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.


Tuesday, December 13, 2016


     Emily doesn't really like math.  She'll tell you that right to your face if you ask her.  Not in a confrontational or disrespectful way; that's not her nature.  But with a scrunched up half- frown and shoulder shrug, and a nearly imperceptible shake of the head, she'll say, quietly, almost apologetically, "No, not really."
     I watch her walk into class every day.  She's tall for a fifth grader, but when she folds herself in at her desk in the back of the room she shrinks down to about half her size, like she's trying to disappear.
     Emily's very quiet.  She never interrupts.  She won't raise her hand.  If you call on her she'll respond, but in a whisper you can hardly hear.  She doesn't cause any trouble.  She follows directions.  Her journal is always turned to the right page.  She always has a pencil and an eraser.  She turns her homework in on time.   If you didn't know any better, if you were just a casual observer, you'd think she knows what she's doing because she looks like she knows what she's doing.  She's one of those under the radar kids.
     I watch her at her desk, elbows flat, head down, pencil up and moving.  She appears to be working away.  What's she doing?  She's managed to get by, doing just well enough to keep one step ahead of the basic skills program.  But if you sit down next to her and really, but I mean really try to get at what it is she actually knows, if you look at her work and try to ask her some questions about it, you'll find that her understanding is very superficial.  She's doing her best to remember and follow some rules she's been told.  She's memorized some things, but they're all fragmented.  They don't cohere.  She studies for the tests, and does her best to hold the pieces together, but when the tests are over the pieces fall back apart.  She doesn't really understand. And despite her best attempts to hide,  I know that she knows she doesn't really understand.  I think it bothers her, which is why she says she doesn't really like math anyway.
     I have to be careful.  I can't poke around too much or she'll shrink down even further, away to a place I might not be able to reach.  It doesn't take much for me to imagine what math experiences she's had to make her feel this way.  I know it's not too late to undo them.
     Justin Lanier, in his brilliant, moving Ignite talk The Space Around the Bar, says the following:

"Students Will Be Able To (And Will Never Want To Again.)  This is how we can write our lesson objectives if we don't pay attention to how kids feel about math." 

Rich and I agree.   Rich and I are trying to create a classroom where it's OK to be wrong; where you can ask and answer questions that you yourself have generated and that serve an intellectual need; where you can solve problems in ways that make sense to you, even if it's not the way they want you to do it in the teacher's manual; where tasks have low barriers to entry and high ceilings and open middles and three acts; where you can move around the room and work them out on big whiteboards; where you can collaborate with your classmates, play games, talk, argue, and laugh. We're trying to create that kind of a classroom.  We don't always succeed, but we're trying.
     Emily's my measuring stick, my benchmark, my litmus test.  Whenever Rich and I  re-work a lesson, or I get an idea for a task or an activity, I ask myself: What will Emily think?  How's she going to react?  I want her approval.  If I can get her approval I know I'm onto something.  The skills and the content are the easy part.  If we can create the conditions in class where she feels it's OK to just be herself, to feel safe and secure enough stand up and stretch out to her full height, I know true learning will occur.
     When class is ending,  I'll find her and ask, "Well, what did you think?"  She doesn't give much away.  It doesn't always go as well as we thought it would.  But sometimes I get a guarded, "Well, that was OK."  I'll do a little fist pump, and she'll maybe even smile.  Baby steps.  Lesson objective: met.

Monday, December 5, 2016

Down the Rounding Rabbit Hole

     Rounding has always been difficult for me to teach, and I suspect I'm not alone.  Here's a look at some of the many gimmicks, tricks, and tools teachers have used in an effort to get this concept across:

Been there.

Done that.

Looks familiar.

Monkeys?  Can't say I've ever tried this, but I'm skeptical.

Not sure what this is.

Seems like a lot to process, and I'll admit I've said similar things.

     Our curriculum likes number lines:

Grade 3

Grade 3 again.  Not a mountain or a roller's a hill!

Grade 4

     I've spent more time than usual this fall thinking about rounding; both the why and the when, but also simply the how.  Using a number line to round is all well and good, but it's not going to help you solve these problems...

...unless you can do the following:
  • Identify place value locations and the value of the digit in a given place value.
  • Identify the next multiple in that place value.
  • Find the halfway point between the two multiples to use as a benchmark.
  • Decide whether the number is more or less than halfway.     
That's one big cognitive load.  No wonder many kids struggle!

    Last year, I tried something different with some third graders who were having difficulty with rounding.  Here's a typical problem from their journal:

An open number line won't help unless you know the multiplies of 10 that come before and after 68.

Start with 68.

Count forward by 1s until you get to a number you would say if you skip counted by 10s.

Same, but this time count backwards.

Find out if 68 is closer to 60 or 70.  This also reinforces complements of 10.

    This was good as far as it went, but to round larger numbers, especially to units represented by a place in the middle of that number, students will need another strategy, hopefully one that doesn't rely on mountains, roller coasters, digits knocking on doors, or monkeys swinging on vines.
     This fall I've been working with a fourth grader who, at the beginning of the school year, showed a complete inability to round.  She had a very limited notion of place value, and we spent several weeks shoring up some basics.

We used this flip board...

...and these place value arrow cards.

   I wanted her to be secure in her knowledge of place value locations and digit values before we ventured off into the world of rounding.  It took several weeks, but once she was able to complete tasks like this:

and this:

without the board or the arrow cards, I moved her on to rounding.  Here's the plan I had sketched out.  Let's take a grade 4 example.  Round 81,866 to the nearest thousand:

Step 1: Write the number out in expanded notation.

Step 2: Identify the place value you want to round to.

Step 3: Identify the next multiple in that place value.  Here I might say something like, "If I was skip counting by thousands, what number would come after 1,000?"

Step 4: Put the 80,000 back.  This can be combined with step 3.

Step 5: Now we're ready for our number line.  We have our two possible responses for rounding 81,866 to the nearest thousand, so now we've got a fighting chance.

Step 6: Where does 81,866 fall on this number line?  Can we benchmark halfway?

Step 7:  This is now a matter of comparing 500 and 866.

We practiced on the big whiteboard in my room, and experimented with color:

Rounding 697,654 to the nearest thousand.  After a while, she started to internalize some of the steps.

Here's an example of the student employing the strategy in her math notebook:

Rounding 818,325 to the nearest thousand.  I would've liked to see her place the mark for 818,325 closer to 818,500, but that's for another day.

     I'm happy to report that she's made lots of progress.  The mistakes she makes when rounding now come when she tries to do too much in her head.  Soon she will be able to visualize the number line in her mind's eye; the two boundaries, the halfway mark, the place where the number belongs.  Soon, but not yet.
     So here's what I found out: 4.NBT.A.3 is a very interesting standard.


    It's stated so simply, and, compared to others, seems so uncomplicated.  But my dive into rounding this fall taught me that there's much more here than meets the eye.  Place value understanding is a complex understanding with many components, and all of them have to be secure and working in concert in order for students to be able to round multi-digit whole numbers to any place.
    As I continued to work with the student, the nagging question of why are we doing this kept running through my mind.  Curious to learn more, I found this on page 12 of the Numbers and Operations in Base Ten, K-5 progressions document:

Rounding to the unit represented by the leftmost place is typically the sort of estimate that is easiest for students and is often sufficient for practical purposes.  Rounding to the unit represented by a place in the middle of a number may be more difficult for students (the surrounding digits are sometimes distracting).  Rounding two numbers before computing can take as long as just computing their sum or difference.  (emphasis mine)

Here's what I want to know:
  • If rounding to the unit represented by the leftmost place is often sufficient for practical purposes, then why are we asking students to round, for example, 287,347 to the nearest hundred?  What headache do I have that rounding 287,347 to the nearest hundred will cure?
  • Are tasks asking students to round numbers to units represented by places in the middle of a number simply exercises in place value?  Is there a way to make this task compelling?     
  • What distinctions do the progressions make between rounding and estimating?  
  • Should rounding be taught as a discrete skill, separate from any context?  
  • What scaffolds, tools, and place value models are helpful in promoting rounding understanding, and which subvert it?
And finally: What are your experiences teaching rounding?

Tuesday, November 22, 2016

More Faculty Lounge Math

     My teaching life pre-MTBoS was certainly different than it is now.  I look at student work differentlyplan lessons and activities differentlyprovide feedback differently, even think of myself as a math learner in a different way than I did before becoming an active part of this amazing community.  But nothing has changed more than the way I look at what I see around me.  My friend Graham Fletcher has taught me that what I'm doing is called mathematizing the world, and now that I've started, I can't seem to stop.  In that spirit, here's another edition of Faculty Lounge Math.

     This is more like a 2-Act than a 3-Act...

What fraction of the container is filled with pretzels?

Does this help?

  This one got tweeted out:

Rainbow Cookie

Fractions anyone?

A Faculty Lounge 3-Act

Act 1

Washing Up from Joe Schwartz on Vimeo.

Main Question: How many squirts will the machine dispense until it runs out of soap?

Act 2

Inside.  According to the company, 0.4 mL of liquid foam soap is dispensed each time it's activated.

The dispenser holds 1 bag.  Each bag holds 1,200 mL of foam soap.  
Act 3

  My first attempt at figuring this out didn't go so good.

I added 3 + 10 + 100.  I don't know... 113 handwashings just seemed too low.

So I tried something different...

3,000 was more like it!

Where did I go wrong?

Thinking additively, not multiplicatively.

 Act 4: The Sequel
The dispenser can be set to squirt 0.7 mL of soap per activation.  How many hand washings per bag?

   A jar of pretzels, trays of muffins, a cookie, a soap dispenser; in my previous life they would have been things to eat and a way to promote good hygiene.  But thanks to DanAndrewGrahamMarilynRobertTraceyMichaelSimonKristinFawnMax,  Annie, Andy, and the rest of the MTBoS crew, I now see them as opportunities to notice and wonder, estimate, spark a number talk, and create and solve problems.  It's a gift I'm thankful for.  Hope they don't mind if I re-gift.