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.

First thoughts on completing Exeter Math 1

I just finished doing the 2010 edition of Math 1 (91pages) today.  Now begins the synthesizing and summarizing, which I will put into blog posts.

Math 1 is an Algebra 1 course that includes an incredibly deep coverage of proportional reasoning, in addition to the usual linear, quadratic, and exponential function topics.

I did Math 1 because most of our incoming students are incredibly bright and hard-working but they were not the math monsters in their middle schools. They have many of the typical middle school gaps, but they are much more sophisticated than most 9th grade Algebra 1 students. So the fact that Math 1 is a REALLY TOUGH course that dives very deep into Algebra 1 material is a great thing because it will give my students the deep rich course they deserve, even though they are placed into Algebra 1 based on their current skill level.

My Algebra 1 learners find themselves stuck in a ZPD no-man's-land: their ZPD as math learners is nowhere near their ZPD as readers. 

This presents a huge problem in the classroom. The math in CPM Algebra 1, for example, is rich and interesting, but the text is written for reluctant readers, discouraged readers, and English Language Learners, which is a huge turn-off for the vast majority of my enthusiastic and highly capable readers.

They feel insulted by it, and they are not shy about expressing these feelings. So my student population tends to dismiss it and resist it, even if they really do need to learn the content. This raises the question of how best to serve a population of learners who need to be challenged with greater nuance in textual interpretation and presentation in an introductory high school math class.

For all of these reasons, Math 1 is going to form a terrific problem-based “spine” for my Algebra 1 classes. The problem sequences are rich and interesting and engaging with sophisticated contexts, though they start from first principles. They develop to a point where even a mathematically sophisticated adult will find them very challenging.

To get started, I printed all pages of the problem sets, answer keys, and commentaries and created a binder with the following sections:

1 - Problem Sets plus glossary at the end

2 - Commentaries

3 - My Worked Solutions (for each page of problems, I have one stapled cluster of my worked solution pages)

4 - Answer Keys

I did all of my work on three-hole binder paper, with each new page from the problem set being its own stapled packet (or "blob") in the Worked Solutions section. 

Whatever problem set I was working on I would take out of the binder along with the relevant answer key page. That way I could work on binder paper without having to carry the whole damn binder around all the time. Much of this work was done on a lap desk with my iPhone/Desmos for graphing, my TI-83-plus (sorry, Eli) for computation, and my monkey pencil case including my mechanical pencil, my ProRadian protractor, and my colored pencils.

A lot of people have asked me why I started at the end and worked from the end forwards, about 10 pages at a time. The answer has two parts: (1) whenever I started from the beginning, I bogged down or got sidetracked; and (2)  it enabled me to see where we were going and where students would end up. By seeing where they would land at the end of the course, I could better understand how things worked from the beginning.

More thoughts coming soon, but I wanted to capture these ideas right away. If you have specific questions you'd like to discuss, please put them into the comments section below.

Composting

In addition to our school vegetable garden and internal CSA (Community Supported Agriculture program), we have a coop with 8 laying hens. The AP Environmental Science classes work in the garden every Monday and harvest the produce that will be provided to faculty and staff subscribers during the school year. In this way, the garden program is self-funding, and students learn from experience exactly how difficult it is to grow food to meet expectations.

The chickens in the garden are our program's local celebrities. The kids love the chickens and the fresh, local chicken eggs are a very prized item for CSA subscribers.

The chickens, however, are a little bitchy. Bitchy and not too bright.

The reason we know this is that the faculty have to take care of the chickens once the school year is over and the kids are out of school.

Taking care of the garden is one thing. It's very meditative to come in to the garden once a week and pull weeds in the sunshine (or more likely, in the fog). Taking care of the chickens, on the other hand, can be a pain. As I said, they are bitchy and often peck at their caregivers as well as at each other. They knock over their feeding bins, poop in their communal water trough, and hide their eggs away from sight.

For this reason, it's always a good idea to tend our chickens on the buddy system.

So during our weeklong training session at school this week, I often brought my lunch down to the garden with my friend and colleague Cathy Christensen to hang out in the rare sunshine and assist with the chickens. One of the other AP Environmental Sciences teachers, Kathy Melvin, was already there between the rows of kale and chard, pulling weeds and listening to Bob Dylan.

Cathy and I ate our lunch, bravely fought off the chickens as we righted the water bins, filled up their food stocks, and gathered eggs.

When we were done, I asked Kathy M whether I should put my compostable fork into the compost pile with my plate and napkin. As a new composter, I am sensitive to the limits of low-tech composting systems, as opposed to our city's industrial strength composting system. I was trying to be mindful.

She said, "It won't decompose in there, but it will spark a good conversation."

That answer told me everything I need to remember about why I love our school and my colleagues there. As Michelangelo wrote to himself in his 80s (on a note found in his studio after his death), "Ancora imparo — I am still learning."