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?

Thx for sharing, I have been noticing and wondering with this group for awhile, and after reading this have a number of new take-a-ways to try this year! I particularly like having the student place themselves in a role , ” What would a scientist notice, etc…”

Ps..just pre-ordered your book, looking forward to reading it!

Thank you for posting this!!! After all the TMC13 hype about noticing and wondering, I have been trying to implement it in one of my classes. But I wasn’t at TMC, and could find a good intro post to noticing and wondering. I still “got” the basics of it, so I have been pursuing it anyway, but I really feel like we are floundering. Now I can see many ways to improve what we are doing. Love the idea of coming at it from many different “lenses”.

My colleague Ellen Clay introduced me to the lenses idea — I love it because it opens the conversation to what is it that mathematicians do, while also letting students bring in some different kinds of creativity and their individual interests. I’m glad that resonated with both of you!

I’m curious to keep this conversation going. I’d love to hear more about what you’re, er, noticing and wondering as you try it with high school students. I’d love to gather more wisdom here and share some different things we’ve tried. Also stay tuned this fall when we’ll add some videos of very different implementations of the activity by different Math Forum staff members.

Hi Max,

Thank you for the great post. I will try noticing and wondering via lenses next.

Can you believe I just did a fabulous ( in that the students did a great job) lesson with my Alg 2 students where we noticed and wondered about Brian and Michelle from thw latest edition of Woman’s Day who lost a total of 294 lbs and I. Didn’t. Take. One. Picture. Wiped every whiteboard clean! Bummer!

I just showed them a graph from Google Docs where I made their weight loss linear. They wanted to know everything from why Brad lost less than Michelle (big discussion about muscle v. Fat) to what was their rates of loss. The coolest part was when we did the next scenario with two divers with the same rate, but leaving from different heights…there was so much engagement ( and great graphics of diving stick figures).

[...] colleague Max recently blogged about Noticing and Wondering in High School and I thought it would be fun to blog about using it at the elementary level. The essence of our [...]

[...] colleagues recently blogged about Noticing and Wondering in High School (Max – @maxmathforum) and Noticing and Wondering in Elementary School (Annie – [...]

I am teaching 9th and 10th graders … they are advanced, taking Algebra 2. After hanging around the MTBoS for a year, I am beginning to incorporate noticing and wondering. I started on Day 1 and captured some of our thoughts in this post: http://algebrasfriend.blogspot.com/2013/08/our-first-noticing-and-wondering.html. I used noticing and wondering with an Estimation 180 warm-up … and today will introduce systems of equations (a review unit) with those questions. I am encouraged by your post … and hopeful that by persisting in asking these questions, noting student responses that my students will grow in curiosity and interest in math!

I just read your sample chapter of your book on the Heineman site. I’m very impressed. You really captured the problem solving process beautifully. I look forward to reading the rest of the book.

-Ihor

[...] think the noticing & wondering that Max Ray and the folks at The MathForum describe can help students feel they “own” [...]

Fantastic post, I will be using this on my return to school after half term. I think it will enable my year 12 students to get their teeth into calculus.

[...] last suggestion came during our ppschat last winter Powerful Problem Solving by Max Ray, his post here. If you aren’t familar with it, you need to look it up! His Ignite talk is great [...]

[...] at problem solving. We discuss how to encourage students to share their thinking (such as through Noticing and Wondering) and how to cultivate classrooms that value those thoughts and ideas as much as answers. But if [...]

[...] at MathForum shares the awesome of the notice and wonder practice in a math classroom. This is what I was experiencing when I saw [...]