Our district is looking at revamping our year map and I would like to suggest a map that has the qualities of Algebra: Themes, Tools, Concepts (ATTC): particularly its ‘integratedness’ and how well it spirals through the topics. I’ve read on your blog about separating topics and lagging homework. But what are some things one might consider when deciding what the next week’s topic would be (or what topics to put in what chapter)?One problem with ATTC is that the spiraling and the fact that the topics are not always explicit makes it more difficult to use. As a department chair, I needed a structure that would be simpler and more transparent, as my colleagues quite rightly wanted to know what was going on. Here is the skeleton of an approach.
Maybe another way of putting my question: how would you go about taking something like your map of themes, tools, and concepts for proportional relationships and turning it into a course sequence?
First decide what the main topics would be. The main topics, not all the topics you can think o. (I recently posted some thoughts on pruning the curriculum.) Even if you're forced to work with an overstuffed list of standards, the first step is to prioritize.
Organize the main topics into units. In general, something that doesn’t deserve a unit or fit in one probably shouldn’t make it into the program. Maybe six to eight units per semester, with related units not contiguous if at all possible. (Also save a stretch at the end for review, or just as cushion if things take longer than expected.)
Important and difficult units should be as early as possible. It is not realistic to leave those for late in the year.
For each unit, what tools are available? (Manipulative, technological.) What great curriculum do you know of? (especially: which books or Web sites have great problems on this topic?) What representations are you familiar with that might throw light on the concept? what “real world” applications can you find? Once you have these ingredients, you map out a way to circulate through them in a way that makes sense mathematically and pedagogically.
If you can, start a unit with an interesting problem or activity (the “anchor"). The anchor should be motivating and memorable. It need not be easy. Examples of anchor activities are Geoboard Squares for the Pythagorean theorem, Rolling Dice for exponential functions, or Super-Scientific Notation for logarithms. They are activities that bring together key content with good practices.
Throughout the unit, go back and forth between easy and hard the whole time. Avoid the trap of starting easy and gradually making it harder, for two reasons. One, starting too easy gives the wrong impression, and students don't launch into the unit with the right attitude. Two, gradually getting more difficult gives the impression to some students that there's a time where they might as well give up. Going back and forth is a way to keep everyone alert and challenged, and to have times where the pressure is lessened.
To guarantee spiraling and integration, make connections with past units when you have ideas for that. But the easiest ways to extend student exposure to ideas are to lag homework, to give cumulative tests and thus test correction opportunities, and especially to separate related topics. If you have the good fortune to work in a block schedule, take advantage of the long periods to always be working on two different units, emphasizing one or the other on different days.
Well, that’s what comes to mind. Good luck! And let me know if you have more questions.