Variation Theory In Practice

A strategy for helping a student who's a bit confused

Here is the key idea of variation theory: Humans pay more attention to things that change than things that stay the same. By keeping many things in the learning environment constant and varying one thing at a time, we can direct the learner's attention to the most important mathematical ideas.

A big part of variation theory is conserving attention. I think it can be especially useful in a situation where the learner is processing something new and needs to focus their attention as much as possible. Here is a simple example I used the other day.

I taught a mini-lesson on rounding to prepare for our upcoming unit on circles where rounding will come up a fair amount. One student was having trouble. After finishing a whole-group mini-lesson and some practice, I pulled up a chair with her and started to talk her through a few examples on a mini whiteboard. (By the way, great tip from Craig Barton — carry a mini whiteboard with you when you’re circulating around the room!) Without considering variation theory I might pick examples more or less at random. Let's say I start with rounding to a whole number: I might pick 8.2, 11.91, 3.5, 0.702, and so on. Here is the sequence I used instead:

3.3, 3.4, 3.5, 3.6, 3.7, 11.7, 11.5, 11.4, 11.45

The goal is to help the student focus on cause and effect. If the example changes from 3.3 to 3.4 and the answer doesn't change, the example reinforces that when the tenths place changes from 3 to 4, you still round down. If the example changes from 3.4 to 3.5 and the answer does change, the student can infer that when the tenths place changes from 4 to 5, you change from rounding down to rounding up.

The number of ideas a student has to hold in their head tends to be invisible to math teachers who are experienced with that concept. I come in with a lot of knowledge I don't realize I'm using that keeps me from getting overwhelmed when I'm thinking about rounding. But put yourself in the shoes of a student who is feeling confused, who struggles to keep "whole number" and "tenths place" and "4 or smaller" and “round up” straight. When every part of the problem changes from one example to the next, it's not obvious which change is important and which is not. Variation theory can also be useful elsewhere. I’ve written about it here as a more general practice tool, and I still use it often during some units. But variation theory is an especially good fit for this situation where I know a student is confused and I want to help them gain confidence and avoid overwhelming their working memory.

The followup steps here matter too. I need to mix in more variety of problem types — more digits, rounding to different places, rounding up from 999.5 to 1,000, other weird edge cases. This student will benefit from some interleaved practice mixing in different types of problems further down the road, once they've gained confidence with the core concept. Variation theory reduces the demand on a student’s working memory by keeping most things constant; I need to increase that demand as time goes on. But my first goal is to build confidence. Structuring a thoughtful sequence of examples varying one thing at a time is a good way to do that.