Why is reasoning about integers so hard?

A few years back, long enough ago that I didn’t have nearly as much figured out about teaching FigureOutAble math as I do now, I was working with some middle school teachers.

Things were generally great. They loved what I was sharing, and I was learning a ton from the experience. Then we hit a sticking point.

They came to me one day and asked specifically about integers. Everything else was making sense, but they couldn’t see how it would be possible to think and reason through integers.

I was a bit taken aback. In my head, what to do seemed obvious. We do what we always do. We think through it. Math is math, and math is FigureOutAble.

To no one's surprise, that answer wasn’t super helpful. Those wonderful teachers remained adamant. Other math is FigureOutAble, but everything changes with integers.

I’ve given integers a lot more thought since then.

In retrospect, it makes a ton of sense that integers would be a breaking point.

Think about it. Think about the extreme basics of the “−” symbol. What does that mean? Subtraction? A negative number? What if there are two next to each other?

Math is always FigureOutAble, but that doesn’t mean the notation we’ve developed to write it down is obvious or intuitive.

Notation isn’t the only issue though. If students have not been building the foundational reasoning to working with integers, there’s more pitfalls than floor to stand on.

How can a student make sense of negative number magnitude if they have no sense of place value with positive numbers?

Can you imagine trying to make sense of a negative fraction’s magnitude if you never mastered relative magnitude with positive fractions?

Fortunately, armed with the knowledge of why integers can seem so alien and impossible to understand, we can adapt our teaching to compensate.

I go into depth in this podcast episode, Integers