Right now my students are taking the final for this course. I thought I’d do it, too, only here on my blog. I’m typing this during the final period (in fact, I had to change to a laptop since the in-class computer’s keyboard was so stinking loud). It’s a little “stream of consciousness” but I wanted to show my students what I was looking for. Here’s the final:

Write an essay addressing the following issues. Use specific standards to illustrate your points.:

- Why was Mathematica critical to this class? How would your learning of the material be affected if you couldn’t use that tool?
- In this class you were encouraged to not forget anything. Why is that philosophy useful in a class on this material?

At first it seems strange to talk about a tool like *Mathematica* as a central tenet of a course. Shouldn’t we be writing about physics concepts instead? Is *Mathematica* just a crutch in this class, or just a way to add colors to images? I don’t think so. I think it’s crucial to the learning that I experienced over the past three and a half months.

Let’s start at the beginning, Chapter 6, the calculus of variations. The book encourages us to do some crazy variable selections to get the brachistochrone problem to be tractable. Instead we chose the obvious variables (x for horizontal and y for vertical) and determined the diffferential equation that the path y(x) needed to be to minimize the travel time for the bead. It was an incredibly ugly diff eq but we knew *Mathematica* could handle it. When we tried, though, we found something weird. We couldn’t start at our beginning point because there was a divide by zero there. So we had to move away from the point just a little to get it to work. The next thing we noticed was that, since it’s a second-order equation, you have to supply both the initial position and the initial slope of the wire. Since this slope was arbitrary, we found that we got lots of solutions. If we wanted to get the bead to land at a particular point, we had to adjust the initial slope until it did so. I don’t think I fully understood the nature of this problem until *Mathematica* allowed me to play around with it.

*Mathematica *played a similar role in almost every chapter. In Chapter 7 we were able to visualize and analyze solutions to problems where the book simply asked us to find the equations of motion. Why stop there?! Making the movies and having *Mathematica* “do the math for us” enabled us to understand the meaning of the various terms in the Lagrangian. I can’t tell you how often a funny looking movie helped me realize the mistake I was making in crafting the Lagrangian in the first place. In Chapter 11 we were able to show why the inertia tensor is useful because *Mathematica *would take something like 50 times as long to do a straightforward calculation without shifting our problem into the frame of the moving rigid object. In that chapter there’s a ton of math that is not that hard to follow but I don’t think it’s motivated all that well. Seeing how the various approaches affect the computation time, on the other hand, was very motivating, for me anyways. In Chapter 12 we found a way to get around our linear algebra deficit and were able to find the dance moves of lots of crazy systems. I learned something every time I watched a screencast from one of the students trying to model something weird.

I think we need to show our students how to use these powerful tools. They can’t do anything without good information on the front side, though, and that’s why the combination of standards like 12.1 and 12.2/3 or 10.1 and 10.2 worked so well together. Forcing all of us to first get our brains wrapped around the ideas and then puzzling out how to code it in *Mathematica* seemed to be a good recipe for learning, in my opinion.

As for the retention question, like most classes I think the material kept building on previous material. An easy example is from Chapter 6 to Chapter 7. The only change is x becomes t and the functional that nature seems to care about is the Lagrangian. Understanding why

is true is only manageable by going back to Chapter 6.

Another more extended (in time) example is Chapter 3 and Chapter 12. It was interesting to me to see how much people struggled with standard 3.1 until we covered chapter 12. It was then that many of the students realized that 3.1 was a super simple version of 12.2. Having to remember the details of Chapter 3 so that they were near-fresh when we got to chapter 12 made teaching and learning chapter 12 easy (ok, easier).

Angular momentum showed up in lots of different guises in this course. We discussed it in chapter 2, showing that it’s proportional to the area sweep rate. Then in Chapter 8 we made use of the lack of torque in gravitational systems to treat Kepler’s second law as a write-me-down. Then in chapter 9 we saw how a system’s angular momentum is a helpful thing to keep track of, and in chapter 11 we used it like crazy. Having all of these various concepts and models of angular momentum in my head at all times really helped my understanding and it was cool to hear my students say things like “just like we showed in Chapter 2 . . .” in their chapter 11 screencasts.

I think one of the major tools used in the course was being able to figure out the variables of the system, recognize the kinetic and potential energies of the system, and integrating the Euler-Lagrange equations from the appropriate initial conditions. We practiced that like crazy in Chapter 7 and then used it to learn and illustrate the main points in future chapters. The best examples of that are in Chapters 11 and 12 but we really used it almost everywhere.

It was especially interesting, to me, to see how different teaching Chapters 2 and 3 were from previous years. This year we decided to do the calculus of variations first and that really made chapters 2 and 3 pretty easy, compared to previous years. If you can write down the Lagrangian, you can get to the equations of motion pretty easily. Once there, you can talk about all kinds of things like stability, Newtonian mechanics, and on and on.

I had a lot of fun teaching this class. Making use of tools like *Mathematica* and keeping all the content fresh in our minds was a challenge, but one that, I think, really impacted the learning we all experienced. My hope is that the students saw *Mathematica* as an integral part of their work, and not just something that made pretty pictures. I’m excited to see what they have to say on this final. I hope, too, that they recognize the value of not binging and purging the material throughout the course. Watching their screencasts of late gives me hope in that regard. Of course, one students put it quite well that the reason they have a sense of all the material here at the end of the semester is a positive by-product of their extreme procrastination (a good title for a reality show, no?).

Writing this has been a fun experience. There’s a lot of sore hands and loud sighs in the room right now but hopefully I can point them to this post if they ask “what were you looking for?” My guess is that I’ll like almost everything they say today but we’ll see.

This reminds me of paper I read called, “When teaching becomes learning” by Miriam Sherin. And also, how a colleague and I were recently talking about how teachers (ourselves especially) enjoy continuing to have transformative learning experiences about the content of the courses we are teaching. It may sound selfish, but I wouldn’t want to teach if it didn’t involve me learning.

I don’t think it’s crazy that mathematica could be foundational piece of your course. I learned so much by exploring and engaging with mathematica as an undergrad.

I hope my students view VPython the same way yours (hopefully) view Mathematica. It’s a great tool. I’d like to extend my VPython assignments to include some more “what if” examples. Thanks for the inspiration.

That sounds cool, Frank. I’ve read about half the finals now and it’s clear to me that my students saw Mathematica as a learning enhancer, and not just an add on that I wanted them to do. In fact, one students talked about how on the first day he thought it was just me wanting them to play with a toy I liked but he really changed his mind when he started playing with the Brachistochrone problem in the computer.

Andy,

This is great—could you post the exam in scribd and embed it in the post—for some reason its only appearing in a tiny window for me, making it hard to read. Also, did you see this—a link to a way of visualizing differential equations on the ipad. How much of this could you reproduce in Mathematica (lots, right?).

Thanks for the suggestions for re-embedding, John, hopefully I’ll get around to that soon. Didn’t Frank recently tweet about being able to embed gdocs into wordpress? I thought it would work better.

I checked out the video and I liked it. What I think my students can do is construct any diff eq they want and the Manipulate command (mentioned by a few students as their favorite tool they used this year) will let them build the sliders any way they like. The problem with most canned solutions is that at some point you hit the limits of flexibility. Teaching my students Mathematica gives them a huge range of things to explore.

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