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

- 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?”
- “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!).

Max,

Questioning is one of my favorite things and these are great suggestions!

This seems like a really important category of questioning to me, Max. Great contributions. I see these questions serving to jumpstart the higher-order questions and to help students to develop a schema for answering the lower-order-but-also-important questions. In addition to your prediction questions, I’ve found these two helpful for those purposes:

“What do you think an incorrect answer would look like?”

“What more information do you need here?”

Outstanding. Will crosspost on our site. Thx :)

Nice post. The holy grail of superb questions is :

Questions and Prompts for Mathematical Thinking. Page after page of *types* of questions with examples of each at various age-appropriate levels.

http://www.amazon.co.uk/Questions-Prompts-Mathematical-Thinking-Watson/dp/189861105X/ref=sr_1_1?ie=UTF8&qid=1379276602&sr=8-1&keywords=prompts+mathematical

Get a copy!

Love these questions! I don’t know if I’d call it the holy grail, but I also love the book that Rob recommends. My copy is dog-eared and repaired with book tape, but I refuse to replace it since the scars it bears are well-earned.

Max’s questions add a different dimension that I really like for use with certain classes and/or age groups. Sometimes I find that I need a fanciful or an imagination-priming question to help students boost themselves to the next level of understanding.

Keep ‘em coming!

- Elizabeth (@cheesemonkeysf)

Won’t get a copy via that link…..

Here is a US version of the link: http://www.amazon.com/gp/aw/d/189861105X

I really like these questions. I too believe that thoughtful questioning is essential to deeper mathematical discussions which in turn leads to greater understanding. I am always looking for new ideas. I like these. I will be adding your ideas to my tool box.

Hi Max,

Great questions. Here are two questions I use to encourage my students to feel an ownership of any mathematical concept we are working on:

1. How would you explain this to a ten year old?

2. How can you explain this using a drawing?

Thanks for your post,

Max

Max Hoegh, I especially like “How can you explain this using a drawing?” I think this will get kids to visualize their mathematical thinking. Excellent.

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Great post, Max. Good questions are important conversation-starters in class. One of my favorites (from Cathy Fosnot) is a simple statement, rather than an question. “Convince me.”

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It is great to see so many teachers getting on board with higher order questioning. Many schools across the country have realized that due to technological advances, students no longer have to think inquisitively. Students have only had to regurgitate formulas and numerical answers, usually given by a calculator, for way too long. Having resources out there to help learn how to properly question the students is so valuable. Teaching students how to think by asking and answering questions in Math helps in all aspects of life. Thank you to those who have provided these resources.