How long do I study on my own before applying to math grad school?

Question

I'm one of those who realized near the end of their Bachelor's in Business Mgmt that they really really love math. I'm interested in research in pure, theoretical math. I am confident in my ability to learn advanced math on my own. I admit this is based on a small sample of experiences I've had so far with self-studying. After a year of working, I decide to do it full time.

Since I'm behind my peers, I was thinking to just learn the fundamentals (Logic, Real Analysis, Abstract Algebra and Geometry) on my own, and then apply for a Master's. The reason I want to do these on my own is that I find I understand things better at my own pace taking the time to solidify the fundamentals. I took a year of math courses on an exchange program at London School of Econ and it felt like learning disconnected facts. Something like saying object x is round, black, with 4 holes etc. vs showing you a picture a black shirt button (which I find more efficient mentally). Concepts were introduced which were too new but no time was given to "form a picture" of the objects. As a result, it felt like very "syntactic". If I understand the basic concepts used over and over in math, rather than things built out of those facts like calculus, linear algebra, applied math and other "less abstract" branches, I feel I'd do well in my Master's course. I love taking the time explore mathematical concepts on my own, seeing how things fit. I find quite often I make my own concepts (simple ones though) where the picture feels incomplete, which is why I want to do research.

I was wondering how long I should continue this approach (i.e. when have I learned enough fundamentals)? More objectively, what should aim to master on my own, before applying to a master's program? (unless you disagree this is the right approach)

PS: I'm 22, and I have financial support at the moment while I get this done. I've started with Logic and Set Theory and becoming good at proofs. I find calculus and linear algebra intro books skip over too many of the more abstract underlying concepts, hence the bottom up approach.

Answer 1

I disagree that this is the right approach. Many grad schools in the US would require you to have a Bachelor's degree in some subject and to make up the courses you missed by not having a Bachelor's degree in math. So your best option is to get started actually completing those courses. I think that unless you have some publications in math with someone who can vouch for your skills even though you don't have the coursework, you're not going to get into a good Master's degree program without the necessary coursework. The further your history is from the required classes, the more work you're going to need to do. Some programs might admit you with the requirement that you take a few semesters of undergrad classes to catch up, but if you need more than a few catch-up courses, you're really looking at a second Bachelor's degree or working on the side through a non-degree-seeking program at your local university.

Answer 2

I don't know how common they are in math, but in physics/astrophysics I know several people who have pursued "post-baccalaureate" work. Here is an example from my own department. The people I know who are a part of this program are from non-physics/astro backgrounds and are using this program to transition into a graduate astrophysics track.

I think if you can find a good post-baccalaureate program (or something similar) in math, it will provide what you need and what you are looking for.

Answer 3

Although perhaps you might slightly discount this advice on the grounds that apparently I'm sometimes perceived as "radical" or some other dismissive + mildly perjorative modifier, I might recommend that you do study for a bit, especially if you have funding, to... drumroll... be a better scholar for the thing that you want to ... be a scholar/researcher/expert/maven.

In on-the-ground practical terms, if you can find a friendly "post-bac", this would indeed be helpful. These are the catch-up possibilities for people in the U.S. to replicate what (more narrowly educated, due to the system) students in Europe have been required to do... crazily-ironically, whether they were interested or not.

Back to specific advice: follow your interests; do not believe people/advice that urge absolute conformity to ... style, content.

The tipping point is making-a-living verus scholarship-or-whatever. Now that the Cold War is over, it is not as easy to make a living proposing ways to defeat that particular "Evil Empire", and, truly things are subtler. But not much, though perhaps even less gratifying for any of us who thought there'd be progress... nevermind...

But this does explain the situation young academics often find themselves-in. So it's not "how long do I study before trying to enter academe", but a different question about one's own practical situation.

Back to an idealized issue: if one has a good job, and spare energy to study, I'd recommend doing more of this pre-study... and put off petitioning for re-entry to academe a bit... Getting some corroboration of one's putative competence would be wise...

So, as often, the issue is not quite (as I would think) what the questioner asked... but/and the question raises the right further questions...