Thursday, December 17, 2015

22? 30? 50? 100?

     Meet Alex.  Inspired by a post from Andrew Gael, Alex (not his real name) and his classmates in first grade have spent three periods over the past month exploring different ways to count collections.

It's been an incredible experience...

...and I fully intend to blog about it.  But not today.

      I spend 4 days a week supporting instruction in Alex's regularly scheduled math class, and see him once and sometimes twice a week for an additional, individual intervention period.  Today was one of those days.
     It was my intention to work with Alex on our identified objective, which is counting forward and back on a number line.  But I thought I'd warm up with a quick counting activity.  I took a bag of small plastic dogs and dumped them out in front of him.

Nothing too crazy.  Just 30 little dogs.

    I asked him first to estimate how many dogs were in the pile.  I could see him squint, and almost hear him counting to himself.  He seemed reluctant to commit, but after a bit of prompting he agreed there were more than 10 and less than 100. He settled on 22 as an estimate, which I had him record on the whiteboard.
    Off to what I believed was a good start, I asked him to describe some of his classroom counting experiences.  After some more prompting (Alex has trouble expressing himself) he was able to relate that he had counted wooden blocks.  He was also able to tell me that he and his partner were successful counting the blocks by 10s.  I asked him how he would like to count the dogs, and he said he'd count them by 5s.
    Taking one dog at a time, he counted (miscounted, actually) by 5s and here's how 14 dogs turned into 100 dogs:

"5, 10, 15, 20, 25, 30, 35, 40, 50, 60, 70, 80, 90, 100."

    He stopped when he got to 100, leaving the other 16 dogs in the pile.  I decided to set aside his miscounting and focus on the set of dogs now in front of us:

Me: How many dogs are there?
Alex: 100.
Me: (Pause.  What now?) Can you count them again for me?  This time one at a time?
Alex: (Counting with one to one correspondence as he touches each dog) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,  11, 12, 13, 14.
Me:  So how many dogs are there?  100?  Or 14?
Alex: Both.  100 and 14.

    I decided it was time to step in with some direct instruction, and I turned our focus back to the original set of 30 dogs.  I tried to explain as best I could that he was counting 1 dog as 5 dogs, and that if he wanted to count the dogs by 5s, he was going to have to first put them in sets of 5.  Which he did.

"5, 10, 15, 20, 25, 30."

Me: How many dogs are there?
Alex: 30.
Me: Count them by 1s now.
Alex: OK.  (Touching each one as he counted) 1, 2, 3, 4, 5, 6, ... 30.
Me: So how many dogs are here?
Alex: 30.

     I had him write that on the board, and when he came back to the table I took his neatly arranged sets of dogs, smushed them all back into a pile, and asked him again: How many dogs are there?  He studied the pile intently, and then, with a little crooked finger, began trying to "air count" them all one by one:

He had no way of knowing which dogs he had already counted,  and which were left uncounted.  He stopped at 50.

Me: So how many dogs are in the pile?
Alex: 50.

     I took a breath. I had an idea.

Me: You told me that when you counted the wooden blocks in class, you counted by 10s.  Try counting the dogs by 10s.
Alex: (Taking one dog at a time and setting it aside) 10, 20, 30, 40,...
Me: (Bad idea. What now?) OK, you can stop.  Let's go back to class.

    We got up from the table, me thinking about how 14 dogs became 100 dogs, how 30 dogs became 50 dogs, and how 4 dogs became 10 dogs.  We walked out the door and started down the hallway, me thinking:  What just happened? and How did things ever come to this?  and, What am I going to do now?  And through all the noise in my head I heard his little voice call out: "One".  
   I looked down, momentarily confused.  He was staring straight ahead with a little smile on his face.
   Again, "One."
   On our walks back to his room, we always play a little game.  We alternate counting by ones, sometimes forward and sometimes backward, and stop when we reach his classroom door.  He wanted to play.
   "One," he insisted.
   "Two," I responded.
   "Three," he said.  We were off, until we got to 88, and he was delivered back into the hands of his teacher.
     So now I'm  trying to untangle this mess.  I know that a lot  was revealed, and it needs sorting out before I can map the way forward.   I have some ideas, but I'll take all the help I can get.



Tuesday, December 1, 2015

That's What He Said

     "I really hope you can see how what we're doing here is taking a compelling question, a compelling answer, but we're paving a smooth straight path from one to the other and congratulating our students for stepping over the small cracks in the way.  That's all we're doing here.  So I want to put it to you that if we can separate these in a different way and build them up with students, we can have everything we're looking for in terms of patient problem solving."
                                                                                       Dan Meyer
March 6, 2010
                                                                                       Math Class Needs a Makeover

      Another example of some surgery, this time in first grade, as Nicole and I do our best to follow Dan's advice.   After an opportunity to explore combinations of 10 with ten frames and red and green counters, the kids are presented with a problem to solve.
     Here was the opening suggested by the manual:

  Could we get a student to generate the question?  We were determined to find out.

  I suggested we take off the question, simply present the table, and ask for some noticings and wonderings:

It's an easy change to make.

The kids came up with some interesting observations, including:

  • The red apples start from low (1) and go to high (10) , and the green apples start from high (9)  and go to low (0).
  • The numbers 4 and 5 are missing from the red apple column and the numbers 6 and 5 are missing from the green apple column.
  • There are some reversed.  There's a 2 and an 8 and an 8 and a 2.
  • All the different numbers (of red and green apples) add up to 10.
And the wonderings:
  • Why are some numbers missing?  
  • Is there supposed to be a pattern?
    OK, the question is not exactly there.  So I combined the wondering about the missing numbers with the noticing about the numbers of red and green apples adding up to 10 to set their task: find all possible combinations of 10.

Here's what the manual wanted the teacher to give the kids:

Too helpful.  First, why a table?  We know that a table is a useful way to organize information, but what might a first grader do?  And if a student felt compelled to use a table, why provide one for them pre-made?  And  besides, isn't it too much of a hint that there are 11 spaces on the table and 11 possible combinations of 10?

     Nicole and I decided to take a page from Tracy Zager's playbook.  The plan was to pair the kids up and let them have at with counters, ten frames, and blank pieces of paper.  We would stop for a mid-workshop interruption that would take the form of a gallery walk.  Seeing the way their classmates organized their work might inspire students to evaluate what they were doing and perhaps modify their strategy or change course altogether.

These two students started by writing the combinations they found as a string of digits across the paper...

...and after getting a chance to look at what some of their classmates were doing during the gallery walk, went back to revise their work.
These students started by writing number models.  After the mid-workshop interruption they went back and color-coded the addends.

These students started out drawing red and green hearts to represent the apples, but then decided it was too time consuming and used letters.

Only one group opted for a table.

Here are some other attempts:

     There were as many variations as there were groups.  But this attempt, from one of our most at-risk students, might have been my favorite:

He wanted to work alone.  Nicole and I simply were glad he was engaged with the task..

Hmmm.  What's he doing?

He was content just drawing apples and counting them.  Was he going to find all the different combinations of 10?  No, and we didn't really care.  "He's differentiating the task for himself!" observed Nicole.

     At the end of the day, no one found all the ways to make 10.  Does that mean the lesson was a failure?  I say no.  There's time enough to talk about the most efficient and effective methods to record and keep track of work.  The kids were engaged in a messy, beautiful struggle, experimenting, devising systems that made sense to them, building intellectual need.  Why rob them of that opportunity?  Why rob ourselves of the chance to discover what's going on inside their amazing minds?
     Close to 6 years, over 2,000,000 views, and 32 languages ago, Dan urged us to, "Be less helpful."  What does that mean?  When possible, let the students generate the question. Give them the time and space to explore the mathematics in ways that make sense to them.  Watch, listen, and learn.