## Wednesday, August 1, 2012

### A quick thought on sequencing within units

Sequencing matters. Sequencing matters within courses. It matters within units, and it matters within problem sets and lectures.

But I think that I've been doing this sequencing thing all wrong. When I plan units I sit down with a blank sheet of paper and try to figure out the best order for kids to learn a topic. I spend a lot of effort trying to turn my unit into a good story.

I think I'm retiring that way of thinking about things. Here's why:

• A unit is not a good time frame in which to tell a story. Any tension you "build" in the first lesson is lost by the fifth.
• Kids don't learn ideas in the order in which it seems natural for me to sequence them.
• Who says that a story has to be matched with a unit? Maybe these big ideas matter to some lesson months down the road?
So instead of trying to craft my units into well-told stories, what I think I need to do is craft every lesson into a good story, and use my units to develop themes and concepts.

---
All of these thoughts came up as I was planning my functions unit for Algebra 2. I was stressing out about whether my kids need to know compositions before they can be exposed to inverses, and how I would help my students understand that functions are a whole lot like numbers. But I was all worried, because, while my kids have had no trouble with things like f(g(2)), they've had a terrible time with f(g(x)). How could I preserve the story while also focusing on the skills?

I think that this way of thinking is wrong-headed. Instead of trying to figure out a way to tackle composition and a way to tackle inverses, I'm going to do a series of good problems that have kids interact with both of these ideas at the same time. An earlier day will throw a bunch of functions with numerical inputs at students so that they notice that functions can undo each other, and then they'll try to figure out how to find an inverse of a given function. Next day, we'll use what we know about inverses to bump up the level of sophistication and handle variable inputs, and talk about undoing as an algorithm for finding inverses. We'll tease out a rule for recognizing inverses, stated abstractly. Then we'll use that as our "in" for talking about composing f and g into a new function, and using the ideas of composition of functions with variable inputs to get a second algorithm for finding the inverse.

I guess what's different about this for me is that there's no particular day when I can say, oh, we handled composition today. Learning of any single topic is being distributed over an entire unit. But those units are also connected to each other, so the big themes, ideas and skills need to be distributed in a similar way over the entire course.