Lots of times when we ask questions in math class, they fall into 2 categories:

1. Procedural/Right Answer questions, e.g. “What do you call the longest side of a right triangle?” or “What did you get for number three?” or “What is the mode of this data set?”
2. “Higher-Order Questions” aka hard questions, e.g. “Why do you think someone might have come up with that [wrong] answer?” Or “Which of these is correct? Defend your choice.”

In my experience, even though we want all kids to be able to answer both types of questions, they’re both tricky. For the first type, kids either know what I’m looking for or they don’t, and so I either get a few loud kids participating or awkward silence, and often devolving into off-task behavior.

For the second type, look out! Talk about awkward silence and devolving into off-task behavior. Kids look at me like I’m crazy when I ask them to synthesize, justify, explain, etc. And they wait. They can’t out-wait me (I am the king of outlasting the awkward silence) but they sure do try.

So I’ve been trying to come up with questions that are good, math-y questions that don’t fit in either of those categories. I want questions that every kid can answer, by virtue of being a human (and therefore reasonably observant, semi-rational, interested in other humans, and decently resourceful). I want questions that kids see some need to answer, or are interested by. And I want questions that get kids doing some intellectual work that will help them do more work. And that doesn’t shut them down. Oh, and that helps me figure out what’s going on with them. And that aren’t questions I already know the answer to. Here are some:

• What do you notice about ______?
• What are you wondering?
• What’s going on in this ______?
• What’s making this hard?
• On a scale of 1-10, how easy is this for you? How come?
• What’s one thing you remember about ______?
• Here are three different ______. Which do you like best? What’s one thing you liked about it?
• Tell me one thing you thought about problem three.
• What’s the first thing that pops into your mind when you see this?
• What’s the fourth thing that pops into your mind when you see this?
• What do you think a mathematician might notice about this?
• If you saw this image/story/statement on a math quiz, what question(s) might go with it?
• If your math fairy godmother appeared right now and offered to give you one helpful hint, what would you ask her for?
• How confident are you in the work you’ve done so far?
• The answer to the problem you’re about to work on is ______. How could someone have figured that out?
• Have you ever had an experience like the one in the story?
• What do you think the person in the story might be feeling?
• Why do you think I showed you this?
• What’s one thing you like about what she just said?
• What’s one thing you’re wondering about what he just said?
• What’s your best guess for the answer to this problem?
• What is an answer that is definitely wrong for this problem?
• Make a prediction. What do you think will happen…
• Without writing anything down or calculating or thinking too hard, could ______ be the answer?
• What’s your gut feeling?
• Do you have a reason or a gut feeling (or both)?

And from the comments/Twitterers:

Dan Meyer:

• “What do you think an incorrect answer would look like?”
• “What more information do you need here?”

This Google Doc from Justin Aion of questions he uses to help his students become better readers in math class.

Max Hoegh:

• “How would you explain this to a ___________?”
• “How would you explain this with a drawing?”

Ed note: In part because some of us teach 10-year olds, but also because I think that explaining math is a constant process of revising and adjusting based on audience feedback, I left the audience of “How would you explain this to…” blank. I like the idea of playing around with different audiences for different explanations. Like, “How would you explain this in a Tweet?” or “Send a friend who missed class today a text message about what they missed.” or “How would you explain this to a friend? How would it be different to explain it to an enemy?” I even know of a teacher who pasted her class picture from 3rd grade on a chair and will drag that chair to the front of the room when she wants kids to explain something clearly and step-by-step.

In general, I’m trying to push myself to ask more questions in which I’m not trying to get the kids to say the thing I need them to say. Instead, I’m trying to find questions that get kids to put into words the things they need to say — to let me know what’s on their mind, what their current working model is, where they’re stuck and what they’re ready for. I can make predictions but I never know exactly where a kid will turn out to be, and so I try to maximize what I can learn about them, while using questions that let them know I value them and really want to hear their ideas (not them stating my ideas for me!).

In the Powerful Problem Solving book that’s coming out this month (!) we included the famous “Make a Peanut Butter and Jelly Sandwich” activity, in which students write instructions for making a sandwich and then their teacher or partner acts the instructions out very literally — e.g. “put the peanut butter on the bread” is interpreted as “place the (unopened) jar of peanut butter on top of the pile of bread.” Or “spread the peanut butter on the bread” doesn’t imply “with a knife” so you’re scooping out globs and globs of peanut butter and smearing it all over all sides of the bread with your fingers.

I got to visit a classroom where the activity was being implemented. The activity is in a chapter on good math communication and focuses on the important of revision. Watching the activity in action, I was struck by the subtle differences between focusing on precision and focusing on revision.

If you focus on precision, this can become a kind of “gotcha” activity. An activity in which the teacher sets up the kids by saying, “hey, this is really simple, everyone knows how to make a PB&J, so of course you can explain it…” knowing that they won’t be able to explain it to the alien the teacher is going to pretend to be, without warning them. The message the kids might take away is “writing in math class means painstakingly explaining your work to someone pretending to be an idiot” which is clearly not fun. There’s a reason the word “pain” is the first syllable in “painstaking!”

Because in fact, giving instructions that teach someone how to do something is NOT easy. The tricky part of giving instructions is figuring out what the other person does and doesn’t know, and tailoring (aka revising) your instruction to meet their needs.

When I watched the “Make a Peanut Butter and Jelly Sandwich” activity, the teacher had a GREAT launch — she showed a picture of an alien and explained that on Bob the Alien’s planet, they’d picked up radio transmissions of “Peanut Butter Jelly Time.” Bob wants to know what this amazing experience of  making a peanut butter and jelly sandwich might be, since the aliens so enjoyed hearing the song!

So right away the students were in a mindset of needing to figure out how Bob the Alien thought. After the teacher acted out some instructions as Bob, the kids started to say, “Wait, Bob has no common sense!” and “Bob is taking these directions SO literally!” and then “Wait, can I change my instructions?” or “I need to revise this part.”

The activity was structured to have lots of revision moments built in — once after seeing “Bob” in action on some sample directions and then again after having a peer pretend to be Bob. The students revised other people’s instructions, not their own, to help make the revision not personal, and more about thinking about what they’ve learned about Bob.

The teacher’s language can help reinforce that we’re revising based on new data, not just recognizing that we should have done better the first time. The teacher can ask, “What are some different ways you think Bob might interpret that? How would you change your instructions if Bob did this instead of that?” The teacher can also ask, “What did you think about Bob before he read the first directions? How did your thinking change after you saw how he interpreted them?”

Writing instructions on how to make a Peanut Butter and Jelly Sandwich for an alien can be a great experience that helps students understand what revision is, why we revise, why feedback from others is an important part of revision, and why explanations might need to take different perspectives into account. The key is to make sure that the expectation is that we will get new information about how the alien thinks and revise based on that. This is not a “haha, you thought you knew how to write instructions!” It’s a “wow, that alien sure does interpret things weirdly, I guess I’ll have to try again now that I know that!”

Finally, it’s nice to have the students be the ones to articulate what they learned from the experience. After the activity, it’s neat to wonder, “What does this have to do with math?” or “What math experiences does this remind you of?” One thing I thought I might do is make an audience-o-meter with “Bob” at one end and “Myself” at the other and try to think about different audiences we might write mathematical explanations for, and the levels of detail, amounts of revising, etc. that we might need to include depending on where we are on the audience-0-meter.

PS — with the rising prevalence of peanut allergies you might want to make cream cheese and jelly sandwiches, or use American cheese slices, or do instructions for brushing your teeth…

PPS — Another cool part of the activity is when the students get to be Bob. They’re playing with the mathematical skill of coming up with a counter-example, of finding other ways to interpret a mathematical definition or instruction… which is so important! It’s sort of like they’re practicing the skills in this amazing play-let based on key moments in modern Geometry.