“Young man, you never understand mathematics. You just get used to it.”

-Von Neumann

I remember as an undergraduate, and even still to this day, trying to optimize any time I put toward learning new material. It is clear from the psychology of education that some methods of learning are better than others, and this is quite dependent on the individual. But questions always arise: when do I say I understand something? Is the kind of understanding I have of this object good enough for this purpose?

This is a difficult question, one that is likely in practice unanswerable. Nevertheless, I think it is worth putting some thought into, just to get an understanding of what the process of learning is for oneself. My interest in this topic was sparked by a general interest in pedagogy, but also by a deep desire to optimize how I spend my time learning new mathematics. As a disclaimer, I am thinking about my own current career path of academic (applied) mathematics. I am sure similar thought processes might work for other disciplines and careers, but possible conclusions may be different.

There is some literature on various ways that students begin to understand maths. Here, I am more interested in advanced undergraduate and graduate learning rather than earlier concepts that a lot of the education literature is focused on. An example would be Terence Tao’s discussion of three stages of mathematical development.

Perhaps with some analogy, I might break learning I’ve experienced and seen in others into four stages. Again, these are rough lines, and I’ll only extend them through the levels of beginning research. That means that rather than discussing entire fields of mathematics, I will focus on particular topics one might learn (e.g. Theorems, or the material covered in traditional courses). I would break these four levels into the following:

**Aware:**This is where you are familiar with the idea of a topic. You may have seen it briefly in a lecture, or read small parts of a textbook, or the Wikipedia page. It is useful to know the words or themes, so if you see them later you can come back and pick up what you would need from the topic.**Learned:**This is what I tend to call “vague understanding.” You’ve had a course in the area or read a book, but it is something you didn’t master. You can follow the proofs well enough, or even reproduce the overall idea, but it isn’t something that you’ve used enough to live and breathe. The majority of coursework done in specialized areas that isn’t related to your research usually falls into this category.**Mastery:**This is the stage where you have really understood the concepts as they stand by themselves. You can recall the proofs with minimal references, you know how to use the Theorems and the concepts in several applications, and you are at least aware of the motivation for the development of the ideas, and their connection to other areas. This is the usual level of understanding required to do well on graduate preliminary examinations in mathematics.**Expertise:**Finally, this level of understanding is beyond just a mastery of a small field. This is where you understand how the field is progressed, and you are capable of contributing meaningfully to the literature in the area. The PhD is usually intended to provide you this expertise in a fairly narrow subfield, and it is usually hoped that you could continue to work in the area without being as reliant on anyone else to contribute novel mathematics.

It is hopefully clear that these levels described are approximations of things, and aren’t in reality so neatly divided. There really is a spectrum from understanding nothing of a topic to becoming an expert in it. Nevertheless, sometimes it helps to think about one’s own abilities in an area, and try and classify them.

It should also be noted that these levels are just the beginning of understanding, and only apply to particular skills or fields. General mathematical maturity, research independence, and the connections between areas of mathematics and the sciences should not be overlooked when deciding what to spend your time learning. Most prelims require such deep mastery of the concepts because they open doors for collaborative and interdisciplinary thinking. This kind of process is crucial nowadays, even in pure mathematics. Expertise in a small subfield seems to be a good way to begin a research career, but it should sit on a broad base of mastery. Thinking of the above four levels in terms of an upside-down pyramid may be a good structure to shoot for when beginning a research career. It is important moving beyond the PhD to see the connections between many different areas. As Stefan Banach once said,

“A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

Of course, I’m just scratching the surface of having a research career myself, so these are all tentative thoughts. Any comments, criticisms, or suggestions are always welcome in the comments.

Some further reading:

The Illustrated Guide to a PhD

A Mathematician’s Survival Guide

I like “Learned” because some vernaculars let you say, “My teacher learned me some real analysis.” I think the main transition is being more active, but it’s annoying (in terms of delimiting levels of maturity) that you can actively learn proofs (i.e. do what you can of the proof; cheat and read some lines of it; do what you can of the proof etc.) at any stage of your career.

The sequence I normally go through is:

1. Have read on Wikipedia or been given a two sentence synopsis.

1b. A first motivating problem (e.g. a construction showing, if you assume translational invariance, countable dj additivity, and some set with finite measure, then there are unmeasurable sets.)

2. Learn the definitions and do some trivial applications.

3. Steadily become more comfortable with the techniques, spending roughly as much time doing unguided problems as reading.

4. Get to the point where the new written proofs are supplemental.

5. Go on to another topic/course lol

Also Tao’s description of getting bogged down in transferring intuition to formalism resonates with my intro analysis course (although my main difficultly was never knowing when my prof would be satisfied with how elementary my reasons were…)

But yeah, I really can’t talk about anything PhD.