## Tuesday, September 5, 2017

### 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.

## Friday, August 18, 2017

### It's Not You, It's Me

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.

## Monday, July 17, 2017

### 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

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:

"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.
• 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?

## Friday, June 23, 2017

### 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

## Wednesday, June 14, 2017

### \$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?
"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?

## Monday, May 22, 2017

### 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.

## Monday, May 8, 2017

### 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

## Monday, April 24, 2017

### "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!"

## Thursday, April 6, 2017

### 4 Students, 1 Question, 1 Wish

58 students on a field trip.  4 students can fit in a van.  How many vans are needed?

I gave this problem to four of my basic skills students, each one individually.  They worked it out on a big whiteboard in my room.  All fourth graders,  just finishing up a unit on division.  Here's who they are and how they responded:

I. Daiba

Daiba has been in our district since 2014, but she's new to our school, one of the several dozen students repatriated back last fall due to the opening of an ELL class.  She's since placed out, and although she does have some trouble expressing herself, her English sounds pretty good to me. She's compliant, sweet, quiet and shy.  She's had trouble making friends.  I never see her interacting with other students.  Sometimes I wonder if it's because she's only one of two girls in the entire school who wears a hijab, or if it's just because she's socially awkward.
Daiba likes coming to my room, and I pull her several times a week in addition to pushing into her class.  We've worked on multi-digit addition and subtraction computation, place value concepts, and rounding.  She asks insightful question.  She started work on the problem right away, using the partial-quotients algorithm she learned in class.

When I  asked her how many vans would be needed, she responded 14.  I asked her about the remainder, and she explained that 2 students were left.  I asked her whether or not they'd be going on the trip, and she said no, they wouldn't be going.  I didn't respond.  After a moment or two a little smile broke across her face.

Her: That's not nice to leave them behind.
Me: No, it's not.  What are you going to do about it?
Her: I could put them in another van.
Me: Then how many vans will you need?
Her: Fifteen.
Me: (smiling) Back to class with you!

II. Justin

Justin's the boy with the buzz cut.  I know him pretty well, having worked with him all of last year, and in my experience he's a happy-go-lucky kid who likes to draw, collect toy cop cars, and ride dirt bikes.  But he's had a rough year.  He's been unable to connect with his teacher, and has spent a good part of class time emotionally shut down and unavailable for learning.  Crying, visits to the school counselor, behavior modification plans, teacher team meetings, parent phone calls and conferences; clearly Justin is working through something and we're doing what we can to help him get through the year in one emotional piece.
As a math learner, Justin has good number sense and an ability to manipulate numbers in his head.  But he doesn't like to write things down, and is easily discouraged.  When I first gave him the problem, he spent several minutes staring at the whiteboard, twirling an expo marker in his hand.  I couldn't guess what was going through his mind.  Was he thinking about the problem?  Would he refuse to engage?  Should I say something?  I waited for what seemed like an eternity.  Finally he uncapped the marker, drew four tally marks, and enclosed them in a drawing of a van.  He continued to do this, keeping a running total on top of each one.  When he got to 56, he drew two more tallies and enclosed them in another van.

After he was finished, I asked him if he could attach an equation to what he had done.  Without hesitation, he wrote 58/ 4 = 14 R 2.  I asked him how many vans were needed, and wrote the answer, 15, on the whiteboard.  I asked him why he didn't use the partial quotients algorithm he learned in class from his teacher.
Him: It would be too much work.
Me: (pointing to all he had written on the whiteboard) This seems like a lot of work.
Him: It's not for me.
Me: What if there had been 158 kids going on the trip?
Him: I don't know.
Me: With that many kids, doing it the way you learned in class would make things easier.
Him: (No response.)
Me: (Deciding to quit while I was ahead, thinking I was lucky he had even engaged with the
problem) OK, you can go back to class.

III. Tashana

Tashana has personality coming out of her ears.  If you were out sick, she could run the class.  She knows exactly where everybody is supposed to be at all times.  But she's a little too much of a busybody, and you can often find her in the hall along with a few other girls and teacher in the middle trying to mediate a dispute.  Like Justin, I know her pretty well, having worked with her last year. When I ask her to come and work with me, which happens several times a week, she gets up very slowly from her desk and looks furtively around the room.  I worry she doesn't like being taken out of class, if only for 10 minutes, but she invariably warms up to her usual bright self.  I know that her outgoing persona hides a very scared, often confused math learner.  In my room she doesn't have to worry about what everyone else in class is doing, which more often than not is something she has trouble understanding.
Tashana and I have spent a good part of the year developing concepts of place value and working on multi-digit computation, especially subtracting.  She likes finding the difference between two numbers by counting up on an open number line, and she's gotten pretty good.  But the standards call for the use of the traditional algorithm, and here we've gotten somewhat stuck.  As I explained the problem to her, she noted the information on the whiteboard.  She drew 58 circles on the board, representing the kids, and then circled groups of four.

Me: So how many vans will you need?
Her: 14.
Me: What are those two circles left over?
Her: That's two kids.
Me: What about them?  Are they going on the trip?
Her: Does a van have to have four kids in it?
Me: Well that's up to you!
Her: They could go in another van.
Me: Nice!  You can go back to class now.

IV. Vanessa

Vanessa squints a lot, but no matter how hard she tries, the math is blurry.  She's another repatriated ELL student.  She's been in the program for years, and her teacher tells us that she thinks Vanessa's learning issues extend beyond just language.  She's quiet.  Her voice never rises above a soft murmur, and she has a habit of responding to questions with answers that themselves sound like questions.  She's in the same class as Daiba, and I've wished they could somehow find and befriend each other.  Two newcomers, outsiders looking in on a rambunctious, lively class, but it hasn't happened.
Vanessa's math learning is hard to pin down.  Just when I think she has a handle on a concept, it mysteriously slips away.  Concepts that I think she'll find confusing, like fractions, she readily grasps. But she'll solve a problem like 13 - 9 by counting backwards from 13 on her fingers and get an answer of 5.  Her first move when I gave her the problem was to start drawing circles with four dots in each.  She did this three times, stepped away from the whiteboard, and after some consideration, erased what she's done.

She then set up a division problem using a partial quotients method I've taught her, useful for envisioning partitive division.  But this is a measurement division scenario, and I was worried that this might confuse her.  Nevertheless, she arrived at an answer.

Me: So how many vans will you need?
Her: 14 remainder 2?
Me: What does that mean?  What's 14 remainder 2 vans?
Her: 16?
Me: (after a few moments of thought) When you first started to solve the problem,  you drew circles with dots in them.  Why did you erase them?
Her: (no response)
Me: I'd like you to go back and solve it that way.

Her: 15?
Me: Is it 15 or 16?

At this point Vanessa stepped back and squinted hard.  The minutes ticked by in silence.  She had gotten the right answer, even though she had made a mistake along the way.  Was she waiting for me to intercede?  Was she still not sure?  I know all about wait time, but how long are you supposed to wait?
My patience paid off.  Vanessa went back to the board and corrected her mistake.  She saw clearly what she needed to do.  She carefully recounted the dots, erased the 8 in 58 and changed it to a 6...

...erased two dots, and was left with 14 vans with four kids in each and 1 van with two kids.

Me: So how many vans?
Her: 15?
Me: Are you asking me or telling me?
Her: 15.
Me: By the way, that remainder two from before?  Those are the two kids in the fifteenth van.  Now back to class.  I don't want you to be late for lunch!

Four students, one question.  Two weeks from now, they'll all take the PARCC.  There will be questions on that test much more complex than this one: bigger numbers, text-laden scenarios, multi-step problems.  How will Daiba, Justin, Tashana, and Vanessa respond?  Will they have enough time? What will they see reflected in the chromebook screen?  Will the best they have to offer be thrown in the trash with the rest of the class's scrap paper?  My wish for them is to come through emotionally unscathed, because I love them all.