean by "learning it". I hope it is no secret that most undergraduate courses aim to give an introduction to the subject they concern. Things in the undergraduate curriculum which need to be learned more deeply are usually addressed as part of multiple courses. In other words, one of the reasons that students take multi-variable calculus and/or differential equations is to reinforce the differentiation and integration stuff they saw in a superficial whirlwind the year before.

But I said it was a good question and I meant it. It's a good question for you to answer for yourself. In general, you should assume that homework assigned in courses is not chosen so as to be the pedagogically ideal amount but is limited by practical considerations such as time and moderate interest. An advancing student of mathematics needs to learn how to judge for herself how well she is learning a particular topic, including how many problems to do, how quickly to move on to the next topic, and so forth.

Solving all the problems in one or two books is probably more than enough reinforcement for most topics beyond the calculus level. (I have fond memories of working calculus problems as a high school student. I loved calculus and I was good at it, but nevertheless I remember working through several hours of problems a night. Most of them did not give me much trouble, but there were a lot of them. And I do think it was good for me...not least because I have solved so many calculus problems in a teaching capacity in my adult life.) At a certain point, many texts stop having problems, or the problems are thin on the ground and obviously not chosen primarily for pedagogical purposes: e.g. they may be either trivial or unsolved. Making up your own problems becomes an important part of learning at an advanced level. Spoiler alert: it makes things much more interesting.

Answer 3

As other posters have noted, the answer varies from person to person. There is, however, some knowledge from education research (much of it conducted specifically in mathematics) that you might find helpful.

One key insight is that the number of problems isn't usually as important as the timing or spacing of your practice. Once you've "worked to criterion", i.e., once you've figured out what you're supposed to figure out, working a bunch more of the same type of problems in the same session probably won't do much for your retention. The ability to remember information depends more on whether your brain has ever had to work to remember it. Your brain doesn't really have to work to remember what to do in problem 25 when 1-24 were the same idea, and so the massed practice doesn't really impact retention as much as you would think.

What will help, however, is going back and working more problems after enough time has passed that you have almost, but not quite, forgotten the ideas. When your brain has to work a bit to remember something, it strengthens the retrieval pathways in response, so your brain will be better able to dig that information out of your brain on command in the future. Thus, working five problems the week you learn something, five more the next week, and five more the week after that will probably help you more than working all fifteen or even more problems all in one block.

So how many do you need? Only you will really know, but if you space them out strategically, it shouldn't be as many as you might otherwise.