Out of the Box

    One of the first things I did when I took over the position of math specialist at my school was take inventory of what had been left in my new room.  There was quite a collection gathering dust in the closets, cabinets and drawers, most of it dating back many years: office supplies, manipulatives, old textbooks and workbooks, files, games, flashcards, and measurement tools.  Some of it I tossed.  Some of it I distributed to teachers and classrooms.  The rest I kept.

A battered survivor of the purge.

     One day, looking for something to fill a few extra minutes with one of my basic skills students, I grabbed the game, opened the box, and gave it a try.

Roll 3 dice, two 1-6 number cubes and another with addition and subtraction symbols.  The die with the operations was missing, so I made a replacement.  Use them to create an addition or subtraction number model.   Find the sum or difference...

...and move a token (snail, frog, turtle, or snake)  around the game board from the start, through the swamp, to the lily pad at the finish.

     Simple.  I grew to love the game, and it soon became a stand-by.  It has many benefits:

• Basic addition and subtraction fact practice
• Counting with one-to-one correspondence
• Identifying numbers as odd or even (If you land on a square that says "even" you roll a number cube.  If the result is an even number you move forward that many spaces; same for an odd number.)
• Social aspects of a good, old-fashioned board game, including learning how to take turns, handle adversity, and be a gracious winner... or loser.  Because I don't let kids win.  Well, hardly ever.

 The game has one insidious aspect, the dreaded "Endless Loop":

Once you enter the loop, you need to keep going around and around until you land exactly on the exit.  Only then can you dash to the lily pad.  It can get very frustrating.

  Last week, I was playing the game with one of my grade 2 basic skills students.  She had traveled around the endless loop a few times, and her frog sat one square before the exit.  She picked up the dice, but before she rolled I stopped her, and asked a question it had never occurred to me to ask before:

   Me: What will you have to roll to land exactly on the exit?
   Her: A one.
   Me: Yes, a one.  How can that happen?  (I knew that a roll of 11 would also work, but that's not where I wanted to go.)
   Her: (silence)
   Me: Show me what the roll would have to look like for you to get a one.

It was fascinating  to watch her think through this task, turning the dice and working with her fingers until she found a combination that worked.

   Her (putting up two fingers and taking one down): 2 minus 1 equals one!
   Me (realizing I might be on to something): What else could you roll?

She continued  manipulating the dice and using her fingers to calculate.  I pulled out a whiteboard and started writing down what she told me.  I was hoping she would see a pattern, and use it to generate the rest of the combinations, but no go.  She got stuck with one more left to find.

She needed a lot of prompting to come up with the final combination...

...but then she saw it, and continued to generate more subtraction number sentences equal to one.

    The period was coming to a close, and we finished the game.  I put the whiteboard aside, with the intention of taking it out again at our next meeting and using it to help her dig through what she had discovered about differences equal to one.   And I understood that there were more benefits embedded in the seemingly simple game than I had realized.  I had been focused on content standards, and had overlooked the opportunities to engage students in practice standards; in this case, a nice combination of SMPs 1 and 7.
     I've written about games before; how they can be used and repurposed.  I'm glad my predecessor left Sum Swamp behind.  More importantly, I'm glad I held on to it.  I never could have imagined what would come out of that box.
   
   

     

Noticing and Wondering: A Sampler

    It's hard for me to believe that I've spent most of my teaching career in complete ignorance of noticing and wondering.  This simple, elegant prompt has become an integral part of my practice, and its use is spreading throughout the school.  Here are some examples from the past few months:

Grade 5:

I snapped this picture in the teacher's lounge the Monday after Thanksgiving break.  Rich  put it up as a noticing and wondering math message.

It was clear that some of the responses were inspired by our continuing use of the pie-eating contest:

• How many degrees are eaten?  How much of the pie is left in an angle?
• How big is the missing piece?
• Is 75 degrees eaten?

But many took a different direction:

• How much did the pie cost?  What would be the unit cost for the one piece missing?
• How many ingredients were used?  How many teaspoons of sugar?
• How heavy is the pie?
• What's the perimeter of the pie? 
• Could you divide it in equal pieces?  Can you make it into a fraction, like 1/6?
• Is it store bought? (Not overtly mathy, but I liked it!)
• Why does it look like pac-man?

Grade 2:

     The second grade teachers have done a wonderful job incorporating noticing and wondering into their math message routine. 

     Instead of asking the kids to fill in the diagram and solve the problem, Maggie provided the answers and let them have at it.  Here's what came up:

• In the two parts (45 and 25), the digits in the ones places are the same, while the digits in the tens places are different.
• The two parts are odd, but when you add them together the total is even.
• If you have a number and you add more to it, you get a higher number.
• The cents symbol looks like a "c" with a line through it.

Grade 1:

     Theresa and I have been spending time with the first grade teachers and students talking about the importance of subitizing.  It made me happy to walk into Jen's room one morning and see what she had chosen for a warm-up activity:

I've been taking photos in the supermarket and downloading them into a folder on the school's shared drive for the teachers to use as subitizing prompts.

Jen recorded their responses,  The kids are getting really good at this!

Grade 5

    Last week I found myself staring down a guided group and a pile of  Magic Squares that I had found to help the kids practice adding fractions and mixed numbers.  I will admit that I had not put much thought into how I was going to present the task.

As I showed them the model, it struck me.  How about a simple, "What do you notice?  What do you wonder?"

      Asking the kids to notice and wonder allowed them all to enter the task at the ground floor.  They noticed the boxes, some empty and some filled; they noticed the two mixed numbers and the one whole number; they noticed the tic-tac-toe-like configuration.  They wondered what might be magical about the 40 1/2; they wondered whether some mathematical operations might be involved.  After only a few minutes of discussion, they were ready to go,

I highly recommend this activity.  It hit some kind of 5th grade sweet spot. 

     Noticing and wondering has many benefits:

• It's easy for a teacher to remember and can be used in many situations.
• There are no right or wrong answers, and nothing to solve.  It's anxiety-free.  
• It's a one-size-fits all prompt. It's as simple or challenging as the individual child that responds.
• It can be a valuable formative assessment tool, and a window into the minds of our students.
• It can be practiced, and kids can get better at it.  
• Kids can utilize it as a protocol for understanding a problem before they go about deciding how to solve it.

Noticing and wondering is all the rage at my school.  How are you using it at your school?

      

Functional Misses

     What do you do when a student in class raises his or her hand...and then proceeds to give a wrong answer?  Do you:

• Call on another child to provide the correct answer under the guise of "helping" the confused student?
• Stop and intervene with the individual student while the rest of the class sits idly by and waits for you to finish?

    These are the two most frequent responses I observe (and have used myself).   As teachers we know that our students will give wrong answers.  And as teachers we know that there is a lot more riding on our response than the misunderstanding of a concept or procedure.  There are emotional consequences as well.  If we're not careful, the way in which we handle wrong answers can effect a child's perception of himself or herself as a learner, cause embarrassment, and lead to an unwillingness to contribute to and participate in class activities.  So what can we do?

    As with many things, we can look for answers to the NBA.  Specifically Houston Rockets extraordinary two- guard, James Harden.

Harden currently leads the league in scoring, and possesses the best beard in professional sports.

      In a recent article in Grantland, Kirk Goldsberry deconstructs Harden's scoring genius.  He finds there's more to Harden's offensive game than his shooting prowess and uncanny ability to draw fouls and get to the line.   He explains:

  But even when his attempts fail, they have a chance of succeeding. The entropy from his slashing drives, which scramble defenses, enables his teammates to slip into great rebounding positions. Not all missed shots are created equal, and sometimes they function a lot like inadvertent passes or shot-clock reset buttons.

 Goldsberry calls these "functional misses".

 
    In Goldsberry's example (above), Harden drives down the lane and misses the lay-up.  A teammate, however, is in perfect position for a put-back slam.  Functional miss.
   Is it possible to put this notion into practice in the classroom?  The day after reading the Grantland piece, I was helping a guided group of struggling fifth graders work through fraction addition at Rich's back table.

Hmmm.  Unlike denominators.

   

The kids dutifully took out their multiplication grids and found common multiples for 2 and 3.

  

We settled on 6.   So far, so good!

We converted 1/2 into an equivalent fraction, 3/6.  

   
     It was at this point that a student, I'll call him Jeremy, raised his hand.  Jeremy is new to our school.  His math skills seem shaky, which is why Rich decided to have him work with me in the guided group.  I was glad he felt comfortable making a contribution, and was curious to know what he had to say.
    "That's a mixed number!" he exclaimed.
    "Where?" I asked, hoping he was referring to the sum of the two fractions.  Was it possible he was two steps ahead of us?
    "Three-sixths," he said, pointing to the fraction on his whiteboard.  "Three-sixths.  That's a mixed number."
    I could sense the unease of the rest of the group.  We had spent several weeks talking about improper fractions and mixed numbers, and I was sure they knew he was wrong.  I decided to press him a little.
   "What makes it a mixed number?" I asked
   "The numerator and the denominator got bigger," he told me.
     The moment of truth.  The time to make one of the hundreds of decisions, both big and small, a teacher has to make each day, every day.  What size decision was this one?  What would you do?
   
     I decided to highlight what was correct, ignore the mistake, and move on.  Perhaps I could turn it into a functional miss.
    "What you said about the numerator and denominator getting larger is correct, Jeremy," I responded.  "Let's work on finishing the problem."

We turned 2/3 into 4/6, and then added the fractions together to get 7/6.

     The kids jumped all over it.   "That's an improper fraction, Mr. Schwartz!"
     "Yes it is," I said.  "Can you say more about 7/6?"
     They didn't let me down.
    "6/6 is 1 whole, and there 1/6 left.  That makes 1 1/6!"

Look at that!

     "There's Jeremy's mixed number!" I said, addressing everyone at the table but looking right at him.  "1 1/6.  A whole number and a fraction.  That's what makes a mixed number."
      Did I make the right decision?  I'm sure there are others I could have made, and maybe some would have been better.  I know I have to spend some time with Jeremy, who is likely in need of some intervention.  But in that moment, Jeremy had the correct answer.  Because inside of every wrong answer, there's a correct answer just waiting to get out.  If we can find the right question, we turn it into a functional miss.
    As fate would have it, that night the Rockets destroyed my sorry Knicks.  James Harden went off for 25 points and 9 dimes in just 30 minutes.  As much as it hurt, I had to smile.
 

A Game, a Decimal Point, and Some Formative Assessment

    Looking for a fun way to have the fifth graders practice their multiplication and addition facts, I mashed up two games, one from Math 4 Love and the other from Malke Rosenfeld.  I did a demo round, then wrote the rules on a whiteboard:

Fact practice, division, some strategy: what's not to love? 

I hoped they could come up with something creative.  In the interim we called it, "The Game With No Name".

The entries came in fast and furious. 

     As I walked into class one afternoon, a little later than usual, I sensed something wasn't right.  The room, normally buzzing with activity, was quiet, save for the sound of suppressed giggles.
    "Something going on I need to know about?" I asked.
     Rich looked at me with a smile.
    "Check the whiteboard."

Look carefully.  I missed it at first glance, too.  Someone had erased the decimal point.

;
     "Oh no!" I cried out, in my best fake-horrified voice.  "How in the world am I going to come up with that kind of money?  I've got bills to pay!  A family to support!"  I turned to Rich, pleading,
     "Maybe Mr. Whalen can help me out?!"
     They couldn't hold it in any longer.  The class exploded with laughter.  It took a few minutes for them to get back on task, but I didn't mind in the least.
    "Well," I said, turning to Rich, "There's a little bit of formative assessment for you.  Someone in here knows the significance of a decimal point."
     And the winner was...

Second place: "Alligators on a Train: Math Edition".  Don't ask.

   

Happy Birthday, Exit 10A

     Exit 10A is one year old today!

Let's celebrate!

     The blog has come a long way from its first post, which featured a video of me peeling a clementine for an estimation180-type activity.  When I started writing, 1 year, 51 posts, and 35,000 pageviews ago, I could never have imagined that it would experience such growth.  I remember the feelings of nervousness and anticipation as I hit the "publish" button for the first time and sent my thoughts hurtling into the internet universe.  Would anyone listen?  Would anyone respond?
     I spent a week obsessively checking, hoping someone would leave a comment.  Nothing. I rationalized by telling myself that teachers were on winter break, and had better things to do then read math blogs, but as the days wore on I grew increasingly despondent.  And then, finally, it happened:

     A comment!  Someone named George Broklaw had taken the time to respond to the clementine post.  Who was he?  A teacher?  Maybe from the UK?  No matter.  I was ecstatic.  It wasn't until several days had gone by that I found out (from my wife) that "George Broklaw" was actually my son, Sam, commenting under a pseudonym.  He felt sorry for me.  What a good boy!
    Comment number 2 came about a week later:

     Andrew Stadel had been the inspiration behind the estimation task that I had described in that first post, and his work, both at estimation180 and his blog, Divisible by Three, was a major catalyst for starting a blog of my own.  His comment, along with Mr. Broklaw's, was all the encouragement I needed.
     Of course it takes a village to raise a child, and I need to take a moment to thank those who have been instrumental in helping make Exit10A a reality.  I have written about the tremendous influence those in the MTBoS have had on my work.  Special thanks to Dan Meyer, who was an early supporter, and brought word of my blog to the attention of his wide audience with this post.  I am blessed to work with an incredible group of teachers.  They have given me their time, their classrooms, and their trust.  Nothing could happen without the support of my principal.  A shout-out to my partner-in-crime and "work wife", Theresa.  And to Barbara, Kay, and Mr. Broklaw: love you guys!
  OK, enough yappin'.  Time to eat!

How many people can this cake feed?
Give me a too low, a too high, and a just right.  On an open number line, of course.

The Most Important Period of My Week

   What's the most important period of my week?

Monday morning, first period.  

     Every Monday morning, beginning at about 9:20 and ending around 10:00, Rich and I sit down at his back table.  There's no formal agenda.  Just two teachers talking, and we cover the waterfront.  For example, during the the past several Mondays we:

• Looked over the next unit assessment and made some decisions about what concepts to give priority focus;
• Checked the calendar and the curriculum guide to make sure we were on schedule, and determined how to arrange and organize the week's lessons;
• Previewed a fraction project we had found on the Georgia Frameworks and started brainstorming how it might be adapted for the class;
• Discussed the most effective way to provide meaningful feedback for an open response problem we had given the kids to work on the previous week;
• Analyzed the work of some at-risk students and began re-formulating intervention plans;
• Debated what to assign for homework;
• Reorganized the room to accommodate the new 4 ft. by 8 ft. whiteboard we had secured to mount on his back wall;
• Commiserated on the woeful seasons our two favorite football teams were suffering through, the Redskins (him) and the Giants (me).

We're very busy, and there's lots to accomplish.

     It may seem counter-intuitive to claim that the most important period of my week is spent sitting at a table with a teacher and not standing up in front of a class.  But everything good that happens during the time we spend with our fifth grade class has its origin in our Monday morning meeting.
     I wish I could do this with every teacher in my building, but there are many obstacles to making this wish a reality.  Teachers are very possessive of their "prep periods", and for good reason.  They don't have many of them, and it's the one time during the day where they can catch their breath: check homework assignments, clean up from lessons just completed and set up for lessons that will follow, touch base with grade-level colleagues, meet with administrators or school support staff, mark classwork, contact parents, fix bulletin boards, catch up on clerical work, and the myriad of other tasks, both important and mundane, that  must be done but cannot be done while the kids are in the room.  And of course there's the issue of finding a time in my schedule that might coordinate with a time in theirs.  So we talk wherever and whenever we can.  Before school (if you happen to come in early), after school (if you happen to stay late), in the hallway while you're walking your class to gym or back from music, in the copy room, by the laminator, in the all-purpose room during morning line-up, in the parking lot.  Small moments, and, yes, they're helpful, but they feel rushed and incomplete.  Not like my Monday morning meeting with Rich.
     Clearly this is a problem, and it's illustrated in a graphic I saw on my twitter feed this past October:

.@Veganmathbeagle via @bkdidact pic.twitter.com/bDkJ9vRgVh
— Ilana Horn (@tchmathculture) October 26, 2014

     I found these statistics astonishing.  It's clear we need more time during the day to engage in meaningful collaboration with our colleagues.  If we use it wisely, it will make the time we spend in class more meaningful to our students.
     What's the most important period of my week?  Monday morning, first period.
     What's the most important period of  your week?

Kindergarten Interlude

    Out of all the grade levels I visit, kindergarten holds the most surprises.  I never fail to be astonished by something I see or hear during the hour a week I split between two kindergarten classrooms in my school.
     Case in point: Several weeks ago I stopped by a room on my regular Thursday morning rounds.  The kids were working at centers while their teacher called students over to her desk one at a time for some individualized reteaching.  I sat down at a table where some students were working with a basket of bears and some number cards.

The activity called for the kids to pull a number card out of the basket and then count out a corresponding set of bears.  

     I watched for a while, paying close attention to their counting strategies.  Were they counting with one-to-one correspondence as they took each bear out of the basket?  When I asked them to recount, were they actually touching each bear as it was counted, or did they point with a finger in the air?  Did they use a finger at all?  Were they moving bears from one set to another as they counted?  Did they keep them in a jumbled pile, or did they arrange them in a row or array?  What did they do if they lost count?  As they counted up from 1, did they encounter trouble with the "tricky teens"?  The diversity of technique and strategy was both fascinating and informative.    After several minutes, I had an idea.
         "Next time you count out a set of bears, try arranging them in a pattern," I told them.  I was curious to see how they would execute two different skills, counting accurately and patterning, at the same time.

The fact that they had to pattern led many to arrange the bears in a single file row.

I challenged them to come up with 8 bears in a different pattern.  No sweat.

No prompting necessary.

Of course not everyone prefers single file.  Here's a 6 by 3 array.

This child made a line of 20 bears (18 are visible in the picture).  

Here's a close-up of part of the row.  Not only did the student alternate color, the bears also alternated sitting and lying on their backs!  Then, instead of putting the 20 back and starting over, he pulled another card from the pile.  It was a 6.  "I'm going to add a 6 and a 20!" he exclaimed proudly.

Now we've got problems! 

     Can you make a pattern with just one bear?  What about 2 bears?  What's the least number of bears you need to make a pattern?  I left them to ponder these questions,  questions they had earned.
     I wasn't always able to work kindergarten visits into my schedule.  Until recently, our district had a half-day kindergarten program.  By the time the little guys and girls had taken off their coats, found their seats, eaten a snack, and gone to the bathroom, it was time for them to go home.  But now we have a full-day kindergarten program, and the teachers have more time to spend on math.  And since Theresa has joined the staff, adding another math specialist to our building, it has freed up some time in my schedule, time I have dedicated to kindergarten.
     Working in our kindergarten classrooms has made Theresa and I much more effective and knowledgeable specialists.  It is important for us to understand what is happening in these classrooms; to observe how our youngest students interact with and create mathematics, to help their teachers find appropriate resources and continue their professional development as math educators, to model lessons, to assist with assessments, to poke, push, and experiment.
     Yes, kindergarten is a wonderland full of surprises.  Plus they have the best snacks, and some really cool things to play with!

Who doesn't love big dice?