** A couple of recent students.** Let’s start with a couple of examples. These are the two most recent young students we worked with.

Kiki Pichini is currently 16 years old. She lives in a small town in the middle of Washington State, far from any big city. A few years ago, she wanted to learn more advanced mathematics but did not wish to leave home at her age. So she enrolled to an online undergraduate program of Indiana University. She will complete her undergraduate degree this upcoming spring.

While working on her degree at 15, Kiki wanted to continue to even more advanced projects, so I started working with her on a research project. This was done purely by email and video chats. Kiki just submitted her paper to a combinatorics journal. It can also be found on arXiv here. Kiki improved the current best upper bound on the number of rectangulations a planar point set can have – a bound that remained unchanged since 2006. She was recently accepted to a graduate program at Oxford.

Our second most recent student is Michael Manta. Michael is a high-school student in Xavier High School in New York City. He was studying more advanced mathematics through an NYC-based program called Math-M-Addicts. We met Michael in the summer after his sophomore high-school year. He worked under the mentorship of Frank de Zeeuw and proved a new result about triangle colorings of the plane. He submitted his paper to a combinatorics journal a few months ago and it is currently being refereed. The paper can also be found on arXiv.

Michael was accepted to many top colleges. Eventually he chose to attend Caltech. After finishing his research project, he wanted to do more math. So he is currently taking a couple of the more advanced courses our department has to offer.

We had other successful projects with young students and also a couple of unsuccessful projects. I still feel like a beginner at this, and am constantly learning.

** My dilemmas.** The main question for me is what young students should pursue such a project. There are two obvious criteria. Over time, I noticed a third criterion that I believe is at least as important.

- One obvious criterion is that such projects fit students who have a strong potential for doing mathematics. This is a somewhat vague notion, and students who don’t satisfy it are not very likely to consider such a project to begin with.
- The other somewhat obvious criterion is that such projects are only for students who already know that they are strongly driven towards advanced mathematics. Why would anyone spend a significant amount of time doing such hard work at such a young age?
- One main difficulty we had is students who fit the first two criteria, but consider the program as just another extracurricular activity. That is, as one of many activities that they spend an hour a week on. This is a very understandable approach for students who never saw math research before. But it will be obvious to anyone who had done research in pure math that this approach cannot work in research projects.

I am still having difficulties explaining this third point to students who approach us. This is clearly more important than, say, how much math a student already knows. It is completely understandable that most students don’t wish to pursue such an intense experience, and should indeed focus on a variety of less demanding activities. But how to properly explain this? And how can one distinguish the students for which such a project is a good choice?

Another question is how exactly to run such research projects. These are very different than REU projects. On one hand, such students start with a more limited mathematical background than REU students. On the other hand, the period of the project is not limited to one summer. We thus aim for a minimum of one year for such a project. That way, we first spend several months on learning, before getting to work on a problem. We choose problems that do not require a lot of background, but the students still have a lot to learn about how mathematical proofs work.

I am experimenting with the best resources for learning proofs. One book I’ve been working with is Proofs from the Book by Aigner and Ziegler. Another is Rotman’s Journey into Mathematics. I’m happy to hear about other good options.

If you have any observations about mentoring this type of research projects, I’d like to hear about those. I know of many high-school-age programs that are more competitions-oriented or work on reproving existing results. I don’t know many programs that are focused on producing and publishing new research. I would be very happy to learn more.

In the next post we’ll discuss toddlers doing math research đŸ™‚

I love this.

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