I had a pretty scary experience today.
Background: I’m a regular 8th grade math teacher, but I have several other small responsibilities, including teaching a section of 7th grade remediation twice a week. It’s a small group, and most days we get materials from their regular class to work on. Students practice a variety of questions and I work with them on specific misconceptions, conceptual gaps and skill fluency. When students are doing well with the current material in class I bring in some problem solving, but most of it is student practice and feedback, and focused remediation where I find skill gaps.
This week we’re working with area, and focused specifically on circles. One student was struggling in particular with moving between different pieces of information. It seemed like he didn’t have a big-picture idea of what radius, diameter, circumference and area meant, but instead saw it like a guessing game — sometimes the answer is divide by two, sometimes it’s multiply by pi, sometimes it’s squared then multiply by pi, and his job is to figure out when to do which one. Very little algebraic understanding of the formulas, and not much of an idea of what they represent either.
My first mistake was not taking this more seriously and drawing pictures and bringing out rope and rulers to get at what all of this means.
Anyway, we found some success finding area, given both diameter and radius, and he was getting pretty good at it — he had stopped telling me that 3 squared is 6, and his decimal multiplication was solid, and he was moving between diameter and radius consistently and appropriately.
After some more successful practice, we moved on to some embedded figures questions — circles in squares, etc. Not bad, although it was still a bit of a guessing game — “so this is subtraction?” — and I really had trouble getting him to reason about the questions.
Anyway, we move on to an embedded figure question with a triangle inside a rectangle. He was pretty frozen at the start, and with some prompting we decided he should find the area of the triangle first. The triangle had a height of 12 and a base of 10. He says, “Ok so the diameter is 12, and the radius is 6, so 6 squared is 36 and 36 times pi is ….” and begins multiplying.
When I told him that that wasn’t the formula for the area of a triangle, I think I broke his world. It’s hard to communicate the look on his face — not of fear, or confusion, but of grim amusement — his attitude was not “oh, that makes sense, what was I thinking”, it was “oh, they changed the rules of math again, I wish I wouldn’t do that, but I guess I have to forget that now and learn something different”.
I think I did something bad to that student’s brain today. I think that what he learned was that to get me to stop bothering him and get questions right all he had to do was square the radius and multiply by pi. Never mind looking at pictures, it takes too long and is just confusing anyway. Then, one random question, I told him that what I told him was right was actually wrong. So never mind math making sense, or reasoning about the world, or being puzzled and unpuzzled. Math is a thing he has to do every day where he guesses what’s in the teacher’s head and sometimes he’s right and sometimes he’s wrong and there’s not much of a way to tell the difference.