# Squaring the Superhero

I teach many different courses in mathematics. All of these courses contain wonderful material. I love math. Last year, my son asked me why some of the students at his elementary school did not seem to share my view regarding the beauty of mathematics. So I started a math club at the elementary school, in an attempt to effect a positive change.

I decided to cover the same material covered in university courses, realizing that I would have to do it in a much different way. Here’s an example:

University version: If a rectangle has perimeter p, what is its maximum possible area?

Here’s what happened in the math club:

I asked them if they knew what a rectangle was, and how to find its perimeter and area. We discussed examples. It turns out they knew quite a bit. Then I asked each of them to produce their own rectangle, with perimeter 12 units. They each had a favorite! We calculated the area of each one. By the end of the meeting, we were convinced that the square provided the maximum area. By the end of the next meeting, we owned the difference of squares formula. In the third meeting, we used this formula to understand the formula for the area of a circle, partly by considering an appropriate annulus—this meeting was only a couple of days after a solar eclipse.

So what does this have to do with superheroes?

Special transformations, accompanied by superpowers, can really capture the imagination. It might be Clark Kent removing his glasses; but probably it’s a tadpole transforming into a frog or a caterpillar becoming a butterfly! In this video (link below), we introduce Rover the rectangle, Squeaky the square and Ellise the ellipse. Rover can change his shape as long as he stays rectangular and keeps the same perimeter—he catches circles by flattening himself! The intuitive connection between superpowers and transformations illustrates how a mathematician may think and feel about the subject. It is a human activity. It is fun, it is personal—it is mathematics!

So far we’ve made three of these videos. Each one represents portions of a different, hour-long meeting. Perhaps the best way to use them is to watch them with children, pause to discuss certain points, and then go off on a tangent. For example, in our first club meeting we eventually changed the fixed perimeter from 12 units to 14 units, which led to a wonderful discussion about fractions.

I tell the math club members how our discussions fit into the context of a university course. They LOVE this! At the end of our first meeting, we wrote down the formula for the maximum area “in terms of p” and they hurriedly copied it down as if it were a treasure. This was one of many pleasant surprises. Perhaps you are already doing something like this; if not, please try it. Become a superhero!