A Tale of Three Classrooms

      If there is a better state curriculum document than the Common Core Georgia Performance Standards, I challenge you to send it to me right now.  I was made aware of it reading a post from one of its developers, Graham Fletcher.  Starting in kindergarten, it includes 3-act tasks, estimation180 challenges, problems with "open middles", and all the goodies an MTBoS groupie could desire.  As my focus turns more towards the primary grades, it has become a go-to source for ideas and activities, and I have barely scratched its surface.
  Looking through the grade 1 frameworks, I was stopped in my tracks by a game called Fill the Stairs.

The premise is simple.  Students take turns rolling  two 10-sided dice and creating a two digit number.  For each turn, the player writes the number created on any stair.  The object of the game is to keep the numbers in order between 10 and 100.  If there is no space to write the number, that player loses their turn.  First to fill in all the stairs wins.

    I felt that the game was better suited, at least at this point in the year, for our second graders.  I made a slight modification to the game board, adding two small spaces at the bottom for place value purposes.  In the first classroom I visited, I explained the rules and started playing a demonstration game against the classroom teacher, with the students acting as "advisers".  After we were confident the kids understood the premise, we set them off to play, first asking them to turn in a sheet recorded in pencil, then asking them to play on blank ones in their SmartPals. 

It was interesting to see their strategies develop as they became more experienced playing the game.  We allowed them to decide which die they wanted to use to represent the tens and which to represent the ones.  So if you rolled a 4 and an 8, you could choose 48 or 84, whichever worked best for you.

   The kids loved it, and as the period came to a close we debriefed with some strategy talk.  I left the game with their teacher to use in a center, or with a guided group of students still shaky with their number sense, but as I reflected on the experience I decided to make some changes as I brought it into second grade class #2.  I replaced the dice with number cards, and further modified the game board.

I removed the rules.  I left a larger space for the cards.  Another modification I wanted to try was to not allow the kids to manipulate the place value of the number.  The first card pulled would always be the tens digit, the second card the ones.  Why?  I was curious to see what would happen.

    I gathered the class in the front of the room and showed them the page on the Smart Board.
    "Today I'm going to teach you a game," I told them.  "But before I do, I'd like you to do something for me."
    Although they had never done it before, I asked them to respond to the following prompts:
   "What do you notice?  What do you wonder?" 
   There was quite a bit of buzzing as they shared their observations with each other.  They noticed stairs, the 10 and the 100, the lines at the bottom of the page, the big space in between the two.  They wondered what the rules of the game might be, if they were going to be asked to count by 10s, why there was a space for tens and ones. 
   Being prohibited from changing the place values of the digits changed the nature of the game.  It took longer to fill the stairs because there were fewer options, but I felt it removed a layer of complexity and reduced the game's potential to build both number sense and understanding of place value.

Many students adopted the strategy of using the second step for the 20s, the third for the 30s, and so on.   What happens now?

   So it was on to classroom #3.  I followed the same procedure, but this time made no mention of the fact that it was a game.  And I asked the kids to write down their "noticings and wonderings" in their notebooks.  Many children wrote that they noticed stairs and numbers, others wondered if they were going to be asked to count by 10s, and whether or not they would be playing a game.
     I need to thank our three amazing second grade teachers for allowing me to turn their classrooms into laboratories.  I am excited to work with them this year, together exploring some different practices and new principles in the attempt to build number sense and make math more meaningful for our second graders.

My Factor Captor Obsession

    Last June I published post detailing my nearly year-long exploration of the game Factor Captor.  Up until the last week of school I had the fourth graders cutting apart the game board and playing around with the numbers:

I was curious: How many multiplication sentences could be made using the 48 numbers on the board as both factors and products.  With three per sentence, could someone make 16?

Close!!  But can you spot the mistake?

     I didn't get any further than this; it was more of a test drive to see what would happen, and if the activity would be worthwhile for next year's class. But as fortune would have it, our grade 5 curriculum opens with a three-week unit on factors, primes, composites, squares, and divisibility rules.  So I knew I would have an opportunity to use the game to extend some learning.
    First, Rich and I let them get reacquainted with the game.  We pulled out the boards they were familiar with.  I was pleased to see that the majority remembered how to play, and for the rest a quick review of the rules was sufficient to get them up and running.  I thought it might make things more meaningful if we explored the unit's vocabulary (prime, composite, even, odd, square), using the numbers from the board.  It seemed like the perfect opportunity to get out the scissors and glue again!
     I thought they might find it helpful to use Venn diagrams.  Many of them chose prime and composite as their first sort.  Of course there was much debate about where to put the number 1.

Some students used reference books.

     Many students finished one sort, and we encouraged them to choose two new labels and try another. And since we're into noticing and wondering, we asked them to write one thing they noticed about their diagram.  These ran along the lines of statements like:

• Most of the even numbers between 1 and 37 are not square, except 4 and 16.
• There are no numbers that are both prime and composite.
• 2 is the only even prime number.

I liked that they were attempting to out into words what they were seeing in their Venns.

And since the obsession shows no signs of abating...

We introduced the advanced grid with a noticing and wondering  "do now".

The kids had some interesting observations, including wondering if playing on the new grid made the game easier or more difficult, noticing that, except for 1, the single-digit numbers are repeated and wondering why that might be so, and wondering whether or not they would get a chance to play.  Well of course!

   I knew that this question would come up, and I knew that, as comfortable as most of them were with the original Factor Captor game board, this one was going to be somewhat intimidating.  I mean, 51?  How would one go about finding its factors?  My hope was to build some intellectual need for divisibility.
   My experience exploring the learning opportunities embedded in the game Factor Captor has me excited about the possibilities that lay hidden within other games, at other grade levels.  It has me thinking more about how we can put games to better use in math class.  In my experience, games such as Factor Captor are used as reinforcements for concepts and skills that have been previously taught. Teachers might provide their students with the opportunity to play them at centers, or when they are done with classwork.  But children who struggle often have limited opportunities to play; it may take them most of the class period to complete their assigned work.  For others, playing the same game in the same way over and over again can quickly become just as dull as another workbook page.  But what if we used the games, not as afterthoughts, or as ways to keep some kids busy while we work with others, but as the vehicles to deliver instruction? Turn them upside down and inside out, take them apart and put them back together?  Not every game lends itself to this kind of treatment, but there are many that will.  I have some in mind; feel free to comment with your thoughts and suggestions.
 
 

Off and Running

  So it's time to get back to work.  I'll be collaborating again with Rich in fifth grade, and we decided to start the year with a problem solving project.  We felt it would set the tone for one of our focal points, which is to help the kids develop their questioning skills.  We would follow the same protocol as last year's movie theater project.

We used this page...

...from this book.

  We gave the kids some time to look at the Birthday Party Basics list, and asked them what they noticed and what they wondered.  We gave them some time to write down their questions.
    While they were working, I overheard two students trying to formulate a question regarding the pints and quarts of ice-cream.  I walked over to listen in, and one student turned to me and said,
   "Mr. Schwartz, what's bigger, a pint or a quart?  I can never remember!"
    I hesitated a bit, and gave her my i-pad.  "Here.  Find it out for yourself."
   "I know how to do that," she said, as she quickly googled "what's bigger a pint or a quart".  (A small moment, I know.  But the fact that she could find the answer herself instead of relying on me or a fellow student empowered us both.  Another reminder of how important those small moments can be.)
 

They chose their favorites to write on chart paper.  Between the AM and PM classes there were quite a few to look through.  Many kids wanted to know how much it would cost to buy everything on the list.  They were also intrigued by the fact that a bag of pretzels cost less than a bag of potato chips but served more people.

A representative sample got posted on Rich's bulletin board.

We decided to have the entire class work on this one.  We liked it because of its openness; we felt it would prompt some interesting discussions as kids prioritized their list, and then re-evaluated decisions  based on how much money was left to spend.

     We weren't disappointed.  The question sparked some good conversation.  There was debate about which size cake to buy, what kind of drinks to serve (one group decided that bottled water was the way to go because kids might be allergic to apple juice and soda, another group agreed but decided to save money by just serving tap water), whether it was necessary to have potato chips and pretzels, whether or not it was important to have money left over, and the difference between 1 pint of chocolate, vanilla, and strawberry ice-cream and 1 pint of chocolate, vanilla, or strawberry ice-cream.
     The AM class worked in groups of three.  Not so great.  One (or sometimes two) tended to dominate while the other(s) were left with nothing to contribute.  You'd think I would have learned this by now.
     We switched it up in the PM.  Rich had them work in groups of two.  As they started in on the problem, I remembered something that I heard this past summer from a teacher at a workshop.  When she has kids collaborating this way, she makes each use a different colored marker.  Her kids know that she is expecting to see a balance of colors.  It helps promote accountability.

I had never tried it before, but it seemed to work!

       Here's what I like about this approach:

• The kids generate the questions.
• The kids get to select which questions they'd like to solve.
• The questions vary in level of complexity and necessitate the use of a variety of skills.
• The kids can work on these at centers, independently, or in guided groups as the unit progresses.

The day the kids were working on their solutions, this tweet came across my feed:

It's like teacher christmas! Except that we buy all the gifts ourselves and they are greeted with subdued acceptance! pic.twitter.com/4Md59dIOW7
— Justin (@JustinAion) September 10, 2014

   I'd like to experiment with big whiteboards for this type of problem solving activity.  Using whiteboards would allow the kids to erase when they make a mistake or want to revise their work instead of crossing out or just tossing the paper and getting another one.  Justin, who knows a lot about whiteboards, tells me that I can get them at Home Depot, and that they'll even cut them to size if I tell them I'm a teacher!
    So school's in.   We are excited about building on all the good work that the fourth grade teachers did with this group last year, and I think we are off to a good start.

My Confession, Part 2: I Am Saved by the MTBoS

    This poster is displayed prominently in my room.

     I tell my students that it's there to provide them with encouragement and hope, but really it's there for me.  It gives me comfort, even though I highly doubt Einstein had the type of dysfunctional relationship that I have had with math.

    Several years ago I was plucked out of out of my elementary classroom, where for over 20 years I taught reading, writing, science, social studies, health, and, yes, math to 7 and 8 year olds.  I put up bulletin boards, made sure everybody got cupcakes and juice on birthdays, planned and took field trips to zoos and museums, drew smiley faces on papers, filled out report cards, did the million things, both large and small, that all elementary school teachers do.  And then one September it was over.
    I was given the title "math specialist", a title that made me cringe, because my mathematical ability is by no means special.   But as long as the math didn't get too difficult, I had no doubts about my ability to help struggling learners.  In fact, I felt I had a small advantage; I could empathize with their struggle and perhaps be able to re-teach and explain concepts in ways that might make sense to them. For inspiration I looked to a familiar world, the world of sports.  I could be like Charlie Lau!

A mediocre hitter himself, Lau (right) is considered the most influential hitting coach in the history of Major League Baseball.  His disciples included Hall-of-Fame third baseman George Brett, a lifetime .305 hitter.

      I spent the first several years in my new position mostly pulling kids, both individually and in small groups, out of their classrooms and back to my room for one-to-one and small group instruction.  We call this "basic skills".  I was patient. I was sympathetic.  I took out manipulatives.  We did journal pages together, maybe got a start on the homework.  We studied for unit assessments.  I did what I could to patch them through a curriculum that raced relentlessly forward and never slowed down long enough for them to catch up, then sent them back into the inferno.
    Two years ago, something happened that changed everything.
   One day,while surfing the internet for math resources, I followed a link to Dan Meyer's TED Talk: Math Class Needs A Make-Over.  There's a powerful sequence when he shows a page from a textbook...

Just looking at it made my eyes glaze over and gave me a familiar sinking feeling in the pit of my stomach.

 ...and then strips everything away until he's left with just the visual of the chairs going up the lift.

   "Which section do you think is the steepest?" he asks.  That was a question I could answer.  It was a question anyone could answer.  You could just eyeball it and make an intuitive guess; you didn't need any "math".  I was hooked: how would we find out?  If there was math that would help answer that question, then that was math I wanted to learn.  Very powerful stuff for a kid who just wanted to crawl under his desk during math class.  I replayed the video over and over, and from there went straight to his blog, started at the beginning, and began reading.  I became convinced that if he had been my math teacher, things would have been much different, and realized that what I had been doing wasn't really teaching.

   Next came Paul Lockhart's  A Mathematician's Lament.

 I remember reading it and thinking, "This must what it would've been like to read a samizdat in the post-Stalin USSR." I imagined math teachers passing worn and dog-eared copies to each other, one step ahead of supervisors waiting to confiscate the manifesto and denounce them as heretics.  I just couldn't believe that a real, honest-to-goodness math teacher would write something that was so damning of his profession and that so accurately captured my learning experience.   I wanted to cry when I read this:

...if I had to design a mechanism for the express purpose of destroying a child's natural curiosity and love of pattern-making, I couldn't possibly do as good a job as is currently being done-I simply wouldn't have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education. 

    It would be hard to overstate what this meant to me.  It meant that I had, somewhere buried deep inside, an ability to do math.  Maybe it was small, but it was something that could be nurtured and, given the right conditions, it could grow.   It had been crushed out of me all those years ago, but with some help it could be found again.
 
   I suppose I followed a well-worn path: from Dan, who taught me about 3-Acts and intellectual need; to Andrew, whose work at estimation180 has had the biggest impact on my practice; to Fawn, whose humor,  humanity, and creativity  has helped me keep my eye on the ball; to Michael,  whose relentless and passionate search for meaning inspire me to dig deep; to Graham, an elementary compadre who keeps me company in a middle- and high school world.  And there are many others.  I took Jo Boaler's course, and learned about Carol Dweck's growth mindset research.  So it was true.  I could learn, not just how to do math, but maybe even to like math.  And if it was true for me, it could be true for all those other strugglers out there: the finger counters and the red x'ers, the fraction flunkies and the long division losers, the times table fist bangers; the confused, the lost, the drowning, and the already drowned.
   So I joined a wild and wonderful community called the MTBoS. I lurked.  I started commenting on other people's blogs.  I started my own.  And I've grown more as a professional in the past two years than in the previous 25 combined.
   There is another, smaller picture hanging in my room.

   This one has been with me since my first years teaching.  But it has taken me all this time to realize that the words apply just as much to me as they do to my students.  We no longer have to use our imaginations to envision what engaging, exciting, and nurturing math classes can look like.  The teachers who are embracing and exploring new ways to make math meaningful in their classrooms are taking no small amount of risk.  But they are doing no less than what they expect of their students.  I am proud of them, and proud to be counted in their number.
    School's in.  It's time to get back to work.