At Play in a Mathematical Sandbox

     One of the highlights of this past summer's Twitter Math Camp was the chance to attend a session led by Federico Chialvo.

Friday afternoon from 4:00-5:00.  It was a tough call because Fawn Nguyen and Matt Vaudrey were presenting Barbie Bungee at the same time.  An example of the difficult decisions one had to make at TMC 15.

     Federico is the Director of Mathematics at the Synapse School in California.  He also plays professional ultimate frisbee for the San Francisco Flame Throwers.  (Who knew there was such a thing?  He even had to leave TMC early because he had a play-off game!)  I had read some of his blog posts, and we tweeted back and forth a few times last year.  I knew we shared an interest in the potential that games and game-like activities have to engage kids in learning.  So I was excited to meet him in person and learn something new to bring back to my school.
     One of the activities that caught my attention was called Subtraction Reversal Mysteries:

 
     Federico gave us time to experiment with the game, and led us through some guided discovery.  He showed us student work samples he had collected from a post he wrote describing his experience using the game with his class.  I was attracted to this activity for several reasons:

• The directions were simple and easy to understand.  Everyone in the class could participate.
• There were some important content standards embedded within.  I saw subtraction with regrouping and place value at work.  
• The data collected from playing the game would lead to some very interesting mathematical discoveries.

      I felt it would be perfect to use with a third grade class at the beginning of the year, both to review multi-digit subtraction and introduce the idea of looking at data and making conjectures.
     Back in school I approached Shannon, whose third grade class I'm working in this year.  She agreed to let me try it out, and armed with some 10-sided dice I gave it a go.

The kids picked it up quickly.

They got about 15 minutes to play.  I collected their sheets, looked them over, and brought them back the following week.  I wanted them to play for at least another 15 minutes in order to collect more data.

One pattern I saw in their mistakes involved regrouping when there was a 1 in the ones place:

Kids who, under other circumstances, could regroup correctly would make this mistake.

Here it is again.  

    
     So we were able to gather some good formative assessment information on their abilities to subtract, and jump in with some direct instruction.  Several days later I brought their papers back and let them continue to collect subtraction problems.
     During the third session, I asked some kids if they noticed anything interesting happening with their subtraction problems.  One girl explained that when she rolled two numbers that were right next to each other, the resulting subtraction problem had a difference of 9. I asked her for an example:

I put this up on the board, and asked the kids to pair up.  I wanted them to collaborate, looking through their data to see if they could come up with any more interesting observations.

I asked them to write a statement and back it up with evidence.

I explained that they should not limit themselves to the combinations they had collected during the investigation.

For their first time, I thought they did well.

This student worked alone.  He didn't want to collaborate and I didn't force the issue.

The same student came up with this: "The difference of the difference added together equals a round number."  Can you see what he means?

At the end of class, one girl proudly showed me this.

   
     During the session, Federico spoke about the importance of providing students with a "Mathematical Sandbox" in which to explore, play, and create:

     I like everything about this, and an activity like Subtraction Reversal Mysteries is a wonderful example.  Everyone in class is playing in the same sandbox.  Some are playing together, some alone.  Some are building very intricate castles, others are simply filling pails and digging holes.  But they're all in there together.  It exemplifies the MTBoS ideal of an activity with a low barrier to entry that scales up high.
     Federico had lots more examples of sandbox activities that he had us try out during his presentation.  I look forward to experimenting with more of them as the year unfolds.

Meatball Surgery

     This summer our district renewed its commitment to Everyday Mathematics, purchasing the 2015 EDM4 for all grade levels K-6.  Theresa and I have spent the past weeks helping teachers navigate their way through many of its improved features, which include: a different online platform, a revised order of units, and new manuals, assessments, games, differentiation options, and manipulatives.  So far so good.
      One morning last week I stopped by to check in on one of our grade 4 math teachers. She was busy at her back table, writing out a bunch of posters:

         She explained that she was copying over problems from the student journal:

Read these problems.  When your eyes glaze over, raise your hand.

And you thought you were done!

    It was her intention to put the kids in groups and have them spread out all over the classroom floor, an idea I supported fully.  But I had seen these problems over the summer, and didn't like them.  Wordy, dense, unengaging, and contrived were four adjectives that came to mind.  Time for some meatball surgery.

I suggested separating the question from the scenario.  Put out some post-its and ask the kids what they notice and what they wonder.  Or tell them it's a number story with the question removed and see if they can guess the question.  Just give them time to process all that information without the anxiety of having something to figure out.  

 
      For an opening math message, I suggested she find a picture of one of the tall buildings and do a simple notice and wonder.  Perhaps a visual and a small discussion might help set the table for some of the comprehension work that would follow.
     When I got to class later that morning, I saw this up on the SMARTBoard:

Trump Tower.  The kids had some interesting observations about its shape and composition, and were very curious about its height.

     She divided them into groups and gave them the surgically altered problems:

They noticed and wondered...

...and came up with their own questions.

When they finally received the actual assignment, most of them were ready to give it a go.

Each student in the group got a different colored pencil.  A way to make sure everyone participates.

   
     We stopped the kids several minutes before the end of class and gathered everyone together to take a look at some of their addition strategies under the document camera.
 
     Later, I went back to take a look at the manual.  This was Lesson 1-6: Guide to Solving Number Stories.  The heart of the lesson, which was slated to take 30-40 minutes, was composed of three parts...

1.  Math Message.  Instead of the Trump Tower notice and wonder, here's what the kids were asked to do:

   Read the Math Message problem on journal page 13.  Be ready to explain what you already know from the problem and what the problem wants you to find out.

(I've voiced my criticism about math messages like this.  I'll just say that many kids in the class would not even make it past the first few sentences before completely shutting down.)  After an unspecified amount of time, the teacher is instructed to move on.

2.  Using the Guide to Solving Number Stories.  Instead of giving the kids the scenarios with the questions removed and asking them to notice, wonder, and/or come up with their own questions, the teacher was to refer the students to this guide...

     ...and use these questions...

• What do you know from reading the story?  
• What do you want to find out?  
• How can you find the number of stories Terrell needs to climb?  
• What will you do first?  
• What strategy or tool can you use?  
• What is the unknown quantity?  
• What number model might we write, using a letter to stand for the unknown, to represent what we want to find out?  
• Are you finished?  Why or why not?

...to guide a whole class discussion on the problem solving process.
     After helping the class come to a consensus regarding the correct solution to this problem, and reviewing some of the different addition strategies employed by students, the teacher is to move on to the next step.
     

3. Solving Multistep Number Stories.  The teacher is instructed to put the students in partnerships to complete journal pages 13 and 14.  Remember them?

page 13

page 14

     Back together as a whole class, the teacher is instructed to ask volunteers to share solutions to the problems.
     This is pretty standard stuff.  And kids need strategies to solve number stories. But it's unimaginative.  There's too much whole class, teacher-directed discussion, which means more opportunity for kids to tune out. My quick meatball surgery was intended to lower the barrier to entry to these difficult-to-access word problems, and to get the kids more involved in their own learning. It's nothing amazing or revolutionary, only the best I could do given the time that I had. The classroom teacher had good instincts.  It's only her second year teaching, and I'm confident that given a little time and some good PD she will be able to make those changes and decisions on her own.
     But it's really not a sustainable model.  Teachers, especially elementary school teachers who are responsible for teaching multiple subjects, do not always have the time to perform the kind of surgery lessons like this require.  They may not even be aware that other types of strategies, activities, and instructional practices exist.  A lesson from a curriculum interested in making math meaningful, accessible, and engaging should deliver them right to their classroom door.  Is that an unrealistic expectation?
 
   
   
   

Can I Get Some Feedback With That Pie?

    My most popular post to date is a description of an activity Jeff and I did with the fourth graders about a year and a half ago, a revised version of the Everyday Math game Angle Race  we called the Pie Eating Contest.  So when it came time to revisit some of the same concepts last year with the fifth graders, I suggested to Rich we try the activity again.

Most of the kids remembered how to play, and they were excited to see the game.

We had the kids cut out their angle pieces and compose them into right angles and straight angles, and write the corresponding number models underneath.

While the kids were working, I thought of a comment that Annie Fetter had written in response to a post describing a game I had introduced to the second graders:

    Annie's comment was intriguing.  Asking the kids to respond to this prompt would be a great way to get them thinking about the inner mechanics of the game.  It would be an opportunity for them to form an opinion and provide justification, something that Kristin Gray had me considering.  I decided to have the kids write about it, but also wanted them to include some type of visual representation.  Using the pie itself seemed like a good idea:

After several false starts, here's what I came up with.  Most kids decided to use angle measures to describe the luck/skill relationship.  But others used percentages, and one student played out a demonstration game.  Most students agreed it was mostly luck.

     The prompt provoked some lively debate, and what happened next is what always happens: we were left with a pile of about 40 papers.  Now what?  How should we respond?  Many of their justifications were vague and unconvincing.  If we wanted the kids to get better at this, what type of feedback would be appropriate?
  Feedback is something that had been bothering me, mostly because I'm not very good at providing it.   Michael Pershan had been blogging about the topic, and inspired me to reflect on a part of my practice that is, quite frankly, lacking.  I decided to experiment with comments based on "noticing and wondering":

I decided that each paper would get one "notice" and one "wonder".   I asked them to revise their explanation, integrating the answer to my wondering question.

    My intention was to push the kids to justify some of their statements with more detail.  What is it about the cards that has to do with luck?  What are the particular skills involved?  The class came up with a list as we debriefed:

Rich noticed something very interesting.  It was the kids who struggled with using the protractor correctly who were best able to identify the component skills.

 After they completed their second drafts, I had another go:

I made a point of noticing what they had added, and chose to poke them on something different.

    Rich and I had a discussion with their writing teacher.  Guess what?  They were working on writing persuasive essays in writing class!  Could we use some of their writing language in math class?  We borrowed a poster of hers:

The kids were surprised to see this hanging in Rich's room.  We had them use it when working on their final drafts.

        As school starts up again, the experience reminds me of two things I want to work on this year:

• Collaborate with teachers to explore ways we can give better feedback.  I believe that couching the feedback in "I notice/I wonder" language has potential.  It  provides a model for kids to give feedback to each other, and is more specific than a star, a smiley face, or a check mark.  I realize that it is impossible to do this for every piece of work the kids hand in, but we can pick our spots.
• Make better connections with our ILA teachers.  We want kids to construct viable arguments. They do that in writing class!  We want kids to make sense of  problems. This is a familiar sight in all the reading classes in my school:

What would happen if we asked kids to use their active reading strategies and "post-it" their way through text-heavy problems in math class?

   Yes, the Pie Eating Contest has come a long way from its humble beginnings as the Angle Race game.  But it took a village to make it happen.  Thanks to Dan for popularizing it, to Annie, Michael, and Kristin for helping it along the way, to my amazing colleagues at school, who continue to let me use their classrooms as places where we can learn and grow, and of course to the kids, who greet all (OK, most) of our new endeavors with enthusiasm and good cheer.

29 Going On 30

     This September marks the start of my 30th year teaching.  What was happening 29 years ago, September 1986?

President Ronald Reagan, midway through his second term, appointed Trenton, New Jersey native Antonin Scalia to the United States Supreme Court.   He is now the court's longest serving justice.

The Oprah Winfrey Show made its national television debut.

"Crocodile" Dundee was released.  It was the second highest grossing film of the year, right behind Top Gun.

.

Ivan Lendl...

...and Martina Navratilova were U.S. Open champions.

Stuck with You, the first single released from Fore!, topped the Billboard 100 for three weeks.

     I was all of 25 years old.  Here are some things my school didn't have:

Chromebooks

iPads

Smart Boards

Whiteboards

Document cameras

Lockdown drills

High-stakes standardized tests

Here are some things my school did have:

A ditto machine

Filmstrip projectors

Opaque projectors

A card catalog in the library

Cursive writing

A pay phone in the hallway

   1 graduate degree, 3 positions, 5 classrooms, 5 superintendents, 4 principals.  Lots has changed.  But the most important thing hasn't: Every September the kids still show up at the door.  How to sum it all up?  I'll leave it to R. Chanina:

     I have learned much from my teachers.  I have learned more from my colleagues than from my teachers.  But I have learned more from my students than from them all.  (BT, Taanit 7a)

Have a great 2015-2016 school year!