Here's a perfectly imperfect model unit of how I use the How People Learn stages and cycles in a typical unit in my Algebra 1 class. I'm documenting this for myself, so anybody else who finds this useful is just icing on the cake! :)
All of the files I use are in this downloadable zip file on the Math Teacher Wiki:
Algebra 1 Inequalities unit
Here's a rough overview of how this works.
ESSENTIAL QUESTION:
How are algebraic inequalities related to our basic number sense concepts of "more than" and "less than," and how can we use this understanding to build a more generalizeable algebraic understanding?
STAGE 1 - hands-on introductory task
(1) Deleted scene (readers' theater activity): groups read the deleted scene about how to do Talking Points, which also contains a review of number sense concepts of more than and less than.
(2) Talking Points - set #1 — more than and less than: what does your group think?
Students follow the protocol they have just learned and do set #1 of Talking Points that active prior knowledge about which is more and which is less, given two sets of quantities.
STAGE 2 - initial provision of an expert model
Each day is different, but usually I spend about 10-15 minutes working with the whole class to "do some notes" (combination of mini-lecture and note-taking and modeling). This is often the last thing we do in the class period.
STAGE 3 - deliberate practice with metacognitive self-monitoring
Practice happens at both a macro- and a micro- level. On the micro- level, each day's homework (which is completely distinct from the day's classwork) gives a chance to practice and review. Then the first activity of the next day's class is comparing answers in your table groups and answering all questions that groups or students are able to answer for each other or for themselves. I take only Burning Questions (like only taking group questions during Complex Instruction problem-solving tasks).
At the macro-level, we are building toward two big days of Speed Dating, which is differentiated deliberate practice with metacognitive self-monitoring and peer tutoring or reteaching as needed.
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We cycle back through all of these for several days, as you can see, with a new set of Talking Points each day that students have to work through and puzzle over. Each day's Talking Points build in a new piece of knowledge that is in students' Zone of Proximal Development so that they can encounter it, wrestle with it, and formalize their understanding of it. Then they get the nightly chance to practice some more.
This is a highly Vygotskian model of learning.
Eventually we need to introduce a new concept into our exploration of greater than and less than. I call this concept the concept of "betweenness." We do this through (5) another deleted scene.
Once again, we are investigating numbers and quantities, but we are extending our investigation to more abstract conceptions of quantities. We go back to Talking Points. We do some problem-solving. We struggle together. We organize our learning.
STAGE 4 - transfer task
I don't have a particularly great transfer task yet for this unit. That's why it's such a good one to use as an introduction or reintroduction to my Talking Points norms and practices.
If you have a super-terrific transfer task for Algebra 1-level linear inequalities (not yet at systems), I'm all ears. Please let us know about it in the comments section.
Let me know what you think if you try any of this!
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A Postscript —
Today was the kind of day that makes all this work worth it. To review for tomorrow's unit test, we did group whiteboarding of some pretty hard problems. And even though many kids were still stumped by some of the harder problems, they felt excited. They were understanding it.
And doing it.
Doing math.
Now that is the kind of thing that makes it worth waking up at 5 in the morning five days a week for ten months of the year straight at near-poverty wages. :)
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