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I’ve been doing this, working with high school students on mathematics research projects, for close to twenty years. The settings have varied from "Independent Study" classes at a local high school - in which up to 20 young people (many of them "average" students) worked in small teams on projects we designed together - to summer programs for very advanced students to informal internet discussions with children of my friends. In spite of wildly different levels of student sophistication across these settings, I continue to be awestruck at the kind of profound mathematical thinking that seems to reside naturally in the intellectual capabilities of almost every student with whom I have worked.

There are dozens of different stories to tell, but almost all of them lead to the same punchline: if you allow students to work with you on a mathematical problem of substance, they will come up with insights and germs of ideas that will make your head spin, and it will become perfectly clear to you that the ability to do real mathematics resides in just about everyone.

Just one example: In the mid-eighties, I worked with three students, Rafael, Ahmed, and Fran, on the problem of describing and explaining the patterns one gets in Pascal’s triangle modulo n. Ahmed was already a mathematical thinker - he was taking all the advanced courses. Rafael was headed for engineering school and was doing "college prep" work. So was Fran (perhaps the deepest thinker of the three), but Fran spent most of his elementary grades in special-ed. His home life was just about non-existent.

The team first wanted to see what the patterns were. So they spent a month learning to program in Logo (this was the eighties, after all), learning enough list processing and efficient programming to produce 150 rows of Pascal’s triangle. This was quite a feat - we were using computers that had 32K of RAM. Because our monitors had only four colors, they looked at the triangle modulo 4. When they had a program that worked, we had to let it run all night in order to get the picture we wanted. The next morning, they were delighted to see what was (to me anyway) an amazing picture, full of self-similarity and replicated patterns. This picture stayed in my mind for years, and it eventually led to some research of my own.

The next month was spent trying to find language to simply describe the patterns they saw. They eventually settled on a system of triangular icons and some binary operations on these that could be used to build the ever-increasing intricacy of what was on the screen. Then, as they said in their write-up, they put away the computer and started the real mathematics: proving what they conjectured. This took them into an excursion into elementary number theory and eventually led to the re-creation of some results of Kummer about 2-adic absolute values of binomial coefficients (I don’t think we ever used this exact language).

With the "research" over, the Ahmed, Fran, and Rafael wrote up their results. It’s a lovely paper that I still have. They published it in a local newsletter devoted to Logo in mathematics education. Then, in the summer of 1987, they came with me to a conference at Concordia University in Montreal to present their work. It was the first time Fran had been out of Massachusetts (and he made the most of the opportunity). They gave an hour talk, complete with computer graphics (all we could do was show the pictures and the code) and mathematical proofs. I like to believe that this was an experience they’ll always remember.

I’ve lost touch with them. Ahmed became an actuary. Last time I talked to him, he said he hated it and was thinking of going back to graduate school in mathematics education. Rafael became an engineer; he and Fran both commuted to college together for the first year. Fran dropped out of college and, last I heard, was managing a McDonalds. But for one year at least, they were all three mathematicians, and they (and I) had a great time.

—Al Cuoco, Director, Making Mathematics

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