What do you notice in the story below? What are you wondering about? Leave a comment to tell us your thoughts!

• some gauge of student reaction to what you did (of course, from your viewpoint)
• some prediction of what students will do during your next session
• some reflection on what you predicted and what you now observed

It turned out that Erin’s quick (5 minutes tops!) reflection on what happened in class helped her work through the process. I found it interesting to read (and now I have something to look back on and refer to for this post) … but … Erin and I both agree that the time she took to write her own “teacher exit ticket” was most valuable for her.

Here are some excerpts:

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!).

I’ve written about it in the past, including in one of our Teaching with the Problems of the Week documents, How to Start Problem Solving in Your Classroom [PDF]. In that, I tell the story of the first time I explicitly asked students (who were “low-level” eighth graders) to tell me everything they “noticed” about a picture. The short version is that the students were awesome and their teacher was amazed at how much math they came up with.

Just as I started composing my post, I got email from my friend Debbie, who teaches at an elementary school school in the district I live in. She described the first lesson she did with a new class she’s co-teaching, in which she asked the students to notice and wonder. I asked her if I could use her story as a “guest post” on my blog, since I think it’s as compelling as anything I could have written. She agreed, so here goes.

Baking Blackberries

I want to bake blackberry cobbler. The recipe calls for a 9″ pie pan. All I have are rectangular ones.

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.

A buddy from Twitter Math Camp asked about the value of noticing and wondering for high school. She knew it could be powerful but wondered if colleagues and students would feel it was beneath them. Here are some things I’ve noticed, and some thoughts about them:

• High school students, especially juniors and up, are the shyest noticers and wonderers.
• By high school, kids (especially kids who are good at school) are very attuned to what the teacher wants them to notice, so they often say, “I don’t notice anything,” or “What do you want us to notice?”
• Noticing and wondering often starts with “what can I get away with” type noticings like, “I notice the graph is blue,” or “I notice your drawing isn’t very good.”
• A good prompt goes a long way with high school students in particular — they have a harder time suspending their disbelief than, say, third graders.
• It’s harder for high school students to make noticing and wondering a habit — they tend to be more likely to compartmentalize and think of it as an activity someone has to direct them to do rather than a skill.
• High school students, like all people, feel valued when their ideas are heard, recorded, and made use of — so they can get a lot of value from noticing and wondering.

Based on my noticing here are some tips for noticing and wondering with high-school students:

• Go multi-media. Start with pictures or videos. Some good places to find pictures and videos are:
  • Some of my favorite pictures for doing math with on the Internet: http://mathforum.org/blogs/max/pictures-for-the-lindy-scholars/
  • http://mathforum.org/blogs/pows/ (search around for the pictures and videos)
  • http://mathforum.org/pow/support/videoscenarios.html (though honestly other than Charlie’s Gumballs and Val’s Values, these are more for younger students. However, you might challenge high school students to make better videos).
  • Any of Dan Meyer’s 3 Act Math Tasks: http://threeacts.mrmeyer.com
  • Any of Andrew Stadel’s Estimation 180 images: http://www.estimation180.com
  • Any of Fawn Nguyen’s visual patterns: http://visualpatterns.org
• Make it clear that everyone has something to say and everyone’s things are valued, by:
  • Not commenting at all on students’ noticing and wondering, just listening with a welcoming expression.
  • Writing EVERY noticing or wondering down, whether it’s “relevant” or “right” or not.
  • Asking, at the end, “are there any noticings or wonderings that you’re wondering about?” and then encouraging the authors to clarify as needed.
  • After solving a problem or doing an activity that you launched with noticing and wondering, ask, “How did we use our noticings and wonderings?” and go back through them to value the contribution of each.
  • When something comes up that a struggling student had noticed, foreground that moment to help give that kid more status. For example sometimes a student notices something “obvious” but then later on that obvious thing turns out to be a key to the solution — value that contribution!
• Be explicit about the skill you’re teaching. Here are some ways to do that:
  • Ask students to notice and wonder with different lenses on. Choose a picture and ask “What would a scientist notice? What would an artist notice? What would an athlete notice?” Then ask “What would a mathematician notice?”
  • After noticing and wondering, once everyone’s voice has been heard, ask, “Which of these did you use math to think of?” and “Which of these could we use math to explore more?”
  • After everyone’s voice has been heard, talk about how as a group they’re getting better at noticing and wondering.
  • Look at noticings and wonderings from another class (people share lists on blogs and Twitter a lot and you can compare your list to theirs).
  • Notice and wonder about an example or image from the text, and then see if you noticed everything that the text pointed out about the image/example.
• Use student wonderings to drive lessons to make the class feel more student centered:
  • Encourage silly, creative, and fun wondering by valuing even off-the-wall wonderings (like when someone wonders “Does Sally have a tapeworm?” when you do a problem about Sally eating a whole pizza, encourage more thinking and discussion about tapeworms and the math behind them).
  • Choose a student wondering to explore, rather than the question you’d originally intended.
  • If student wonderings don’t make sense to explore that day, come back to them later, support the students to answer them on their own, and/or choose a different scenario where you and the students DO wonder the same things.
• Help them remember to use noticing and wondering:
  • When they’re stuck.
  • When they’ve got a possible answer.
  • When someone else is explaining.
  • When they’re reading a textbook.
  • When they’re reading a math problem.
  • When they’re looking at a math image like a table or graph.
  • All the time!

And as for how to help colleagues experience and appreciate noticing and wondering:

• Use your own students as guinea pigs and videotape or record the session. When students notice cool things or wonder something awesome, share that (innocently)!
• Math teachers love noticing and wondering about math-y images like this: http://mathforum.org/blogs/pows/free-scenario-filling-glasses-wcydwt/ so get them doing it as a fun exercise, and then thinking about how it can help students.
• Send your colleagues to http://101qs.com to get them wondering about math images and videos.
• Share Annie Fetter’s Ignite talk about noticing and wondering: http://www.youtube.com/watch?v=WFvYZDR4OeY
• Share some of these blog posts about noticing and wondering or with examples of noticing and wondering:
  • http://blog.mrwaddell.net/archives/808
  • http://kalamitykat.com/2013/02/19/intro-to-projectile-motion/
  • http://resolvingdissonance.wordpress.com/2013/02/15/noticing-and-wondering/
  • http://oldmathdognewtricks.blogspot.com/2013/02/noticing-and-wondering.html
  • http://justyourstandarddeviation.blogspot.com/2013/02/notice-and-wonder.html
  • http://blog.constructingmath.net/2013/02/analyzing-student-questions/
  • http://mathreuls.pbworks.com/w/page/63615099/Business

So, Math Twitter Blog o Sphere — if you’ve noticed and wondered with high school students, what have you noticed and wondered about them? What’s unique about the high school experience, and what helps high school students and their teachers value noticing and wondering?