Tuesday, December 23, 2014

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.

Saturday, December 13, 2014

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:

     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?

Saturday, December 6, 2014

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?


Thursday, November 27, 2014

The Return of Practice Standard #6

    Thanksgiving means many things...

Turkey and trimmings...

the Macy's parade...

Detroit Lions football...

and, at least at our house...

     I have documented the importance of Practice Standard 6, Attend to Precision, in a previous post.  In that case, the failure to demonstrate proficiency led to disastrous consequences.  Here, it's a different story.
    Early on in the movie, Santa comes to Toyland to check on his Christmas orders.  The irascible Toymaker asks Stan and Ollie, two of his bumbling employees, to bring out the wooden soldiers Santa has requested:

Stan, Ollie, and a wooden soldier.

Toymaker (turning to Santa): Isn't it wonderful?  It does everything but talk!

Santa: Wonderful, yes, but not what I ordered.

Toymaker: What do you mean?

Santa: I ordered 600 soldiers at 1 foot high!

Toymaker: What? (turning Stan and Ollie, disapprovingly): You took that order!  What was it?

Stan (sheepishly): I thought you said 100 soldiers at 6 foot high.

"Ho!  Ho!  Ho!  You got the order all wrong!  I couldn't give those things to my children to play with!

"Put it back in the box and get out of here!  You're through!"

   Lack of attention to precision leads to their getting fired from their jobs!   However, as we all know, the 100-strong regiment of 6 foot tall wooden soldiers save Toyland from the invading boogeymen.

The boogeymen are no match for the relentless wooden soldiers.  

     Is the lesson that sometimes our mistakes lead to providential outcomes?  I'll leave that up to you to decide.  In the meantime, Happy Thanksgiving everybody!

Monday, November 24, 2014

Mr. Whalen Goes to Chicago: A Problem in 3 Acts

     By now you all know that Mr. Whalen is a talented teacher.  But what you don't know is that he is an accomplished long-distance runner!   I suggested he use his recent participation in the Chicago Marathon as the basis of a 3-act with our fifth graders, and, well, he ran with it.

ACT 1:

    • What do you notice?  What do you wonder?

There he is, in the middle of the photo, with the black headband.
The chatter started immediately:
  • Is that Mr. Whalen?
  • When was the race?  How long was it?
  • Where did he run?
  • Was he running for charity?  How long did it take him?  
Rich explained that they were looking at a photo taken of him running the Chicago Marathon, and that for the next few periods they would be working on finding the answer to one of their questions.  That set the stage for...

Act 2:

This was the question that would inspire the problem solving activity.
He asked both classes what they thought they would need to know in order to figure it out:

What the AM class wanted to know.

What the PM class wanted to know.
    Rich and I looked over their requests.  We decided to give them most of what they wanted, and would wait to see if they could separate the essential from the non-essential.

Rich divided the class into partnerships and provided each with this information sheet.
   But before they got started, we wanted them to make an estimate.  In  order to give them some frame of reference, he showed them the following: a time-lapse video of the entire 26.2 mile course.

The kids sat back and watched.  As the video progressed, the kids began to make some observations and comments...

  • Why would someone want to do this?
  • You went to Chicago?
  • I'm getting tired.  Did you really run all that way?
  • It's 26 miles?  I can't survive 10 miles without falling asleep!
...and when it was over they had a much better sense of the length of the course, which they were able to translate into some quite reasonable estimates.  One student justified her thinking by explaining that her father had run a half-marathon in 1 1/2 hours, which she doubled to get an estimate of 3 hours.  Another student explained that he had run a mile in about 10 minutes, so figured Mr. Whalen's time at about 260 minutes, which he converted into 4 hours and 20 minutes.
   We let them work for a while, then called for a mid-workshop interruption.  We had noticed that most groups had started in on the problem by attempting to find the elapsed time between his start and finish, but we were interested to know what information they felt was not useful.  Most students agreed that the total number of runners (40,802) was unimportant, but one student disagreed: "Maybe they all got in Mr. Whalen's way and made him run slower!"  A minor dispute erupted over the significance of his coming in 7,230th place overall. That's when I overheard one student whisper to his neighbor, 
     "That's bad.  But don't tell Mr. Whalen."
     "No it's not," his neighbor responded.  "Actually it's good!"
     We let them get back to work, and the answers began to roll in:


     Clearly they could not all be right.  The most common mistake was treating his times, 11:16:58 and 7:35:08, as whole numbers that could be subtracted following the normal regrouping rules:

We had covered this last year.  Obviously the lesson did not stick.  Although some groups were able to convert 3:81:50 into 4:21:50, it still did not help.  Further complicating matters was the fact that all these times seemed reasonable given their estimates.  Except, that is, for one group, who somehow came up with a time of 34 hours and 14 minutes.  This elicited a comment of, "What the heck?!" and a return to the drawing board.  Rich and I were pleased that the group had used their estimate to realize their answer was unreasonable.

     After about a period and a half of work, the kids began to realize that the elapsed time number line they had explored last year was the better option:

Something like this...

...or this.

     Understanding why operating with time does not always yield the same results as operating with whole numbers was a very difficult concept for many of the students to understand.  Even after repeated explanations, I could tell that their knowing nods and comments of, "Oh yeah, now I get it!"  were not sincere.  Rich and I decided to revisit this in the future, perhaps with an "Always, Sometimes, Never" activity, or another problem solving project that would have elapsed time embedded within.
    Yet overall we were pleased with the 3-act, and there were some side benefits to the project, including:

  • Two more estimation activities, one for the first place time and one for the last place time;
  • Some spirited counting circle activities centered around counting by tenths from 0 to 26.2;
  • An attempt by some students to find his time by multiplying his average speed by 26.2;
  • A lively class discussion about whether coming in 7,230th place was good or bad.
And in case you're interested:

Eliud Kipchoge won the Chicago Marathon with a time of 2:04:11.  


Thursday, November 13, 2014

Number Grids and Number Lines: Can They Live Together in Peace?

     "Numbers do not just evoke a sense of quantity; they also elicit an irrepressible feeling of extension in space."  
     Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics, pg. 69

     There's one hanging in every kindergarten...

first grade...

second grade...

and third grade...

classroom in my school.  Yes, the ubiquitous "Class Number Grid".  These grids hang out with big number lines on classroom walls:

Sometimes right next to each other!

And are often seen in smaller versions going hand in hand on desks:

This type of name tag appears on  student desks in many first and second grade classrooms in my school.  It includes a hundreds grid on the left and a number line running across the top.
     But how do number grids and number lines exist in the minds of the students?  How do students see the numbers arranged and extended? Can multiple conceptions exist in their minds at the same time without conflict?
     The kids use number grids to do exercises like this:

Grade 1

Grade 2


And to play games like this:

This game, along with the activities above, are designed to reinforce facility with a number grid.  The grids make patterns appear more apparent and help build place value concepts.  When they first play, you may see kids counting each space as they move closer to 110.  Most catch on quickly to the idea that to move 10 or 20 they can move their token "down" one or two spaces.  Of course there are the few that continue to count by 1s, unconvinced that the moves down the grid actually work to advance them them 10 or 20 spaces on.

But most kids lose interest quickly.

Last year I tried a small innovation:

Play on a blank grid and fill in the numbers as you go along.  After three or four trips from 0 to 100 the grid begins to fill up.
       I had tried this out last year with one of my basic skills students, and liked the results.  This year I had the entire second grade give it a spin.  The more they played, the more their grids filled up with numbers, and those numbers became benchmarks for empty squares.  Looking for a way to help them see the relationship between number grids and number lines, I had them try something different:

Turn the number grid into a number line by cutting the grid into strips and taping the strips together.
Play on the number line.
This was a different kind of experience.  Instead of simply moving "down" to go +10 and +20, the kids had to move to the right.  It was interesting to see how they worked.
This student was on 76, rolled a 1, and jumped 10 to 86.  How did he know where to land?  One more than 85!
I was happy with how this activity played out, although the number lines were long, and finding room to play became an problem (some kids went into the hallway).  Then there was the issue of storage:

We hung them from a clothesline in the room.

    I decided to try the experiment out in third grade with Capture 5:

First we gave them some time to get familiar with the game.

Next, cut out a number grid into strips and glue together to make a number line.
However I couldn't resist adding in an estimation challenge:
How long do you think the number line will be once we put it together?

Normally I would put these estimates on an open number line.

The number line measured 64 inches.  We needed the 39 inches of a meter stick, plus 25 more.  I turned this addition problem into a little number talk.  There were many ways the sum was calculated mentally, and reinforced moving by 10s, 20s, and 30s in the Capture 5 game.

Playing Capture 5 on a number line.  

  I asked the kids to comment on the difference between playing on a number line as opposed to a number grid.  Some reflections:
  • "I had to use my my mind more because I couldn't go up and down."
  • "I had to do more thinking, but it got easier the more I played."
  • "I like the number line better.  The grid is more confusing."
So what's next?  Graham Fletcher has written about the potential benefits of "upside down"  grids, another way for kids to conceptualize how numbers can be arranged.   How about playing on a vertical number line?  Could we adapt games like Capture 5 and the Number-Grid Game for play on a yardstick, tape measure or meter stick?  A thermometer?  Will this cause confusion or help build number sense?  And is it possible that some of our students see numbers in ways we cannot imagine?

Number Forms

    "Though a majority of people have an unconscious mental number line oriented from left to right, some have a much more vivid image of numbers.  Between 5% and 10% of humanity is thoroughly convinced that numbers have colors and occupy very precise locations in space."
     Stanislas Dehaene, The Number Sense: How the Mind Creates Mathematics, pg. 71