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?

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