Property Values

     One of the first things I do when I walk into a classroom is look for vertical whiteboard space.  Ever since being introduced to the work of Peter Liljedahl, I've counted myself as a member of the #VNPS movement.  Liljedahl's research, along with the efforts of chief practitioner Alex Overwijk, combined with experience watching students use them in my own practice, has convinced me that every available inch within student reach is precious.

     However this prime real estate, which should be preserved as open space for student work, is often taken up by some dubious development.

SMART Boards are one culprit.  Think of all the wasted space underneath!

Big Ideas, Essential Questions, Objectives

 

What's the opportunity cost of all that lost whiteboard space?

     In her book Becoming the Math Teacher You Wish You'd Had, Tracy Zager  summarizes Liljedahl's work, as well as her reaction.  She writes:

     "When I first read Liljedahl's research, a whole series of narratives and images from the history of mathematics buzzed through my brain.  Mathematicians frequently talk about standing around blackboards or whiteboards together, thinking and talking.  This particular kind of collaboration--standing, talking, thinking, and writing--is so inherent to doing mathematics that many math buildings are designed around it.  Given that mathematicians work this way, and that educational research has revealed there are tremendous benefits to vertical, non-permanent surfaces in classroom settings, it seems we have ample reasons to set them up." (pg 322-323)  

     She offers options for teachers in classrooms with limited wall space, including hanging whiteboards on cabinets, closets, and bathroom doors, and allowing students to write on windows with vis-a-vis markers.

Mirrors work, too.  just ask Will Hunting.

     But before we resort to those measures, let's take stock of what's on our existing, classroom-wall  mounted whiteboards.  What's there?  What purpose does it serve?  Who benefits?  Who's it for?  If it's necessary information, can it be moved someplace else?  When I first started teaching back in the mid-1980's, objectives went in our plan books.  We didn't plaster them all over our chalkboards. (No one had a whiteboard back then.)  We didn't know from Big Ideas and Essential Questions.   Is there research similar to Liljedahl's that shows how advertising them promotes a thinking classroom, enough to sacrifice empty whiteboard space?
     "We all have real constraints on the size and layout of our teaching spaces,"  Zager writes.  "Nevertheless, it's worth thinking about how we can work within those constraints to provide students workspaces that promote thinking partnerships." 
     After all, it's their room too.
         

     

Anything's Possible

     In an attempt to make peace with the Exeter problem set, I followed Jasmine Walker's advice:

   Middle school math?  Yikes.  Anyway, here's page one:

     Problems 7-11 came pretty easy.  (Hopefully I didn't get any wrong!)  For #11 I didn't feel like writing an explanation using words and complete sentences, so I just drew a picture.

   Problems 1-6 were another story:

Problem #1
     I didn't think my answer made sense.  I had the feeling that it took a lot longer than 500 seconds for light from the Sun to reach the Earth.  So I googled it.  I was right!

Problem #2
    I knew it would take a really long time, and that I would have to consider that you have to breathe, eat, sleep, and go to the bathroom, and that different numbers take different amounts of time to say out loud.  It all seemed kind of overwhelming!  There's some interesting stuff about it on the internet, though.

Problem #3
    I used what I knew about the speed of light from problem #1 and got an answer of 232,500 miles.  Maybe I was wrong; according to google, the moon is actually 238,900 miles away, but what's 6,400 miles when we're talking about the solar system?  Close enough.

Problem #4
   Nope.  Couldn't do this one.  And I like baseball!

Problem #5
   This one felt a lot like problem #2.  However I buckled down and gave it a go.  I figured on a step being about 2 feet.

Problem #6
    Offshore pipelines?  Cylindrical mechanisms?  I nearly gave up on this one, but at the last moment before publishing this post I figured I'd give it another try.  But 1.36 miles per hour seems kind of slow, so I think I'm wrong.

To sum up:

•  The math was a barrier in problems 4 and 6, although I did understand what was being asked  of me.  I don't feel real good about it, but that's on me.  
• Problem 2 just didn't seem worth the effort.  
•  I had the math skills for problems 1, 3, 5, 7, 8, 9, 10, and 11.  I can't say I enjoyed solving them, although I'll admit to a feeling of accomplishment when google confirmed my answer to problem #1.  And, in all fairness, I didn't get the emotional rewards of sitting down with a peer group and discussing my answers and attempts.  That's also a part of the Exeter experience.     

     Will I print out page 2 and start working on those problems?  Probably not.  If I'm going to solve problems like these, I need a little romance.  I need some help building intellectual need.  I can't always just gin it up on the spot.   I did my homework, but it felt like just that--homework.
     This could be the heart of the MTBoS project: how do we (as teachers) take problems like these and turn them into something that a student might want to solve?  Not because it's assigned for homework, or because it's going to be on the test, but because he's curious to know the answer.  Is that putting a premium on answer-getting over sense-making?  Does this mean that deep down I'm an answer-getter, not a sense-maker? Do I have to be one or the other?  Or can I have some of both in me?  If needing to know the answer is the motivation for learning the math, does that mean the answer-getting impulse is something that should be cultivated?
     Michael Pershan says that just like there are different genres of literature, there are different genres of math problems.  OK, so Exeter problems are not my genre.  But what genres do I like?  And what does that say about me?  The answer may lie in an experience I had at TMC '17.
     Since the host site was not within walking distance of lunch options, on several days it was arranged that food trucks be brought to campus.  Feeding 200 people out of 2 food trucks was quite an operation; the lines were long and the Atlanta heat was unforgiving.  My hunger got the better of me, and I ignored the advice from my friend (and Georgia resident) Graham Fletcher to wait inside in the air conditioning until the line dwindled down:

Big mistake.

    For the first few minutes it actually felt good to be outside after spending all morning indoors.  I got a chance to talk to Annie Perkins, who was standing behind me in line, about her experience in the Playing With Exeter Math morning session.  She explained that it was a good opportunity to just work on problems, something she had trouble finding the time to do during the school year.   As we spoke, I started to feel the famed southern heat and humidity start to kick in, and I realized that the line had barely moved.   I began to have second thoughts.  Should I go back in?  Or have I reached the point of no return?  How much longer am I going to have to wait?  Will I get sunburn?  Heatstroke?  Maybe if I started to keep track of how long it took someone to receive his lunch, I would get an idea of how much longer I'd have to stand in line.  So, in part to pass the time and in part because I was just plain curious, I asked John Stevens, who had just placed his order, to guess how much time it would take for him to get his food.
     "4 minutes and 29 seconds," he ventured.
      As I looked down at my watch to check the time, I heard a voice behind me.  It was Annie.
     "No, it won't take that long.  I've been standing here for twenty minutes, and..."
     And here Annie launched into a very cogent mathematical explanation about why it wasn't going to take John 4 minutes and 29 seconds to get his food.  My brain, baking in the heat, couldn't quite follow, but it made perfect sense.  I forgot to look back at my watch, and by the time John had his food I was back inside, in the air-conditioning with Graham, who was kind enough to refrain from saying, "I told you so."
      Should I have waited outside with Annie?  How long would it have taken me to get lunch?  How did Annie know?  It would have to depend on how long it took a person to get his food.  Would that depend on what he ordered?  Whether he paid with cash or a credit card?  If I could find the average time, and then multiply that by the number of people ahead of me in line, I could get a good idea.  It was nothing if not a math problem, but a problem that, unlike the Exeters, I was curious about.  I imagined what the problem might look like, written out and embedded inside a page of the Exeter problem set.  Would I have cared about it then?  Probably not, because it didn't come from inside of me.
     Maybe one day I'll feel like Annie, and jump on the opportunity to solve problems like those in the Exeter set.  Right now I kind of doubt it, perhaps because there's too much scar tissue.   I'd like to be able to have the choice, and the measure of how bad I want it will be my willingness to suck it up, buckle down, and learn the math.  I won't rule it out, because if there's one thing I've learned it's to never say never.  Given where I've come from, who'd have ever thought that I'd be standing outside in the hot Georgia sun, in line to get lunch from a food truck, in the middle of the summer,  at something called Twitter Math Camp, on my own dime, and turning the experience into a math problem?  Believe me when I tell you: If that's possible, then anything's possible.
       
          
    

It's Not You, It's Me

 Twitter Math Camp, 2017:

     In the end there were two: Mathematical Yarns with David Butler and Megan Schmidt or Math Coaches Huddle with Chris Shore.  Veteran Twitter Math Campers know that your AM session choice is all-important, because the one you select is the one you commit to for all three mornings.  That's a total of six hours, so best choose wisely.  As much as I would have loved to spend that time exploring mathematics through the medium of hyperbolic crochet with David and Megan, it was Math Coaches Huddle that ultimately got the nod.  I'm going to be doing quite a lot of coaching in my new role, and who better to help me up my game than Chris Shore.
     One thing for sure; I wasn't going anywhere near Room 711.  That's where Wendy Menard, Danielle Reycer, and Jasmine Walker would be running a session called Playing With Exeter Math. What I knew about Exeter Math I learned from Joel Bezaire: people sat around something called a Harkness Table and solved problems created at the very exclusive Phillips Exeter Academy that looked like this:

 

     Who in the world could possibly want to solve these? And voluntarily too?!  Playing with Exeter Math?  Really?  To me it looked about as much fun as a root canal.  You want playing?  Playing was what I did in David Butler's Friday afternoon session One Hundred Factorial: Playful and Joyful Maths; a smorgasbord of puzzles and activities we could, according to the session description, "Follow down any rabbit hole that looked interesting."  And before we all made our way down to the cafeteria, which David had set up to look like a math amusement park, he explained that the goals of the session were: learning something, making or seeing something beautiful, understanding someone's thinking, and last, but definitely not least, sharing joy together.  Now there were some SWBATs I could get behind!  We weren't disappointed:

Megan Schmidt and Steve Weimar 

Taylor Belcher and Doug McKenzie

Jasmine Walker, Jim Doherty, and Maureen Ferger

     The contrast, in my mind, was stark.  One the one hand, Exeter Math problem sets: walls of text, a whiff of elitism, for the smart kids in the honors class.  On the other, David Butler's One Hundred Factorial: visually engaging, democratic, no experience necessary.  They were diametrically opposed.  I felt the two sitting on my right and left shoulders like an angel and a devil.
    But if it's nothing else, Twitter Math Camp is a place where one can work out his or her math issues with no fear of judgment.  Which is how dinner on Saturday night turned into something of a therapy session.  Out with a bunch of TMCers at the Cowfish Sushi Burger Bar, I found myself seated next to Jasmine Walker.  I had worked with Jasmine on a Skyscrapers puzzle in the One Hundred Factorial session.  Jasmine had also helped facilitate the Playing With Exeter Math session.  Perfect! Looking for confirmation and affirmation, I asked her to compare the two.
   "They're completely different experiences, right?" I asked.
   "Not at all!" she exclaimed.  "I find the same kind of emotional happiness working on the Exeter problems as I do when I work on problems like Skyscrapers."
    I was stunned.  It sounded impossible to me, but Jasmine went on to explain that, despite their outward dissimilarities, they actually had much in common.  Like the activities in One Hundred Factorial, the Exeter Math problems could lead to unforeseen and exciting places, and engender the same amount of joyful, raucous back and forth that we had experienced together in David's session. In fact, according to Jasmine, all of the goals David had set for his session got checked off in the Exeter Math session.  From his seat across the table, Jim Doherty, another One Hundred Factorial participant and himself an Exeter problem set veteran, did his best to come to my aid, but in the end pretty much echoed Jasmine's reaction to my question.
    I've thought a lot about the conversation I had that night with Jasmine and Jim.  And what the experience has caused me to do is take a good look in the mirror.  I was angry with the Exeter problem sets.  Angry because they looked like all those word problems I had so much trouble with when I was in high school.  Angry because they reminded me of how inadequate they made me feel back then, and how inadequate they make me feel in the here and now.  Angry because they seemed so intractable, so cold, so bloodless.  I wanted them to change.  I wanted them to lower their barrier to entry, to turn themselves into something I could access, to be something I could share with Jasmine and Jim and Joel.  I wanted to be able to walk into Room 711, and I was angry because they were blocking my way.
     But what I've come to realize is something I've known all along but been unwilling to truly admit: Exeter Math isn't to blame.  The one deserving of my anger is me.  I have to take responsibility for my own learning.  If I want to play with the big kids in Room 711 I have to learn how to do the math.  And to do that, I mean to really do it, I'd have to start back in middle school and go all the way up through Algebra 2.  Not quite Billy Madison, but pretty darn close. After I did that, I could decide for myself if solving Exeter Math problems was a rewarding activity or a waste of my time.  And I don't think I'd care either way, because the decision would be mine.  All mine.
   
 
   
       
     

     

How Do You Get to School?

"The standards do encourage that students have access to multiple methods as they learn to add, subtract, multiply, and divide.  But this does not mean that you have to solve every problem in multiple ways.  Having different methods available is like having different means of transportation available to get to work; flexibility is good, but it doesn't mean you have to go to school by car, then by bus, then walk, then bike--every single day!"

Bill McCallum

Misconceptions About Multiple Representations

   

     Last month I stopped by a second grade classroom where the teacher was administering an end-of-year math assessment.  I paused by the desk of a student who, with a look of frustration on her face, was puzzling over this question:

     "What's the matter?" I asked.
     "I forgot how to use an open number line," she responded, head down, staring at the blank page.
     "Do you know another way to solve the problem?"
      She looked up at me.  "Partial sums?"
     "Could you show me how you would do that?"
      Here's what she produced on a piece of scrap paper:

And wrote the answer, 79, in the space provided.

Question: Do you mark this wrong because she couldn't show her thinking on an open number line?

     Continuing to make my way around the room, I came upon this response:

     "Tell me what happened here," I asked.
     "I got confused about using the number line."
     "Do you know another way to solve the problem?"
     "I could draw base-10 blocks."
     "Could you show me how you would do that?"

Question: Do you mark this wrong because he couldn't show his thinking on an open number line?

     I spent the next several days looking through other end-of-year assessments for examples of questions where students were being commanded to solve problems using specific representations and methods.  Here's a sample from grades 1-4:

• Use the break-apart strategy to solve each problem.
• Use the turn-around rule to solve.
• Explain two different ways you could use doubling to solve 6 x 8.
• Explain how you can think addition to solve 14-7.
• How can you find the sum using a number grid?
• Explain how you can use the near-doubles strategy to find the answer.
• Use base-10 shorthand.
• Use an open number line.
• Solve using partial-sums addition.
• Solve using U.S. traditional addition.
• Use partial products or the lattice method to solve.
• Use U.S. traditional subtraction (this for 38,000 - 23,177.)

     As I recall, in my math classes growing up there were no multiple methods or representations.  You memorized your facts and used the traditional, standard algorithm.   I'm sure I had classmates clever enough to devise alternate strategies on their own.  As for me, I was out of luck.  That's too bad.  I wish I had the exposure to the multiple methods and representations that are now considered essential components of math education today.  If I had, maybe this wouldn't have happened.
     But in the leap from standards, to curriculum, to assessment (especially assessment), something has gone awry.  We want to expose kids to multiple representations and methods, and encourage them to experiment with, explore, connect, and analyze them.  But do we want to force kids to use them on summative assessments? For a grade?  The two students wrestling with question 16 above each had their own way of thinking about 43 + 36.  But the directions to the problem, which instructed them to show their thinking on an open number line, only served to shut their thinking down.  How did it make them feel?  And how will they feel when they get their test back and see that a problem that they can find the answer to is marked wrong because the way they want to show their thinking is not what the test maker wants?
     Providing access to and connecting different models, methods, and representations for students as they find their way to computational fluency is very important.  But I think that in forcing the issue we run the risk of doing more harm than good.  How kids get to school is dependent on many variables, none which are under their control.  The ultimate decision rests with us adults.   How about we let the kids decide for a change?    
 
           
      

     

Goodbye

   

     After 31 years, 23 in the classroom and 8 as a math specialist,  I am retiring from public education.  I've spent them all in the same K-5 elementary school, off the New Jersey Turnpike in East Brunswick, NJ.  Over half my life.  It's where I got my first teaching job, where I met my wife, where I lost my wedding ring on the big playground, where my kids came to visit on Halloween and Field Day and Bring Your Child to Work Day, where I shared all the ups and downs of life, both professional and personal, with my colleagues, where I had the privilege of getting to know so many amazing students and their families. What is a school if not an intersection where lives meet?  What is a school if not a place filled with life, in all its very beautiful, very messy, and very human complexity?

     While I will continue to be active in the world of math education and write about my experiences here at Exit 10A, I'm going to miss my brick and mortar school and the family I found within its walls.  The noise in the all-purpose room during afternoon dismissal, the bustle in the hallways when periods change, the groans when the announcement that, "Recess today will be indoors" broadcasts over the intercom, the faculty room and copy room teacher chatter.  And the small, intimate moments.  The little kindnesses.  The inside jokes. The whispered gossip.  The hushed, secret corner conversations.  The tears and the laughter.  Those are the things that seem to matter most to me now, and I can already hear their echoes.

Goodbye, Room 10A

Goodbye, Chittick School

For all those facing transition, in this season of transition:

We shape our self 
to fit this world
and by the world
are shaped again.
The visible
and the invisible
working together,
in common cause,
to produce 
the miraculous.
I am thinking of the way
the intangible air
passed at speed
round a shaped wing
easily 
holds our weight.
So may we, in this life
trust
to those elements
we have yet to see
or imagine,
and look for the true
shape of our own self,
by forming it well
to the great
intangibles about us.

David Whyte
Working Together

$167.36 On the Nose

From estimation180, Day 161:

What's the value of all the coins in the bowl?

  Before you read on, take a moment and come up with an estimate.

   Andrew was gracious enough to provide us with the receipt, so Rich and I decided to use the prompt as Act 1 of a 3-act task.  It would provide the kids with multi-digit addition, subtraction, and multiplication practice, and since the bank takes one-tenth of the total amount as a fee for non-members, we would also receive some formative assessment information on how the students thought about decimals and place value.

   We set the kids up in random groups on whiteboards, and asked them first for estimates. Nicole, the ILA teacher next door, poked her head in.  She asked what was going on, and one of the kids explained that they were estimating how much money was in the bowl. After a few minutes of thought, she started looking around for a scrap of paper.  Finding none, she pulled a tissue from a nearby tissue box, wrote something down, and handed it to me.  I folded it up and put it in my pocket, distracted by all the activity in the room as the kids finalized their estimates and began figuring out how much 1/10 of 519 quarters, 898 dimes, 719 nickels, and 917 pennies was worth:

Emptying my pockets at the end of the day, I came across Nicole's estimate:

     What began to fascinate me, what I wanted to know, wasn't how she came up with the number, but why, having been asked for an estimate, she came up with something so exact!  Not $160, or $170, or $200, but $167.36.  We've been playing around with estimation for years now, and we're continually encouraging the kids to choose friendly, round numbers as estimates, numbers that tell about how many or how much, not necessarily exactly how many or how much.  But we've met with reliable, obstinate resistance.  I looked back at some of the pictures I had taken of student work, looking for a record of their estimates, and while I did see estimates like $300, $40, $50, and $60, I also saw $63.12, $5.57, and $312.10.  Pointy numbers.  Precise numbers.  Numbers that spoke of exact amounts.  Not round numbers.  Not in the general vicinity numbers.  Not numbers that spoke of about how much.  Why?  Is there something hard-wired into our human nature that, when presented with a task like this, makes us want to be exactly right?  Not close enough, but closer than any of our classmates?  Have we been so conditioned by the "Guess How Many Jelly Beans in the Jar"  challenges that we treat every estimation task as a chance to win a prize?
     Several days later I asked Nicole about her estimate.
    "Why so exact?  Were you trying to guess the exact amount?"
    "No,"  she explained,  "I was trying to estimate.  But I guess in my mind they're the same thing."
    A few days later, eating lunch in the teacher's lounge...

A leftover party favor from a week-end birthday party.

 ...four of my non-math teaching colleagues found themselves unwitting participants in an experiment.
     "I want everyone to estimate how many gumballs are in the container."
     The group was willing to cooperate, and within several seconds one piped up:
     "Are we going to count them?  We have to find out who won."
     I quickly got up and searched for a piece of paper and a pencil to write down the quote.  She had, on her own, without any suggestion from me, injected an element of competition and challenge into the task.
     After a few minutes I asked for their estimates.  I received 3 pointy numbers, 84, 74, and 78, and one round number, 190.  (This teacher had estimated 192 but rounded down to 190.)  Although I wasn't so much interested in their reasoning, they all wanted to explain their thinking, and carefully listened to one another as they each shared their strategy in turn.  I explained my purpose in asking.  I was curious, I explained, why the three hadn't chosen round numbers as estimates.
    "The answer is never a round number!" one explained.
    That statement gives me a clue as to what may be at work here.  During these kinds of  estimation tasks, I'm asking kids to engage in sense-making, not answer-getting.  Maybe the line between the two is blurred, but it's there.  $167.36 is an answer, not an estimate.
    Answer-getting is stubborn.  I know this is true, because my teacher's lounge colleagues were just dying to know, couldn't wait to find out exactly how many gumballs were in the container.  They couldn't let it go:


gumball1 from Joe Schwartz on Vimeo.

Can you?

In Orbit

     A number, I'm told, is like a Russian doll.

  This is because the quantity that a number represents contains the quantities represented by all preceding numbers.  This is called hierarchical inclusion, and its understanding is a very important stage in the number sense trajectory.  I thought I knew this.  But something happened recently that caused me to wonder: Do I really understand?
 
   It started in fifth grade with a Fraction Talk:

 Two different attempts caught my eye, and I turned them into a notice and wonder activity:

     I was trying to draw out the idea that the first response, the one on the left, was correct because the student had accurately labeled each small square as one-sixteenth of the whole.  However in the second response, the small squares were labeled incorrectly.  That student was counting by sixteenths, labeling each successive square as if it included within it all the preceding squares.  Was that an example of hierarchical inclusion?  I went back to this picture, which has helped me understand the concept:

From Early Childhood Mathematics Education Research: Learning Trajectories for Young Children by Julie Sarama and Douglas H. Clements

   
     It certainly seemed the case, but then I noticed something in the caption that I hadn't really noticed before; that each cardinal number includes those that came before.  What about other kinds of numbers?  Wait a minute.  There are other kinds of numbers?
    A quick search led me here, where I learned about three different kinds:

• Cardinal numbers.  They tell us how many of something there are. 

• Ordinal numbers.  They tell us the position of something.

• Nominal numbers.  They are used as names, or to identify something.

     I realized that I already knew about ordinal numbers; nominal numbers were new to me.  So I started collecting numbers around my school and mentally trying to classify them.  Here's a sample of what I found:

I.

  Nicole has her first graders count the days in school, and my first instinct is to call this a cardinal number.  Many primary teachers accumulate tokens, such as Popsicle sticks or unifix cubes to represent each successive day, suggesting that days are a set of things that can be counted.  However does each individual day include within it all preceding days?  Or do we get a fresh start each morning?  I don't know.  Here's Sarama and Clements, again in Early Childhood Mathematics Education Research, "When the topic of 'ordinality' is discussed, even by some researchers, it is often assumed that all ordinal notations must involve the terms 'first, second, third...' and so forth.  This is a limited view.   A person who is 'number 5' in a line is labeled by a word that is no less ordinal in its meaning because it is not expressed as 'fifth.' (p. 85)   So here Day 124 is the 124th day of school.  It is in position 124 of a sequential count of the days of school that started at 1 and will end at 180.
Conclusion: Ordinal.  Maybe cardinal.  Definitely not nominal.

II.

   This storage room doesn't include within it all rooms numbered 1 through 139, and there's no reason why this specific room should be identified as the 139th in a sequence of rooms.  In fact, there are not even close to 139 rooms in my school.  139 here seems to function as a signifier or name for this particular room.
Conclusion: Nominal.

III.

    The hooks in Wendy's grade 3 classroom are in sequential order.   The polka-dot backpack is on hook number 3, or the third hook.  In one way it seems like the number 3 is also acting here as a name for that hook.  Can a number be both ordinal and nominal at the same time?
Conclusion: Ordinal.  Maybe nominal.  Definitely not cardinal.

IV.

     The thermostat in the office must be broken.  No way it was anywhere near 76 degrees.  Regardless, this number is definitely not nominal.  Thinking of the temperature as I would see it on a mercury thermometer...

...helped me see that the number of degrees did include within it all preceding temperatures.  So did this, which I recorded one night testing my chicken pot pie:



thermometer from Joe Schwartz on Vimeo.

Conclusion: Cardinal.

V.

 The back of a fourth grader's basketball jersey.  We're not counting or ordering anything here.  The number 3 is just identifying Clippers point guard Chris Paul.  Maybe Paul picked it because it was his favorite number.  Maybe he wanted another number but it was already taken.  Anyway, the student sporting his jersey isn't even a Clippers fan.  He likes the Cavs.  Front runner!
Conclusion: Nominal.

VI.

   This is how the kindergartners in Kelly's room keep track of who's in school.  This one was easy.
Conclusion: Cardinal.

VII.

     10:36:16.  What kind of number is this?  I've come to think of nominal numbers as having a randomness about them; a phone number, a driver's license, an account number, my zip code.  I don't get that feeling here.  Could this be ordinal?  Ordinals have a sequential, positional feel to them, and of course there's always the th.  Could we say it's the 16th second of the 36th minute of the 10th hour?  Is 10:36:16 a Russian doll, nesting right between 10:36:15 and 10:36:17?  Maybe it's like what T.S. Eliot wrote:

    Time present and time past

Are both perhaps present in time future

And time future contained in time past.

Conclusion: Cardinal.  (Pretty sure.)

     Sarama and Clements cite research indicating that, "It is not until age nine that most master the hierarchical inclusion relationship."  (ECMER, p. 339)   If I did master the relationship way back when, then I guess my path through the world of number sense is more like an orbit than a trajectory.  It's taken me 46 years, but I've circled back around to find that numbers, for me, remain enigmatic and mysterious, in need of continued and constant rediscovery.

  

Cousin Ben

     He appeared in our lives out of nowhere.  Just showed up one night, invited to dinner by my mom. A distant relative, and for the life of me I would never be able to remember exactly how we were related.  In his mid to late-thirties at the time, a bachelor, with a receding hairline, a fu-manchu mustache, and a big ring of apartment superintendent keys dangling from his belt loop.  Except he wasn't a super.  He was a middle school english teacher in South Bound Brook.
     Cousin Ben became a regular dinner guest.  I was a geeky 1970's middle-school bookworm working my way through the entire Ray Bradbury catalogue, and we connected through a mutual love of science fiction.  He took me into the city, and we wandered around the Village, poking our noses into used bookstores.   He introduced me to Piers Anthony's Macroscope and Larry Niven's Ringworld and Ursula Le Guin's The Dispossessed.  I ate it up.  All of it.  He talked about his house, which he was constantly fixing up, and told us stories about his classes; how the kids would tease him because of his balding head, how he tried hard to connect with them and get them engaged, how he argued with his supervisors in the english department over the assigned novels, how he worked for his union local as a member of its negotiating team.
     I come from a family of businessmen; builders and real estate, insurance and finance.  A few lawyers thrown in.  Those were the kinds of jobs you got when you grew up.  But a teacher?  Cousin Ben was the first teacher I knew outside of school, the first teacher I thought of as someone who taught to make a living, the first teacher I knew who talked about the job of teaching.  Looking back, I realize it was Ben who first put the idea in my impressionable mind that teaching might actually be a career opportunity.
     After I graduated from college and moved back to New Jersey, we met several times for dinner at a place on Route 22.  He was still working, and a little jaded.  I was just starting my first teaching assignment, and he was curious to know how I was making my way in the profession.  Of course he talked union, but I didn't really pay attention.   (It wasn't until later that I realized how important that work was too.)  Then we lost touch.  But I did see him several years ago, again out of nowhere, invited by my mom to dinner.  He'd retired, and lost whatever hair he had left.  He was still reading sci-fi, still fixing up his house in South Bound Brook.  And I still couldn't remember exactly how we were related.
     Cousin Ben helped improve the lot of many teachers by fighting for fair working conditions and compensation, and touched the lives of countless middle schoolers with his passion for reading and literature.  And he showed one awkward teenager that teaching could be a life's work. So before I bring the curtain down on 31 years in an elementary school somewhere off the New Jersey Turnpike, I want to take this Teacher Appreciation Week 2017 to say:
Thanks Ben.

  Those who build walls are their own prisoners.  I'm going to go fulfill my proper function in the social organism.  I'm going to go unbuild walls.
--Ursula K. Le Guin, The Dispossessed
                                                           
   

"Who Wants To Count My Windows?"

     Just back from spring break, with two days to fill before the start of PARCC testing.  What better time to have the fifth graders dig in to one of our all-time favorite projects: Ant Hotel.

     Ant Hotel is a modified version of the Fawn Nguyen/Andrew Stadel classic Hotel Snap.  The goal is to build the hotel that, given certain constraints and parameters, yields the highest profit.   (Read Fawn's post to get all the details.) We took one of her suggestions, and reduced the number of hotel rooms (snap cubes) from 50 to 12.  With an adjusted tax table and new recording sheets, we were ready to go.

     We divided the class into teams of 2 and 3, and explained the task.  We had each team build the exact same hotel, and modeled how to calculate costs, including how to count square units of land, windows, and roofs.  We unpacked the tax table, and explained how to compute the property tax.  (This is New Jersey, after all.)  On the profit side, we led the class through the process of counting and classifying rooms and then figuring out each one's value.  Finally, we determined the net profit by subtracting total cost from income.  All told, this took about 30 minutes.  The kids caught on quickly, and immediately set off to work.

IMG_6971 from Joe Schwartz on Vimeo.

     There's a ton of great math packed into this activity, including: area, multi-digit addition, subtraction, and multiplication, fact extensions, decimals/percents, geometry in three dimensions, and counting.

counting rooms from Joe Schwartz on Vimeo.

      Counting?  Counting is a skill that doesn't get much play in grade five, but it's a critical element here.  An accurate count of hotel windows (exposed vertical faces of snap cubes) can be the difference between a hotel making and losing money.  There were disagreements among team members, which led to one of my favorite scenes so far this year: a student, whose team could not agree on an accurate count, walking around the room, waving his hotel in the air, and pleading,
   "Who wants to count my windows?"
    Audible groans when the period ended, and the following day they couldn't wait to start back in.

     Teams built multiple hotels, and recorded their best efforts on the whiteboard:

     Over the course of the week, the kids clamored to continue the project.  Despite the schedule being upended by the standardized testing timetable, Rich found the time to accommodate both AM and PM classes.  To the sound of howls and groans, the whiteboard standings changed as top-earning hotels were toppled from their perch due to mistakes, most of them careless, in calculating tax and room values, as well as errors in counting windows and roofs.  (This is why a sketch is imperative: so the hotel can be recreated and checked.)  In one class, competition got a little too heated, and there were some hurt feelings when errors were exposed.  Note to self: give teams the opportunity to reexamine their work before submitting it to the scrutiny of the class.
     This task has many things to recommend it.  Besides the math (both content and practice standards), it's hands-on and low-tech, and requires little prep.  So from all the kids (and their teachers): Thanks Andrew and Fawn!

"We love it!"