This pair of questions was on the multiple-choice test my students took:

A rock is thrown vertically upward, slowing as it rises until it reaches it high position, where it stops momentarily before falling back to the ground. Take the positive y-direction to be upwards.

1. Immediately after the the rock is thrown, its y-component of acceleration is _______

A. Negative

B. Zero

C. Positive

D. Not enough information to tell

2. Just before the rock hits the ground,  its y-component of acceleration is _______

A. Negative

B. Zero

C. Positive

D. Not enough information to tell

For all 250 some students,

30% of students answered negative for both (NN): To me, the answer pair might serve as a proxy for understanding acceleration as a change in velocity and an understanding of signing conventions for vector quantities.

25% answered negative on the way up and positive on the way down (NP): To me, answer pair might serve as a proxy for understanding acceleration as change in speed, with student thinking that negative signs mean slowing down and positive signs means speed up.

30% answered positive on the way up and negative on the way down (PN): This pair serves as a proxy for not having disentangled velocity (speed) and acceleration

15% for all other combinations: This serves as a collection of students who either misunderstood the question or have other confusions about concepts.

As is, each questions was graded independently. This makes it so that answering negative on either question gets you one answer correct. To me, this make no sense, because students’ answer to any one question is meaningless in terms of what they might know about acceleration. From the MC-question alone, we have evidence to support a claim that about 30% of students understand the concept fully, but 55% of them are getting some points. Additionally, we have evidence to support a claim that 25% understand the concept partially, but 55% got partial credit.

How might I grade this question?

I might grade this in terms of the pair combinations, and give more points to the student who answered NP than the student who answered PN. My reason for this would be that thinking of acceleration as denoting a change in speed is further along than someone who still has to figure out that acceleration is different from velocity (or speed).

The other option I would consider treating the two questions as one question, and only give credit for NN pair combinations, and give no credit for any other answer pairs.

Whether I did one or the other would depend on what I was trying to assess precisely about their understanding of acceleration.

What do you think?

1. September 27, 2011 5:43 pm

Is it Joe who says “BE THE BALL!” ? Method acting in physics? – if so, then it’s natural to associate your forward direction as being the positive direction, and answer NP. This seems a really fruitful “embodied” kind of way of thinking about the problem. The only thing incorrect about that reasoning is that it’s the wrong convention (we keep axes fixed, usually!, in intro physics) – but to make a mistake about a convention is a pardonable sin in physics. So I agree. NP is a better answer than PN.

2. September 27, 2011 8:11 pm

I think this is a cool idea thinking in pairs like that.

But I’d probably agree with ljatkins here. For my students at least, the most likely explanation is that they didn’t read the question stem fully and missed “Take the positive y-direction to be upwards” and assumed a “direction of the ball” point of view. Assuming you’ve ruled that out, I don’t think of these as separate questions so if forced to choose, I don’t think I’d treat them independently.

3. September 27, 2011 8:43 pm

I suppose one problem with the “direction of the ball view” is that it fails in the case when ball isn’t moving (i.e., the top). Thus, the question pair should be a triplet with a question about the very top.

Here is my prediction of how these degenerate levels might split with 3rd question introduced (as middle one)

NN splits to become NZN and NNN
PN doesn’t split, it merely becomes PZN

But what does NP become? I’m not sure. One part of me thinks you’d get more spread in answers, because the reasoning for NP itself can’t be used to produce an answer for top. Another part of me says that NZP would be dominant, just because it’s so salient that the ball has “nothing left”. But, a student really committed to the “direction of the ball view” (or a robot who only knew that rule) should be at a loss to say anything about the top.

There are many other logical problems with the “direction of ball view” for acceleration, including that observers don’t agree on when and where objects are speeding up, slowing down, and turning around. Turning around is not an “event”, in this sense. It also causes problems for calculus–speed is not differentiable everywhere, but velocity is.

I suppose it’s best to ask 6 questions! Ask them to draw acceleration vector everywhere and then state whether the component would give a positive or negative sign for each location. I’m kidding, sort of.