I'm interested in what liking and being good at a particular mathematical topic area may suggest about likely future academic and career paths. In particular, high school students who excel and delight in geometry, especially when other mathematical topics (algebra, etc) don't merit the same level of success or devotion.

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Most of the variance in what you like and/or are good at at the high school level can probably be explained by what you've been exposed to / trained in, so I wouldn't expect it to be predictive of anything in particular if you expect to be exposed to more things in the future. –  Qiaochu Yuan Jan 13 '13 at 5:43
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I heard many times the complaint "I used to be good in math when I was in high school, but now I don't get it". It's because K12 math is taught as a way to solve a puzzle. To fit the pieces together, whereas math-undergrad level courses are taught in a way which requires inventive thinking and boxes and whatnot. If you want to be a mathematician don't let your high school experience to intimidate you, math is not what you thought it was. –  Asaf Karagila Jan 13 '13 at 7:00
    
@asafkaragila I'm the concerned Mama rather than the student in question. My daughter, however, may be a future mathematician. She has amazed me by independently discovering a particular property of the triangle, not described in her textbook - and was disappointed to find that it was (of course) well known. She drew 6 iterations of Sierpinski triangles on the back of her latest exam - she learned about them, by name, on her own. I wonder if I should encourage her to consider becoming a mathematician - or trust she'll figure out on her own. We don't personally know any mathematicians. –  marfarma Jan 13 '13 at 18:23
    
@marfarma: Allow me to toot my own horn here, and tell you that my dad tells that as a four years old toddler I understood that one can divide things into halves indefinitely; and when I was 17 I noticed that two parallel lines on paper meet at "infinity" when viewed through a magnifier glass. I had a supportive family but no one pushed me towards mathematics, and it was really quite a lucky accident that I ended up studying mathematics and rediscovering this passion of mine. I think that you should encourage her, but not push her beyond her wishes. [cont.] –  Asaf Karagila Jan 13 '13 at 18:26
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[...] If you have some university nearby with a mathematics department try to find out if they have some program for gifted youngsters and find out if your daughter wishes to enroll. I was offered something similar in high school and I rejected it (and I have very glad I did, because it allowed me to have a pseudo-normal teenager experience which I appreciate). You might want to sit with her and discover sites like Khan academy or other sites which offer mathematical education, and try and help her discover advance mathematics on her own. If anything, seeing someone learning can be lots of fun –  Asaf Karagila Jan 13 '13 at 18:28
    
She qualifies for the Johns Hopkins CTY program. But she's not interest in more formal coursework. She's graduated from Vi Hart's YouTube channel to Numberphile's. I never heard of Graham's number before she excitedly made me watch. I'm not about to push her - just wonder what I might want to help her discover. –  marfarma Jan 13 '13 at 18:47
    
I suppose that you can post a question on this site detailing the story a bit more. What she did so far, whether or not she has any programming skills (which may be useful if one attempts to discover mathematics via sites like Euler Project). Perhaps a good recommendation for a book with problems is useful. If she doesn't want the formal coursework then it's probably a good idea to let her have her fun while she can. –  Asaf Karagila Jan 13 '13 at 18:55
    
That's an excellent suggestion, albeit not the same question I started with. Thank you. –  marfarma Jan 13 '13 at 19:01
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If she was excited by Graham’s number, her interests go beyond geometry; she might well find a lot of fascinating material in The Book of Numbers, by Conway & Guy. I’d have loved it when I was a kid. It doesn’t sound to me as if she needs any pushing: she’s pushing herself just fine in the areas that interest her. I’d just make sure that she realizes that a career in mathematics is possible. –  Brian M. Scott Jan 19 '13 at 20:11
    
Tell her she's always welcome to ask questions here about any topics she finds interesting, so long as they have a definite answer! I only mention the last bit because this particular community isn't really for discussion questions usually, but there are other areas that are, like Dr. Math. –  Robert Mastragostino Jan 22 '13 at 0:17
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You should definitely encourage her to consider becoming a mathematician at least insofar as letting her know that "mathematician" is an actual choice of career. This is more already more than many children ever hear. (My grandfather was good at math, so he was encouraged to consider becoming an accountant...go figure!) –  Trevor Wilson Jan 24 '13 at 3:15
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