If you’ve spent any time around the Math Forum folks, you’ve heard of “I Notice, I Wonder” — two little phrases that we use to start students talking mathematically. We’ve seen the questions “What do you notice?” and “What do you wonder?” used to launch lessons, get kids doing careful reading during problem solving, help kids give each other constructive feedback, support students to look for patterns after doing an activity, and encourage reflection and extension after a project or activity. They’re powerful questions because *everyone * has something they can notice (note that it’s not “what do you know” or “what do you think” — it’s a much more fundamental level than that!). And wondering is plain old fun!

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?