All four of these kids appear to have slightly different models for understanding how this graph relates to Barbie falling. I’m assuming that we are just asking for a rough sketch here, as per your previous post.

#1 seems to indicate some important understandings of the relationship between the two variables. It is hard to come up with that graph by accident. My feedback to this kid would be to ask her what else could be modeled with this graph.

#2 seems to know that the more rubber bands there are, the longer the distance is. This is a pretty key understanding. I am curious about why they chose to start their graph at the origin, and I would ask them to explain their reasoning behind their creation of this graph. Either they will notice their mistake themselves, or I will have more information with which to ask a better question. One possible response would be to ask kid #2 and kid #3 to justify their graphs and defend them.

#3 seems to be confusing the graph as a map of the actual fall itself, but there could be other explanations for their choice of graph. For example, they could be interpreting distance fallen as just distance, in which case they might be thinking that this means the distance from the ground. I need more information about their thinking, and so I would ask them to explain to me what they have done, and then depending on their response, I ask another question.

#4 did not do the question. There are many reasons why this could be true. They could not be able to read, they could not have a starting place for figuring this out, they could be unwilling to make a mistake, they could be still thinking about the problem by the time I get near them, and more. I need to know more information. Is this a typical pattern from this student? Have they produced similar graphs in the past? What socio-emotional concerns do I need to be aware of? Based on my understandings of these questions, I would ask a question like “Can you explain to me what the problem is asking?” Ideally I have already spent enough time clarifying the problem before everyone started that this particular question will not give me much information (eg. the student does know how to explain the problem) and I will likely need to ask another question. Maybe I need to ask them to describe the relationship between rubber bands and falling bands in words first.

I’m working on a post about clearing working memory in order to create space for conceptual processing. I’ve also been exploring the literature on math anxiety. Student “D” might have this condition. When I was in the classroom, I told the kids, “I know what you think you can do. I’m pushing you to do just a little bit more. Make a drawing first. Show me how you think.” I followed through by grading based on growth, which infuriated my superiors but endeared me to my kids.

I’m an engineering professor, not a math teacher, and my courses are built around design projects. What I’d tell the students is probably what I usually tell the students in the lab: “Try it and see!”

I’ve found that answer particularly valuable when students go through some calculation and then ask me “is this right?” They need to learn that the real world is the arbiter of correctness, not the pseudo-authority of a teacher.

My approach may not be suitable for math classes, as a lot of math is not about the real world. I was a math major interested in the purest of pure math before I got diverted into a PhD in computer science and eventually into being an engineering professor, and I can see big differences between “math” approaches and “engineering” approaches. But bungee Barbie is clearly an engineering problem, rather than a pure math problem.

Guilty as charged. Last year, the motion graphs unit was the most loathed unit of the year—by both me and the students. I have to do a unit on the same topic again with the same kids this year—can’t say I’m looking forward to the experience. Sorry to be a downer, but my kids hate drawing graphs—and to be honest, I’m not a huge fan either. I’ll keep reading and maybe you experienced-types can help out me and my hapless 10th graders.

These 4 sketches, plus Dan’s “sketch first” strategy, made me realize that this task is about having kids experience a clear, structured connection between the number of rubber bands and the height of the fall. So, instead of reacting to these sketches, I redesigned the activity in hopes of focusing it more on that goal. Here’s the redesign…

Part 1

Have students: 1) create a slider in GeoGebra going from 0 to 12 counting by 1’s; 2) create a vertical line segment of length 2*r+3; 3) slide the slider and describe what is happening; 4) Answer the question: what could this be a model of? Take answers looking for a variety.

Part 2

Have students: 1) create another vertical line segment of length 2*r+6 right next to the original one, noting the change of the constant term from 3 to 6; 2) Slide the slider (which will animate both segments) and answer the question: what effect did changing the 3 to a 6 have?; 3) Answer the question: if this were modeling (pick a phenomena they brought up in step 4 of Part 1), what would the change of 3 to 6 mean in that context?

Part 3

Repeat Part 2, but with a third vertical line segment of length 4*r+3, noting that only the multiplier is different from the original.

Part 4

Introduce the Barbie context and have students do the drop, complete with fall height measurement, with 2 rubber bands so they have the opportunity to see and handle the props in context. Then explain the connection between the model and the situation. Specifically tell them two things: 1) The “r” in the expression for the length of the vertical line segment will stand for the number of rubber bands (so right now, r=2); and 2) the length of the line segment represents the distance from the point where the rubber band chain is attached to the support at the top to the tip of Barbie’s head at the bottom of the fall (in whatever units you want, let’s say inches).

Part 5

Have the students slide the slider to r=2 and determine which of the three line segment models predicts the length of the actual fall the closest (none will be very close, likely, especially if you work in millimeters! The purpose of this part is simply to get them to perceive the fact that different values for the multiplier and constant produce models of better or worse value for predicting the fall height.) Then challenge them to create a fourth segment, by changing the multiplier and constant, that comes as close as they can. There will be many strategies here from “playing with different numbers” all the way to making very accurate measurements of Barbie’s height and the length of single stretched rubber bands.

Part 6

Have them use their new and improved models to predict the fall height for 4 rubber bands. Then have them check their model’s effectiveness by actually doing it and seeing. Have them re-alter their parameters to get their model to make a more accurate prediction for 4 rubber bands without losing much accuracy for 2 bands.

Part 7

Have them repeat this “use your model to predict, then actually do the Barbie drop, then improve your model” procedure with 6, 8, 10, and 12 rubber bands. By the end, they should have a fairly fine-tuned model.

Part 8

Discuss strategies. In the discussion, bring out the connection between the multiplier and the length of a single rubber band, as well as the constant and the height of Barbie.

Part 9

Do a linear regression fit on the data and compare the student parameters with the ones generated by the fit.

Boom. There it is. Oh, and by the way, during the data gathering and function tailoring stages play Hideaway by Kiesza, then Black and White by Parquet Courts then Control by Olympic Ayres (at 70 decibels, cuz that’s what the research says is loud enough to stimulate creativity but not loud enough to distract). People will be bobbing their heads a little feeling like they’re doing something cool and inventing stuff.

Notice this version still uses Dan’s wisely proposed “sketch first, measure later” strategy, but there are a few key differences here: 1) The “sketch” is a line segment instead of a height vs. time graph. This is a much less abstract visual representation of the fall and easier to process; 2) The visual properties of the sketch are controlled simultaneously by your hands (sliding the slider with your finger on an iPad) and by mathematical processes (slider values entering into functions and coming out as line segment lengths). It’s hard to describe how satisfying this is. 3) The strategy is used multiple times throughout the activity instead of once, with immediate feedback following each instance. This allows students to “carve out” the linear model, slowly fusing their intuitive, informal knowledge about the situation with the formal mathematical notation and measurements.

So Dan, what is that one question you would ask all four students?

For me it might be one invitational comment for all: “Okay. Talk me through that…”

This can even work with the Opt Out learner – especially if you unpack and prompt with Polya question stems such as “do you understand the problem?” If that fails to elicit a response, maybe try asking them to talk through someone else’s graph.

It’s important for me to get in the habit of using the same script whether the solution is correct or not – that develops rigour in thinking and learners don’t come to hear “Talk me through that..” as “you’re wrong, but I’ll humor you.”

BTW we *did* ask learners to sketch them on a mini whiteboard *before* the group work or class discussion didn’t we? [Lemov, Take a Stand]

I should say, my comment assumes that the student’s problems in understanding were significant enough to need scaffolding beyond the initial “So what you’re saying is…” discussion. With my particular students, I found that multi-day scaffolding was often needed for gaps in understanding. Ideally, it’s done without becoming a time-suck.