long chains of reasoning-making them real
The trick here, I think, involves a few things, but one is working memory. I think that my “regular” physics students would have about the same issues and successes on both of these problems, and that not all of it is about what or how we teach them. Cognitive ability has a big dependence on working memory, and most of these students that I have always had in these classes simply can’t hold several concepts in their minds at once. They can remember the facts or even apply the concepts to different problems, but only one at a time (or two). Having to juggle competing or complementary concepts and synthesize them is the hardest thing for them, and even though we push (and should push) them towards this, a significant fraction simply can’t hold all of those ideas in their heads at once. Probably as a result of this, I’d answer your question “is this the way that my kids see math?” with a “yes,’ because that’s all of the concept flow that they can really get into their heads at once. The trap that lots of conceptual/regular/intro classes fall into is to remove lots of the symbolic manipulation (good idea) and to replace it with more complex reasoning that’s really dependent on understanding and applying the heart of the math all at once. In lots of ways, that pendulum “graph” is really really hard. Even though it’s elegant and awesome, lots of them can’t appreciate the argument, because they can’t understand the whole thing, but only pieces at a time.
This is awesome, and it makes me want to go and do a bunch of reading about working memory to see how to develop deliberate practice to help students improve working memory.
If following long chains of reasoning really is a core skill of science, necessary for understanding everything from the photoelectric effect to global warming, and I think it is, how can I help kids to build this skill?
My idea is that when we next try to make an argument like how the floor knows how hard to push you up, or how we are able to measure the acceleration of a mass spinning around a circle using a stopwatch, students practice writing each step of the argument on a stip of paper and connecting them in a paper chain, like so:
Here’s an idea I had recently. What if we practice building real chains of reasoning out of paper?
Would this help them see how they are linking ideas in physics? I guess the only way to tell is to try it. I’ll let you know.