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?

Perhaps because I’ve watched every step of this saga of Michael’s Journey Into the Complex Plane with hawk-like attention, I’m totally down with what he’s trying to do in his blog post about how he might introduce students to complex numbers. He’s looking for a genuinely perplexing, easy to formulate question that students have the werewithal to begin to answer.

I think he’s nailed easy to formulate, and his 17 pages of work show that students do actually have the arithmetic know-how to answer this. I think students will be perplexed by this, but I do wonder about where are the parts where students need specific mathematical habits of mind & skills to be enabled to persevere.

Some things that need to be in place for this to work:
Students need to be able to hold on to the ambiguity of multiplication as an operation on the plane and the (shorthand) idea of multiplication of a point by a point. Or we need to have a language that unambiguates that. E.g. 3 * -2 is “where does 3 go under the transformation that takes 1 to -2?” [and then commutativity is NOT obvious].

If we keep using the shorthand of multiplication of a point, by a point, students need to be comfortable with having multiple physical representations of the same operation, or we need to train them in one that we want them to use. Again, I’m not sure which is better, but I’m leaning towards really hammering and making both sensible and automatic the idea of a twisting, scaling slide rule kinda thing (i.e. multiplication of real numbers is rotating and dilating the real number line, and adding real numbers is translating the number line left or right).

Also, depending on your definitions of dilation of the number line based on points, you don’t need the rotating idea until you introduce complex numbers, because the signs of your points will take care of that (dilating by a negative ratio includes a rotation in GeoGebra or Sketchpad, but you can define your dilation based on length, not position, and then you do need a rotation. I made a collection to help play with that idea here: http://www.geogebratube.org/collection/show/id/5056. It’s not fun and visual, but it is mathematically intriguing to see how the points are defined).

Here’s where the habits of mind really come in. If we ask students to extend their understanding of 1D operations on the real number line to 2D representations, they need to be able to:

• Understand that generalizing means making a coherent system that doesn’t “break math”
• Decide on the rules that we want to define not breaking math to be
• Generate conjectures about what a generalization might look like
• Test those conjectures
• Persevere through multiple conjectures and tests
• Accept a definition of multiplication that is not their initial intuition and may even trouble their 1D understanding of multiplication
• Persist through defining a generalized multiplication to mastering said multiplication, both geometrically and algebraically.

Most students have never been asked to conjecture possible definitions for an operation, and have never been exposed to the idea that mathematicians posit the existence of objects and operations and then test to see if they break or not. Which is too bad because that’s a lot of what mathematicians do, and something students are capable of, but getting students to the point where they’re willing to define mathematical operations or objects for themselves and then persevere through playing with possibly broken objects long enough to find one that works, is hard.

[It's sort of like giving a kid a huge pile of boxes to open on her birthday, with the caveat that most of the toys she'll find are missing pieces and will never work, but once she's done opening & testing them all she'll have found some AMAZING working toys and learned a lot about how toys work. This is why math class is not a birthday party, it's HARD fun, much more like learning to ride a bike (ouch!) then opening birthday presents.]

Students also need a robust enough understanding of operations that they get what it means to not break math. They need to expect commutativity and associativity and the distributive property (which most kids don’t understand, let alone value!). They need to compute fluently with positive and negative numbers, including distributing. A robust understanding of the geometry of transformations would be nice too.

All images from http://www.ics.uci.edu/~eppstein/junkyard/spiraltile/

Finally, the transformation of the plane that relates closely to complex multiplication is the beautiful Spiral Similarity, which results in lovely spiral tessellations. Could a launch perhaps be based on telling some technology to make spiral tessellations for you, and then making the connection among algebraic and geometric definitions of transformations, and finally generating a robust set of algebraic rules for exploring and defining spiral dilations and 2D translations and then connecting that to the transformation composition that takes 1 to -1. See more about Spiral Similarity here: http://www.ics.uci.edu/~eppstein/junkyard/spiraltile/

PS — on contexts, perplexity, motivation, etc. see Riley Lark: http://larkolicio.us/blog/?p=787

Update — I’ve been meaning to share this article for a while; it’s not completely relevant here but I like it: Research Mathematicians as Learners And What Mathematics Education Can Learn from Them — it’s about the doing of mathematics as mathematicians see it and the opportunities for students to do that kind of thinking. The second is a book about elementary math concepts and how they relate to higher math, as told through the eyes of what used to befuddle research mathematicians when they were elementary students: Shadows of Truth: Metamathematics of Elementary Mathematics.