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.