Guilty. I can do a lot of procedures, but when asked to put it in a context or explain WHY it works…I may or may not be able to do it. Or at least at that point in time.
For example, a colleague who subs in our building is finishing up an education certification. As part of that program, they are currently enrolled in a math methods/problem solving course. They had a question about a problem…
2/3 divided by 4/5
Getting an answer was not the issue. We both knew to invert the 2nd and multiply. They were asked to put it in a context and then follow up with a different interpretation as well.
I thought for a bit and said, there is a lesson sequence in the Vn de Walle book but mine is at home. “Oh! I’ve got that book!” So they ran to get it. By the time they returned, I had found an article I remember running across a couple of summers ago. Measurement and Fair Sharing Models for Dividing Fractions by Gregg & Gregg. It was essentially the same lesson, modeled after the Van de Walle lesson. Christopher Danielson has a very nice post here that references it as well.
Basically, I went through the lesson sequence. Just as when we teach procedures, if we don’t provide opportunities for student thinking to develop, they are stuck with meaningless, rote steps. As part of this sequence, I realized how much easier it was for me to divide with like denominators. Honestly, I had never considered it before. Shameful, I know. But noone had ever led me to do it that way and I had never taken the time to consider it.
Along with Jo Boaler’s course, I wonder how spending some time thinking about how elementary and middle grades teachers develop concepts would impact my own teaching. I realize how important it is to allow time and provide a structure for students to make those meaningful connections on their own. I am there as the support to help them along their way. But in order to be successful, I might need some of my own productive struggle in order to answer the WHY and provide some meaning.