The constant acceleration particle model is so tough!
We started the day talking about the two readings; more from chapter 2 in Arons and the Hammer article "Two Approaches to Learning Physics".
The Hammer article highlighted two students in a college physics class; one who wanted the formulas so that she could plug and chug the problems, and another who wanted to reason them out verbally. Hammer asserts that these two students represent stereotypical approaches that he has experienced as a physics professor.
The participants identified with each of these students - but more the equation user than the big picture students. I find that with physics teachers it is often difficult to get away from a traditional lecture, equation, story problem methodology because;
1. That is all they have ever seen
2. They were successful and don't see any reason everyone shouldn't be
This causes heart ache and frustration and is why they identified with one student more than another. Knowing and using the formulas was - for a long time - viewed as success in introductory physics. Now we know that there is little correlation between being able to use the formulas correctly and actual conceptual knowledge of the physics!
To be truly successful, it is important to be able to do both.
I would never say that the equation and mathematical part of the the constant acceleration particle model wasn't important. In a traditional setting the equations were the end all be all of the "chapter" on acceleration. However, it is only one part of the "model". One of the hallmarks of the modeling method is the use of multiple representations to understand a concept. And the equations used to represent the acceleration of an object are only one of the representations.
So although the equations aren't more important than the other representations they are still important and I want to make sure they are given their due.
The problem is that in a traditional setting they are rolled out in the beginning of a chapter without much explanation. And because we were good at using them we never asked, "where do they come from?"
The first thing I did to develop the equations for the unit was to have the participants run another cart down the ramp and find the equations, again. This time we put them on white boards making sure that they recorded the equations that the computer gave them. This was like the 4th time they'd done this so it was pretty easy.
Then I went to the big board and wrote out the structure of the equation that went with each graph. From there I asked them to tell me their data and I filled them in the appropriate spots. I asked them to look for patterns in their numbers and they found them easily.
The basic idea is that the velocity graph is linear and can be modeled with a y=mx+b equation. From there everything else comes. When I was done I turned around and looked at about 15 jaws on the floor. I had apparently blown their minds! I got the usual, "I never knew where those came from!" and "I've never even thought to go about it like that!"
Seeing where the equations come from puts the participants and students in a position to use them way more effectively and see how they relate to the rest of the representations to build the full model.
I have seen this be a turning point in a lot of participant journeys - the first time they really see the power of the method. I hope that this works to help some of the last few get on board!
With this knowledge in hand the participants embarked on completing the worksheets from the unit. I had them pay close attention to worksheet 4 because it looks like a traditional, story problem worksheet. But I asked them if they could do the whole thing without using any of the equations that we developed earlier in the day but by only using the velocity vs. time graph. We then had them put various problems on white boards. After that Laura facilitated a board meeting, again modeling how we would do it with the students.
Finally we got started on the practicum for the unit. In this practicum we rolled a steel ball bearing down a ramp and it had to pass through the seats of a buggy as it passed in front of the ramp. They, as a group, did much better this time than last unit.