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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 don’t know what to do next if their students can’t sketch a graph. But they know their students can follow mechanical instructions, so they’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:

Kate Nowak:

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.


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!”

David Wees:

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.

Denis Roarty:

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.

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

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

Here is a resolution: ask your students for a sketch first.

I’ve been a bit obsessed with “Barbie Bungee,” a lesson on linear regression which you’ll find all over the Internet. It’s the kind of lesson that doesn’t seem to have any original mother or father, only descendants. (Here is NCTM’s version as well as a video from the Teaching Channel.)

Search the Internet for “Barbie Bungee handouts. I have. Invariably, the handout asks students to collect data for how far Barbie falls given a number of rubber bands tied around her ankles and then graph the results precisely. Often times those handouts include a blank graph with precise units and labeled axes.


Developing the question means starting from a more informal place. It means asking the students, “What do you think the relationship looks like between the number of rubber bands and Barbie’s distance? Sketch it.”


Asking students to sketch the graph serves so many useful purposes.

  • It helps us clarify assumptions. What do we mean by “distance”? Barbie’s distance off the ground? The distance Barbie has fallen?
  • Predicting the relationship makes it easier to answer questions about it later. This is from Lisa Kasmer’s research. It’s productive for students to decide if they think the relationship is linear, constant, increasing, decreasing, etc. What is its general shape? How do these quantities covary? As rubber bands increase, what happens to distance? Later, when students start to graph data precisely, the fact that the shape of their data matches their sketch will help confirm their results.
  • It’s great formative assessment. Do your students even know what a graph represents? Find out by asking for a sketch. If they can’t sketch a graph, their later precise graphing is likely only going to be mechanical and instrumental. (ie. “First number right, second number up.”)
  • Comparing informal sketches, which may vary widely, will likely make for better debate than comparing precise graphs, which will largely look the same. And controversy generates interest.

Which would make for a more interesting classroom debate? These three precise graphs?


Or these three imprecise sketches?


If the answer is “make a precise graph of a real-world relationship,” then developing the question means asking for a sketch first. That’s my resolution.

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

I’m proud of Graphing Stories. That was the first math lesson that drew in any serious way on my video editing hobby. That was the first math lesson that alerted me to the enormous value in sharing curriculum with teachers on the Internet.

I’m unhappy with the project now. I look at it and see the product of a math teacher who is eager to get to the answer of how to graph a real-world relationship and less interested in developing the question that leads to that answer.


If you watch Adam Poetzel’s graphing story, in which he slides down a playground slide, here’s what you’ll see:

  • A title announcing the quantity we’ll be recording: “height of waist off ground.”
  • A gridded graph that shows the scale you’ll use. It runs up to 10 feet.
  • A video of Adam sliding.

None of this is Adam’s fault, of course. That’s my editing.

Here’s how I’ve been doing a better job developing the question lately in workshops.

  • I play the video of Adam sliding.
  • I ask participants to tell their neighbors everything they saw. “Don’t miss a detail,” I say, and I’m always surprised by the details participants recall.
  • I play the video again and I ask the participants to tell their neighbors their answer to the question, “What quantities could we measure throughout the video?” People suggest all kinds of possibilities. Speed, distance from the left side of the screen, height, temperature.
  • Then I tell them I’d like them to focus on Adam’s height. I ask them to tell their neighbors in words what happens to his height over time.
  • We share some descriptions. People compliment and critique one another. Then I point out how difficult it is to describe his height over time in words alone.
  • Only then do I pass out the graphs.

The difference is immense. It takes an extra five minutes but participants are much better prepared to make the graph because they’ve spent so much time thinking about the relationship in so many informal ways. So many more participants walk away from the experience feeling like valued contributors to our group because the questions we’ve asked require a wider breadth of skills than just “graph relationships precisely.”

That’s the benefit. Again the cost was only five minutes of class time.

The most productive assumption I can make about any question I pose to a student is that a) there are questions I could have asked earlier to develop that main question, b) there are interesting ways I can extend that main question. In other words, I try to assume the question I was going to ask is only a thin middle slice of the corpus of interesting questions I could have asked. Tell yourself that. Maybe it’s a fiction. Maybe you use the entire question buffalo every time. It’s a useful fiction in any case.

Next: Let’s make a resolution.

BTW: Kyle Pearce got here first.

Featured Comment

Everything Harry O’Malley said.

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

Curmudgeon has taught math and science for thirty years and runs the Math Arguments 180 blog, an indispensable source of interesting prompts and questions.

Here are three images he’s posted in the last month:


In nine classes out of ten, you’ll find a teacher ask her students to calculate the area of those shapes. Maybe Curmudgeon would ask his/her students to calculate their area also. That’s a fine question. But Curmudgeon does an excellent job developing the question of calculating area by first asking:

  • What is an easy question we could ask about the shape? A medium difficulty question? A hard question?
  • What is the best way to find the area of the shape?
  • What combinations of addition or subtraction of figures could you use to find the area?

Each question develops the next question. Earlier questions are informal and amorphous. Later questions are formal and well-defined. They all develop the main question of calculating area. They all make it easier for students to answer the main question of calculating area and they make that main question more interesting also.

This technique runs back to my workshop participant’s advice that “you can always add but you can’t subtract.” Once you tell your students your question, you can’t ask “What questions do you have?” Once you tell students what information matters, you can’t ask them “What information matters here?” Once you tell them to calculate area, it becomes very difficult to ask them, “What shapes combined to make this shape?”

Tomorrow: Why Graphing Stories does a pretty lousy job of developing the question.

Preparation: If the main question is “sketch this real world relationship,” what are ways we could develop that question?

In his book Why Students Don’t Like School, Daniel Willingham writes:

One way to view schoolwork is as a series of answers. We want students to know Boyle’s law, or three causes of the U.S. Civil War, or why Poe’s raven kept saying, “Nevermore.” Sometimes I think that we, as teachers, are so eager to get to the answers that we do not devote sufficient time to developing the question. But as the information in this chapter indicates, it’s the question that piques people’s interest. Being told an answer doesn’t do anything for you.

Developing a question is distinct from posing a question. Lately, I try to assume that every question I pose is more precise, more abstract, more instrumental, and less relational than it had to be initially, that I could have done a better job developing that question. If I do a good job developing a question, my students and I take a little longer to reach it but we reach it with a greater ability to answer it and more interest in that answer.

Over the next few days, I’d like to offer an example of someone doing a good job developing the question and somebody else missing the mark. I’ll be the one who misses the mark with my Graphing Stories lesson. Math Curmudgeon will be the one who gets it right. After those entries, I’ll encourage us all to make a couple of resolutions for the future.

2014 Aug 13. Daniel Willingham weighs in:

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