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.

Monday, March 27, 2017

To Each According To His Need

   One class.  Three groups.  Three different classwork assignments.

This group gets two questions.

This group gets three questions.

This group gets five plus a challenge.

Here's what I'm wondering:
  • Is this differentiation?
  • What does it mean to differentiate?
  • As a general rule, is it bad practice to put kids in ability-based groups or give them ability-based assignments?
  • Are there situations where ability-based groups or assignments are appropriate?

     I believe there are times when an ability-based skills group is appropriate.  If there are five students in the class having difficulty multiplying multi-digit numbers, collecting them in one place for some further instruction makes sense to me, at least more sense than going back over the skill with the entire class.  Homework assignments might also look different for different students.  Those same five students shouldn't be working on a page of multi-digit multiplication problems at home until the skill is secure.
     In the above example, the three worksheets certainly are different, but I believe there's a better way to achieve a differentiation objective.  Here's my suggestion:

1.  Start everyone off on equal footing with just the table and a notice and wonder prompt.  This will provide time for everyone to take a breath and process the information.

What do you notice?  What are you wondering?

2.  Have students write their own questions.  Stealing vocabulary from our ILA teachers, we can ask the kids to come up with both thin questions and thick questions.  Or, from Vacca's work on Question-Answer Relationships, questions that are right there and think-and-search.

This was from a different prompt, but you get the idea.  And here's a catch: you have to be able to solve the questions you write!  

3.  Post an assortment of questions.  Let the kids decide which ones they want to solve.  But be sure to vet them first!

Again, these questions were generated from a different prompt.  They vary in difficulty.

    This model has many benefits.  For one thing, it eliminates the stigma of being in the group that got only two questions.  It also transfers ownership of the question-asking from the teacher (or teacher's manual) to the students.  Of course you can learn a lot about students from the way they solve problems, but you can also learn a lot about them from the questions they write as well as from the questions they elect to solve.  Will a bright student take the easy way out?  Will a struggler try to punch above his weight?  Yes, and these choices are very telling.
     The reality is that students in any given classroom will have a wide range of abilities and needs.  We want to preserve a sense of whole class community, and we also want to make sure each individual student is receiving what they need and working at tasks that engage them in an appropriate productive struggle.  It's a very difficult balance to maintain.  Enlisting the help of our students may make it just a little easier.

Tuesday, March 14, 2017

"It Depends on the Meaning of Almost."

     My favorite statement so far this year came from a student via a recent tell me everything you can about... prompt.  This time it was:

Tell me everything you can about
4 2/5 and 3 1/2 

  The cards poured in...

...and provided many options for agree/disagree/not sure statements.  One statement in particular caught my eye:

The only thing this student could think of to say about the two mixed numbers.

I set a limit of five, but that one had to make the cut!

The final five.
     After giving the kids some time to work independently, Rich and I put them in groups with instructions to hash things out.  We listened in, and I settled down with one group and asked them to explain what they had decided about 4 2/5 and 3 1/2 being almost the same.  Agree?  Disagree?  Why?  Not sure?  Why not?
     The first student to speak up avowed that yes, they were almost the same.  Her explanation:

After converting both mixed numbers into improper fractions, she found they were 9/10 apart.  "That's pretty close."
   The next student said that she believed they weren't almost the same.  She started drawing on the whiteboard:

"They're almost a whole away from each other.  That's not almost the same."

     A third student chimed in:

"They're not almost the same.  They're 9/10 away from each other."

A fourth student stood up in support of the first student:

"I rounded 4 2/5 to 4.  And 3 1/2 is right in between 3 and 4.  If I round it up than they will both be 4.  That makes them almost the same."

     The debate lasted for several minutes, until the first student, in an exasperated voice, asked,
     "So Mr. Schwartz, are they almost the same or not?"
     Before I had time to even formulate a response, a student in the group, silent the whole discussion, piped up.  "It depends on the meaning of almost."
     Couldn't have said it better myself.
     Taking a look at the whiteboard as the kids left the room to go to their special, I felt gratified by the different ways they had thought to compare the mixed numbers.

But there was one key representation missing.

Back in my room, I played around with some number lines:

Are they almost the same?
How about now?

     Would these help, or just add fuel to the fire?  Talking about it the next morning, Rich and I realized that, in the end, it comes down to context and units.  There are situations where the difference between 3 1/2 and 4 2/5 seems insignificant (if I arrive at school 3 1/2 seconds before the bell rings, and you arrive 4 2/5 seconds before the bell rings, we've arrived at school at almost the same time), others where it's not (if I run 3 1/2 miles, and you run 4 2/5 miles, we haven't run almost the same amount of distance), and still others that seem debatable (if I have $3.50 and you have $4.40, do we have almost the same amount of money?)  In some cases it can mean the difference between winning and losing (as in a race), a delicious or inedible dessert (measuring ingredients), or even life and death (medicinal doses.)  In other cases the difference makes no difference at all.
     So I'm left wondering: What experiences and contexts could our students bring to the table?   Have we tried hard enough to free the numbers we work with from lives of lonely isolation?  From a dreary existence in the land of pseudo-context?  How can we make them come alive?  For a brief moment in time,  3 1/2 and 4 2/5 ran wild in the classroom.  The next day they went back to the black and white of the journal page and the worksheet, but man, they had fun while it lasted!

Wednesday, February 22, 2017


One of the most important habits a math teacher can develop is to do the problem first, always.

pg. 46

     During one of our weekly planning sessions last month, Rich and I decided to roll out a new 3-Act task.  The idea came to me in the faculty lounge, a place where I find inspiration on a regular basis.  The Act One video showed four teachers in succession going up to the sink and washing their hands, each one receiving an automatic squirt of liquid soap.  The question I had was: How many squirts would be dispensed before it emptied?
     I had our custodian open up the dispenser.

She's gotten used to these kinds of requests.

She showed me a refill, and I found out that each bag held 1,200 mL of liquid soap.

But how much soap came out in each squirt?  A call to Georgia Pacific and I had my answer.

The dispenser has two settings: one for 0.4 mL and another for 0.7 mL.

We couldn't determine where the dispenser was set, and I worked out the answer for the 0.4 setting.
After a false start...

113 is wrong!

...I figured it out.  3,000 squirts!

   I was curious to see how Rich and Megan (our grade 5 resource room teacher) would attack the problem, so I asked them to have a go:

Rich's solution.  While I thought of four tenths as a decimal, he used a fraction.

Megan also thought in fractional terms. 

    I thought it was interesting that the three of us took three different solution paths, and we imagined we'd see a variation in the class responses.
    For Act 4, we thought we'd ask the kids to find out how many squirts it would take to empty the bag if the dispenser was set at 0.7 mL per squirt.  Megan had to leave, so that left Rich and I to work it out.

Rich got an answer, but he wasn't happy.  He was in a rush because he had to leave to pick up his class, and I told him I would save his work.  He came back the next day and found where his thinking had gone off the rails.

I went back to my table.  It had worked before.  Why not again?
My problem was that 0.7 didn't fit nicely with 1,200.  I narrowed things down until I got to 1,714 pumps, which would empty 1,199.8 mL from the bag, leaving just .2 of a mL left.  Now what?

     As I started to puzzle it out, the phone rang.  It was Rich.  He had gone back to his room, divided 1200 by .7 and gotten a wacky decimal.  "It's 1,714.285714," he said, "And it goes on."  I got out my phone calculator, punched in the numbers, and got 1,714.28571429.  I knew that the answer lay somewhere between 1,714 and 1,715, but where did that ugly decimal come from?  What did it mean?  If there was 0.2 mL left in the dispenser, wouldn't the answer be 1,714.2?  Or maybe it was 0.2 of 0.7?  But then that would leave an answer of 1,714.14 squirts.
     No kidding, I stared at the whiteboard for the better part of a week, thinking about decimals, automatic soap dispensers, and Robert Kaplinsky on productive struggle.

The decimal was killing me!

   I finally broke down and called my supervisor.  He was fascinated by the way I went about solving the problem, and he picked up a marker and started working in the bottom right corner.

For one thing, he wasn't crazy about my equal signs.  Then he explained that if 0.7 mL was a whole squirt, and only 0.2 mL came out, then the last person to wash their hands would get 2/7 of a squirt, assuming that an amount of soap that minuscule would even come out.  And what's 2/7 as a decimal?  Light bulb time!

     It was one of those knock yourself on the head kind of moments, but I didn't feel bad.  I had enjoyed the process.  I always tell the teachers I work with that if they have questions about the math they're teaching to please come to me and we'll figure things out together, but I know that some feel embarrassed.  They think that asking for help is a sign of weakness, but really it's a sign of courage. So it's important for me to put my money where my mouth is, and I reach out to our math supervisors for help when I have questions.  It's how I learn and grow, and it's a good habit to develop.
    So how did the 3-Act play out with the kids?  Stay tuned for a report.  But because we had worked out the problem beforehand, Rich and I felt ready for anything.

A postscript:
  Curious about the decimal, I did a little research.  Turns out it's a repeating decimal, and the part that repeats is called the period or repetend, and there are cool patterns that occur with the digits.  I tried dividing 2 by 7, first by hand...

...and then on different calculators:

 They were all different, because some of them rounded and some of them didn't!  I started to like the decimal, and had to revise my opinion.  What I first thought of as something ugly turned out to be rather cool!  

Tracy gets the final word:

     Playing with the problem ourselves activates our identities as mathematicians.  Just like reading teachers need to read and writing teachers need to write, math teachers need to do math.  Noticing which mathematical concepts and techniques come into play while we work focuses our thinking on content.  Rather than jumping right to planning activities students will do, we spend time thinking about the mathematics students may learn