Preservice Performance Assessment, standard B2c, question 1:

Does the candidate appropriately balance activities for developing conceptual and procedural learning activities to understand mathematics?

Does anyone? At the end of a year of student teaching, I know a lot of the ways I want to be better, and I’ve seen (or read! yay blogosphere!) what better looks like, so I at least know which direction to go. But procedure vs. concept is still chicken and egg. Which comes first?1

Item: My students can approximate distance traveled from a velocity table. They can approximate the area under a velocity curve. But they don’t recognize these as the same computation.

Item: My students can describe why their paper boxes that are too shallow or too tall hold fewer jelly beans. They can also tell me that to find a critical point, take the derivative, set it to zero, and solve for x. But they can’t get from one to the other.

I’ve lectured. I’ve modeled. I’ve Socratic-ed. I’ve left them alone in groups with thought-provoking questions. I’ve worked one-on-one. What I consistently see is that “what am I supposed to do?” [with the symbols] lives in a different neighborhood of their brains than does “what is this about?” If I make the connection, they don’t come with me. If I leave them to make the connection, they don’t. Is this the dark side of playing to multiple learning styles? If kids do a physical or manipulative exercise and then do an analytical exercise, but don’t make the connection, then I wonder if the mode-switching just confuses them.

So it seems there are two big design challenges in a lesson. First of course, and biggest, what’s the genuine problem that leads to the burning question?

But next, what has to happen so they see their question in its symbolic setup? All my students could tell me that they’re looking for the peak of a graph and that the slope is zero at the peak. They could all tell me that the derivative measures the slope. But fewer than 1 in 10 could go from “where is the peak?” to “f’(x)=0; solve for x.” That’s where I lose them. It happened in harmonic motion, related rates, optimization, integration. Is there a design principle for getting this right?

1I’m fully bought in to context before content. The procedure-and-concept issue arises as we move into content.