Exeter Math 1 Reflection 3: A Course in Advanced Proportional Reasoning

This is the third in a who-knows-how-many-part-series I am doing on my experience and practice of doing and using Exeter Math 1 in my Algebra 1 classes. The three labels I am using for this series of posts are: Exeter Math 1, Algebra 1, and metacognition.

As I see it, there are two core developmental strands in Exeter Math 1 that are woven together throughout the course. One strand concerns advanced proportional reasoning. The other involves what I would characterize as Exeter's method of "micro-modeling"—an ongoing spiral of frequent, small, subtle modeling tasks that provide extensive both variety and depth of practice in modeling. Many variations are explored so that students get a lot of practice in making sense of similar and differing contexts.

What I love about this blend of proportional reasoning and micro-modeling is that it occurs at the intersection of advanced textual interpretation and advanced proportional reasoning. This means it is an immersive experience in relentless sense-making and meaning-making as students explore modeling. In this course, mathematical modeling is a full-contact sport. Having worked all the way through the entire course, I can see how it is going to develop great fluency and confidence in modeling for Algebra 1 students, regardless of where they are starting (assuming, of course, that they have the basic prerequisites for Algebra 1 success).

The opening problem sets are deceptively simple, although page 1 problem 2 (from here on out, I'm going to use the Exeter-style notation of 1#2 to mean "page 1 problem #2), would be a fantastic Day 1 in-class rich task that drops students right into a hard micro-modeling problem with whatever tools they have.

But other than 1#2, most of the problems in the first 7 pages are deceptively simple. They're clearly written to review prior knowledge and to establish individual and group norms of work, with the major themes being work on rates, distributive property, order of operations, functional thinking, notation, number line, negatives and opposites, #unitchat, fractions, reciprocals, and rational numbers. Major concept development focuses on distance = rate - time, distributive property, working with various kinds of graphs and graphical representations, and micro-modeling.

And then you arrive at 8#1, and BLAMMO.

This is what I'm thinking of when I talk about a truly rich task blast-off.

I'm not going to give away the punch line here, but this "box within a box" problem is an excellent example of what I mean when I say the course focuses on advanced proportional reasoning. The problem requires a very advanced analysis of many distinct moving parts, along with an ability to track back and explain your thinking. By my count, this problem requires the learner to navigate and articulate issues of area, volume, footprint, a difference of footprints, layering, and negative space. Perhaps you can see other ideas here as well.

So if you're just getting started with Math 1 and wondering what the heck all the fuss is about, I encourage you to hang in there. 

I imagine that we will get to page 8 around the middle to end of the second week of school. And when we do, students should know that the fun is just beginning.  :)

Exeter Math 1 Reflection 2: Growing Up as a Mathematical Thinker

This is the second in a who-knows-how-many-part-series I am doing on my experience and practice of doing and using Exeter Math 1 in my Algebra 1 classes. The three labels I am using for this series of posts are: Exeter Math 1, Algebra 1, and metacognition.

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.

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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.