Reflection for Day 3: I’m thinking about what I’d like to do for my lesson to present to the group on Friday. Chances are, I’m working by myself, since I don’t really have time to try to coordinate with someone else between workshops, driving and tutoring (not to mention wedding stuff). I’ve got a couple options on the table–first, I could refine a lesson I sketched out over the last year, creating a stronger activity for something I’d like to go deeper on; second, and more likely, I can put together something that will help my students with something I feel I didn’t do justice with last year. Big Hitters for this category: FTC, or Volumes using Cross-Sections. Since FTC seems to be a theme for these reflections, let’s go with it.

Here are my thoughts: Last year in my calculus class, we focused on the FTC as a means to an end–how can we solve certain problems using the FTC–students did OK in the time we spent on it, but then we moved on. The FTC wasn’t viewed as a thread woven throughout calculus, which it really should be.

Let’s brainstorm what that might look like:

(1) I want my students to understand that derivatives and antiderivatives are inverse functions.

(2) I want my students to connect the summation form of the FTC to what they know about linear functions.

(3) I want my students to be comfortable applying the FTC to totally symbolic problems.

(4) I want my students to see the FTC in area problems.

And to do all this, without spending weeks on a separate FTC unit. Cool. Now to take this thought process and articulate it in a more meaningful way. But, you’ll have to wait for that.

Another note–again, nicely situated in the thematic neighborhood of the rest of this post–getting better. I think it’s easy to lose track of the big picture of mathematics, or mathematics education, or on a more biological level, mathematical processing, and it’s nice to step back and take a look at what’s out there. I found it particularly poignant today, taking a look at the cognitive leaps we make in math from the time we begin to understand that quantity is something that we can identify and compare, to the point where our solutions are no longer numbers, but functions or families of functions in the case of general solutions to differential equations. I think sometimes it’s also useful for our students to see how far they’ve come, even if they don’t think it’s as cool as I do.

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