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.

      

      

     

    

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.
   

Fill the Stairs, Redux

     Last year we had an adventure in second grade with the game Fill the Stairs.

I stole it from the Georgia Frameworks.  It became one of the most popular games in the grade level.

   So when I saw that the second grade teachers had brought it back this year, I was delighted.  Turns out there are students who can compare numbers in traditional tasks like this:

...and correctly answer questions like this:

...who aren't always successful applying the skill in a different context:

Same student as above.  The game is another way to assess number sense.

     I knew the second grade teachers were working on having their students get better at constructing viable arguments and justifying their thinking, and had an idea about how to use the game to further that goal.  I ran the idea by Kristin, one of our second grade teachers, and she helped me come up with the following task:

We wondered if any students would choose to use 42.  None did.

Most kids had variations on similar answers.

Even our "strugglers" managed to get something down.

       My next thought was to have them actually play out a game, starting with the number 24 on the stair they selected, and then evaluate their choice.

We asked them to trace over their 24 with a marker to ensure it would not be moved before they completed the game...

...and write their reflection on the back of the paper.

 
   Next, a comment on the original post left by Joshua Greene inspired me to experiment with our first graders, some of whom are still working on counting and ordering numbers between 0 and 20:

I modified the staircase to run from 1 to 20, and decided to use an icosahedral die.

I tested it out with one of my basic skills students:

I gave her no hints or help of any kind.  I filled in my staircase first because she had limited her chances by placing 10 on the stair just below 20.

I was curious to know if she would learn from this experience, so I suggested we play another round:

This time I provided her with a number tape that ran from 0 to 20.  The first number she rolled was a 1, which she placed on the stair directly above 0.

     

She next rolled a 7, and then a 17.  Based on where she placed the numbers, I felt that she had learned from the previous game.

Next came 6, followed by 10.  And she had a nice spot between 1 and 6 to place the 4.  After the experience, I knew the game was ready to be rolled out to the grade level.

  Here are Joshua Greene's ideas:

A bunch of possible variations to play:
(1) each player has a different color to write their number and claim a stair. Player who claims more stairs is the winner.
(2) players have hands with more than 2 cards (5 is often a good number, reasonable amount of choice, but not too much) and get to choose which cards they play on their turn. Could be played head-to-head as in (1) or parallel
(3) different stairs have different point values and/or last stair claimed gets a bonus
(4) different stairs have multipliers that multiply the value entered (for kids who are ready to do some 2 digit by 1 digit multiplication)
(5) A 1-digit version with fewer than 10 steps with or without 0 and 9 already marked


And I'll add to his list: (6) a decimal version for the fourth and fifth graders.  Feel free to continue the list in the comments!
     
 

Scenes From the Revolution

   Here's a MTBoS story:
   At TMC '15 this past summer, Lisa Henry told me that her high school students had a field day with a Which One Doesn't Belong task I had submitted:

Credit also goes to my dad and his extensive collection of license plates.

 
     Knowing that something I had created for elementary school students here in New Jersey sparked a positive experience for a class of high school students in Ohio gave me feelings of pride and empowerment, feelings that up until a few years ago I would never have associated with math.
     Here's another MTBoS story: Mary Bourassa, a high school teacher in Ottawa, Ontario, once spent a winter break creating a website to house wonderful Which One Doesn't Belong tasks, tasks themselves inspired by the work of Christopher Danielson, a college teacher in Minnesota.  And no teacher has to pay any other teacher for the privilege of using the site or the tasks collected there.
     Yet another MTBoS story:
     Last week I brought one of Andrew Gael's Which One Doesn't Belong tasks to Nicole Rocha's first grade class:

     The task provoked a lively class discussion, and Nicole was so excited she stayed in school until after 6 PM that evening working on one of her own.  Later that night she e-mailed me this picture:

She used Andrew's as a template, and could hardly wait to use it with her class.  Now you can use it with yours.

   One final story:
   You all know our accomplished fifth grade teacher, Rich Whalen.  He created a 3-Act based on his own experience running last year's Chicago Marathon.  He regularly uses images from 101qs, another crowd-sourced MTBoS treasure trove, to spark interesting noticings and wonderings.  Several weeks ago, one of his students e-mailed him this picture:

The student was at the mall one Saturday, saw this display, and thought his teacher might want to use it in class.  Of course he did!

His classmates came up with some great questions, including:

• How many cans did they use?
• Is it hollow or does it have volume?
• How heavy is it?
• How long did it take to make?
• How many bags are on the floor?

   I'm not sure who was more excited: the student whose picture inspired his classmates, or Rich, who used his influence as a teacher to inspire one of his students.

    Stories like these are being written every day.  They exemplify what Jo Boaler has called, "the mindset revolution", and what I like to think of as, "the MTBoS revolution."  It's a revolution against an order that believes some people are math people and others are not, an order that has sat by while generations of kids are made to feel humiliation and shame while their classmates look on in helpless silence.  I'm proud to be a foot soldier in that revolution, and it's less about arithmetic than it is about attitude and agency, less about rigor than it is about wonder, less about college and career than it is about collaboration and curiosity.    And in case you're wondering, it's personal.

This One's For Robert

        Robert Kaplinsky is one of my MTBoS role models.  He is the creative force behind the wonderful collection of problem-based lessons found at Glenrock Consulting, a thoughtful and provocative blogger, and the co-founder, along with Nanette Johnson, of one of the quintessential MTBoS sites: Open Middle.  His lessons have inspired some of our most successful projects, including highway signs and movie theater, among others.
     So it was a thrill for me to finally get to meet him at TMC '15.  I was excited to tell him about another one of his ideas that had sparked a lesson so successful it had even our most math-averse students plugging away for days.
    "Thanks for blogging about it," he said.  I was confused.  This was one that had escaped a blog post.
    "But I didn't..." I started, and stopped when I saw his face and realized he was being sarcastic.  I admit I'm a little slow on the uptake.
     Well, we did the project again last week, and it's not going to get away this time.

     What became known to us as "The McDonald's Project" started with this tweet last September:

Hey #MTBoS, I need your feedback. Skim this vid https://t.co/BGhr3obhtG. I'm thinking about turning it into a lesson on equations. Thoughts?

— Robert Kaplinsky (@robertkaplinsky) September 9, 2014

     I loved the video, and Theresa and I immediately started thinking about how we could use the idea.  After some false starts, here's what we came up with:

• A list of foods would be provided, and the kids would be challenged to put together combinations to equal 1,500 calories (2,000 being the adult requirement.)
• Since we knew most kids liked to eat at McDonald's, we decided to limit the food selection to items on the McDonald's menu.
• We'd use the project in fourth grade, where the kids were working on multi-digit addition.

     Our first hurdle came when we printed out the list of nutrition facts for McDonald's menu items:

The PDF ran 27 pages and contained well over 400 items, with information ranging from the % daily value of vitamin A to total grams of fat.  It was overwhelming.

      The list would need to be whittled down.  Our first idea was to pick out the most familiar items.  But that's not very Danielson, is it?  Why not survey the kids, and see what they like to eat at McDonald's?

Here's the final list.  We added in a few items for balance.  And I know Coca-Cola is misspelled.  That one slipped by the editors.

  What happened next may have been the most important part of the lesson.  Before even explaining the directions, the teacher distributed this sheet...

...and asked the kids to do some noticing and wondering.

    During the discussion that followed, the wondering overwhelmed the noticing because the teacher did something very smart: she left out any units or labels.  So the kids were left with a very pressing question:  What did those numbers mean?
     The first volunteer offered price.  But there was no decimal point or dollar sign!  The class agreed it was unreasonable for a Big Mac to cost $540.00, but certainly $5.40 seemed about right.  Did the teacher leave out the decimal point on purpose?  One student thought the numbers might stand for the amount sold in a year or a month.  Another thought the numbers could stand for the amount left over at the end of a day.  Many students offered their opinion that the numbers stood for calories.  And although they weren't quite sure exactly what a calorie was, they did know that healthier foods, like apple slices and side salads, would have less of them than vanilla shakes and french fries, a wonderful application of inferencing skills that would have made their reading teachers very proud.  Listening to their thoughts as they tried to puzzle this out was fascinating.  It reminded me of this wonderful Graham Fletcher activity, and I made a mental note to try to do more of what Graham calls "undressing tables".
     The class agreed that calories did make the most sense, and after a very brief detour into the world of nutrition, they were asked this question: How many calories does a fourth grader need every day?  We got answers ranging from 1 to 30,000.  We didn't wait long to tell them the recommended daily amount and get them working on the project.

We put them in pairs and gave each a different colored pencil.

There was a lot of trial and error...

   

...along with a lot of addition.

The engagement level was high.

   We wrapped up the lesson by gathering the kids together and asking for strategies.  Many had started with the largest calorie items like the Big Mac and milk shakes, gotten as close to 1,500 calories, and tried to fill in from there.  It was again interesting to listen to their observations about the foods and their calories; for example they were intrigued by the 30 calorie difference between the chocolate and vanilla shakes.
   The kids revisited the project in the following days, which gave me time to prepare an extension:

Some found this difficult, and needed more direct help from the teacher.  But they plowed ahead with gusto:

 

     And this work left me with material to work out a problem set:

Here are some of the problems I was able to generate by removing an item from each equation.  I also added the name of the child who had created the original problem in the margin. 

   I'm sure there's a lot more gold we get out of this task, and Robert has great ideas for using the calorie lesson with middle school students over at his site.

     So thanks, Robert, for your inspiration, encouragement and polite but firm way of pushing me to think and work outside my comfort zone.  Looking forward to more collaboration, and to connecting with you again at TMC '16!