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)

  

     

Second Grade for the Win

  One morning, a few weeks after publishing this post detailing the struggles some of our third and fourth graders were experiencing with the traditional algorithm, I found myself in Maggie's second grade classroom.  A unit progress check was looming, and the kids were hard at work on a review page.  As I walked around observing their efforts, I found myself admiring the varied ways they were solving this addition problem:

I asked the three other teachers on the team to save the work their students had done.  Here's a representative sample of what I saw:

Number lines

Starting with 12 + 8 was common.

This student decomposed the 50 into 5 jumps of 10...

...while this one took the entire 50 in one jump.

Although it came last in the sequence of addends, this student started at 54.  A great question to ask him would be, "Why did you do that?"  He also broke the 12 up into a 10 and a 2 with the 8 in between.

Partial Sums

This student added the tens, then the ones, then put them together.

This student also started by adding the tens, then put the 8 and the 2 from 12 together to make another 10.  She added that ten to get 70, then added the 4 left from 54.

This student started by putting the 12 and 8 together to get 20.  Then added 20 and 50, and tacked the 4 on at the end. 

Same as above, but he used words to describe what he did.

Can the traditional algorithm be far behind?  It's close, and when it comes it will have a conceptual underpinning.

Base 10 Block Representations

This student is transitioning from the representational to the abstract...

...while this one is not quite ready yet.

This example would provide a nice opportunity to discuss equations.  

Another student in transition.

    One after another, these beautiful examples of student thinking passed before my eyes.  And out of about 70, I saw only one traditional algorithm.
     I have not been shy about calling out our curriculum when I feel it's been lacking (see here and here.)  But I have to give credit where credit is due.  The work I saw was a result of the focus that the new, Everyday Mathematics 4 has placed on computation strategies that include finding number bonds for 10, decomposing multi-digit numbers into tens and ones, and using open number lines and base 10 block representations.  The teachers have maintained fidelity to the program, and in addition have started to explore number talk routines.  The results are plain to see.  I've been teaching elementary math for 30 years, and I've never seen anything like this.  The second grade teachers haven't been at it that long, but neither have they.  Does it represent a sea change, or is this an isolated phenomenon?  What will the third grade teachers make of these students?  Will they notice a difference?  How will they bring them forward?  It's questions like this that keep me excited to come to work every day.  Stay tuned and we'll find out together.

Teacher Appreciation Week

"All the best teachers in my life have been grace-givers."

Francis Su

The Lesson of Grace in Teaching

     December 8, 1980.  Monday morning, last week of classes in the fall semester of my sophomore year at the University of Pennsylvania.  I'm heading to my Intro to Law class in Deitrich Hall, but torts and contracts were the last things on my mind.  Truth be told, I was taking the course only as a favor to my dad, who at the time harbored a misguided notion that I might one day go to law school.  But nothing about the subject interested me.  My grade was barely passing, and I had cut the previous two weeks.  My plan was to make the last session, find out when the final was, get the notes from someone in class, squeak by, and call it a day.
      As I said, the class was the last thing on my mind.  After months of waiting, the day had finally arrived.  That night I was going to see Bruce Springsteen in concert for the first time.  A New Jersey boy born and raised, I was (and remain) a big Springsteen fan.  The double-album The River, his long anticipated follow-up to the intense and moody Darkness on the Edge of Town, had just been released.  I remember sitting with my roommate Cliff in our dorm in High Rise South, listening to WMMR as they previewed each cut.  It was a new Bruce, with a new sound: Hungry Heart and Cadillac Ranch; The Ties That Bind and Out in the Street.  We loved it all. We wore out the grooves in the vinyl.  He was touring behind the album, and we got tickets to see him at The Spectrum.  December 8, 1980.

     From the moment I walked into the hall, I knew something was wrong.    There was a tension in the air, not the normal, casual talk of students getting ready for another lecture.  The professor, a practicing attorney named Frederick G. Kentin, was helping some TAs pass out blue books.  Blue books?
     "What's going on?" I whispered to a nearby classmate.  She told me that the final had been pushed up to today.  Professor Kentin had announced it during class two weeks ago, and the past two weeks had been spent in preparation.  Two weeks of classes that I had cut.
     I started to panic.  I was totally unprepared.  Taking the final at that moment meant certain failure.  But what to do?  As the blue books made their way through the room, I approached the professor and told him that I needed to talk to him.  Maybe the look on my face told him all he needed to know.  He told me to go outside and wait.
     Sitting on a bench outside Deitrich Hall, waiting for Professor Kentin, I decided on a course of action.  I wasn't going to make up any excuses.  I would come out with the truth, put my fate in his hands, and deal with the consequences.  Life was still good, right?  No matter what happened I was still going to see Bruce!
     I'm not sure how long I spent sitting on that bench, but eventually Professor Kentin joined me.  Embarrassed and ashamed, I explained everything.  He spent a few moments looking over a grade book.
     "You know you haven't done all that badly this semester,"  he told me.  "It would be unfortunate if your grade was ruined because you were unprepared for the final.  Why don't you take a few days to study, and then come to my office in Center City.  You can take the final there."
     I breathed a tremendous sigh of relief, and thanked him profusely.  That night I saw Bruce for the first time, and it was all I could have hoped for and more.  The evening, however, was a tragic one.  When we got home from the show we found out that John Lennon had been shot and killed outside his apartment building.  December 8, 1980.
     Later that week, sitting at a big table in a conference room at his downtown office, I took Professor Kentin's Intro to Law final.  I passed, and got a C for the class.  I never saw him again.
     I've thought about that moment often, and I've retold the story many times.  But it wasn't until last year, when I came across Francis Su's profound meditation The Lesson of Grace in Teaching, that I understood its meaning, and the important lesson that Professor Kentin taught me.

Su is a math professor at Harvey Mudd College in Claremont, California.

     In the essay, which is derived from the lecture he gave upon accepting the 2013 Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics, Su describes the life-altering experience he had when, as a struggling graduate student in pursuit of his PhD, he changed advisors.  Even though it meant starting over in a new area, he chose Persi Diaconis, a teacher who had, several years before,  shown him a small act of grace.  While taking one of his classes, Su had asked for an extension on his work because his mother had died.  Diaconis offered his condolences...and then took him out for coffee.  He never forgot this, "simple act of kindness, of authentic humanness."  And when it came time to choose a new advisor, he picked Diaconis.  Su writes,
   "Knowing my new advisor had grace for me meant that he could give me honest feedback on my dissertation work, even if it was hard to do, without completely destroying my identity.  Because, as I was learning, my worthiness does NOT come from my accomplishments.  I call this : The Lesson of Grace."
     Professor Kentin had shown me grace.  And while the act did not transform my life in the way Diaconis's act transformed Su's, I want to honor Professor Kentin this year during Teacher Appreciation Week for reminding me how important it is to show grace to our students.
     "Sure, good instructional techniques are necessary for good teaching, Su explains,  "But they are not sufficient.  They are NOT the foundation.  Grace-filled relationships with your students are the foundation for good teaching because it gives you freedom to explore, freedom to fail.  Freedom to let students take control of their own learning, freedom to affirm the struggling student by your own weakness.  Grace amplifies the student-teacher relationship to one of greater trust in which a student can thrive."
     I wish a wonderful week to all my amazing colleagues: in my school, in my district, in my virtual faculty lounge, and all over the world.