$167.36 On the Nose

From estimation180, Day 161:

What's the value of all the coins in the bowl?

  Before you read on, take a moment and come up with an estimate.

   Andrew was gracious enough to provide us with the receipt, so Rich and I decided to use the prompt as Act 1 of a 3-act task.  It would provide the kids with multi-digit addition, subtraction, and multiplication practice, and since the bank takes one-tenth of the total amount as a fee for non-members, we would also receive some formative assessment information on how the students thought about decimals and place value.

   We set the kids up in random groups on whiteboards, and asked them first for estimates. Nicole, the ILA teacher next door, poked her head in.  She asked what was going on, and one of the kids explained that they were estimating how much money was in the bowl. After a few minutes of thought, she started looking around for a scrap of paper.  Finding none, she pulled a tissue from a nearby tissue box, wrote something down, and handed it to me.  I folded it up and put it in my pocket, distracted by all the activity in the room as the kids finalized their estimates and began figuring out how much 1/10 of 519 quarters, 898 dimes, 719 nickels, and 917 pennies was worth:

Emptying my pockets at the end of the day, I came across Nicole's estimate:

     What began to fascinate me, what I wanted to know, wasn't how she came up with the number, but why, having been asked for an estimate, she came up with something so exact!  Not $160, or $170, or $200, but $167.36.  We've been playing around with estimation for years now, and we're continually encouraging the kids to choose friendly, round numbers as estimates, numbers that tell about how many or how much, not necessarily exactly how many or how much.  But we've met with reliable, obstinate resistance.  I looked back at some of the pictures I had taken of student work, looking for a record of their estimates, and while I did see estimates like $300, $40, $50, and $60, I also saw $63.12, $5.57, and $312.10.  Pointy numbers.  Precise numbers.  Numbers that spoke of exact amounts.  Not round numbers.  Not in the general vicinity numbers.  Not numbers that spoke of about how much.  Why?  Is there something hard-wired into our human nature that, when presented with a task like this, makes us want to be exactly right?  Not close enough, but closer than any of our classmates?  Have we been so conditioned by the "Guess How Many Jelly Beans in the Jar"  challenges that we treat every estimation task as a chance to win a prize?
     Several days later I asked Nicole about her estimate.
    "Why so exact?  Were you trying to guess the exact amount?"
    "No,"  she explained,  "I was trying to estimate.  But I guess in my mind they're the same thing."
    A few days later, eating lunch in the teacher's lounge...

A leftover party favor from a week-end birthday party.

 ...four of my non-math teaching colleagues found themselves unwitting participants in an experiment.
     "I want everyone to estimate how many gumballs are in the container."
     The group was willing to cooperate, and within several seconds one piped up:
     "Are we going to count them?  We have to find out who won."
     I quickly got up and searched for a piece of paper and a pencil to write down the quote.  She had, on her own, without any suggestion from me, injected an element of competition and challenge into the task.
     After a few minutes I asked for their estimates.  I received 3 pointy numbers, 84, 74, and 78, and one round number, 190.  (This teacher had estimated 192 but rounded down to 190.)  Although I wasn't so much interested in their reasoning, they all wanted to explain their thinking, and carefully listened to one another as they each shared their strategy in turn.  I explained my purpose in asking.  I was curious, I explained, why the three hadn't chosen round numbers as estimates.
    "The answer is never a round number!" one explained.
    That statement gives me a clue as to what may be at work here.  During these kinds of  estimation tasks, I'm asking kids to engage in sense-making, not answer-getting.  Maybe the line between the two is blurred, but it's there.  $167.36 is an answer, not an estimate.
    Answer-getting is stubborn.  I know this is true, because my teacher's lounge colleagues were just dying to know, couldn't wait to find out exactly how many gumballs were in the container.  They couldn't let it go:


gumball1 from Joe Schwartz on Vimeo.

Can you?

In Orbit

     A number, I'm told, is like a Russian doll.

  This is because the quantity that a number represents contains the quantities represented by all preceding numbers.  This is called hierarchical inclusion, and its understanding is a very important stage in the number sense trajectory.  I thought I knew this.  But something happened recently that caused me to wonder: Do I really understand?
 
   It started in fifth grade with a Fraction Talk:

 Two different attempts caught my eye, and I turned them into a notice and wonder activity:

     I was trying to draw out the idea that the first response, the one on the left, was correct because the student had accurately labeled each small square as one-sixteenth of the whole.  However in the second response, the small squares were labeled incorrectly.  That student was counting by sixteenths, labeling each successive square as if it included within it all the preceding squares.  Was that an example of hierarchical inclusion?  I went back to this picture, which has helped me understand the concept:

From Early Childhood Mathematics Education Research: Learning Trajectories for Young Children by Julie Sarama and Douglas H. Clements

   
     It certainly seemed the case, but then I noticed something in the caption that I hadn't really noticed before; that each cardinal number includes those that came before.  What about other kinds of numbers?  Wait a minute.  There are other kinds of numbers?
    A quick search led me here, where I learned about three different kinds:

• Cardinal numbers.  They tell us how many of something there are. 

• Ordinal numbers.  They tell us the position of something.

• Nominal numbers.  They are used as names, or to identify something.

     I realized that I already knew about ordinal numbers; nominal numbers were new to me.  So I started collecting numbers around my school and mentally trying to classify them.  Here's a sample of what I found:

I.

  Nicole has her first graders count the days in school, and my first instinct is to call this a cardinal number.  Many primary teachers accumulate tokens, such as Popsicle sticks or unifix cubes to represent each successive day, suggesting that days are a set of things that can be counted.  However does each individual day include within it all preceding days?  Or do we get a fresh start each morning?  I don't know.  Here's Sarama and Clements, again in Early Childhood Mathematics Education Research, "When the topic of 'ordinality' is discussed, even by some researchers, it is often assumed that all ordinal notations must involve the terms 'first, second, third...' and so forth.  This is a limited view.   A person who is 'number 5' in a line is labeled by a word that is no less ordinal in its meaning because it is not expressed as 'fifth.' (p. 85)   So here Day 124 is the 124th day of school.  It is in position 124 of a sequential count of the days of school that started at 1 and will end at 180.
Conclusion: Ordinal.  Maybe cardinal.  Definitely not nominal.

II.

   This storage room doesn't include within it all rooms numbered 1 through 139, and there's no reason why this specific room should be identified as the 139th in a sequence of rooms.  In fact, there are not even close to 139 rooms in my school.  139 here seems to function as a signifier or name for this particular room.
Conclusion: Nominal.

III.

    The hooks in Wendy's grade 3 classroom are in sequential order.   The polka-dot backpack is on hook number 3, or the third hook.  In one way it seems like the number 3 is also acting here as a name for that hook.  Can a number be both ordinal and nominal at the same time?
Conclusion: Ordinal.  Maybe nominal.  Definitely not cardinal.

IV.

     The thermostat in the office must be broken.  No way it was anywhere near 76 degrees.  Regardless, this number is definitely not nominal.  Thinking of the temperature as I would see it on a mercury thermometer...

...helped me see that the number of degrees did include within it all preceding temperatures.  So did this, which I recorded one night testing my chicken pot pie:



thermometer from Joe Schwartz on Vimeo.

Conclusion: Cardinal.

V.

 The back of a fourth grader's basketball jersey.  We're not counting or ordering anything here.  The number 3 is just identifying Clippers point guard Chris Paul.  Maybe Paul picked it because it was his favorite number.  Maybe he wanted another number but it was already taken.  Anyway, the student sporting his jersey isn't even a Clippers fan.  He likes the Cavs.  Front runner!
Conclusion: Nominal.

VI.

   This is how the kindergartners in Kelly's room keep track of who's in school.  This one was easy.
Conclusion: Cardinal.

VII.

     10:36:16.  What kind of number is this?  I've come to think of nominal numbers as having a randomness about them; a phone number, a driver's license, an account number, my zip code.  I don't get that feeling here.  Could this be ordinal?  Ordinals have a sequential, positional feel to them, and of course there's always the th.  Could we say it's the 16th second of the 36th minute of the 10th hour?  Is 10:36:16 a Russian doll, nesting right between 10:36:15 and 10:36:17?  Maybe it's like what T.S. Eliot wrote:

    Time present and time past

Are both perhaps present in time future

And time future contained in time past.

Conclusion: Cardinal.  (Pretty sure.)

     Sarama and Clements cite research indicating that, "It is not until age nine that most master the hierarchical inclusion relationship."  (ECMER, p. 339)   If I did master the relationship way back when, then I guess my path through the world of number sense is more like an orbit than a trajectory.  It's taken me 46 years, but I've circled back around to find that numbers, for me, remain enigmatic and mysterious, in need of continued and constant rediscovery.

  

Cousin Ben

     He appeared in our lives out of nowhere.  Just showed up one night, invited to dinner by my mom. A distant relative, and for the life of me I would never be able to remember exactly how we were related.  In his mid to late-thirties at the time, a bachelor, with a receding hairline, a fu-manchu mustache, and a big ring of apartment superintendent keys dangling from his belt loop.  Except he wasn't a super.  He was a middle school english teacher in South Bound Brook.
     Cousin Ben became a regular dinner guest.  I was a geeky 1970's middle-school bookworm working my way through the entire Ray Bradbury catalogue, and we connected through a mutual love of science fiction.  He took me into the city, and we wandered around the Village, poking our noses into used bookstores.   He introduced me to Piers Anthony's Macroscope and Larry Niven's Ringworld and Ursula Le Guin's The Dispossessed.  I ate it up.  All of it.  He talked about his house, which he was constantly fixing up, and told us stories about his classes; how the kids would tease him because of his balding head, how he tried hard to connect with them and get them engaged, how he argued with his supervisors in the english department over the assigned novels, how he worked for his union local as a member of its negotiating team.
     I come from a family of businessmen; builders and real estate, insurance and finance.  A few lawyers thrown in.  Those were the kinds of jobs you got when you grew up.  But a teacher?  Cousin Ben was the first teacher I knew outside of school, the first teacher I thought of as someone who taught to make a living, the first teacher I knew who talked about the job of teaching.  Looking back, I realize it was Ben who first put the idea in my impressionable mind that teaching might actually be a career opportunity.
     After I graduated from college and moved back to New Jersey, we met several times for dinner at a place on Route 22.  He was still working, and a little jaded.  I was just starting my first teaching assignment, and he was curious to know how I was making my way in the profession.  Of course he talked union, but I didn't really pay attention.   (It wasn't until later that I realized how important that work was too.)  Then we lost touch.  But I did see him several years ago, again out of nowhere, invited by my mom to dinner.  He'd retired, and lost whatever hair he had left.  He was still reading sci-fi, still fixing up his house in South Bound Brook.  And I still couldn't remember exactly how we were related.
     Cousin Ben helped improve the lot of many teachers by fighting for fair working conditions and compensation, and touched the lives of countless middle schoolers with his passion for reading and literature.  And he showed one awkward teenager that teaching could be a life's work. So before I bring the curtain down on 31 years in an elementary school somewhere off the New Jersey Turnpike, I want to take this Teacher Appreciation Week 2017 to say:
Thanks Ben.

  Those who build walls are their own prisoners.  I'm going to go fulfill my proper function in the social organism.  I'm going to go unbuild walls.
--Ursula K. Le Guin, The Dispossessed
                                                           
   

"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!"

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.

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.

"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!