I teach a class called Mathematics Content and Pedagogy; it’s a course for Elementary pre-service teachers. Most days are pretty fun. Today was amazing.
We began with introducing numbers in base 5. Why on earth would we use base 5 with elementary teachers? I wondered this myself the first time I encountered numbers in different bases in the course syllabus. It immediately occurred to me as the students were working…now I can get them to think like first graders.
In this course, we often ask the students to think like a student and imagine how their students would approach the task. The problem with that is…once you know the standard algorithm for the various arithmetic operations, it’s very tricky to un-know it. It’s difficult to think like a person who doesn’t know the standard algorithm.
Think like a student who doesn’t know the standard algorithm for addition. Until I had a child of my own who has built from counting all to counting on to derived fact, I didn’t really understand how young children built their understanding of arithmetic (See Children’s Mathematics, Carpenter et al., 2015 for more information).
But if I put them all into base 5 (or any other base), now they can think like 1st graders. First graders who do not yet know how to add 4 + 3 (in base 5).
We started with representing the numbers with unifix cubes. At first they were resistant to the cubes, but then they realized they were first graders. They NEEDED the cubes.
Then, we agreed on some common language for hundreds, tens, and ones (flats, longs, ones).
Here’s the beauty: They needed to model the problem. Our brains are so wired to think in base 10, that base 5 required a physical model, whether that was the number line or the cubes or a picture. They were truly thinking like a first grader. And it was beautiful.
Here are some of the models they came up with for subtraction:
Can you see the borrowing in the picture? Can you FEEL the borrowing?
I’m grateful for teaching moments that make me want to stand up and cheer. Today was one of those days.