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

Extending Children's Mathematics: Fractions and Decimals

     

     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.