A while ago, I posted a entry about how I changed one of my favorite lessons on the nature of the normal force, from a lecture discussion format, to more of a discovery format. And at the end of that post, Jerrid Kruse in the comments challenged me to devise a way to measure the learning that took place in this lesson. Since I’d been mulling Frank Nosche’s excellent post, SBG to Nowhere, where he ponders the idea of a group assessment, and since I’d just seen in action in my colleague A’s classroom, I thought it was worth a shot.

Here’s what I came up with. I gave each group of stundets 20 minutes to look the assignment over and discuss how they’d go about answering it. Homework was to spend another half hour digging deeper into one of the questions on the assignment.

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Then, on the next day, I gave kids a clean copy of this assessment and 20 minutes to work on it. 20 minutes isn’t enough time for one student to do all the work, so they had to split it up, and trust that their classmates to show the group’s best understanding on each part. I then collected the assessment they completed, along any work they had completed the previous evening or class. I was amazed by some of the things students work on as they completed this assignment. One group asked, “if you double the weight on the table, does the deflection of the table double too?” Then they set out to do an experiment to test the idea. It was a gratifying sight to behold.

Here’s what their work looked like:

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Overall, this is great work. These kids have not studied oscillatory motion yet, and yet they are developing many of the key points of this model. But there are also places where they are applying incorrect models, not fully developing their explanations, and not seeing the connections between Newton’s laws that is necessary to fully understand the phenomenon of the normal force.

You’ll also notice there’s no grades on these assignments. As I was making comments, I fully intended to go back and write in what level of understanding their work showed of each of the various standards in our course, but that turned out to be pretty hard to do, since in this one section, a group might have shown a serious misconception about a concept, that in a later section, the group showed mastery. And I do think this might be a weakness of SBG—when you ask students big, open-ended questions the kind of questions you want a course to be about, it is very difficult to then go and assign fine-grained measures of understanding of a long series of concepts.

In the end, I took Frank Noschese’s advice he recently wrote to the AP Physics listerv in a wonderful post about enlightened grading:

There is no need to grade everything.

But there were still gaps in my students’ understanding that I wanted to probe further. So I told them we’d do a follow up oral discussion about their work. In particular, I wanted to see if they could begin to explain, in terms of free body diagrams and a velocity graph, what happens when you you release a brick onto a very wobbly table, and it oscillates up and down. This is one of the first times I’m trying to test their ability to work through long chains of reasoning.

So while my class was taking another assessment over the last chapter of the semester, I decided to pull each group out for a 20-30 minute discussion of their group assessment work. I started by walking them back through the questions asked on the assessment, but this time asking them to draw a free body diagram and velocity graph for each step of the way.

## Model outline of discussion

If you aren’t interested in the blow by blow outline of the discussion I had with the students, feel free to jump down to the video.

I start with drawing a picture of my hand holding a brick on the table, and say it is supporting half the weight of the brick (instant A).

Some important questions:

• What is the velocity of the brick right now?
• It’s at rest, so the velocity is zero.

• What model(s) can we use to describe the brick?
• Since the velocity is constant, we can use the constant velocity particle model (CVPM), and by Newton’s 1st law, we know that the net force must be zero, so we can use the balanced forces particle model (BFPM).

• What does a free body diagram look like for the brick?
• The forces must add to zero (BFPM), and there are only three forces acting on the brick: the normal force of the table, the gravitational force of the earth and the force of my hand

• Should the table be bent right now?
• The brick is touching the table, which means it is exerting a force on the table (we also know this by N3). This force causes the table to bend downward slightly.

The table is slightly bent when the hand is holding the brick, and the velocity is zero.

Next, I ask them to think about the moment when the hand releases the brick (instant B).

Relevant questions

• What is the velocity of the brick at this instant?
The velocity is zero. The brick still hasn’t started to move. However, it’s acceleration is not zero, it points downward in the negative direction
• What models apply to this situation?
Just the unbalanced force model, since there’s a net force. The acceleration will change as the board deflects, so this won’t be CAPM.
• What does the free body diagram look like? What is the net force, and what forces have changed since instant A?
• The free body diagram for the brick at instant B, when the hand releases the brick. There is a net force pointing downward.

The force of the hand has disappeared, but otherwise the forces aren’t the same. Since the brick’s velocity is still zero, it hasn’t pushed the table downward to increase the normal force.

Now the brick has begun to move downward. (Instant C)

• What does the Free Body Diagram look like now? How have forces changed since instant B?
• The free body diagram for the block when it is beginning to compress the table further (instant C) The normal force has increased from its value in instant B, causing the net force to decrease.

The brick has begun to move downward, which means it will compress the table more. This will cause the table to exert more upward (normal) force on the brick.

• What is the velocity at this instant, and how is it changing?
• The velocity is negative. It is increasing in the negative direction, but because the net force is decreasing as the normal force increases, the acceleration is getting smaller (closer to zero).

• How should the velocity graph look now?
• The velocity graph will look concave down. The slope represents the acceleration, which should be steepest in the beginning, when the table is compressed least, and as the table compresses more, the acceleration should get closer to zero (slope becomes less negative).

Now, as the brick continues to descend, and the normal force increases, eventually, the normal force will be equal to the gravitational force. (Instant D)

• What is the velocity of the brick at this instant? How is it changing?
• The velocity here is it’s maximum negative value. Students often think it is zero, but looking at the velocity graph and free body diagrams can help them to see that the velocity must increase between instants C and D (get more negative).

• What about the free body diagram? Net force? What forces have changed?

Free body diagram at instant D. The net force is now zero, since the normal force balances the gravitational force.

The net force is now zero, since the normal and gravitational forces are equal.

• What models apply to this situation?
• Since the forces are balanced, BFPM. This is also technically CVPM, since the velocity at this instant, is not changing.

• What does the velocity graph look like?
• The velocity has now reached its maximum negative value, and the acceleration is zero, so the slope of the tangent line at this point should be zero as well.

The brick continues to move downward past the point where the normal and gravitational forces are equal (Instant E).

• Describe the motion of the brick now.
• The brick still has a downward (negative) velocity, but it is slowing down. The acceleration at this point is opposite the velocity, so the acceleration must point upward.

• Describe the forces acting on the brick.
• A free body diagram for instant (E), when the brick has compressed the table past its equilibrium point.

The normal force is now larger than the gravitational force on the brick. This means the net force on the brick is upward.

• Describe the models that apply to the brick.
• The forces are not equal, so UBFPM applies. The acceleration is not constant, so CAPM does not apply.

• What does the velocity vs time graph look like now?
• The acceleration is positive (upward) so the graph should have a positive slope now

Now the brick reaches the maximum downward extension. (instant F)

• Describe the motion of the brick now.
The velocity is zero at this instant, however it is not at rest. It is accelerating upward.

• Describe the forces acting on the brick.
The normal force has now reached its maximum size. The net force is larger than it was at instant E, and it still points upward.

Free body diagram for instant F. The net force points upward, and the normal force is at its maximum value, since the table is maximally compressed.

• What models apply to this situation?
The acceleration is not constant, so CAPM doesn’t apply. The forces aren’t balanced, so UBFPM applies.

• What does the velocity vs time graph look like now?
The velocity is now zero. We can also tell we’ve reached maximum negative displacement becuase the area under the graph is largest.

So what happens now?

The brick will accelerate upward with an increasing positive velocity. As the brick rises, the normal force on the brick will decrease, causing the net force to decrease. The brick will then pass through equilibrium, and following that point, the gravitational force will be larger than the normal force. The net force will then point downward, causing the brick to slow down, until it eventually reaches the position where it was released. In reality, this won’t happen, since some energy is lost as the atoms in the table also oscillate, so the brick wouldn’t make it quite as high, and eventually the oscillations would dampen until the brick comes to rest at the equilibrium point.

The velocity vs time graph for the brick oscillating on the table.

## What this looks like in reality

Here’s a video from one of the discussions. I apologize in advance for the poor quality audio, as we were trying to be quiet so that the other students who were taking an assessment were not disturbed.

So in the end, I’d give myself very mixed results. If you watch the video, you’ll see there are places where students are deeply confused, and when they resolve this confusion, I’m not sure how much of that is due to them achieving understanding and insight, or simply being led to the answer by my questions. And I still wasn’t able to come up with any sort of grade to stamp onto their work. I do think the students got more out of this than they would a series of questions on a test, and I do think this is helping them to see assessments as more of an opportunity to learn, rather than to simply be “tested.” So that must be a good thing.