First, a video I made with the help of some workshop friends at Eanes ISD. They provided the video. I provided the tracking dots.

To develop the question you could do several things your textbook likely won’t. You could pause the video before the bicycle fades in and ask your students, “What do you think these points represent? Where are we?”

Once they see the bike you could then ask them to rank the dots from fastest to slowest.

It will likely be uncontroversial that A is the fastest. B and C are a bit of a mystery, though, loudly asking the question, “What do we mean by ‘fast’ anyway?” And D is a wild card.

I’m not looking for students to correctly invent the concepts of angular and linear velocity. They’ll likely need our help! I just need them to spend some time looking at the deep structure in these contrasting cases. That’ll prepare them for whatever explanation of linear versus angular velocity follows. The controversy will generate *interest* in that explanation.

Compare that to “rushing to the answer”:

How are you supposed to have a productive conversation about angular velocity without a) seeing motion or b) experiencing conflict?

See, we originally came up with these two different definitions of velocity (linear and angular) in order to resolve a conflict. We’ve lost that conflict in these textbook excerpts. They fail to develop the question and instead rush straight to the answer.

**BTW**. Would you do us all a favor? Show that video to your students and ask them to fill out this survey.

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

**Featured Comment**:

Bob Lochel, with a great activity that helps students *feel* the difference between angular and linear velocity:

]]>I keep telling myself that I would love to try this activity with 50 kids on the football field, or even have kids consider the speed needed to make it happen.

Without some physical activity, some sense of the motion and what it is that is actually changing, then the problems become nothing more than plug and chug experiences.

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 nextif 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**:

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!”

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.

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.*

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.

]]>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**

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?

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:

@ddmeyer Devoting sig time to explicating why something matters to the field, and doings so in a way that gets student buy-in.

— Daniel Willingham (@DTWillingham) August 13, 2014

@ddmeyer Prob may be made interesting via practical app, everyday life examples, but developing=deeper interest by situating 2days problem..

— Daniel Willingham (@DTWillingham) August 13, 2014

@ddmeyer ….. relative to what else we’ve learned.

— Daniel Willingham (@DTWillingham) August 13, 2014

]]>@ddmeyer Ideally, "developing" means getting s's to understand why it's important to know the answer & not just for an practical reason.

— Daniel Willingham (@DTWillingham) August 13, 2014

For instance, Dandy Candies tells students they’re going to package up 24 cubical candy boxes. It asks them, “Which of four packages uses the least amount of packaging? Which uses the least amount of ribbon?”

This is the usual house style. Concrete imagery. No abstraction. Contrasting cases. Predictions. Students make their guesses. Then they get the dimensions from the video. They calculate surface area and ribbon length. (Ribbon length is a little bit more interesting than perimeter but not by a lot.) They validate their predictions with their calculations.

But then we ask them to find out if *another* package dimension will use even *less* material.

So now the students have to think systematically, tabling out their work so they don’t waste effort finding the surface area of a lot of different prisms.

Contrast that against a worksheet like this, which is practice also, though rather less purposeful:

Where else have you seen purposeful practice?

I’d look to:

- Robert Kaplinsky and Nanette Johnson’s Open Middle project.
- Malcolm Swan’s tasks: impossible points & break 25.
- Bryan Meyer’s scientific notation task.

**BTW**. Given any number of cubical candies, what is the best way to minimize packaging? Can you prove it? I can handle real number side lengths but when you restrict the sides to integers, my mind explodes a little.

A workshop participant gave this algorithm. I have no reason to believe it works. I also have no reason to believe it doesn’t work.

- Take the cube root of the volume.
- Floor that to the nearest integer factor.
- Square root the remainder factor.
- Floor that to the nearest integer factor.
- With the remainder factor, you have three factors now.
- The smallest of all three factors is your height.
- The other two are your length and width. Doesn’t matter which.

Note to self: test this against a bunch of cases. Find a counterexample where it falls apart.

]]>A person lives inside a room that has baskets of tokens of Chinese characters. The person does not know Chinese. However, the person does have a book of rules for transforming strings of Chinese characters into other strings of Chinese characters. People on the outside write sentences in Chinese on paper and pass them into the room. The person inside the room consults the book of rules and sends back strings of characters that are different from the ones that were passed in. The people on the outside know Chinese. When they write a string to pass into the room, they understand it as a question. When the person inside sends back another string, the people on the outside understand it as an answer, and because the rules are cleverly written, the answers are usually correct. By following the rules, the person in the room produces expressions that other people can interpret as the answers to questions that they wrote and passed into the room. But the person in the room does not understand the meanings of either the questions or the answer.

This is a pretty perfect parable.

**Featured Comment**

Joel Patterson, talking to commenters who’d send this parable along to their students’ parents:

]]>I like this parable. Before you condense it, and give it to all your parents, consider whether those parents have science/engineering backgrounds. It’s a pretty complicated picture to envision. I think S/E people would grasp it (if they haven’t heard of it already) and would get your point. But if the parents have less of an S/E background, the complicated parable is likely to bore them and not convey your point.

Have an explanation at hand that is more like the guitar players who can improvise, not just repeat the 5 songs they’ve memorized. Or cooks who can put together a soup without the recipe because they know which spices and foods have good flavors together.

**tl;dr**. I made another digital math lesson in collaboration with Christopher Danielson and our friends at Desmos. It’s called Central Park and you should check out the Walkthrough.

Here are two large problems with the transition from arithmetic to algebra:

**Variables don’t make sense to students.**

We give students variable expressions like the exponential one above, which they had no hand in developing, and ask them to evaluate the expression with a number. The student says, “Ohhh-kay,” and might do it but she doesn’t know what pianos have to do with exponential equations nor does she know where any of those parameters came from. She may regard the whole experience as one of those nonsensical rites of school math which she’ll forget about as soon as she’s legally allowed.

**Variables don’t seem powerful to students.**

In school, using variables is harder than using arithmetic. But what does that difficulty buy us, except a grade and our teacher’s approval? Meanwhile, in the world, variables are responsible for anything powerful you have ever done with a computer.

Students should experience some of that power.

**One solution.**

Our attempt at solving both of those problems is Central Park. It proceeds in three phases.

*Guesses*

We ask the students to drag parking lines into a lot to make four even spaces. Students have no trouble stepping over this bar. We are making sure the main task makes sense.

*Numbers*

We transition to calculation by asking the students “What measurements would you need to figure out the exact space between the dividers?” This question prepares them to use the numbers we give them next.

Now they use arithmetic to *calculate* the space width for a given lot. They do that three times, which means they get a sense of the parts of their arithmetic that *change* (the width of the lot, the width of the parking lines) and those that *don’t* (dividing by the four lots).

This will be very helpful as we take the next big leap.

*Variables*

We give students numbers *and* variables. They can calculate the space width arithmetically again but it’ll only work for *one* lot. When they make the leap to variable equations, it works for all of them.

It works for sixteen lots at once.

Variables should make sense and make students powerful. That’s our motto for Central Park.

**2014 Jul 28**. Here is Christopher Danielson’s post about Central Park on the Desmos blog.

**Featured Comment**

In thinking further about your complaint about “Write an expression” I think what is also going on in this app is a NEEDED slowing down of the learning process. The text (and too many teachers) are quick to jump to algorithms before the students understands their nature and value. Look how long it takes to get to the concept of an appropriate expression in the app: you build to it slowly and carefully. I think this is at the heart of the kind of induction needed for genuine understanding, where the learner is helped, by scaffolding, to draw thoughtful and evidence-based conclusions; test them in a transfer setting; and learn from the feedback – i.e. the essence of what we argue understanding is in UbD.

]]>One reason I like this activity so much is that it hits the sweet spot where “What can you do with it?” and “What does it mean?” overlap.

In one case, a teacher was teaching a lesson about division with remainders and the example was packaging meatballs in pack of 4. When faced with the problem of having 13 meatballs and needing 4 per pack, one student’s solution was “I would eat the extra meatball and then they would all fit.” It was so funny and joyful to see that all thinking was welcomed and the teacher artfully led them to the general thinking that she wanted by the end of the lesson.

I can trace my development as a teacher through the different reactions I would have had to “I would eat the extra meatball,” from panic through irritation to some kind of bemusement.

**BTW**. The comments here have been on another level lately, team, including Simon’s, so thanks for that. I’ve lifted a bunch of them into the main posts of Rand Paul Fixes Calculus and These Tragic “Write An Expression” Problems.