You might recall that in my multivariable calculus class (four students), we’ve been turning a really badly written section in our textbook on Kepler’s Laws into a great learning experience. The section was really unclear, the authors didn’t motivate any algebraic work, nor did they relate the equations to any conceptual understanding of what was going on.

We decided — well I decided, but my students agreed to play along — to rewrite the textbook to make it clearer. We wanted to focus on motivating each step of the derivation, we wanted to organize the derivation in a more sensible way, we wanted to be explicit with each of our calculations so the reader isn’t left wondering “where the heck did that come from?”

My students got into the project. Heck, I got into the project. We spent about 5 class days working on it. Most nights I didn’t assign homework. One night, I asked students to each individually outline how they thought the rest of the paper should go. Another night, I asked students to proofread what we had written for stylistic and conceptual inconsistencies. We finally came up with a formula describing all conic sections — which describes how the earth moves around the sun. We didn’t get to actually derive Kepler’s Laws (see below for why).

The students are really proud of this paper. They want to send it to the math department head who left last year, the publisher of the textbook, and their calculus teacher who retired last year. (We will send it to all three!) We embarked on this together (I didn’t know anything about this section; I was going to skip it but the students really wanted to cover it), and I let them do a lot of the thought work themselves. It’s hard to let go as a teacher, because I have this drive to explain and clarify everything when someone doesn’t “get it.” But these kids are advanced enough that they can grapple with the material, ask each other questions, and be okay with getting stuck. And I suspect it is precisely because of this, because they did it, that they feel ownership of the paper.

Two of my favorite parts of the paper:

Our current draft is here.

Their next “problem set” isn’t like the others. I pretty much said: “we learned how to read the book, and make sense of that which we thought we could never figure out. Now your task is to each individually finish his paper off. We’ve gone 2/3 of the way to the end together. Go try to do the last 1/3 yourself. With this formula we came up with, your textbook, the Internet, and your wits, write the final part of this paper. That’s right: you derive Kepler’s Laws.”

Their problem set it due on Friday. I’m excited to see what they do with this!

If you’re wondering what our class looked like when we were working together on this paper:

Imagine four students, sitting at a square table. Each has photocopies of the relevant textbook section in front of them. I am sitting at the front of them, laptop on, with Lyx (my LaTeX editor) open. The screen is being projected so the students can read what I’m typing. I prompt: “so what do we want to write?” and we’re off. The students talk with each other about the section — not only asking questions and answering each other on the mathematical content, but about how the section should be presented. One might say “I think we should say something about how vector b will actually be crucial to understanding everything. The book introduces b and then forgets about it and never really explains it.” Another might respond, “Yeah, we should devote a whole section of our paper to explaining b.” And they’re off. I sometimes interject to ask questions, or to get them on the right track, but it’s rare that I’m directing. Finally, when I see they’re coming to some sort of consensus, I say “so what am I going to type — what’s my heading? The introductory paragraph explaining what you’re planning on doing?” And then one of them will say “In this section, we will introduce a new vector, b, which will end up being unchanging over time. This constraint on the motion will …” And then another student might say “maybe after ‘unchanging over time’, we should say “no matter where the earth is or what speed it is moving at.”

And we’re off to the races. This goes on for 50 minutes. Which always seemed too short. Each day we got about 1 to 1 1/2 pages (single spaced) written. At the very end I scanned in the images that one student drew as we worked our way through the material.

This project was hands down the best thing I’ve done in any of my classes all year, in terms of student learning.

In my multivariable calculus class, we spent last Friday reading the textbook as a group, trying to understand the section on Kepler’s Laws. We got done showing that if there is a sun-like object and another object with a particular initial position and velocity, it will either fall into the sun, be an circle, be an ellipse, be a parabola, or be a hyperbola.

Today we were going to move onto using this result to derive the three Keplerian laws of planetary motion.

But then I decided to scrap that. Because even though we read the book and followed the text, line by line and equation by equation, we lost sense of what we were doing. We lost sense of the conceptual underpinnings for each equation. We didn’t know what motivated the book to make the moves it made. It’s largely the book’s fault, which is really unclear — if you’re a high school student and not used to having your book say “we leave this as an exercise to the reader.” (Seriously, it did that.)

One of the things you’ve heard me say is that I want to foster the skill of students learning to communicate math well.

So, I decided to scrap the plan of moving forward, and we’re devoting two or three days to

WRITING OUR OWN TEXT EXPLAINING THE DERIVATION OF KEPLER’S LAWS.

We started out the class outlining a basic structure to it (Part I: What we want to show; Part II: Initial Conditions; Part III: Gravitational Pull; etc.). Then the four students started talking about what they wanted to say. (One agreed to draw the diagrams we’re going to include in our text.) I just sat up front, and when they decided, I typed it up in my LaTeX editor — projected so that students could tell me to fix or reorder something. Sometimes I prompted them (“you told me to write but you never told the reader what that is” or “does it matter if the initial velocity weren’t orthogonal to the position vector?”). And it took us 50 minutes to get about a third of the way done.

But you know what? It is working. They’re talking, they’re thinking, they’re arguing with each other, they’re asking questions. And they’re learning to work through things, and explain them to someone else.

I was so pleased. Hopefully the next few days go as well.