If I ever am lucky enough to get a sabbatical, I can think of few better ways to spend it than traveling around the world visiting the classes of the many incredible teachers I’ve met through twitter/blogging.

This past Friday, I got a chance to watch one such teacher, Kelly O’Shea (although I knew her long before twitter and blogging).

Before I get too far into describing Kelly’s class, I want to offer a theory about good teaching—it’s that it doesn’t really look like your preconception of good teaching—you know, fireworks—the teacher delivering a spellbinding presentation, students rapt with attention in their seats, answering questions on cue with everyone on task.

Truly excellent teaching looks nothing like this, and it is what I saw Friday. I walked into Kelly’s class, five minutes into class, and the eight(!) students in the class were working in pairs on solving some problems related to circular motion and universal gravitation. When I arrived, students were deep into calculations and Kelly was sitting in the back of the room. Students were asking questions to each other, and working totally independently. Throughout this work, Kelly was standing back, students would occasionally ask questions, which more often than not, she’d answer with a question.

Soon, students were ready to whiteboard, and students quickly whiteboarded the problems, and explained their process in less than 10 minutes, ending with Kelly asking the class if they agreed with the work that was presented. It was a reminder to me that whiteboarding can be fast—it doesn’t need to involve rearranging the furniture and bringing everyone together in a circle—you can simply have students stop what they are doing for a moment to listen to a peer’s presentation. Think simple.

After this whiteboarding, Kelly led a short mini lecture on Gravitational Potential Energy (her students call it interaction energy) away from the surface of the Earth. This is a knotty topic for first year physics students—deriving the formula for the interaction energy requires calculus, something almost none of the students have.

Here’s how Kelly does it. She reminds them of the big question they had been thinking about before—how fast will a student in he class need to run in order to leave the earth and never come back? She tells them that to answer that, we need find a formula for the gravitational interaction energy of an object far from the surface of the earth. First, she writes $U_g=mgy$ and asks why won’t this formula works. Students very quickly figure out that it doesn’t make sense because g will get smaller as you move away from the surface of the earth, so this expression can’t work when g isn’t constant.

Then she asks students how they got the idea for interaction energy in the first place—what did we graph? Students start to think it through and say it was a force vs distance graph. And then someone says that the area in the graph was the work, or the negative work, or the negative of the change in energy, they aren’t quite sure. This is a key turn—Kelly doesn’t sit there (like I often do) and interrogate them to think out what the right answer is—she knows students are 90% there with the idea that area and energy are related. So she draws the graph.

Then she talks about how you could find the area, and a student (not in calculus) says you could draw a trapezoid approximate the area, and someone else says if you drew multiple trapezoids the approximation would be better, and Kelly adds that if you draw an infinity of trapezoids, it would be perfect. All this is quick—a couple of minutes at most (again, resisting my urge to go in and underline all the math by showing these successive approximations).

So she then asks a student who is in calculus to step up find this area for us—calculating the area under the curve between some initial and final position.

$Area=-Gm_1m_2\left(\frac{1}{r_f}-\frac{1}{r_i}\right)$

Then they recognize that this looks like the change in a some quantity, since it is a final value minus an initial value. So we get an equation for the gravitational interaction energy

$U_g=-\frac{Gm_1m_2}{r}$

Then Kelly reminds them that for the old interaction energy expression ($U_g=mgy$) the gravitational energy is zero at the surface of the earth (y=0). With some questioning, the students realize that this new expression must be zero when the object is at infinity. This is puzzling, but the students wrestle with it and realize this also means the gravitational interaction energy must always be negative—to which Kelly asks whether the negative sign tells us something about the value or the direction of this quantity, and students realize that energy doesn’t have direction, and so it must be about the value. This was a pretty critical question—Kelly is stripping away some of the unnecessary vocabulary (scalar or vector) and getting students to focus on the more important idea of interpreting the meaning of the symbols they’re working with.

And just like that, Kelly’s “lecture” was over in less than 10 minutes. She paints the physics with light brushstrokes, and lets the students fill in the details. The students are quickly back on task trying to calculate the answer to the jumping off the surface of the earth question, and Kelly is going around and answering individual questions as they come up. It isn’t too long before someone blurts out “11,000 meters per second! That’s not that fast…”

## An interesting comparison

For comparison, I’d like to show you how I used to teach this same topic, at the very same school, 7 or so years ago. Back then, we were trying to get students to appreciate the power of numerical integration and computational modeling, but at the time, the tool we used wasn’t VPython, it was Excel. Students would generate elaborate spreadsheet models to predict the motion of objects by integrating the motion of the object over thousands of tiny steps in Excel.

When we got to energy, we framed the numerical integration question with the question—if you drop a ball from 2 earth radii away, how fast will it be going when it hits the ground. And we gave them a spreadsheet that looks something like this:

Yes, that’s a 15×150 spreadsheet. And you can imagine the reaction that students had to looking at a page filled with numbers. It took days to get students to understand what was going on in this exercise (we were crazy enough to get them to figure out the 1/r relationship by linrearizing the interaction energy vs separation distance data), and even after most of them had gotten the “right” answer (likely by copying a peer) almost none of them understood it. What’s more, there was still extreme confusion about why the energy was negative, and why it was zero at infinity. Fruitless exercise indeed.

## The takeaway

To me, watching Kelly’s class was an excellent reminder some characteristics of great teaching.

• It isn’t about you. The longer you lecture, the less students get to wrestle with the ideas themselves where they really learn. Skip the embellishment to give students more time to practice.
• Let students do the heavy lifting by solving problems and building their own understanding.
• Cut away unnecessary cruft and focus on the essential details and connections to past ideas.

And I think I’m starting to see how Kelly is able to be so far ahead of me in terms of coverage, with no drop (and probably even an increase) in student understanding. Now I just need to put more of these ideas into practice.