Here’s one thing that’s been going well in my summer physics class that I don’t want to forget about.

**Warm-ups that Emphasize Mathematical Strategies**

Let’s say later in the day, I plan on discussing and having students practice using a particular approach to solving constant acceleration problems–using the average velocity to figure out how far an object goes. For this, I’m going to need students to be able to have a strategy for finding the velocity that’s in the middle of the initial and the final velocity.

The warm up for class goes like. What number is directly in the middle of 10 and 20? Share you strategy for how you figure it out? How can you check to make sure that number you found is precisely in the middle?

Now try out your strategy for the numbers 320 and 20. Check to make sure that number is precisely in the middle. I keep all the conversation focused on the strategies and the checking strategies.

Next part of the warm-up involves trying our strategy for situations we might not be sure will work–non whole-numbers and negative numbers. Use the numbers – 32 and +12. Use the numbers 9.2 and 2.8.

Another example was strategies for finding the area of trapezoid. We talked strategy. Later in the day, we spent time talking about and practicing use the area under a velocity vs. time to find displacement.

There are a couple of things to note:

- I want students to feel like they invented or at least own the strategies. For this reason, problems with multiple strategies or strategies for checking will be most useful.
- I think it works best when you start with something intuitive (what’s in between 10 and 20?) before moving to the less intuitive (9.2 and 2.8).
- That said, building up to the abstract is made possible by emphasizing the strategies (not answers) and strategies for checking. So you as teacher have to emphasize the strategies students are using. Give them names if need be. The idea is to formalize the intuitive strategies.
- I found it useful to play up the possibility that the strategies might not work in new situations even when I knew they were going to (e.g., Do we think this strategy will work with negative numbers? Or will we need a new strategy? Everyone try out the strategy first, and then check to see if it works.” This will be important later, because later students will likely try to find average velocity later in ways that won’t work (e.g., averaging two constant speeds that were maintained for different times). Since you’ve already built up the notion that strategies may only have limited usefulness, students are poised and ready. When it happens (and it will if you provide them an opportunity), you can say, “Cool, you just discovered a situation where our strategy doesn’t work? … How did you do it? How is this situation different that the ones where it does work? Why didn’t it work? I wonder if there is another strategy that would work… etc.).
- In practice, I have been explicitly having students strategize with math problems that will involve same numbers in our physics problem. So our trapezoid problem had the same dimensions as the one in the physics problem. The initial and final velocities in our physics problem were the same as one of our math problems. I’m not sure if this is necessary, but it seems to have some benefit. A few students notice right away as you are setting up or working on the problem. They go, “Oh, this is just like the trapezoid we saw earlier.” I think it makes those students feel insightful and, for others, it makes the problem a little less intimidating. I suppose it may even reduce cognitive load.

I imagine a lot of people do things like this. I certainly have, but there’s also a sense in which I haven’t. I’ve done it before, but I haven’t necessarily done all the things necessary to pull it off: designing good problems that build on each other and toward physics, managing the discussion to focus on strategies not answers, and celebrating accidental discoveries of finding out that a strategy didn’t work in a certain context.

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