I make an effort to practice mathematics on a regular basis. I think the experience of challenging myself and confronting my own mathematical misconceptions and areas of weakness is important to my teaching — and I enjoy it.
Two problems I’m working on right now:
Problem One: I was digging through old PCMI problem sets, and found some problems from the 2009 set that interested me:
I learned a bit about the sigma function in college, and I’m really enjoying these questions:
For what values of n does A(n) produce an integer?
What is the largest possible value for B(n)?
I’ve also been digging through one of my favorite books: Euler: The Master of Us All, by William Dunham, a survey of Euler’s greatest mathematical work. If you’re looking for a good math read, check it out! The first chapter covers his work on perfect numbers, and has helped me with some of my insights into the sigma function.
Problem Two: I learned that Euler was the first person to find four numbers such that any two of the numbers add up to a perfect square. I spent some time playing with such numbers, and found that it’s pretty simple to find three numbers where any three add up to a perfect square. A few small ones are 2, 23, and 98, or 48, 73, and 96. I’m trying to find an elegant formula to generate these triples, and maybe go from there to find a set of four numbers to meet Euler’s challenge. This has been a bit intimidating because I have no idea how hard it is, but it’s been fun to play with the algebra.
Please don’t ruin these for me in the comments!
As I’ve worked on these problems, I’m reminded of the experience of discovering new mathematics — in particular, that it’s a bit exhausting, and often requires me to look up information or use a variety of tools. Discovering math doesn’t mean kids in my class have to build all of their knowledge from scratch. I think the most important element is that the question they’re trying to answer has a real need that makes them want to find the answer, and that they are given the tools to be successful as often as possible.