## Leaving a gaping hole

This past week in my optics class I think I made a mistake. We were talking about how light interacts with a system with multiple parallel interfaces, and we started with analyzing a single interface that didn’t happen to be in the plane z=0 (which is how the previous chapter did it). I asked the students to re-derive the work from the previous chapter, just with a different location for the origin. I had primed this pump by asked a student just this question during an oral exam last week. His standard for the exam was “I can derive the laws of reflection and refraction (angle in=angle out and Snell’s law)” and he, as expected, did it with an interface at z=0. So in this class they knew how to get started.

The first thing I asked them to do was to find where in the derivation the z=0 idea was made use of. That didn’t take long, so then they could focus on how the next few steps would go. My favorite part of the class was when they realized that they could still relatively easily show that the color shouldn’t change upon reflection and refraction, that angle in equals angle out for reflection, and that Snell’s law or refraction ($n_i \sin\theta_i=n_t\sin\theta_t$) is true.

The next part was the hard part. Could they figure out the ratios of outgoing light to incoming light for both reflection and refraction (these are typically called the Fresnel Equations)? The added complexity of z not being zero was making a mess of the pretty equations in the previous chapter. But here we talked about how they didn’t really expect any differences for the ratios given that we hadn’t really changed the physics of the situation, just the math approach.

Here’s a quick analogy: when analyzing a falling object from an energy approach, you are free to choose the zero point of potential energy. The two most useful choices are where you drop the thing and where it lands. However, you can pick any height and still get the same results for the time of the fall, the speed at the bottom, etc. Now, to be sure, the intermediate calculations to get there will be different, but the “physics” should be the same.

So my students became relatively convinced that eventually the math should work out to give the same ratios. Only they couldn’t see how to get rid of the extra ugliness. Ok, the table’s set. Now I step in and make my mistake.

I gave them three choices:

1. Figuring out this “gaping hole” in our understanding would be the standard for the day. Something like “I can fill the gaping hole that we dug today.”
2. We could move on without filling the hole (I wanted to talk about how further interfaces changed things) and I’d fill the hole with a screencast outside of class.
3. I could fill the hole right now (in class).

It led to a good conversation, with my pushing how number 1 might lead to the most learning of this topic. They got that idea, but I think they were nervous they wouldn’t be able to do it. I cautioned that they might see the fix and think it was “a trick I’d never think of!” (I really like to avoid that line coming out of my students).

In the end we all settled on number 2. We moved on to the case of multiple interfaces, making note that each new interface would provide two more boundary condition equations (each of which would have the new ugliness due to the fact that they weren’t at z=0) and showing how the number of equations would always match the number of unknowns (typically the +z and -z traveling waves in each region).

Why do I call this a mistake? At the end of class I asked one student about the vote. He admitted that he was nervous about number 1, but that if we weren’t going to do number 1, I should have done number 3, since he was invested in the problem at that point and was ready to hear/explore the solution. Moving on and pushing the “filling of the hole” to outside of class wasted an opportunity to have the students contribute to the solution.

So, what do you think? Would you put it to a vote? Would you add more options to the vote? What would you vote for? as a student? as a teacher? Has this happened to you?

Here are some comment starters for you:

1. I’m in this class and I liked the whole process because . . .
2. I’m in this class and I hated this whole process because . . .
3. I would definitely do the vote, just with this small change . . .
4. I would definitely not do the vote, because it irreparably damages the students in this way . . .
5. I would add this to the choices . . .
6. I would vote very differently as student or instructor and here’s why . . .
7. Boy you’re brave! I would never admit to such a terrible mistake.
8. Your analogy is dumb. Energy approaches don’t tell you about the time of travel!
9. Here’s a better analogy for you . . .
10. I love it when my students say “it’s a trick I’d never think of!” It shows how smart I am and gives them something to shoot for.