What I'm Looking For

     Last year Jo Boaler's multiplication task How Close to 100? caught my eye. I decided to bring it to our grade 3 teachers, and it quickly found its way into the rotation.

Roll two dice.  Use the numbers as factors and generate corresponding rectangular arrays.  Place them on a 10 x 10 grid, and see how close you can get to filling it up.  I liked color coordinating the equation with the array.  Colored pencils worked best.

     Although designed to build fluency with basic multiplication facts, we found the activity had other benefits.  In order to maximize your chances of filling up the grid, rectangles needed to be placed strategically.  Also, in order to calculate your final "score", you needed to do some computation.

Students experimented with different computation strategies, from adding up all their products to subtracting the total number of blank squares from 100.

The activity provided inspiration for a number talk:

     And when we experimented with a 20 by 20 grid, and dice that would generate higher factors, things got a bit more difficult...

And a little messier...

   

We also challenged the kids to find different ways fill up the grid using exactly 10 rolls:

     The rectangles came in quite handy when we got to studying area and perimeter:

There's really no escape from the scissors.

We had them cut out the rectangles, find their perimeters and areas...

...and sort them based on their relationship.

     I'm often asked, "What do you look for when choosing a game or activity to bring to a class?"  Once it passes through the first, most important test of might a kid who doesn't like math find  this engaging, I look for its potential to be extended or repurposed.  There's something about the idea of  becoming familiar with the way something works, then using that familiarity to build on or connect to something different, that appeals to my sense of how we grow and learn.  How Close to 100? is a good example of an activity that checked off those boxes.

"I Like This Game Because You Have to Think Hard."

    A Monday afternoon, working with one of my second graders, helping him fill in a blank hundreds grid.  Suddenly it hit me.  What if...

...using different colored pencils, players alternate capturing squares on a hundreds grid...

...while trying to get 4 squares in a row, column, or diagonal.

It would work  just like tic-tac-toe.

Each tic-tac-toe is worth a point, and you can keep track with tallies.  Play until the board is filled up.

   Theresa and I played a practice round:

Initially we tried 3 in a row, but things got very confusing and we decided that 4 in a row was better.  We also tried a game where a square could be used in multiple tic-tac-toes, but didn't like that either.  We also decided that the 0 and 100 would not count towards any player's tic-tac-toe.

And then it was time to take newly christened Number Grid Tic-Tac-Toe for a test drive.  (Apologies if someone holds the patent on this!)  Jane's class was first, and we decided to get crazy and use a grid that went from 1 to 120.  It took only about 5 minutes playing a demo game with a volunteer under the document camera and they were ready to rumble.

"Hmmm.  Where should I go?"

"I think I can figure this out in my head."

Playing it safe.

Using a number grid to help is so SMP 5.

A barn-burner.  You can see here that 34 was used twice, as was 46. 

   Over the next several weeks I tried it out with 3 more second grade classes and 1 third grade class.  I asked each to write down their reflections and reactions on the back of their game boards.  Here's a sample:

• I liked this game because it was fun to figure out where the numbers go.
• I like this game because you need strategy.
• I loved it so much but hard.  So hard!!!
• I think the game helps because to put the right number in the right place you have to count by numbers and learn to do that in the game.
• I liked it because I outsmarted (my opponent) when he tried to outsmart me.
• I like this game because you have to think hard.

The kids came up with some awesome suggestions for modifying the game, including:

• If someone gets stuck, the other player gets a point.
• If you get three in a row you get half a point.
• Add a few numbers scattered around the board.  (Love it!  This child wants it differentiated!)
• If you get five in a row you get 2 points.

Jane's class experimented with a 3 player game:

They found that 3 in a row was better with 3 players.

Maggie's class tried something a little more difficult:

"What goes here?"

Many possibilities...

If it's green's turn, where should he go?  Why?  Defend your answer.

     I'm happy to report that the game is wildly popular; for some reason it hits that elusive kid sweet spot.  I suspect it's because the cognitive demand is just right. (I like this game because you have to think hard.  Unspoken: But not too hard?)  And because the kids already know how to play tic-tac-toe, they can expend their mental energy on strategy and on figuring out which number goes in what square without worrying about a bunch of rules.   The teachers love it too because it's easy to explain and takes almost no effort to prepare.
   So grab a blank grid and some colored pencils and have some fun!
 
   
   
      

Standards for Mathematical Practice and the Cinema, Part 3

    In the first two installments of the series Standards for Mathematical Practice and the Cinema, I  explored how SMP 6 (Attend to Precision) played out in two classics: This Is Spinal Tap and March of the Wooden Soldiers.  I'd like to turn my attention to a different practice standard, SMP 5 (Use Appropriate Tools Strategically), and one of my favorite movies of all time, Hoosiers.
   The film, released in 1986, is loosely based on the true story of tiny Milan High School's Cinderella run to the 1954 Indiana state basketball championship. It stars Gene Hackman as Norman Dale, a disgraced, temperamental ex-college coach seeking redemption...

Barbara Hershey as Myra Fleener, a small-town teacher with unfulfilled dreams...

And the late, great Dennis Hopper as the alcoholic assistant coach "Shooter" Flatch.

   The Hickory High basketball team responds to Dale's unorthodox coaching methods.  They make an improbable march to the state basketball championship, and face increasingly more difficult competition along the way from schools much larger than their own.  The team is used to playing in their small, claustrophobic high school gym...

...and are visibly shaken upon entering cavernous Hinkle Field House, on the campus of Butler University in Indianapolis, where the championship final is to be held.  Afraid that his team will be intimidated by the size of the arena and become mentally psyched out before the game even begins, Coach Dale employs a wily teaching move:



So was the tape measure responsible for Hickory's last second 42-40 win over South Bend Central?  If so, it played a very small part.  Because we all know who the real hero was:

Jimmy Chitwood!

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!