Here is a picture of a fountain from Pearson’s Common Core Geometry iBook. (Full disclosure: I consult with Pearson.)

Given ten tries, you’d never guess the question connected to that image: “What is the measure of the arc of the circular basin of the fountain that will be in the photograph?”

Same with this line from problem 23 on page 351:

Campers often use a “bear bag” at night to avoid attracting animals to their food supply.

It is followed by:

Are angle one and the given angle alternate interior angles, same-side interior angles or corresponding angles?

Not only will those questions fail to interest many of my students but they’re also unnatural and disconnected from the context to which they are attached. The fountain doesn’t want that question. The bear bag has no use for its question. Students notice that disconnect. Some have fully internalized that disconnect and concluded that math is some alien, otherworldly thing they’ll survive and then forget as quickly as possible.

What Do We Do?

Over at Danâ€™s site people have been discussing these last set of questions and we find, naturally, Dan promoting his brand of â€œMake the prompt scream the question you are looking forâ€ …

I hear it too often in emails, tweets, and conversations after conference sessions:

“It took some time but I prompted them a little and they asked the question I wanted them to focus on!”

“They guessed the question I wanted them to ask!”

Just ask the question. My point has never been that you should never ask questions rather that you should ask questions with some certainty they will be interesting and seem natural to your students.

How can you tell in advance that a question will be interesting or seem natural to your students? Ideally, I’d have a room full of students I could run ideas past â€” an on-call focus group. I’d punch a button and they’d snap to attention. Then I’d introduce a context and a question and they’d give me a thumbs up or down. (Standard disclaimer: math is a context.) Maybe they’d suggest other, more interesting questions. That would be great â€” all of it â€” but I don’t have those students on call. I have you guys instead, and that’s way, way better than nothing.

But just because the football player runs through tires on the scrimmage field doesn’t mean he runs through tires on game day. See? 101questions is our scrimmage field. It isn’t the game itself.

BTW: Avery Pickford has some smart writing along these same lines.

Previously: Unnatural Currents

Featured Comment

Inquiry-based science teaching sometimes gets bogged down in similar games of “guess what the teacher wants you to say.” Â Almost as frustrating as known-answer questions are these, which I shall start calling “known-question answers.”

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Different Dave

May 7, 2012 - 1:39 pm -

I nearly lost it when I saw the “bear bag” problem describe the trees in the picture as parallel when they obviously aren’t. Why not find a better picture (metal poles, maybe)? Why not have students solve for the angles using the non-parallel reality? It’s like the book’s stomping on my hand and yelling “don’t forget, you’re playing by MY rules, kid.” Math’s too beautiful for this. Tell me REAL angle of the trees, book! I can handle it!

2. #### louise

May 7, 2012 - 5:36 pm -

Oh, I was thinking of slamming a pic up on the projectors, and asking for questions. Lots of them. Not more than 4 questions from the room? Not enough: pick another photo. Then have the kids pick a couple of the questions that they want to pursue alone or with a group. They can first find out what they think they need to know. Then they can go find it out. Then they can see if it leads to any more questions. Like Science.
Example: Bear bag? If a bear climbs the fat tree on the left, how far does she have to reach to get the bag? Does this seem possible for a bear? (this from my daughter) Leading to – how many bags are opened by bears every year? How stupid do people think bears are? What if the bear left this bag as bait for campers? I can’t answer some of these.
Of course, my school will never let me do it. I’m supposed to lecture to those awestruck rows of teenagers (sure they are merely fantasizing about the boy/girl next to them, but at least they’re quiet).

3. #### Paul Salomon

May 7, 2012 - 6:01 pm -

Dan-

I’m glad to hear you condemning, “make the prompt scream the question you’re looking for.” Thanks.

4. #### Janet Abercrombie

May 8, 2012 - 5:59 am -

I’d want to know the volume of the person (in cubic cm) who decides to relieve heat exhaustion with a fully body dip into the fountain.

I think I see palm trees in the photo…

5. #### Sean

May 9, 2012 - 7:22 am -

The top ten first acts on 101qs with their most frequent questions:

1. How long?
2-10 How many/how much?

Timon implied that this may obscure the underlying variation in topics addressed by the photos, and I vigorously agree. This isn’t strictly a rates/ratios party. There’s volume of irregular solids, volume of a pyramid, area of an annulus, probability, and currency conversion in there. It could be that how many/much/long is a logical pathway to rich application.

That said, I’m curious if the ideas in this post are in tension with the idea of a perplexity score. For teachers who have used 101qs: how well do perplexity scores predict student perplexity?

6. #### Dan Meyer

May 9, 2012 - 8:17 am -

Sean:

That said, Iâ€™m curious if the ideas in this post are in tension with the idea of a perplexity score. For teachers who have used 101qs: how well do perplexity scores predict student perplexity?

I dunno if they’re in tension. For sure, I wouldn’t go so far as to say they’re in agreement yet. Maybe I’ll copy and paste an e-mail I sent last night to another reader. A little too research-y for the main floor I suppose but if you have any ideas here, I’d be obliged:

I’ll be showing 101qs to the other math ed doc students in a couple weeks, trying to see if I can set the hook and get some interesting questions / criticism out of them. As I think about that presentation, I think the toughest part will be laying the theoretical groundwork that leads to 101qs.

Stuff like, 1) visuals are under- and mis-used in a lot of print curricula, 2) tasks should lead with a concise summary of their task, not end with it, 3) all other things being equal, we’d rather students be interested in the questions we ask than not, 4) this perplexity measurement accurately measures student interest even though 5) the sample (math teachers) doesn’t generalize to the population that interests us (math students) … and it just go on and on like that in my head.

Someone in my MCTM breakout called me out for papering over what is (in his words) a large philosophical shift in how we think about math instruction. He didn’t say whether or not it was a good shift (and I’m still uneasy about some of this stuff, myself) but it seems like I need to be more explicit about a lot of this which (unfortunately) means I need to think it out a lot more before sharing it in my professional circles.

7. #### Sean

May 9, 2012 - 9:06 am -

Quick take:

1) Visuals are under- and mis-used in a lot of print curricula.

Your work (on this project and throughout the years, really) is clear and persuasive here.

3) All other things being equal, weâ€™d rather students be interested in the questions we ask than not.

Easy sell.

4) This perplexity measurement accurately measures student interest even though the sample (math teachers) doesnâ€™t generalize to the population that interests us (math students).

From a measurement perspective, this might be an area of pushback/concern. For one thing, your sample doesn’t generalize to math teachers, much less math students. Teachers engaged in 101qs differ mightily from teachers in the general population, likely in unobservable ways (motivation, eagerness for math ed reform, etc.). The problem here is identifying which way the bias goes. Are scores, on average, biased upwards because we think this stuff is cooler than kids will? Or are scores biased downwards because we have a more discerning, critical eye? Are there certain kinds of photos/videos that are more stimulating to this particular cadre of teachers than others? I’d wager you have a few theories but it might be tough to say.

The other issue might be the measurement instrument itself. While a binary indicator is obviously useful in some respects, it might constrain you in others. For one thing, “skip it, I’m bored” is different from “skip it, I’m not perplexed” or even just “skip it.” Others have made the point that they often skip for reasons other than boredom. I gather that refining the instrument by adding a Likert-style button system is not what you want, but there are tradeoffs with your current system.

Additionally, that prompt- “what’s the first question that comes to your mind?” – may subtly encourage users to not sit and work with an image, which is likely an outcome that math teachers are interested in for their students. I recall in your Ukiah presentation the image of the water graph in Edmonton during the Olympics. That image (to this user) was wildly perplexing. I wonder how well it would fare with the instrument.

In short, get a multi-million dollar grant and run a 3 year RCT on this.

8. #### Jason Dyer

May 9, 2012 - 12:08 pm -

What I’d really like to see is a 101qs-type deal where rather than prompting for an immediate question, the question the poster had in mind is rated as interesting/not interesting. I’d have a hard time constructing an image where the immediate question is “how fast do you have to be driving to outrun a speed camera?” but I can attest that students are very interested in the result.

Or pulling from something I wrote for the Illustrative Mathematics project, I have a map of the United States with the question “if you started anywhere and walked in a straight line until you hit the border, what’s the longest walk you can take? How?” (It’s rewritten slightly for the 2nd grade crowd.) Would the question occur to most people just looking at the map? Probably not. But I’d love to know how interesting people consider the question to be.

9. #### Dan Meyer

May 10, 2012 - 4:40 am -

Thanks for your thoughts here, Sean. I’m starting to realize the trade-offs I’ve made between usability (from the average web users POV) and validity (from a researchers). I may need to create a separate, more nuanced instrument if I have any questions worth studying here.

I recall in your Ukiah presentation the image of the water graph in Edmonton during the Olympics. That image (to this user) was wildly perplexing. I wonder how well it would fare with the instrument.

Me too. My guess is it’ll get clobbered. Let’s find out.

10. #### Mylene

June 5, 2012 - 3:03 pm -

Thanks for this point —