I broke math

A few months ago, a student ran into class early one morning with an excited look on her face. “I broke math!” she exclaimed.

[The day before, we had just learned about imaginary numbers and explored a dimension where square roots of negative numbers were possible. What an unjust name for that number set – they deserve to be acknowledged as real! see more here]

I asked her to explain how she had broken math; she grabbed a space on the whiteboard and started to show what she thought was a loophole in the reasoning we had explored the day prior. Alas, mathematicians had already accounted for that loophole and she did not break math that day, but what a powerful moment for her.

In other subjects and classes, students sometimes (hopefully often) get to create. Quite often, these creations take shape explicitly in the products of their thinking – paintings in art class, poetry and essays in language arts, lab results in science, and so on.

How do we help kids see math? Do they see it as a construct that was created by other people and only to be received? Or do we help them see it as ideas that they can create themselves, bend, and even break? I was excited by the fact that this student had viewed mathematics as a subject that she had a role in forming; to even think that she could play with it in order to find a loophole is incredibly empowering.

It made me reflect on how I can help more kids approach mathematics with this mindset.  I thought of an idea that had been in the back of my mind for a long time: teaching mathematics concepts by exploring what they are not.

I learned early in my career that one of the worst ways to introduce a new term was to provide the definition. For example, the definition of a polynomial is “an expression of more than two algebraic terms; the sum of several terms that contain different powers of the same variables”.

This does not help someone who is learning this term for the first time make any sense of what it means. I always wondered what would happen if instead, we talked about what a polynomial is not. We would still draw pictures, talk about examples, but what if we also spend just as much time exploring visual models and expressions that are not polynomials.

A foundational math concept that I can see this working well would be when students first learn about place value. We spend a lot of time talking about what each place means and represents, but don’t really explore different number systems. To get to the core idea of how our base-10 number system works, it would be helpful to explore systems that are not base-10.

James Tanton from G’Day math has developed a construct called exploding dots that does an amazing job laying the foundation for this exploration. I encourage you to watch it regardless of what level of math you teach. Here, students get to truly see what happens when we run out of digits to express values in different places. It’s intuitive, based on an intriguing story, and engaging to play with – it invites learners to bend the rules.

When I ask teachers why they don’t let kids explore the negative space around a new concept more often, I always hear that “the kids aren’t ready” or that “it will confuse them”. I encourage teachers to give it a try; consider that new concepts are new to kids anyway, providing a clear cut definition doesn’t paint the whole picture. Building experiences that let them see both what something is and isn’t will help them construct a deeper understanding. It will hopefully show them that they have a role in creating new ideas in mathematics.