Specializing early

Question

Topic: this is a mathematics education question (but applies to other sciences too).

Assumptions: my first assumption is that most mathematical concepts used in research are not intrinsically more complicated to grasp than high-school and undergraduate maths, the main difference is the amount of prerequisites (and hence time and experience) involved. My second assumption is that some undergraduate topics currently taught compulsarily are a bit of a burden for someone focussed on a particular topic.

Now of course cognitive development is a constraint, but upon reaching the age of high-school, I would think that a fairly large proportion of the scientifically-enclined students could really understand things usually taught much later and indeed become active at research level within a few years, provided some shortcuts are introduced.

Early specialization: I'm wondering if a balanced curriculum already exists (or is planned) to provide such early specialization. What I'm looking for is this: a one-week panorama of maths (or physics, or biology) would be organized at the beginning, and then the students would decide which subtopic to study. For example someone interested by group theory (or quantum optics, or genetics) would thus start with basics at the age 15 or 16, and gradually learn more stuff and skills, but for a few years with a strong emphasis on things directly relevant for the chosen subtopic.

So for example the student specializing in group theory would only learn differential calculus and manifolds in passing in the context of Lie groups, and would skip most undergraduate real and functional analysis until it becomes relevant for his/her research topic, if at all. Of course other general courses would still be taught (history, sciences, programming, foreign languages...), but at least 50% of the student's week would be devoted to the research topic, ensuring satisfying progress.

Question: do you know of any active or planned educative curriculum (at a high-school or university, or maybe a specific home-schooling program) as outlined above? As an example of successful early specialization see e.g. the winners of the Siemens Foundation Prizes, but I haven't been able to learn much about their specific curriculum if any.

Note: Skipping grades in school to enter university earlier is not the point, I'm really interested in a subtopic-oriented curriculum.

I do not know about such programs, but really hope they do not exist. Mathematics is big and beautiful, with lots of connections between its remotest branches. Why be cruel to the children and deprive them of that? — Boris Bukh, Feb 27 '10 at 12:24
Of course the aim is not to deprive them of these connections forever, but just during the first few years to help them actually do something. Once they are mature in some area, of course they would be curious about other things. To me it is a bit of a pity that these years 16 to 20, when intellectual power is already near its height, are spent by most people learning lots of basic well-known material. — Thomas Sauvaget, Feb 27 '10 at 14:22
Thanks for spelling out your assumptions. I vehemently disagree with them, and with your conclusions. Far from ensuring progress, I think this would be disastrous for future researchers, and even worse for the vast majority of students who do not become researchers. — Douglas Zare, Feb 27 '10 at 17:52
I like your idea. Not the whole of it, but at least the spirit. I feel that our education system focuses more on breadth than depth. Implementing a program like yours could be a reasonable counterbalance. But maybe not in high-school, and maybe not in the States, where inequities between people are already huge (I feel your program would only increase those). — maks, Feb 28 '10 at 04:14
@TS: You say that intellectual power is near its height at ages 16-20. Could you a) explain what you mean by "intellectual power" and b) provide some evidence for this? A little googling shows: (i) Youngest world chess champion: Kasparov (22); (ii) youngest Fields Medalist: Serre (27), youngest Nobel Prize winner: Bragg (25) [shared with his father!]. Kasparov retained the championship for many years and, based on his ratings, was significantly stronger in his late 20s and early 30s than in his early 20s. — Pete L. Clark, Feb 28 '10 at 07:48
@Pete: since this is perhaps turning into a too lenghty discussion for MO I've put up a blog post where this will be easier http://thomas1111.wordpress.com/2010/02/28/discussion-section-for-mo-question-16587/ Basically I meant that at age 18 all the folks you mention were not less intelligent than at age 25, just less knowledgeable and experienced, and that with earlier and more focussed training they would have performed just as well. — Thomas Sauvaget, Feb 28 '10 at 09:12
Thomas, nothing stops a bright high school kid from sitting in courses at a nearby college. You just have to convince the instructor that you can handle the load, and usually you will be let in. And if you are really good, someone will gladly supervise your research. This is what happened to Siemens winners. Having said that, if you are serious about becoming a research mathematician, sooner or later you will need to study basic math subjects. — Igor Belegradek, Mar 02 '10 at 05:28
Igor: precisely, it's on top of the high-school classes, which can then be quite heavy and cumbersome (commuting to college...). This is the reason why I thought a different kind of curriculum is needed, but apparently most people here disagree ;-) — Thomas Sauvaget, Mar 02 '10 at 07:30
Thomas, a common solution is to home school; then you have all the freedom to take the classes you want. Quite a few bright kids take this route. — Igor Belegradek, Mar 02 '10 at 13:49
Incidentally, Thomas, you idea of 1-week panorama of math is very naive, I think. In a typical upper level undergrad course (say on group theory) students spend 3-5 hours per 1 hour in the class solving challenging exercises. Say you are extremely bright and only need 2 hours. A 12 week course is 36 lectures, so your time commitment is over 100 hours, i.e 10 days minimum, just to get a basic idea what group theory is about. It probably takes several times as much to start doing research in group theory coming from high school. — Igor Belegradek, Mar 02 '10 at 14:23
Igor, there's a misunderstanding about the word panorama then: I mean something informal, for example group theory could be presented in a 4-hour session mixing intuitive ideas, key examples, re-phrased theorems, computer programming and open problems. (By the way, I'm asking about specific curriculae as a teacher, I myself finished grad-school long ago.) — Thomas Sauvaget, Mar 02 '10 at 15:54
Thomas, when I said "you" I of course did not mean you personally; sorry if this was unclear. A four hour session can be done in math camp or math circle format, but I doubt that anyone is able to learn what group theory is about without solving hundreds of exercises. — Igor Belegradek, Mar 02 '10 at 16:34

score 25 · Answer 1 · answered Feb 27 '10 at 12:51

I'm not sure how good an idea this would be. I happen to be in a position where I read many applications of students wishing to do a PhD in my group. Some applicants have a very definite idea of what it is that they want to do, but this is usually for lack of exposure to other topics. It is not unheard of that they end up doing their PhD is a completely different area.

I can speak from my own personal experience. At every stage in my life I was sure I knew what I wanted to study, but as I learnt and become exposed to new topics this changed; though not my certainty about my choice. Until about age 14 I wanted to study Molecular Biology. Had I specialised then I would not have seen any of the Physics and Mathematics which have become such an important part of my life.

The point I'm trying to make that is that early specialisation might be depriving the student from finding what it is they truly like. Freedom of choice is only ever meaningful if one can understand (or at least be aware of) the alternatives.

Of course, you could argue that there is a lot more information readily available to school children than when I went to school, so perhaps a more conscious choice can be made at an early age. There is however still a danger even within one discipline, say, Mathematics.

The research council which funds mathematical research in the UK commissioned an international review some years ago. A panel of respected non-UK mathematicians analysed the state of UK mathematical research. One of their conclusions was that due to the short length of the UK PhD (36 months at the time) students were forced to specialise earlier than in other countries (though not as early as the OP suggests) in order to complete their PhD in time. This then made it harder to switch fields later in their career and made them less competitive in the long run.

I will refrain from commenting here on the half-measures that were introduced to try to solve this problem, but I simply want to point out that even such a late "early specialisation" as this one is not desirable.

I'm in complete agreement with José here, and want to point out that there are problems even with early subspecialization within your subject. I know that I'm glad I kept learning about other things over the first few years of grad school, because though I'm still a complex algebraic geometry person, I've almost traversed the entire length of the subject (even coming back from a very different angle!) compared to what I thought I was interested in at first. — Charles Siegel, Feb 27 '10 at 13:29

score 11 · Answer 2 · answered Mar 02 '10 at 03:55

I agree with José's comment above: I do not think early specialisation is a good idea. Did I understand correctly that you want to give a one week to a 15-year old to decide on which area of mathematics to specialize?

I want to add something different, however. I fail to see how "some undergraduate topics currently taught compulsorily are a bit of a burden". Mathematics is not a set of disconnected areas. They are all highly related. Most research problems, while staying in one area, may be related to another, motivated by another, applicable in another, or steal ideas or techniques from another. One general course in, say, real analysis, complex analysis, abstract algebra, differential geometry, discrete mathematics, or topology is not a burden, but I dare say an actual necessity for anybody wanting to do research on any topic in pure math. To use your own example, someone doing research in Lie theory will benefit from, rather than be burdened by, a solid understanding of basic differential geometry. Or to use my own case, I am a Poisson geometer, but I have used ideas or results from all the above topics in my research.

To answer your comments: yes, there would be a panorama of maths with lots of examples, open problems, and interactions with various researchers. I agree that math areas are connected broadly speaking, but I'm convinced that for a young person who loves particularly one subject there's a way to learn things as he/she goes along when it's needed, rather than start with a lot of imposed breadth. For example some decades ago plane curves like conics were taught in high-school, not anymore today. It is not so easy to say what is fundamental and what isn't, depends very much on the goals. — Thomas Sauvaget, Mar 02 '10 at 08:25
Well, it is not easy to say what is fundamental and what isn't because you do not know what you are going to need when you are doing research in area X. If it simply were the case that, when you are doing research in area X, it becomes obvious that you need to learn about area Y, then you could go ahead and do it. In practice, however, it may be the case that you will only notice that you need to use area Y if you are already familiar with it behorehand. Hence the need for a "broad" education. And honestly, the standard curriculum in the US or the UK is not that broad in the first place. — Alfonso Gracia-Saz, Mar 02 '10 at 16:28

score 7 · Accepted Answer · answered Feb 27 '10 at 13:09

7

As far as getting high school students involved in research by learning rapidly a narrow range of mathematics but in some depth, this is actually done in the mathematics section of the Research Science Institute program at MIT for students who have completed their junior year. Last year there were four projects in representation theory; I recall that one of them did not now linear algebra until some two weeks before the program (but learned quickly and completed a very successful project).

Sadly, I do not know of many other opportunities- RSI is a small program, and only a portion of it is for mathematics. I believe the PROMYS program supervises some research projects, but it is primarily for learning mathematics. Incidentally, many of the winners of competitions such as Siemens begin their projects at RSI.

Also, alumni of the RSI program do not necessarily end up specializing in the same fields that they did their projects (if they do eventually choose to pursue a career in mathematics, which does not always happen). It does give an exposure to a certain field, though.

answered Feb 27 '10 at 13:09

Akhil Mathew

25,291

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Not quite as research oriented, but still something for high schoolers: Math Camp http://www.mathcamp.org/ – Charles Siegel Feb 27 '10 at 13:26
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Certainly the RSI program is very successful, and it seems like a good idea for there to be more programs like it. However, this is a key point -- RSI is a summer program which excellent, enthusiastic students take in addition to their normal coursework. These students aren't missing out on any of the standard, non-specialized coursework by participating in the program. So there is no tradeoff here, as the OP is suggesting. – Pete L. Clark Feb 27 '10 at 17:14
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I participated in the PROMYS program as a student in its first two years, and think it was great. There was no research component, and presenting a motivated, rigorous college course in number theory to high school students is reasonable whether or not the students want to study number theory or even mathematics later. – Douglas Zare Feb 27 '10 at 17:57
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Mathcamp, RSI, and PROMYS (of which I have experience with the latter two) all share the property that Pete mentions - they are an enrichment experience to be taken in addition to the standard curriculum. While I'm on the subject, I have to bring up an issue with this paradigm: because much of the "subtopic-oriented" material at Mathcamp and PROMYS is taught very quickly by counselors, students sometimes confuse exposure to material with mastery, and may enter college feeling that they know more about e.g. elementary group theory than they actually do. – Qiaochu Yuan Mar 02 '10 at 04:00
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@Qiaochu: I never went to a summer math program before RSI (probably a mistake on my part), but I at least can say that the RSI program made me acutely realize how distantly the peaks of Mount Bourbaki lay from my accumulated knowledge, and how much more climbing awaited me in college. – Akhil Mathew Mar 02 '10 at 23:11
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@Qiaochu @Akhil I too participated in RSI (I think in the year between you two) and "completed" a math project - I say "completed" here because I worked purely in graph theory at RSI, but later found it much easier to extend my results if I started developing them from a topological and group-theoretic standpoint. That being said, I reversed a lot of other things, such as the RSI-Siemens road - I did well at Siemens before doing RSI.
Another good thing for my understanding of how much math was out there beyond what I knew happened at RSI - I met Eric Larson.
– dvitek Aug 13 '10 at 22:08
@Akhil: Dear Akhil there is a typo in your statement "; I recall that one of them did not $\text{now}$ linear algebra" – C.S. Jul 06 '12 at 02:13

Specializing early

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