Last week, students worked through a lab using photogates to investigate how the speed of a cart changes as it descends a ramp. We didn’t have much time to talk about it too much because my broke ankle situation meant having to cut class early to see the doctor. Most students, however, were still able to make good interpretations of the intercept and slope of their linear equation for velocity vs. time. This was helped by having students to a “quick and dirty” experiment to figure out how much speed the cart gained in traveling 1 second (all groups adjusted the location of the 2nd photogate until it measured 1 second later from the first). It was also supported by asking students to think about what value was typical for the first photogate each trial, and why that stayed relatively constant. Both of these gave them something to hang their hat on when interpreting slope and intercept.

In this lab case, the intercept and slope are both positive, so today we ventured into talking more specifically about the sign of acceleration in various cases. Here’s how the day started:

- Warm up to review the photogate lab: Question was “if your linear equation from our last lab had been v = 15 cm/s/s t + 10 cm/s, (1) how fast was the cart moving through the first gate, (2) how much speed does it gain each and every second? In asking students to say how they knew, we ended up drawing the graphs and talking about slopes and intercepts.
- A very brief mini-lecture to review definition of acceleration from their reading, how it connects our lab and the slope of velocity vs time, and some modeling of how to interpret the meaning of acceleration.
- A clicker question where students find acceleration at a particular time from a v vs t graph. Lots of discussion was needed here. Since the graph had an intercept, many students merely calculated v/t rather than dv/dt.
- A similar clicker question where students find acceleration for a graph but with negative slope.
- A very quick review of sign conventions for velocity vectors that we’ve established. (Knight in the algebra-based text always has positive be to the right).
- A clicker question with a motion diagram: the object is on the right side of the origin, slowing down as it approaches the origin. Students are asked to identify sign of velocity and position. Some discussion here, but pretty good here.
- 2nd clicker question with same motion diagram asking about sign of acceleration. About 80% said velocity was negative, which is incorrect.

What do you do when 80% of students have the wrong answer in a clicker question? My move here is to draw specific attention to a tool. I didn’t ask students to discuss. I didn’t lecture. I asked students to work in groups to draw a velocity vs time graphs for the situation, and to use the idea that the slope of velocity vs time. Many groups needed help with correct velocity vs time, but most of it involves reminding them that we had just said in the previous clicker question that the velocity was negative.

We drew a consensus velocity vs. time graph at the front and agreed that since the slope was positive, the graph implied that acceleration was positive. Now and only now I asked students to tell me why 80% of them had answered the acceleration was negative… it was the first time in class that students really opened up about wrong ideas… here’s what we got.

- “I thought that since the velocity was negative, the acceleration had to be negative”
- “I was thinking acceleration is v/t, so a negative velocity divided by a positive time is negative”
- “I was thinking that slowing down has to mean a negative acceleration, it’s taking away speed”

Then only then did I ask for, “How can we make sense of why the velocity vs. time time graphs says that acceleration is positive.” We got lots of good ideas here

- Slowing down should be negative acceleration, but your velocity is already negative… its like the two negative, means the acceleration must be positive to counteract the negative velocity.
- You can’t think of acceleration as v/t, it’s about the change in v; we can see in the graph, that even though the velocity is negative (below the axis), the change in velocity is always positively going “up”
- If it had like -30 m/s velocity to start and later you have -20m/s velocity, it’s almost like it gained +10 m/s velocity… it’s less in velocity debt, and a positive acceleration helped get it less in velocity debt.

I added my vector interpretation of how acceleration “changes’ velocity vectors by either “widdling them down” or “pulling them out”, and how the sign of acceleration is just about which way the acceleration vector points.

I think “order” matters here… while one-on-one in office hours, I have and still would ask student to explain to me their thinking about what I know to be the wrong answer. Often times, I say back to them their idea and why it makes sense, and then say, “I want to tell you about another way of thinking about it. Hear me out, it’s different than your idea, but I want you to understand my idea like I think I’ve understood yours.”

In class, however, I want students to get in the practice of using tools we’ve developed. We spent all that time talking about acceleration as slope of velocity vs time, so I want us to use that. Once we were pretty sure we were almost all wrong, sharing your wrong thinking was less risky. I encouraged students to share their previous wrong thinking through an analogy I learned (I think) from David Hammer. To tell students to, “You know when you sometimes meet someone new, and you immediately don’t like them, but you don’t know why?” I tell them that having ideas in physics is like that… sometimes you have an idea, but you aren’t sure why you thought it. Once you figure out, “why” you don’t like the person (e.g., maybe they remind you of someone), you can let go of not liking them. We need to figure out “why” it’s so tempting to think that acceleration was negative.

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