Tuesday, December 1, 2015

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.


  1. Great post, Joe. So much good food for thought in here! I've been writing about that mid-workshop walkaround lately. So exciting to see the cross-pollination of ideas take place!

    I also love the detail of noticing that the number of rows in the chart was a dead giveaway for the number of combinations. I want students to decide they've found them all when they're sure they've found them all, not when they get to the bottom of the worksheet!

    1. Thanks Tracy! Now that I'm more aware, I see "dead giveaways" all over the place. One I saw the other day in a fourth grade journal was a problem that said "Name all the factors of 40" and then provided 8 blanks. The kids get used to that at a very early age.

  2. There's little more valuable to me than seeing how pedagogical idea x, y, or z translates up and down from K to 12. Thanks for the write-up, Joe.

    1. Thanks for the comment, Dan. It's truly exciting to see it all unfold, especially with primary age kids. Keep the pedagogical ideas coming, and I'll keep experimenting with them in elementary classrooms.

  3. I'm beginning to recognize the look of "taking a page from Tracy Sager's playbook::" A worksheet that attempts to tightly control the students' response becomes transformed into something which allows the students' exploration to burst on to the page.

    I particularly enjoyed how one student started drawing apples then switched to using math symbols (numbers)to made the work flow more easily.

    1. Thanks Paula. That blank piece of paper can be liberating, but also very scary if you're too used to that tightly controlled worksheet. I think these first graders were too young to have been conditioned like that!

  4. I am starting to realize how many answer I give away to my students just by providing this with blanks to fill in. This will help me reform my 6th grade math teaching.

    1. Thanks for the comment Randi. Once you start, it's hard not to see the helpfulness everywhere!

  5. Love the less is more approach here Joe. The less you give, the more insight we're able to gain as it relates to student understanding.