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!
   
     

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.







 
   

Emily

     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.
   
 

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 coaster...it'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?