Open ended activities have never been my strong suit. In the past it devolves into answering a million questions and a clear sense that the students aren't doing the heavy lifting. Many things blew up in my face in the my first couple years because of this. I moved to pseudo open-ended tasks. Students weren't following a recipe of steps, but they weren't wholly responsible for creating the work at hand.

With a slightly different population, it turns out my students can be easily bored with that approach. Very slowly, I've forced myself to get them to invest more effort into their work, and I have stopped explaining every little scenario.

A couple recent examples.

Composite Functions

The last little while we were discussing adding, subtracting, and multiplying functions together. Previously, I treated it as a wholly mechanical lesson. I wanted more context this year. We discussed perimeter, area, and volume as parallels to the various operations we study. On the relevant assessment, I defined length(x) and width(x) and asked for this:

We never did these as class work, but we had discussed their relevance to composite functions. Reading the answers to this gave me a really good idea of who gets function combination on a conceptual level. I've noticed that my little group of nerds is really good at rote mechanical work, making differentiators like this important. The mind blowing part is that no one gave me that "what do I do?" look that I've seen before. Win.

Rational Functions

Division of functions is an excellent segue to rationals. I had a progression last year that was ok, but clunky. After various discussions about holes, asymptotes, and how to identify each, I gave them a closing task. Design some. If they really get how an asymptote or hole is created, they can design a ratio that meets whatever specs I give.

Kids had questions, and tons of other kids were eager to share the answers. I didn't do much other than stick an errant iPad back on the wifi so they could print. Watching it work made it very apparent how and when direct instruction fails. For many students it was obvious that they were absorbing all of this passively and nothing "clicked" until it was time to make something. Assuming direct instruction is working for your students is foolish.


There is still work to do. But I'm so happy to discover this early, so many other things I have planned can become that much more interesting as a result. I always tell people to challenge their students, that you'll be surprised what they can do, and something forget to do it myself. It's paying dividends in Calculus as well, where you really don't have the time to explain every minor scenario.

Neither of these activities involve "real world" problems, which may bother you. But reality is in the eye of the student. If a student can create a host of things that validate their understanding of how an asymptote is created and defend their method, is that not real? Has that math become something more than words in a textbook?

AuthorJonathan Claydon