Kate Nowak, on my recommendation that teachers ask for informal sketches before formal graphs:

I agree with everything you say here. However, I think you will get silent resistance on this because teachers

donâ€™t know what to do nextif their students canâ€™t sketch a graph. But they know their students can follow mechanical instructions, so theyâ€™ll fall back on that.

Waitaminit. Is that *you*? Is Kate talking about *you*? Let’s talk about this.

Let’s say you’re working on Barbie Bungee. You’re tempted to jump your students straight to the mechanics of collecting and graphing precise data but you decide to develop that question a little bit first. You ask them for a sketch and the results come back:

A is (basically) correct. With zero rubber bands, Barbie falls her height and no further. Every extra rubber band adds a fixed amount to the distance she falls.

So what would you do with each of these sketches? Me, I think I’d say the same thing to each student.

**BTW**. Kate is back in the classroom after a short hiatus so there’s never been a better time to watch her think about teaching.

**Featured Comments**:

Iâ€™d need to think about it in context of the lesson and course flow. What happened before? What was done to orient them to the problem; do they have any concrete experience of the situation or is this more like just get something down, and then what kinds of things would they be basing their response on? What were your reasons for anticipating these 4? Are these kids in Algebra 2 or 8th grade? So I have more questions than answers.

Iâ€™m an engineering professor, not a math teacher, and my courses are built around design projects. What Iâ€™d tell the students is probably what I usually tell the students in the lab: â€œTry it and see!â€

All four of these kids appear to have slightly different models for understanding how this graph relates to Barbie falling. Iâ€™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 â€œCan you explain to me what the problem is asking?â€ 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.

My second reaction, when I read a few of the Barbie PDFs is that these things are so longgg …. I was a middle school science teacher and my ideal worksheet was a one pager. We did a lot of context building by talking through the prompt, what we needed to know and the experimental design. I didnâ€™t always pull it off well, but I also didnâ€™t have kids mechanically following my directions.

*This is a series about “developing the question” in math class.*

## 18 Comments

## Kate Nowak

August 21, 2014 - 10:51 pm -Hi! Thanks for the shout out. I’m not sure what I’d say to each student or all of them. I’d need to think about it in context of the lesson and course flow. What happened before? What was done to orient them to the problem; do they have any concrete experience of the situation or is this more like just get something down, and then what kinds of things would they be basing their response on? What were your reasons for anticipating these 4? Are these kids in Algebra 2 or 8th grade? So I have more questions than answers. But I like the direction of your question, to think about the how along with the what. In the context of a particular class, I think it would be a productive discussion for teachers to have.

## David Wees (@davidwees)

August 22, 2014 - 3:50 am -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.

## Howard Phillips

August 22, 2014 - 4:45 am -You cannot sketch a graph unless you have a clue as to what a graph is. I would imagine that before this experiment the kids have done other, simpler experiments, with data collection and simple graph plotting with some numbers. It is not enough to have plotted graphs of specified functions (y=3x+1) as there is no physical feel to this. Re Kate Nowak’s suggestion that one might have to ask for a description in words, I would say that there is no “might” about it. If they don’t do this they need prodding.

I like the Barbie experiment, but it appears to be isolated, there seems to be an unwillingness to look at anything “mechanical”.

## Ron Fischman

August 22, 2014 - 4:49 am -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.

## Denis Roarty

August 22, 2014 - 5:23 am -My first reaction to this post was to argue with the statement that graph A is correct. I had gone with graph C…

Then I read a couple of the Barbie PDFs and saw that I had assumed a different problem…. that they are chaining rubber bands together, not adding them in parallel as I had assumed. So the student who sketched graph C could have been exactly right but simply responding to a different problem (how many rubber bands in parallel)… this stuff is very valuable to see before diving into the experiment. I like this notion of having them sketch first.

My second reaction, when I read a few of the Barbie PDFs is that these things are so longgg……. I was a middle school science teacher and my ideal worksheet was a one pager. We did a lot of context building by talking through the prompt, what we needed to know and the experimental design. I didn’t always pull it off well, but I also didn’t have kids mechanically following my directions.

## gasstationwithoutpumps

August 22, 2014 - 5:48 am -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.

## Tere Hirsch

August 22, 2014 - 6:42 am -In answer to not knowing how to sketch a graph. I would fall back on an old nugget, The TI Motion Detector Activity, where students need to follow a projected graph by walking in front of the motion detector. At first it’s couner-intuitive, but they quickly orient themselves and are able to do it. (perseverence) The original activity came from NCTM Middle School article in the ’90s. It might have been called “Walk This Way”.

The question is: Can you match the given graph?

I had to experience it first and I was able to see how the kids felt. Great TPR Activity! Very accessible.

## Celeste Sinclair

August 22, 2014 - 8:05 pm -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.

## Chris Painter

August 23, 2014 - 5:43 pm -Completely agree with the premise that students need to try and predict what the graph will look like first. I used to make the mistake of not having students do this while teaching basic trig graphs… like you said, much more robotic and far less interesting discussion. Having said this, I can’t help but question whether we would be judging these graphs prematurely. In the last post, you said that asking them to sketch a graph will help lead to a clarification of the problem, particularly the variables. If such clarification has yet to be done, can we say that graphs B and C are inaccurate? It seems to me that having such graphs come up before clearly defining the variables could be beneficial as it paves the way for some awesome questions. One such question, “could you determine circumstances under which each graph could be right?” Granted, this convo. may need to be postponed until afterward or as an extension for some classes, but others could handle it up front I think.

## Dan Meyer

August 24, 2014 - 5:38 am -Thanks for your thoughts,

Chris. Quick assumptions check: in your ideal lesson here are students using different frames of reference on the y-axis or do they all converge at a certain point?## Chris Painter

August 24, 2014 - 7:34 am -Honestly, I don’t know that I would have an ideal here as both could work out very well, it depends on the class and students. I know that sounds like avoidance, but there are so many different situations that could play out. I think that student responses to questions regarding what they assumed the y-axis to represent as well as how well individual groups are able to respond to both the initial task and probing would guide my actions as I worked through the lesson. Maybe it is just me, but it always seems as though my most rigid type of plans, the ones where I have a plan that I really want to push, seem to go the worst.

As a teacher, convergence can be easier and quicker but if we can get twice as many miles out of the same lesson by making our jobs a tad more difficult, seems like a worthy trade off (not that I think this is possible with all classes).

## Harry O'Malley

August 25, 2014 - 2:04 pm -These 4 sketches, plus Danâ€™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. Hereâ€™s the redesignâ€¦

Part 1

Have students: 1) create a slider in GeoGebra going from 0 to 12 counting by 1â€™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 â€œrâ€ 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 Barbieâ€™s head at the bottom of the fall (in whatever units you want, letâ€™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 â€œplaying with different numbersâ€ all the way to making very accurate measurements of Barbieâ€™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 modelâ€™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 â€œuse your model to predict, then actually do the Barbie drop, then improve your modelâ€ 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 thatâ€™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 theyâ€™re doing something cool and inventing stuff.

Notice this version still uses Danâ€™s wisely proposed â€œsketch first, measure laterâ€ strategy, but there are a few key differences here: 1) The â€œsketchâ€ 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). Itâ€™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 â€œcarve outâ€ the linear model, slowly fusing their intuitive, informal knowledge about the situation with the formal mathematical notation and measurements.

## Harry O'Malley

August 25, 2014 - 2:13 pm -Oh, and to make things a little more clear, here’s the GeoGebra file with the slider and three segments created in parts 1, 2, and 3.

https://www.dropbox.com/s/zsofsyyb71h7j1s/BarbieBungee.ggb?dl=0

## Noah

August 26, 2014 - 6:34 pm -So Dan, what is that one question you would ask all four students?

## Dan Meyer

August 26, 2014 - 7:12 pm -Could’ve sworn I posted it. Dumb blog comments.

I’d translate a student’s graph back to them in words and ask them if that’s what they wanted to represent.

ie. For A:

“So you’re saying here with one rubber band, the Barbie has fallen a little bit. Is that right?”

## Pete Capewell

August 30, 2014 - 2:45 am -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]

## LMacfarlane

September 10, 2014 - 8:15 am -To benefit the student who made no sketch at all, I might collect the graphs at the end of the activity and choose a couple to show the next day. I’d probably choose ones that weren’t quite right, but in different ways. (Sometimes it’s good to address non-understanding without delay, but sometimes it isn’t. There can be an advantage to letting the student try to wrap his brain around it [within your framework] on day one, then coming back to it the next day. The student has struggled, you’ve observed the struggle, there’s been some wait time, and now you’re more prepared for Take Two.)

Next day, I’d start with one of those graphs, and maybe say, “Every point on a line communicates information. For example, THIS VERY POINT says that …” Have students explain another point, work on interpreting the graph, maybe throw up the second student graph I chose to reinforce the ideas, then move on to another activity to build on these same ideas. Time permitting, of course. (ha ha)

In brief, I’d want to return to the meaning of a point (in context) for a student unable to sketch a graph.

*Unless the student didn’t sketch merely for fear of being wrong, in which case you’re looking at a different kind of problem.

## LMacfarlane

September 10, 2014 - 8:28 am -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.