Open Set

Math education is always a loaded topic. Most people shudder when they think about their own experiences with math education, clearly a sign that something went wrong. Other people are sure that “back to basics” is the right way, whatever that means. (A romantic memory of a time that never existed? An incomplete memory of what happened in your own childhood, before you realized how complex the work was that your teachers were doing?) There are fights over inquiry vs direct instruction, teacher-centered vs student-centered. Liljadahl entered the arena talking about “thinking classrooms” vs “non-thinking classrooms” – along with the implied conclusion that everyone who was not following his 14-step method must have a classroom where the students don’t think. (Oh, I was insulted when I read that? Just maybe. Just maybe.) 

Where do I fall in all of this? I am a huge fan of inquiry, and yet, plenty of my teaching is not inquiry-based. For example, sometimes I will put painter’s tape on the floor to map out a coordinate grid. Then I’ll tell a student to start at the origin, and then walk, say, three steps to the right and then two steps forward, so that they understand what (3,2) means. That is not inquiry, but it’s very important for them to experience where things are on the coordinate plane. Similarly, I don’t know if that activity is teacher-centered or student centered. The teacher is giving clear, specific directions, and all the student needs to do is to follow them. 

Then there is this distinction between student-centered and teacher-centered. For years, I was encouraged to have students come up to the board more often to share their thinking, to put solutions up, to make my classroom more student-centered. But I hate giving up control of the whiteboard marker! I like to hear what students have to say, and I want to share their ideas with the class. But students take so long to write it down, and then the class can’t quite figure out what they are getting at. They need a practiced adult there to help get through all the ideas clearly, slow them down at points and speed them up other times, and make sure the writing is actually clear. Of course, those are students’ ideas coming out, as I annotate on the board and manage the flow of the conversation, so it feels more student-centered that way. I guess I don’t really have a good definition of teacher-centered and student-centered, so it’s hard to be sure if I’m doing it or not. 

Then you get conceptual vs procedural math. Again, do we have a definition? I was once working with a 5th grade teacher on this, and tried to explain to him that his work was more procedural and it would be good to also include some conceptual work. “No, I worked on those decimal concepts,” he told me. “You need to move the decimal when you multiply, that’s the concept.” Now, I’m pretty sure “move the decimal place” is a procedure, not a concept. “If you multiply by a number less than 1, you get something smaller than you started with,” is a concept. But it’s also a fact, and you can teach facts through memorization, which doesn’t feel so conceptual to me.

So, how do we know if a lesson is “conceptual”? Maybe, when it’s visual? When it’s connected to other concepts? Let’s go back to that first example of walking on the coordinate plane to plot a point with your feet. It’s a very procedural activity. I tell the students exactly how and where to step. And yet, I’m helping them build a concept. “First you need to move left/right, and then after, you need to move forward/back, and those two motions together describe where a point is on the plane.” Kids who don’t understand that there are two types of motion, and that’s what the two numbers represent, those kids can’t plot the points. But you need a procedure to get them there. 

Maybe we could also talk about “traditional” math teaching vs. “modern” math teaching? I hate to use the word “traditional,” since it usually harks back to a time not all that long ago, but implies that things were always like that even before. If you look at this painting by Nikolay Bogdanov-Belsky, “In the Public School of S. Rachinsky” (1895), it is clear that “traditional” teaching did not always mean students sitting in rows listening to the teacher.

But, let’s put that aside for the moment. One of the best teachers I know was a colleague back when I worked at Horace Mann, an independent school in Riverdale, NY. (I’m sure he’s still there, he was a lifer.) His classroom was set up with rows of students facing the teacher. He would lecture for 20 minutes or more at a time. And then he would throw the students a challenging problem to get them to apply all the new information he had just shared with them. There was very little in here that was not “traditional,” but it was really great teaching. I’ve seen other teachers doing similar things, and I’ve done it myself when that was what my students needed. For example, when I derive the Quadratic Formula for my Algebra II classes, that’s a 30 minute lecture. And often, when the formula is born at the end, kids are ready to cheer, it feels so satisfying.

For all that I’d like to say my teaching is inquiry-based, student-centered, conceptually focused, and “modern,” there are plenty of ways and times in which it is not. So, what separates my work from those teachers that I had back in high school? The ones who told us to copy down their work as they did three example problems on the board, and then practice by doing # 1 – 39 odd out of the book. Anyone have a way to sum that up in a word or a phrase?