Math Research for High Schoolers

[ShoMoJo]

Does anyone here have any experience in mathematical research since high school? Or even if you only started as an undergrad, do you have any advice?
1. How did you secure the research position? Who did you talk to?
2. What area of math did you do research in? Was it necessary to know a lot of high-level math just to be able to cope?
3. If you didn't know the mathematics required, was the research assistantship/internship still manageable? Did your mentor support you? How did you make up for the lack of knowledge of higher math?

Please share your experiences. Thanks alot!

 

[micromass]

I'm sorry but I really can't conceive how a high schooler can do relevant mathematical research. I would be very glad if people responded to me to prove me wrong, but I just can't see it at all.

 

[ShoMoJo]

@micromass Have you heard of Noah Golowich? Or basically anyone else who ever did their project in math in the Intel Science Fair?

 

[micromass]

No, I haven't.

 

[micromass]

Oh well, it doesn't surprise me at all that the paper is in combinatorics. But sure, this guy is obviously incredibly talented. Don't think everybody is able to pull something like this off.

Anyway, I think you should be looking to learning as much mathematics as you can, instead of doing mathematical research. I don't think high schoolers and undergrads are supposed to do research. They're supposed to learn a lot of new things and techniques instead.

 

[Dr. Courtney]

I'm sorry but I really can't conceive how a high schooler can do relevant mathematical research. I would be very glad if people responded to me to prove me wrong, but I just can't see it at all.

Look at the abstracts from the Math category at ISEF the past few years. There is a lot of chaff, but there is some good work also, considering they are high school students. My daughter won a top award in Math at ISEF from Mu Alpha Theta in 2015. The paper on her project is here:

http://arxiv.org/pdf/1507.01832.pdf

Of course, it is applied math with an experimental approach. Nothing fundamental, but her code has been dowloaded thousands of times since she posted it at SourceForge last year. She corresponded in some detail with math faculty at Pitt refining the paper and getting it ready for publication. They were pretty impressed, and even wrote her a letter of recommendation to the effect that they wished their grad students were as careful and diligent and as responsive to direction as she was.

 

[Dr. Courtney]

Come to think of it, I also mentored an ISEF project in 2014 that was really a math project under the guise of a project about fish. The title was, "Improving Bioindicators: A New Weight-Length Model for Fish to Provide More Accurate Ecosystem Condition Assessment." The idea was that least squares fitting weight-length models in fish to the functional form W(L) = (L/L1)^b produces smaller uncertainties in the best fit parameters than the usual functional form W(L) = aL^b. The project won a grand award (4th) and also a special award from an ocean group and was published in a fisheries journal.

In the brainstorming phase, we discussed whether the project should focus on the more general applied math question: does f(x) = (x/x1)^b tend to reliably provide smaller uncertainties in the best fit parameters than the traditional power law, f(x) = ax^b. But the student wanted to frame the question around an ecologically important problem, take advantage of all the fish data sets we have on hand, and compete against lightweights in ecology or environmental science rather than the heavyweights in math.

In brainstorming with other students, we've discussed returning to the more general math problem a few times. A student or two has done some pilot work on a data sets from the physical sciences, and a student even published a paper using it for analysis in Review of Scientific Instruments (https://arxiv.org/ftp/arxiv/papers/1506/1506.02986.pdf ), but it has not been picked for a student project in math yet testing how general the idea is.

We must brainstorm about several math projects each year with students when discussing projects for ISEF-affiliated fairs and JSHS (Junior Science and Humanities Symposium). Due to our experimental mentality, most of the ideas focus in applied math that tests an idea or improves some method of data analysis. Another least-squares idea that generalizes the "better power law" idea is a similar approach for best fit polynomials: f(x) = a + (x/x1) + (x/x2)^2 + (x/x3)^3 + (x/x4)^4 + ... instead of f(x) = a0 + a1x + a2x^2 + ... I'm fairly confident that this will improve on machine rounding errors in some applications, but some numerical experiments would be needed to see if non-linear fits to the "improved" polynomial form would produce smaller uncertainties in the parameters than fits to the traditional polynomial form.