First and foremost, Exeter Math 1 is a course in growing up as a mathematical thinker. It is about leveling the playing field between and among rising 9th grade students.
Here's how I would would frame this journey for students: This is a course in developing your own mathematical self-reliance and resourcefulness as a learner. Your Essential Question is always: How can I exhaust everything I already know before I ask the teacher for help?
You already know an enormous amount of mathematics. In this class, you will need to step forward with that and be willing to attack problems with the best thinking you already have. You may not know everything, but you always know something, and since that something is the best thing you know, you show up and start there and give it everything you've got.
Then, when you have struggled as much as you can and as hard as you can—both by yourself and with your table group—and when you can no longer do anything more with what you've got, that is the appropriate point at which you can ask the teacher for help.
That is the best use of the teacher.
If you are passive in this work or mess around, you are going to suffer.
This course works at two levels. At the content level, we are going to do all of the usual content work in an Algebra 1 class. But the more important work we will do always takes place at a metacognitive level. It is designed to help you learn how you learn advanced mathematics.
OK, back to the teacher perspective.
I have a Post-It on the inside-front cover of my binder on which I wrote this:
Exeter discovery is about guided sequential flailing.I think this is true. The Exeter Math 1 path definitely involves guided, well-sequenced flailing. It also integrates continual spiraling designed to activate prior knowledge. The purpose is always to discover how much math you already know and can put into service with the problems that are directly in front of you.
There is mathematical content and metacognitive content on each page.
This leads to the issue of practice. In Exeter Math 1, there is a very specific theory of action in the practice problems that are given and in how they are used. There are none of the usual taking-up-time, too-easy practice problems. If students need extra practice on certain specific procedures, then you have to source them yourself from someplace else, such as (for us) the Holt Algebra 1 textbook.
But that is OK because at this point in my career, I can do that in my sleep.
The Exeter Math 1 approach to practice problems is to provide juicy, meaningful, gimmick-free practice problems that are (a) always of medium difficulty or above and (b) integrated with metacognitive reflection and discussion. For this reason, I would be inclined to use these inflection points in the curriculum as opportunities to use Talking Points to solidify conceptual understanding and to get students exploring and articulating the subtle misconceptions and potential pitfalls inherent in practice problems of a medium level of difficulty or above.
This is a very deep teaching idea to me — to keep practice problems at or above a medium level of difficulty and to have students explore and give voice to these subtleties as rich opportunities to make meaning in their work.
More thoughts coming soon.