If there is an idea that I repeat over and over and over more than any other, it would be that procedural fluency must build from a conceptual foundation. National Council of Teacher’s of Mathematics explicitly state this in Principles to Actions as one of eight mathematics teaching practices:
Then begins third trimester with a class called Algebra 2 Concepts. These students don’t all hate math. But most of their experiences with school mathematics have made them feel defeated by a subject that has too much power in their lives. And I’m faced with conceptually putting my money where my mouth is as we begin quadratics and factoring.
Factoring is tough for a lot of students, and most of the struggle I see boils down to a couple of things:
- Factoring is introduced using rote procedure with no connection to multiple representations.
- Students never remember the procedures.
And speaking of procedures, I’m as guilty as anyone. I’ve tried them all: Slide and Divide, Guess and Check, The Pull-Out Method (don’t ask) and every other poorly-named, easily-forgotten factoring freak show on the planet.
And then I come across the Box Method. I’ve seen (and taught) a modified version of this method before with some immediate success but no long-term staying power. However, the progression I’m using with this class helps kids make the connection of a factored quadratic to an area model using length times width. Every time I help a student, I am asking about the length and width with respect to area. Procedurally, they are dividing out common factors. But what they are representing is the length and width of a rectangle, which they solidly understand, when multiplied, is the area of that rectangle.
We took a short quiz today, and we had some tremendous results:
Interesting story about that 2nd one with the sign error: this student originally learned factoring in algebra 1 using guess and check. She fought me a little on trying the area model. You can kind of see here that she originally tried the guess and check again. Then realized she couldn’t quite get it and fell back on the area model. I feel like she made a great connection today with it. So proud of her.
I’m so proud of these students. They challenge my resolve every single day, and I’m lucky to get an opportunity to help them learn.