One of the things I’m doing this year is trying to provide more opportunities for deliberate practice (with feedback), while trying to support students in building and articulating explicit strategies for solving problems (not me just telling them strategies), and selecting class problems to work on that can be solved with current ideas but also put us in contact with puzzles that later ideas will help to resolve or bring insight into.
Here is an example from earlier this week:
This week my students are going to be evaluated (by another instructor) on their understanding of projectile motion and Newton’s laws (including circular motion).* Afterwards, we are going to be diving into energy. So, how do I give them more deliberate practice and feedback with projectile motion and forces while also putting us in contact with energy puzzles? Here is what I tried:
First, I asked students to predict which of two ramps would result in a block of ice having more speed at the bottom. Both ramps were from same height, but one was shallow one is quite steep. No numbers were given. They think and vote peer instruction style. We were pretty much evenly split between all possibilities, so they discuss in small groups and then I collect arguments at the front board. The arguments were basically the following:
- The steeper ramp has a greater acceleration, so it will be faster.
- The shallow ramp will provide more time for the ball to speed up, so it will be faster.
- The opposite effects of acceleration and time will balance out so that they take the same speed.
We’ve gotten pretty good at doing this, so I can mostly stand to the side and just write down arguments and do some re-voicing. After hearing the arguments, I have them revote. There were some shifts, but still not near any consensus.
I now tell them that I want to help settle this by applying some of the skills we’ve learned over the past couple of weeks. I add some information to the scenario. Block of ice has a mass of 25 kg. The ramps are angled 30 degrees and 60 degrees. The height is 5m. I split the class in half, half the groups work the 30 degree problem while the others work the 60 degree problem.
Before sending them off to work the problem, however, I tell them to talk strategy with their group–what will you need to figure out to answer the question, what skills and ideas might be useful, what might you do first, second, etc? They talk for a minute or two, and then we collect strategy ideas at the board. They say most of the things they need to–drawing free body diagrams, using Newton’s laws to find acceleration, finding the length of the ramp using trig, using kinematics ideas / equations to determine the final velocity, etc. Now they are off, and the board is there to help remind them of things they can try if they get stuck.
Doing this together makes me free to monitor for progress rather than helping students get started. I’m checking free-body diagrams for bizarre combinations of Normal and Weight fores, if and how they are finding components, whether they are using a rotated coordinate system and using that consistently with forces and kinematics, etc. I point out things that they are doing which are very “physics-y”, like drawing careful diagrams with labels, starting from big ideas rather than launching into equations, etc. If groups finish early, I ask them to solve for other things that came up in our arguments. For example, I might ask students to solve for the time on the ramp to see if its true that the larger acceleration was paired with less time to accelerate, etc. As multiple groups finish, I have them check with each other on their answer and check with people across the room.
Once we are done, I do a quick summary of what we found, highlighting that its odd that both ramps end up giving the block the same speed. I restate the arguments we heard, and I emphasize that the argument for the right answer made it seem plausible that it could balance out, but why it exactly balances out seems like a puzzle to me. It didn’t just balance somewhat, it balanced out exactly.
I tell them that I want to consider another problem where we compare final speeds, but this time not with ramps. In this problem a baseball player throws a ball with same speed. In one case the ball is angled upward, and in the other case the ball is angled downward. The question is about the speed’s of the two ball’s just before impact, and how will they compare. Students vote. This time there is a split between two answers. Most students vote they will be the same, but don’t have good arguments. They are banking on it being similar to the last problem. Intuitively, it makes sense that the one thrown down will have more speed, and I support this argument a bit. If you are throwing it down, in the same direction of gravity, and its got a real direct path to the ground, isn’t it going to be a lot faster when it hits. There are some other really awesome arguments for why it should be the same, about why it must balance out, including consideration of what the one that goes up is like once its on the way back down. The best argument came from a student who had never spoke up in large discussion, so I spent some time re-voicing that argument and giving it space for consideration.
Once again, I turn the conceptual question into two problems to solve, adding angles and heights and an initial speed. We talk and collect strategy at the board. They solve the problems. I monitor progress, give extension questions, ask them to check with each other. Finally, I summarize and make connections at the end. I still try to keep the puzzle open: Why is it that when the two blocks fell through same distance, and ball’s fell through same distance that their final speeds were the same? Our current skills help us to calculate that this is the result that should happen, but it doesn’t help to explain why.
An interesting outcome of asking students questions to compare, and then asking them to compare pairs of questions, is that they start doing more and more comparing. Several groups started re-thinking the shooter-dropper experiments. Looking for connections across phenomena is something I want to promote and this kind of activity seems to promote more of it.
Anyway. So later this week, we’ll revisit these same two problems from an energy perspective, but I’ll also introduce puzzles for us to resolve that further our understanding of energy and kinematics. Namely, this time we’ll do a problem where two balls rolls down the same ramp, one with an initial speed and one from rest. In this case, they will neither end with same speed nor gain the same amount of speed. Rather they will gain the same amount of kinetic energy…
* This used to bother me, having someone else test my students. But I now love it. My relationship with students is not of evaluator or judge. I am a learning coach. Sure, some of the evaluation is not meaningful. Sure, my students are learning things that aren’t evaluated. But my students do well on the evaluations for the most part, and students are constantly getting feedback from me on a broad range of their learning.