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



Wednesday, February 1, 2017

Ball Don't Lie

Adapted from Urban Dictionary:

     A phrase commonly used by former professional basketball player Rasheed Wallace; once famously yelled by the late coach Flip Saunders
    "Ball don't lie" is said when a player misses one, two or all three of his free throws after a questionable (read as: unwarranted) foul call is made by an official. The ball is, essentially, the unbiased judge who will not reward the player by going in if the apparent foul was indeed unwarranted.
     Recently I visited a fourth grade classroom where the teacher was conducting a lesson on comparing fractions.  She explained that the task would be made much easier if the fractions in question had common denominators, and she was reviewing the method they were to use:

       After several worked examples, the students were divided into two groups.  One group was directed to work on Chromebooks.  Their task was to look at two fractions written side by side and choose a comparison symbol from a drop down menu to make the expression true.  The second group had a similar task.  They were sent to a table with a stack of laminated cards.  Each card had two fractions with a blank box between them.  They were asked to copy the fractions onto a worksheet and select the correct comparison symbol.
     After giving the kids a few minutes to settle in, I started to circulate.  I happened upon a student working on a Chromebook.  The screen displayed two fractions, 7/8 and 3/4.  She had selected the correct sign and was ready to click to the next screen when I asked her to take a minute to explain how she arrived at her answer.  That's when she pulled a whiteboard from her lap.

She restated the teacher's explanation almost word for word.

  I decided to leave the multiplication error aside and press her understanding a little.  I wanted to see of she could compare the fractions by using 1 as a benchmark.

Me: Do you know how many eighths make a whole?
Student: Eight.  Eight eighths.
Me: Good!  And so do you know how far away 7/8 is from 1 whole?
Student: One.  
Me: One?
Student: One eighth?
Me: What about fourths?  Do you know how many fourths you need to make a whole?
Student: Four.
Me: And so how far away is 3/4 from a whole?
Student: One.
Me: One?  
Student: One fourth?
Me: And so which fraction is closer to 1 whole?
Student: They both are.  They're both one away.

I decided to try another line of reasoning.

Me: Let's look back at 7/8.  You said that eight eighths makes one whole.  So is 7/8 less
       than, more than, or equal to a whole?
Student: Less.
Me: What about twenty-eighths?  How many of those would make one whole?
Student: Twenty-eight.
Me: (pointing to the 7/8 and 28/28) So then these two fractions are equal?
Student: Yes.  See?  I did the butterfly multiplication.

     Leaving the student to continue her work in peace, I crouched down next to a student looking at a card with the fractions 4/6 and 8/18.  He had implemented the butterfly method, and written the expression 72/108 > 48/108 on his worksheet, and was ready to move on to the next card.  Not so fast.  I wanted to see if he could compare them using 1/2 as a benchmark.

Me: Nice!  You did some fancy multiplication there!
Student: (no response)
Me: I want to talk about these fractions for a minute. Let's look at 4/6.  Is that more or less 
       than 1/2?
Student: (silence)
Me: Well do you know how many sixths you'd need to make 1/2?
Student: (more silence) 
Me: (picking up a nearby pencil) Well, say I drew a rectangle and divided it into sixths...

Me: ...Could you color in half the rectangle?

Me: Great!  Now how much of the rectangle is colored in?
Student: Three thirds?
Me: Carry on.

     A teacher, looking at the Chromebook data and the turned in worksheet, might conclude that both students have a firm grasp of the relative size of fractions.  Like the basketball referee making the questionable call, he sees what he sees.  But the truth about what these students understand about fractions won't be found in the colorful charts and impressive graphs generated by the computer program.  And it's not on the worksheet.  So where is it?  Sometimes the truth is on a whiteboard. Sometimes it's scribbled in the margin.  Sometimes it's written on a piece of scrap paper, and if you look hard, sometimes you can even see it in the faint trace left after it's been erased.  The truth is in the minds of our students, sometimes out in the open, sometimes hiding in a dusty corner. That's where we need to look.  So let's keep our eyes on the ball, because ball don't lie.

Monday, January 23, 2017

Natural Resources

0.4 and 0.75
Tell me everything you can about them.

    As described here, I've fallen in love with this very simple prompt, and it's how Rich and I decided to kick off a unit on decimals.  We followed the previously established protocol, first giving each student time to work on a card individually, then giving them an opportunity to pass their cards around and take notes.  Upon receiving their cards back, they could then cross out or add information.  When all was said and done, between the AM and PM classes we had 35 five by eight index cards filled with fifth grader thoughts about decimals.  Or, as I like to think of it, gold.

An embarrassment of riches.  Not to mention a low stress formative assessment.

     But what now?  What to do with the cards?  We had found a way to uncover all this thinking, but if we didn't use it, capitalize on it somehow, then it would all go to waste.  We had opened up a chest, and found treasure inside.  Now we had to find a way to spend it.
     My idea was to use their thoughts to create an agree/disagree activity.  First, I poured through the cards and selected some statements:

On target and misguided, commonly held and unusual.

     Rich and I decided to wait a week or so, long enough for the kids to get some decimal work under their belts.  I wanted to hold off on anything to do with operations and focus on place value, fractions, and comparisons.  I whittled the list down to five items:

  • 0.4 is closer to 0 then to 1/2.
  • Both 0.4 and 0.75 equal to a fraction with the same denominator.
  • 0.4 is 40 because the 4 is in the tenths place.
  • In 0.75 the 7 is in the tenths place and the 5 is in the ones place.
  • 0.75 is closer to 1 than 0.4.
     They could choose agree, disagree, or not sure, and I made certain to give them room on the paper to justify their decisions.  We gave them time to work on their answers individually, and then meet with classmates to discuss and, hopefully, argue and hash things out.

   As Rich and I circulated around the room, listening in, subtly nudging and facilitating, we found that some statements provoked more discussion and disagreement than others.  Those were the ones we decided to highlight in a whole class discussion.
    First was the statement that both 0.4 and 0.75 equal to a fraction with the same denominator.  I included this one because I wanted to both emphasize the equivalency between 0.4 (4/10) and 0.40 (40/100) and also because I wanted the students to convert the decimals into fractions.  These sample responses are illustrative of how I was thinking:

After first disagreeing, this student changed his mind.

Not all students were able to respond.

A few saw it differently:

Strongly worded!

Well, that's true!

     It came down to a question of interpretation.  It was true that 4/10 and 75/100 couldn't be converted to equivalent fractions with the same denominator, however was that what the author of the statement meant?  I had no ready access to the original cards, and really didn't think it mattered anyway.  The ambiguity made for a much more interesting discussion.  I decided to leave it at: "Whether you agree or disagree depends on how you interpret the statement."  That seemed to satisfy everyone.  Well, all except one kid.  Last in line walking out of class as the period ended, he sidled up to me and asked, in a conspiratorial voice, "Mr. Schwartz, so which is right?"
     The following day we tackled this one: 0.4 is 40 because the 4 is in the tenths place.  I included this because it had an element of truth (the 4 is in the tenths place) but was inaccurate due to the important distinction between tens and tenths that was still confounding some students.

This was representative of the agree faction, however it directly contradicts the original statement.

The disagree faction came on strong:

I like how this student underlined the th in tenths!

This argument convinced the remaining few in the agree camp to change their minds.  Money talks.

     Some observations:

  • In terms of content, the experience drove home how important SMP 6 (Attend to precision) is when talking about decimals.  Minor changes, both to the location of the decimal point and to how we write and talk (tens vs. tenths) have major consequences.
  • The activity also provided an opportunity to exercise the SMP 3 muscle: Construct viable arguments and critique the reasoning of others.  Pictures and models, like number lines, proved especially effective.
  • From start to finish the activity provided a nice balance between individual work, group collaboration, and whole class discussion.  The resolution of the disagreement provided a natural context for direct teacher instruction.
     Above all, I was gratified that the thoughts collected on the cards didn't go to waste.  As I reflected on the experience, I thought about something from my days as a third grade classroom teacher.  I had always enjoyed the unit we taught on Native Americans, and remembered how fascinated the kids were studying the Plains Indians tribes and the way they used the buffalo.  Meat, bones, hide, hair, tail, hoof, brain, stomach, bladder, intestine; they found practical uses for every part of the animal, even its dung!  Were the cards more like the buffalo than like gold?  Rather than think of them as currency that we needed to spend, maybe it was more useful to think of them the way the Plains Indians thought about the buffalo, as a natural resource that we could use not only as a formative assessment for determining who knew what and a way to uncover some misconceptions, but for other things as well.  Like giving everyone a headache.  Like starting an argument in math class.  Like providing a reason to get together and talk things through.
    Student thinking.  A precious natural resource.  And it's endlessly renewable!

Thursday, January 5, 2017

Then and Now

     As young teachers, we believed our job was to carefully explain what we knew about mathematics to our students.  We asked questions and listened to our students' answers but our listening was aimed at assessing whether our students got what we had explained rather than uncovering their understanding of the content.  We now see that we missed valuable opportunities to develop students' understanding because we did not elicit their ideas or relate their ideas to the content we were teaching.
                                  -Susan B. Empson and Linda Levi

     Even though I spend most of my day in classrooms, it's been nearly eight years since I've called myself a classroom teacher.  And while in my position as math specialist I continue to teach my share of lessons, they're really just one-offs.  The responsibility for planning and delivering the math curriculum on a daily basis rests with my classroom teacher colleagues.  I can offer help, support, guidance, and advice but, ultimately, they're in charge.
     When I look back on my classroom days, it's with a certain sense of regret.  Regret that I didn't know then what I know now.  As well-intentioned as I was, I'm not sure I was a very good math teacher.  So I often wonder: If I were back in the classroom, working my way through the curriculum lesson by lesson, how would things be different?  
     The opportunity to explore this scenario arose last month, when Rich told me he needed to take a day and asked what I thought he should leave for the sub.  Ordinarily I would use the time to either pull my basic skills kids and work with them back in my room, or maybe do a 3-Act or some estimation activities and games with the whole class.   But we had just finished a unit assessment, and were a little behind on the calendar.  His grade level partner was starting the next unit.  So I told him I would do the same.  Grabbing the grade 5 manual, I told him not to worry.

Time to put my money where my mouth is.

     Unit 3 was titled: Fraction Concepts, Addition, and Subtraction.  And the first lesson in the unit, Lesson 3-1, was called Connecting Fractions and Division, Part 1.  For the main part of the lesson, the manual called for the following sequence of activities:
  •  A Math Message involving three friends dividing a pizza equally, which the students were to model using fraction circle pieces, explaining how the pieces helped them solve the problem. The sample answer given in the manual was 1/3 pizza, and it was indicated in the teacher notes that the students should use the red circle to represent the whole pizza and the three orange pieces to split the whole into three equal parts.
  • Under teacher supervision, students were then to work with partners to model four fair share number stories, using the fraction circle pieces, or drawings if the whole items were not circular.  The teacher was instructed to go over the solutions with the students.
  • After this guided practice, the students were to complete three journal pages independently: one consisting of some fair share number stories, another with some multiplication and division practice, and a third page of assorted skill review.
   As I began scrubbing for some meatball surgery, some thoughts about the lesson started to emerge:
  • It was too teacher directed.
  • Students would spend too much time sitting at their seats.
  • Based on what I had learned over the summer from Empson and Levi in my CGI fraction book study, I wanted to keep the fraction circle pieces in their bags.

Nope, you guys aren't getting out today.

 Here's how I broke down the hour:

1.  Open with a Which One Doesn't Belong?  (5-7 minutes)

     After some internal debate, I decided on this one.
I didn't want it to have anything to do with fractions.

2.  Randomly assign partnerships at whiteboards around the room. (10-15 minutes)
      Give them (orally) one of the guided practice problems to work on.  Encourage them to solve it in as many different ways as possible.  Do some debriefing with a mini gallery walk.  Add in some direct instruction. 

Leila brought 6 graham crackers for a snack.  She wants to share them with 3 friends.  If they share the crackers equally, how many will each person get?
I chose one with a mixed number answer to reinforce counting wholes and then fractional pieces.  Another nod to CGI.

3.  Assign the journal page of fair share problems.  (15 minutes)
        Allow them to collaborate if they choose, and work out the problems on the whiteboards before recording their solutions in their journals.  Explain that if they finish, they should get to work on the multiplication and division review.

4.  Mid-workshop interruption.  (15 minutes)
      Gather the class back together.  Go over one of the fair share problems on the journal page.  Look at some work under the document camera.  Do some direct instruction.

5.  Back to work.  (10 minutes)
     Continue on the fair share problems and the multiplication and division review. 

6.  Close.  (5 minutes)
     Gather the class back together again.  Choose some more fair share work to put under the document camera and discuss.  Wrap up what it means to solve a fair share story and begin to draw out connections between fractions and division.
     The first problem on the journal page involved three people sharing a pizza.  It was the exact same problem that I would have guided the kids through had I elected to include the Math Message.  As I walked around the room, looking at their work, I got a big surprise.  

No, not this.  I expected that most would just divide the pizza into thirds.  It was what the answer key said they should do.

So far, so good.

Wait a minute!  This student divided the pizza into eighths, because pizzas come with eight slices!  She gave each person 2 slices, then divided the remaining two slices into thirds and gave each person 2/3 of a slice.

Another eight slice pie.

This student's pizza came with six slices.

Another pizza with six slices.

This student decided to give each person 1/4 of the pizza, then divided the remaining fourth into thirds.  What's 1/3 of 1/4 anyway?

A ten slice pizza.  Each person gets three slices and 1/3 of the remaining slice.

This student thought the three people were sharing one slice and attempted to divide it into thirds.

He wasn't alone.
   The wide variation in responses gave me some rich material to dive into during our mid-workshop interruption.  I chose some models to look at under the document camera, and had kids explain what they did.  We talked about the difference between answering how many slices and what fraction of the pizza each person would receive, something I didn't anticipate being a problem.  We made sure we understood that the three children in the story were sharing an entire pie, not just a single slice.  And we discussed the importance of making our fair shares as equal as possible.
      The kids went back to work, drawing models for the rest of the problems.  The sub and I circulated around, observing, listening, asking questions, offering guidance if needed.  Some were able to complete the page, and turned to the multiplication and division review.  For our close, I decided to go back to the pizza problem.  I wanted to explore their notions of equivalency, and we discussed whether or not receiving 2 2/3 slices of an 8 slice pizza was the same as receiving 3 1/3 slices of a ten slice pizza was the same as receiving 1/3 or 2/6 of the whole pie.
     Reflecting on the lesson, I understood that the old me, the pre-MTBoS me, would have run through the lesson as it was written in the manual.  I would have guided the kids through the math message, then congratulated myself on the fact that they all solved the first problem in the journal the way the answer key said they should.  I would have done four worked examples with them, leaving precious little time for them to explore solving fair share problems on their own.  They would have done the work at their seats, rather than all over the room on whiteboards.  I would have front-loaded all the direct instruction, rather than spreading it out over the lesson, waiting until I saw how the kids approached the work and letting that inform what I wanted to highlight.
     So how are things different?  Like Empson and Levi, I used to believe my job was to explain, and then listen to assess whether or not my students got what I explained.  I listened, but I didn't really hear.  I looked, but not always for the right things.  How have I changed?  Searching for an answer, I came across this quote from Dr. Ruth Parker, who answered it for me:
     I used to think my job was to teach students to see what I see.  I no longer believe this.  My job is to teach students to see; and to recognize that no matter what the problem is, we don't all see things the same way.  But when we examine our different ways of seeing, and look for the relationships involved, everyone sees more clearly; everyone understands more deeply.

      They may not be perfect, but I like my messy pizzas.