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Let's say that you're a 27 year old autodidact who has gained enough knowledge in a particular field of the formal sciences - like mathematics, for instance - to be able to take the highest course number from the selection of graduate courses for math.

Now this autodidact wants to enter university for the first time. However, from his perspective, he thinks that it would be a waste of time to restudy what he already knows.

I'm personally wondering if there is a way to demonstrate that he is capable of taking graduate courses so that he could apply for a graduate degree without having to study all the courses at the undergraduate level again.

Also, I'd like to add that this question is really hypothetical. It's just something I pondered on my way to school today.

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All it would take is a 30 minute conversation with a math professor in the department. The mathematical maturity of this autodidact should be clear just from that. –  Chan-Ho Suh Jan 8 at 17:18
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4 Answers

up vote 13 down vote accepted

If you're asking about graduate school in the US, probably the single most significant thing you can do is to take the GRE subject test for math (not the math section of the general test!), and score well on it. This test covers a broad spectrum of material that is taught in a typical undergraduate math curriculum, and so if you get a good score, it strongly supports your claim that you have the level of mathematics knowledge required to enter grad school. Without that key piece of evidence (i.e. a good GRE subject score), graduate admissions committees are likely to look at your statement that you have the knowledge to take graduate courses, contrast it with your lack of an undergraduate degree, and conclude that you're full of hot air, so to speak.

Now, of course there is more that has to be done to actually get yourself admitted. You can't get into grad school based on even a 990 (the top score) on the GRE alone; you'll probably have to present some other sort of evidence of accomplishment that could be viewed as equivalent to an undergraduate transcript. You'll also need recommendation letters and various sorts of essays and forms. But a lot of that can vary from school to school, and is more likely to be negotiable if you talk to someone in the department. The GRE subject score is the one thing you really need to get your foot in the door, so to speak.

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If you get a 990 on the GRE subject test in math, send me your scores and I will get you admitted, no questions asked. –  David Ketcheson Jan 6 at 12:18
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Most universities' mathematics departments have placement exams for lower-division math courses, so, for example, you could "test out" of two years' calculus, most likely. Testing-out of upper-division courses is rarer, so there're not going to be formal procedures in place, but if you simply talk to departmental advisors, they can get the ball rolling to have your self-education appraised. It's just that there probably won't be formal procedures in place, due to the rarity of the event. Math faculty are more-than-happy to see someone who's had the interest and motivation to read on their own.

It's not clear that doing the GRE math subject test (nevermind the other parts) is a good avenue for everyone. I'm not such a fan of that exam as a predictor or appraisal, in any case. One aspect is that it is a very superficial exam, in the sense that it tests test-taking ability almost as much as substantitive knowledge, since it is multiple-choice. Second, it presents a very stylized picture of "undergrad/pre-grad" mathematics... unsurprising insofar as it has to be shoe-horned into a timed, multiple-choice exam.

I think it's also not clear whether you should "try to do research" in order to "make an impression". _Being_interested_ and being curious is one thing, but the further element of presumption involved in too-easily believing that with modest preparation one has unraveled mysteries untouchable by experts... blah-blah-blah... will not make a good impression. That is, honest curiosity and drive are unqualifiedly good things, not presumptious, not silly, all too uncommon, ... and are the features the math faculty would look for.

One last small point: it is very helpful to have in mind the authors of the books or notes you've read, whether they're physical books or on-line notes, if only because the titles of most such things are toooo generic, while the best authors are well-known, and the specific virtues of their books/notes similarly so.

In summary: just talk to the math department undergrad-intake-advisors first, and they will steer you to the right people to talk to in order to have your situation appraised. Some of the people may be a little skeptical, but mostly they will be happy to encounter someone who has the interest who's taken the initiative to study and think about things on their own.

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Many graduate programs are geared toward publishing research, much of which doesn't really require the highest-level graduate education to conduct, so if one could achieve a publication record, a history of research experience, or could secure letters of recommendation, that would probably help for certain programs. Getting to know one's potential advisors in advance might also help circumvent some of the usual hurdles (i.e., they might be able to help).

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Many graduate programs are geared toward publishing research — "Many"?? Not "all"? –  JeffE Jan 7 at 2:01
    
Probably not all, and even "most" would be an empirically falsifiable claim; I have not done the research, so I don't care to go there, and I don't see why it would make any difference if I did. –  Nick Stauner Jan 7 at 2:03
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You can't start changing rules without at least playing by them first. Fortune rewards the patient and as such, all brilliant people who thought they can start in life without going through the system, have landed short. Especially in Mathematics, deductive reasoning and clarification is necessary for others to relate. Imagine if a GRE990 person who hadn't such training, how would others relate to this hypothetical person? You need to be a team player, no solo on this one I am afraid.

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I have no idea what this is trying to say. –  JeffE Jan 7 at 2:02
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