We Noticed, We Wondered. Now What?

     Is there a corner of the known math education world that doesn't know about noticing and wondering?  Introduced by Annie Fetter, developed at The Math Forum, and popularized in Max Ray-Reik's book Powerful Problem Solving, this versatile prompt delivers tremendous value for minimal investment.
     I've seen it happening all over my school, and have been pleased with the results.  But it's time to take it up a notch.  Noticing and wondering is a means to an end, not an end in itself.  It's a problem solving strategy.  After all, it appears in a book called Powerful Problem Solving, in a chapter with the focal practice of SMP 1, Make sense of problems and persevere in solving them.  Here's Max:

    Noticing and wondering activities are very open-ended, and at first can lead to noticings and wonderings that are off-topic and even silly.  The initial process of writing noticing and wondering lists can take a long time, and students will notice details that they won't end up using as they solve the problem. (pg. 49-50)
 
     In my experience this is certainly the case:

Here's an example of some wondering a grade 3 class did recently about Graham Fletcher's 3-Act Share the Love.  By this time of year I would've hoped not to see wonderings like Why am I showing the video? What's the dad's name? 

     Accepting all student responses without judgment is an essential principle of noticing and wondering.  However this can become a source of frustration for teachers, who would like to gently nudge their students into more mathematical waters.  So what can we do?  Max again:
   
       Noticing and wondering is something that students get better at over time:  more focused, more relevant, more efficient, and more automatic.  Once students have become prolific noticers and wonderers, one simple prompt we've found to be helpful in focusing students is simply asking: 
     Which of these noticings have to do with math?
     Which of these wonderings could we use math to help us answer/prove? (pg. 50)

 Here are two adventures in trying to get better at noticing and wondering.

Grade 2:

     Last month I took a slightly altered version of Andrew Gael's 3-Act task Trail Mix on a tour of four second grade classrooms. 




  
As part of the Act 1 protocol the kids noticed and wondered:

   

After the lessons were over, I sorted through all the noticings and wonderings, choosing some that were overtly mathematical, some that had nothing to do with math, and some I felt might start an argument:

I came back to all four classes with the following task:

First the kids worked individually, then I put them in groups to discuss:

 They agreed, they disagreed, they defended their causes.  Some kids changed their minds, others stuck to their guns.

     We met back as a class to debrief.  I wanted to keep things moving, so rather than talk about each one, I asked if there were any that we could all agree on.  What do the Chex taste like?  and I wonder if that's Mr Schwartz? were unanimous no's.  How many pieces are in the bag? (which in fact is the focus question of the 3-Act) was a unanimous yes.  There was an unexpected controversy over I noticed the guy had a watch on.  I had included it as a no, but one student argued that a watch is for telling time, and telling time is math.  Nice!  All except one agreed that I noticed that there were 3 boxes was mathematical.  The lone dissenter argued that, since it couldn't be attached to a number model or equation, it was a no.  One student pointed out that you could add 1 box + 1 box + 1 box and get 3 boxes, and that made a number model.  Another said, "It's counting, and counting is math." She nodded, and changed the N to a Y.   

     

Grade 3

     Shannon's third graders got a number story to notice and wonder about:

No, not this!

This!

I collected and sorted their responses, and came back a few days later with this:

     The most interesting discussion arose from the very first notice, the one about Delilah's bus stop being far away.  All but one student had classified it as non-mathematical, primarily due to the fact that, "It had no numbers."  But one student argued that the notice had something to do with distance, and since distance can be measured, it should be classified as a yes.  That one student managed to convince all of his classmates to change their answers!

Some observations and reflections:

•  At some point we need to call out the elephant in the room.  It's math class, and our noticings and wonderings need to be mathematical. However...
• ...When it comes to number stories, especially ones with a lot of text, there are non-mathematical things kids may notice and wonder about that may help them understand the narrative of the story (such as it is.)  These observations may aid them as they make attempts to work towards a solution.
• Kids have definite notions about what makes something mathematical.  Seeing numbers is a tip-off, but with enough experience I believe they can expand their ability to mathematize.  Conversely, students can help teachers see math in ways they never thought of before.
• Having students classify the noticings and wonderings, and then have to defend their decisions, is a great way to start a fight in math class.
• Having our students notice and wonder about pictures, videos, and number stories is a wonderful way to lower the barrier to entry and engage all learners in math class.  But if we want to leverage that engagement into improving problem solving skills, we need to up our game.  After our initial forays into the practice, we need to carefully guide our students into becoming better mathematizers, and then show them how to apply that habit of mind to problems they encounter in class.  Max, one last time:

     Adding especially mathematical noticing and wondering skills (noticing quantities and relationships, wondering strategically) to students' repertoire increases the usefulness of noticing and wondering.  As students get better at targeted, mathematical noticing and wondering, and as they begin to notice and wonder automatically (as mathematicians do), they may find that all of the other problem-solving strategies become easier to learn as well. (pg. 55)