Sunday, May 22, 2016

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!

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)


  1. I'm inclined to agree with your points, but feel it is case I'm really liberated when I can act as a parent vs a classroom teacher. When we do Notice & Wonder at home, we can move fluidly between the mathematical and non-mathematical sub-topics. Often, we end up getting to an even deeper and, to me, more interesting mathematical conversation through a surprising side-trip.

    For example, in the Trail Mix activity, the question "Is it healthy?" stands a solid chance of being classified as "non-mathematical." However, it could easily lead to conversations about (a) understanding the ratio of mixed ingredients, (b) calculating nutritional values, (c) exploring the degree of variation in nutritional requirements (by people, by time of day, by the activity they are engaged in at the time).

    Because I'm in the mood to be contrary, let me take another example. In the bus example, "I noticed that her name is Delilah" is likely to be seen as "non-mathematical." Where could I go with that one? What about analyzing the availability of public transport to different demographic groups? Suddenly, we could be talking about measures of quality ("last-mile" distances, reliability of service, time taken on a trip, number of transfers, etc), statistics, and geometry. To say nothing of the rich and vital conversations about city planning, politics, history, and the distribution of public services.

  2. Great points Joshua. You've made me think about this more. As teachers, we tread on dangerous ground when we act as the ultimate arbiter or authority in a lesson like this. Our pre-conceived notions get in the way of us really hearing what the students have to say. I will admit that when I chose the examples I had in my mind an idea of what was and wasn't mathematical. But I didn't create an answer key, and I let the students drive the conversation and decide for themselves. Creating viable arguments, like you did for the examples you cited, is a wonderful exercise in and of itself. While it would be unlikely that a second grader would come up with a defense of "I notice her name is Delilah" like yours, as they get older and have more experiences their conceptions of what's math and what isn't will expand and might eventually encompass what you've suggested. That's an exciting prospect!

  3. I love everything about this post. We've done numerous Notice/Wonder tasks this year, and I, too, get frustrated when students come up with responses like the ones you mentioned (see my post about precisely this topic I constantly make the mistake of assuming that since we are doing these investigations in math class, that students would just automatically know to think like mathematicians (I know, I know, we want students to think like mathematicians all the time, not just in math class). I do, however, notice a considerably visible shift in thinking as soon as I ask them, "What would a mathematician notice?" or "What questions would a mathematician ask?" We actually did the Trail Mix task this morning. I had every intention of analyzing their responses and doing the post-task activity ("Which statements have to do with math?), but a student actually brought it up during our discussion! We kept track of the wonderings that we answered or could answer, and she noticed that almost all of the questions we couldn't answer had nothing to do with math! This observation led to another interesting discussion of what constitutes a math-y question. I'll blog about it soon!

  4. Thanks for your comments. I love what you've done with your class (there's no one you could emulate better than Simon Gregg), and it's wonderful that your students are driving the discussions about mathematical vs. non-mathematical noticing and wonderings. I'm looking forward to your next post!

  5. Thanks Joe. I have found it hard to be open to all answers but at the same time wanting mathematical noticings and wonderings. One popular wondering I always hear is, "Why are they making this video?" I love the Y/N checklist idea.

    1. Right on Matt! It was hearing that one question again and again that got me moving towards this activity. I hope I don't hear that question again, at least in the classes that have had this experience. Let me know how it works for you.