Always back to Skemp…

This morning I got up and read my post from last night about the TA in my class, and realized what I was writing about: Skemp, again, and the difference between relational and instrumental understandings.

The TAs who have been in my class have, for the most part, only instrumental understandings of physics. They know some algorithms for solving problems. In addition, they probably also value those instrumental ways of understandings as they have been rewarded for acquiring such understandings. They have been successful with those algorithms: What more could there be? They also, I think, do not tend to have particularly strong relational understandings of the physics, nor do they immediately see much value in such ways of understandings. I think a common response to my ways of approaching problems  is something like, “I have my way of solving the problem; and Brian has another way.” The difference isn’t that I have another way; the difference is that I have many inter-connected and related ways of making sense of the problem; for making sense of the relationships that exist among these various approaches; and for making sense of connections to other problems we might encounter. Both the kind of understanding I have is hidden from plain sight, but also why I value it is also hidden from plain sight.

I’m also not quite convinced that the future teachers I work with truly value relational understandings yet either, which is evidenced by their saying that I don’t like them to use equations. I do believe they are developing better relational understandings, but I think they still see much f this as a “school” thing–something they must do to do well in Brian’s class. I’m saying everyone is this way, or even that any single person is uniformly this way. I’m just saying that part of the ways in which they still make sense of it is in terms of what is expected from Brian. In that sense, I’d venture they are much more likely to think through problems when I’m around, then when I’m not.

Ways of Knowing…

In our physics department, every physics major has to serve as an undergraduate TA. Most of them get assignments in our algebra-based introductory physics course.   Because of the manner in which most of these students were taught (i.e., find an equation and substitute numbers), they can easily find themselves feeling a bit lost in my class, especially if they think they are supposed to be an expert of the content.

For example, here’s a question discussed in class. A bowling ball is dropped from a height of 45m, taking 3 seconds to hit the ground. How fast is it moving the very moment before it hits the ground? The problem is intended to draw out the following answers and arguments, which we hash out.

10 m/s, because all objects fall at the same rate

15 m/s because you can calculate the velocity as 45m/3s = 15 m/s

30 m/s because it gained 10 m/s in each of the 3 seconds

Other more idiosyncratic answers come up as well, but not with high frequency.

The first answer points to the ways in which students haven’t yet teased apart clearly the meaning of acceleration and velocity. The second answer points to the ways in which students haven’t yet teased apart clearly the meaning of average and instantaneous velocity. The third answers is consistent with the idea of constant acceleration. We hear arguments, and counter-arguments, and at some point I help clarify the right reasoning, and what’s both so tempting and subtly wrong about the other answers.

So, here is the way the TA solved it, before class began.

xf = (vf + vi)/2 * t + xi

0 = (vf + 0)/ 2 * 3 + 45

0 = 3/2 v + 45

-45 = 3/2 v

v = – 30 m/s

While the TA could solve this problem, they didn’t have a rich set of ideas for thinking about. It didn’t seem obvious that 30 m/s makes sense, because of the idea that its 10 m/s/s, or because final velocity sould be twice the average velocity (since it accelerated from rest). For other questions without numbers that we discussed, the TA seemed just likely as students to give answers inconsistent with the concept of acceleration. I’m perfectly OK with that, but my suspicion is that the TAs aren’t prepared for this. They aren’t prepared to be wrong about so many things or confused about so many things. I wonder how I can better position them as learners in the class–learners who just know somethings that the first-time students don’t, but not everything.

Of other interesting note is this. In my physics content course for future physics teachers, the students that have had me for a semester or two are pretty rock solid on having a repertoire of ways of think about kinematics problems, and also for avoiding common pitfalls. The others are pretty much falling for all the pitfalls. The difference is pretty striking. The thing that I like is that the range of expertise we have allows for peer-coaching, but also some, “Hey, it’s OK. We were making those exact same mistakes 4 months ago,” and, “Yeah, get used to it. Brian isn’t too into solving problems by putting numbers into equations.”

Time and Speeds from a Given Height

I’ve been thinking a lot about how to think about problems where you drop a ball from a certain height (let’s say its 45 m) and we want to know how long it spends in the air and how what speed it has.

The two big ideas I like to go to are

Δx = Vavg Δt


a = Δv /Δt

For the drop situation, assuming g = 10 m/s/s, these big ideas become

10m = vf/ 2 * t    (The average velocity is halfway between the final and the initial, which is zero)


10 m/s/s = vf /t   (The velocity change is vf -0)

So to solve this equation, we are looking for two numbers (and time and a velocity), and those numbers must multiply to make 20, and divide to make 10. (Or plot them and ask where does this line intersect with this hyperbola?)

My first strategy is guess and tweak. I decided since the numbers need to divide to make ten, I can easily satisfy that by picking a pair of numbers where you the second number is simply a decimal shift of the first one. For example, I picked

10 * 1 = 10 (too low)

20*2 = 40 (too high)

15 * 1.5 = 22.5 (still too high)

14 * 1.4 = 19.6 .. that’s pretty good

The ball will take about 1.4s and end up with a speed of about 14 m/s.


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