Fascinating conversation with a student in my calculus-based physics class today.

We're doing the Interactions chapter in the Mazur textbook, which introduces the relationship between inertia and acceleration of two interacting objects by

we used the first half of that chain of reasoning back in the momentum chapter to arrive at a definition of inertia (the ratio of two inertias is the flipped ratio of the respective velocity changes after an interaction), so this really just adds the idea of taking the time derivative to turn the velocity change into an acceleration.

The student is having trouble remembering how to set up the proportions, because they don't quite see why that last proportion is true, and why other possible proportions they could set up *aren't*.

In particular, they really want \(\frac{m_1}{a_1} = - \frac{m_2}{a_2}\) to be true, and keep coming back to it as the default, in analogy to setting up a proportion in algebra or geometry, where as long as you keep the corresponding "things" in the corresponding place on the proportion, you are fine.

So they wanted that proportion to be true, and, failing that, they wanted it to be obvious why it wasn't true -- and so were hitting hard against their self-worth (something along the lines of "I must be **bad** at this because I don't see why it isn't true. And Mark is sitting down talking to me a really long time, so I must not be good at this -- he shouldn't have to spend so much time on me.")

So of course I'm super proud that this conversation is happening at all, and I'm sitting down for a long time because it's *interesting* and I don't know how to answer and I'm trying to think through all the proportions that could be set up and whether I can find some reason to reject them out of hand -- and let me just throw in that I'm hitting my own wall of self-worth.

I tried several tacks:

Go back to the momentum principle every time, don't try to memorize the proportion. This is probably the "correct" thing to do as it is closest to how actual physicists work, and the momentum principle is the deepest piece with the most value and the most implications, therefore starting from it is always a good thing to do.Check the units. Notice that the "right" proportion is unitless while the "wrong" ones have units. This is admittedly a bit unsatisfying since the units are stillconsistent.Think of it as an empirical question-- "I can imagine a whole slew of possible proportions I could set up, some of which are true and some of which are not. To decide which are which, I need to connect those proportions to more satisfying physical interpretations (like the momentum principle or the definition of inertia), or I need to test them experimentally and see which ones are actually true. This is good scientific method and it supports the notion that it'snotalways obvious whether a given proposition is true or not, which values my student's hard work in trying to puzzle it out.

None of these was enough to completely satisfy my student so we ended on a "outstanding work, let's see what you think of it tomorrow, after it's percolated a little longer", but I think it did help, and I tried to really reinforce that the conversation was awesome -- isn't this how it's supposed to work? Student tries a problem, isn't satisfied with their reasoning, pushes at it, pushes some more, asks the question, still isn't satisfied that they get it.... I love that. How do I make room for more of that? Why didn't I do more of that myself?