The misconception that makes math miserable

Looking to add another math teacher to my team, I put out a job ad.

Virtually the only question on the application was this: How might you coach a student who is struggling to solve this word problem: "You have 2/3 of a can of paint and use half of that to paint a room. What fraction of the can do you have left?"

The answers were disappointing. The typical response went like this: “To find 1/2 of 2/3, all you have to do is multiply the two fractions across.”

In other words: “To help a struggling student, I will simply teach them the procedure, and they’ll be good to go.”

This doesn’t work, because math isn’t procedural. It’s conceptual.

Once you understand mathematical concepts, you don’t need procedures. You can take your understanding of how numbers work and apply it to anything.

On the other hand, if you don’t understand the concepts, procedures won’t save you. You’ll be able to, say, “multiply two fractions across,” but you won’t know how to translate a word problem into the familiar procedure you’re clinging to.

A procedural approach to math is like cooking according to a recipe but without checking to make sure your ingredients are fresh. It’s like following directions from Google Maps and ignoring the stoplights and stop signs along the way. It’s like going on a date and reading your conversation off of cue cards. You’ll be okay going through the motions for a little while, but at some point everything is going to fall apart.

So how do you build a conceptual understanding? The first step is to let go of the procedure. If you think in terms of "You’re supposed to multiply these two numbers together,” or “They said to take half of this and multiply it by that,“ you’re in procedure-land. There is no “they,” there is no “supposed to.” It’s just you and the numbers and how they behave.

How do they behave? To find out, play with them. make a model. Try stuff.

To solve the paint can problem, I might draw something like this for my student:

Then I’d just sit there for a moment. The student might see right away that half of two thirds is one third.

If they don’t get there on their own, I’d say, “How much of this paint are we going to use to paint the room?” then wait for their answer. If they need more support, I might ask them to put an X through half of the paint in the can.

There are a lot of different ways to arrive at the answer — and the “multiply across” procedure is only one. A useless one, if you don’t already understand the underlying concept.

If someone you love hates math, it may be because they have been taught that math is a series of complex and seemingly unrelated arbitrary procedures. It’s not. If nobody ruins it for you, math is fundamentally elegant and simple.

If you would like to detour around the hellscape of procedural math, check out backwardmath.com.