# Conceptual Complexity and Computational Complexity

I taught expected value a few months ago. This is one task I used, from Illustrative Mathematics, that represents what I thought of as an important learning outcome for my students:

Here are a pair of graphics from a neat article on shooting in the NBA, via FiveThirtyEight:

There’s a ton of math here! But I’d argue that the math is different than the math in the first task. It could still be used to teach expected value, but this second task values a different type of thinking. I’d call the second type of thinking “conceptually complex.” It takes some background knowledge and reasoning to parse, but it doesn’t involve extensive calculation, just some multiplying by two and three. It does lead to some great thinking about decision-making and takes the abstraction of expected value and maps it into a context where the math matters.

I’d call the first type of thinking “computationally complex.” The first question is unlikely to matter to students; I don’t think they care very much about Bob’s bagel shop. It gets at some useful mathematical ideas, but in Bob’s bagel shop, calculation is an obstacle between students and the math I want them to learn. They need to parse probabilities written as decimals rather than percentages, pay careful attention to which values to use as the price, and multiply and add decimals.

I often find myself valuing computational complexity in math. When students struggle calculating complicated things, I often feel a need to support them at managing that type of complexity, and prioritize practice and explicit instruction that supports their computational reasoning. It’s how I was taught math. It’s what I was good at in school math. I’m probably a math teacher in part because of the messages sent to me that being fast meant being good. But I wonder what math class would look like if conceptual complexity was valued equally as computational complexity. What would class look like? Which students would feel smart? Who would pursue math beyond high school?