How to study mathematics with rigor, depth and at a good pace

Question

During my university studies I have always had the feeling that mathematics is poorly taught. And I don't want to be misunderstood, I love mathematics and I'm not a very bad student, so I don't complain about mathematics, only that it is taught badly. In particular, I think that in most of the textbooks that I have used and in most of the subjects we are shown very little of the real context in which mathematics arises. And now I will give two examples, although I have many:

1. In a basic group theory course, the teacher introduced us to the concept of "group action". After the definition, we were shown some results and some examples. But that concept always seemed very arbitrary to me, until I finished the course and had time to do some research on my own and came across examples of group actions, like the dihedral group. And that's when I understood the importance of this concept and its motivation.
2. In the usual definition of topological space there are three conditions that the topology must satisfy. If one think about it a bit, those conditions are very reasonable, but I once found a reference that introduced a topological space with a much more intuitive alternative definition based on neighborhoods of each element in the set on which the topology was defined. I found this definition very useful and it was very strange for me not to have seen that definition in the rest of the references on that field.

I believe that it is possible to learn mathematics in a correct way using all the resources that exist, but it seems to me that the path could be much simpler if mathematics were explained with much greater emphasis on the context and motivation of each step.

In this situation, I have two questions to ask. First, am I a very slow learner or, in general, is it that people with great talent for mathematics also encounter, nowadays, the difficulties that I have mentioned? Second, can anyone give any study tips to someone who is trying to learn math with rigor and insight, but is a bit frustrated that the process is so slow?

Thanks for your help.

Answer 1

I sort of agree. I often felt the math courses I took were taught backward. And with no motivation. I don't mean "motivation" like a coach getting you psyched up, but "motivation" as in "motion." Which way are we going and why?

Often, a topic would start with "We might want to classify all finite Abelian groups...."

Why throw students out into the middle of the abstract ocean and hope they swim back to shore? Much better to introduce matrices, linear algebra, modular arithmetic and then, in a systematic fashion, work towards more abstraction. Just throwing "an example of a group" at a student doesn't accomplish much. But if the student has been, maybe unknowingly, working with groups, rings, subspaces, and has developed some facility, then much is accomplished.

I was once of the opinion that instead of courses like "Algebra, Analysis, Topology,...", we'd be better off teaching a course called "Fermat's Last Theorem." Introduce modular arithmetic, then easy ring extensions, ideals, etc. Do the math in the order it was discovered and have a clear goal. I'm not sure I still have this opinion, but I think it's worth thinking about.

So to try to answer your question, I used to do this with my higher math courses: There would be a text appropriate to the level of the course. I would find a much easier, lower level book. The kind my colleagues would roll their eyes at. My instructions were: Find the corresponding material in the baby book because it gives you the big picture without the clutter of all the (necessary but) pedantic details.

So if I were teaching the Algebra sequence out of Hungerford or Lang, I might suggest Lindsey Childs' gentle introduction. (I think the title was "A Concrete Introduction to Abstract Algebra.") Students who get stuck on a point in Hungerford can get a simple, more concrete discussion in Childs. This was pretty effective for the middle level students. The superstars don't need much help, but not everyone is a superstar.

So that's my bit of advice. Find a much easier textbook and read it in parallel.

Answer 2

(1) There's no such thing nowadays as a "slow" learner. Everyone has a different learning pace. Some people learn quicker than others and vice verse due to multiple reasons, so don't feel bad about not being able to absord the material quickly enough. For me, I'm also on the slower side of learning. For me to at least pass a basic calculus with at least a C, I would need to, over and over again, read the textbook, do the homework, raise my hand during class, email my professor on why my proofs make no sense, and keep attending office hours. I've known plenty of people like that yet they still wouldn't pass the class, but that's fine as long as they don't give up on themselves. I would always get C's and D's in high school algebra, but I never gave up. And now I am a recent math graduate from a university despite me failing some geometry and proof classes.

(2) There's no doubt that math is going to be difficult no matter what. Even people with math PhDs find math difficult, especially the topic they're researching. This might be cliché, but my advice is to kick your foot in the door and embrace all the proofs and calculations math throws at you and keep practicing them because that's the only way you're going to get better. No professional football player got better by simply watching others play football. They had to exercise, work out, sweat out in the sun, and practice all day. And the same thing goes for math and anything that requires skill. (Also, asking your professors for help all the time helps too.)

The Math Sorcerer on YouTube has some tips too.

Answer 3

If you feel as if a lot of the mathematics you're learning isn't well-motivated, then you're definitely not looking in the right places. There are plenty of books out there which are written to motivate a large number of notions that you would encounter relatively early on.

It seems like you've tethered a huge part of your learning to the teaching that is provided by your institution. Let me just say this; there's no hope of you always getting the motivation that you want or the insights that you want if you're just going to rely on that. Every course that you take on a given subject is just going to be a viewpoint of that subject being presented by that lecturer.

Have you considered the possibility that you may just be a "slow" learner because the way you've "grown up" mathematically just doesn't vibe well with the viewpoint that your lecturer is presenting? Don't get me wrong; there are definitely large chunks of mathematics that are just straight up extraordinarily hard to grasp and it takes a LONG time to understand them. But it is definitely worth considering the idea that you may just not be learning a given subject in an optimal way.

There are some viewpoints that will be easier for you to reconcile with. There are other viewpoints that will be harder for you to reconcile with. A big part of learning math is to reflect hard on the kind of person you are and to be well aware of what helps you learn a given topic well.

Are you someone who prefers dealing with concrete objects first? When seeing a topic for the first time, do you prefer working through a whole number of examples or do you want to spend time developing the abstractions of the theory first? Do you prefer just seeing the material developed in one way first or do you want to be given three or four different definitions of the same thing & start comparing them first?

You need to start reflecting on the answers to these questions & you need to do it constantly as you pick up more & more math. You need to understand that mathematics is a human activity and, as such, you need to recognize that a big part of it involves reflecting (constantly) on:

1. why you're doing what you're doing, and
2. how you're doing what you're doing.