A Tale of Three Classrooms

      If there is a better state curriculum document than the Common Core Georgia Performance Standards, I challenge you to send it to me right now.  I was made aware of it reading a post from one of its developers, Graham Fletcher.  Starting in kindergarten, it includes 3-act tasks, estimation180 challenges, problems with "open middles", and all the goodies an MTBoS groupie could desire.  As my focus turns more towards the primary grades, it has become a go-to source for ideas and activities, and I have barely scratched its surface.
  Looking through the grade 1 frameworks, I was stopped in my tracks by a game called Fill the Stairs.

The premise is simple.  Students take turns rolling  two 10-sided dice and creating a two digit number.  For each turn, the player writes the number created on any stair.  The object of the game is to keep the numbers in order between 10 and 100.  If there is no space to write the number, that player loses their turn.  First to fill in all the stairs wins.

    I felt that the game was better suited, at least at this point in the year, for our second graders.  I made a slight modification to the game board, adding two small spaces at the bottom for place value purposes.  In the first classroom I visited, I explained the rules and started playing a demonstration game against the classroom teacher, with the students acting as "advisers".  After we were confident the kids understood the premise, we set them off to play, first asking them to turn in a sheet recorded in pencil, then asking them to play on blank ones in their SmartPals. 

It was interesting to see their strategies develop as they became more experienced playing the game.  We allowed them to decide which die they wanted to use to represent the tens and which to represent the ones.  So if you rolled a 4 and an 8, you could choose 48 or 84, whichever worked best for you.

   The kids loved it, and as the period came to a close we debriefed with some strategy talk.  I left the game with their teacher to use in a center, or with a guided group of students still shaky with their number sense, but as I reflected on the experience I decided to make some changes as I brought it into second grade class #2.  I replaced the dice with number cards, and further modified the game board.

I removed the rules.  I left a larger space for the cards.  Another modification I wanted to try was to not allow the kids to manipulate the place value of the number.  The first card pulled would always be the tens digit, the second card the ones.  Why?  I was curious to see what would happen.

    I gathered the class in the front of the room and showed them the page on the Smart Board.
    "Today I'm going to teach you a game," I told them.  "But before I do, I'd like you to do something for me."
    Although they had never done it before, I asked them to respond to the following prompts:
   "What do you notice?  What do you wonder?" 
   There was quite a bit of buzzing as they shared their observations with each other.  They noticed stairs, the 10 and the 100, the lines at the bottom of the page, the big space in between the two.  They wondered what the rules of the game might be, if they were going to be asked to count by 10s, why there was a space for tens and ones. 
   Being prohibited from changing the place values of the digits changed the nature of the game.  It took longer to fill the stairs because there were fewer options, but I felt it removed a layer of complexity and reduced the game's potential to build both number sense and understanding of place value.

Many students adopted the strategy of using the second step for the 20s, the third for the 30s, and so on.   What happens now?

   So it was on to classroom #3.  I followed the same procedure, but this time made no mention of the fact that it was a game.  And I asked the kids to write down their "noticings and wonderings" in their notebooks.  Many children wrote that they noticed stairs and numbers, others wondered if they were going to be asked to count by 10s, and whether or not they would be playing a game.
     I need to thank our three amazing second grade teachers for allowing me to turn their classrooms into laboratories.  I am excited to work with them this year, together exploring some different practices and new principles in the attempt to build number sense and make math more meaningful for our second graders.

My Factor Captor Obsession

    Last June I published post detailing my nearly year-long exploration of the game Factor Captor.  Up until the last week of school I had the fourth graders cutting apart the game board and playing around with the numbers:

I was curious: How many multiplication sentences could be made using the 48 numbers on the board as both factors and products.  With three per sentence, could someone make 16?

Close!!  But can you spot the mistake?

     I didn't get any further than this; it was more of a test drive to see what would happen, and if the activity would be worthwhile for next year's class. But as fortune would have it, our grade 5 curriculum opens with a three-week unit on factors, primes, composites, squares, and divisibility rules.  So I knew I would have an opportunity to use the game to extend some learning.
    First, Rich and I let them get reacquainted with the game.  We pulled out the boards they were familiar with.  I was pleased to see that the majority remembered how to play, and for the rest a quick review of the rules was sufficient to get them up and running.  I thought it might make things more meaningful if we explored the unit's vocabulary (prime, composite, even, odd, square), using the numbers from the board.  It seemed like the perfect opportunity to get out the scissors and glue again!
     I thought they might find it helpful to use Venn diagrams.  Many of them chose prime and composite as their first sort.  Of course there was much debate about where to put the number 1.

Some students used reference books.

     Many students finished one sort, and we encouraged them to choose two new labels and try another. And since we're into noticing and wondering, we asked them to write one thing they noticed about their diagram.  These ran along the lines of statements like:

• Most of the even numbers between 1 and 37 are not square, except 4 and 16.
• There are no numbers that are both prime and composite.
• 2 is the only even prime number.

I liked that they were attempting to out into words what they were seeing in their Venns.

And since the obsession shows no signs of abating...

We introduced the advanced grid with a noticing and wondering  "do now".

The kids had some interesting observations, including wondering if playing on the new grid made the game easier or more difficult, noticing that, except for 1, the single-digit numbers are repeated and wondering why that might be so, and wondering whether or not they would get a chance to play.  Well of course!

   I knew that this question would come up, and I knew that, as comfortable as most of them were with the original Factor Captor game board, this one was going to be somewhat intimidating.  I mean, 51?  How would one go about finding its factors?  My hope was to build some intellectual need for divisibility.
   My experience exploring the learning opportunities embedded in the game Factor Captor has me excited about the possibilities that lay hidden within other games, at other grade levels.  It has me thinking more about how we can put games to better use in math class.  In my experience, games such as Factor Captor are used as reinforcements for concepts and skills that have been previously taught. Teachers might provide their students with the opportunity to play them at centers, or when they are done with classwork.  But children who struggle often have limited opportunities to play; it may take them most of the class period to complete their assigned work.  For others, playing the same game in the same way over and over again can quickly become just as dull as another workbook page.  But what if we used the games, not as afterthoughts, or as ways to keep some kids busy while we work with others, but as the vehicles to deliver instruction? Turn them upside down and inside out, take them apart and put them back together?  Not every game lends itself to this kind of treatment, but there are many that will.  I have some in mind; feel free to comment with your thoughts and suggestions.
 
 

Off and Running

  So it's time to get back to work.  I'll be collaborating again with Rich in fifth grade, and we decided to start the year with a problem solving project.  We felt it would set the tone for one of our focal points, which is to help the kids develop their questioning skills.  We would follow the same protocol as last year's movie theater project.

We used this page...

...from this book.

  We gave the kids some time to look at the Birthday Party Basics list, and asked them what they noticed and what they wondered.  We gave them some time to write down their questions.
    While they were working, I overheard two students trying to formulate a question regarding the pints and quarts of ice-cream.  I walked over to listen in, and one student turned to me and said,
   "Mr. Schwartz, what's bigger, a pint or a quart?  I can never remember!"
    I hesitated a bit, and gave her my i-pad.  "Here.  Find it out for yourself."
   "I know how to do that," she said, as she quickly googled "what's bigger a pint or a quart".  (A small moment, I know.  But the fact that she could find the answer herself instead of relying on me or a fellow student empowered us both.  Another reminder of how important those small moments can be.)
 

They chose their favorites to write on chart paper.  Between the AM and PM classes there were quite a few to look through.  Many kids wanted to know how much it would cost to buy everything on the list.  They were also intrigued by the fact that a bag of pretzels cost less than a bag of potato chips but served more people.

A representative sample got posted on Rich's bulletin board.

We decided to have the entire class work on this one.  We liked it because of its openness; we felt it would prompt some interesting discussions as kids prioritized their list, and then re-evaluated decisions  based on how much money was left to spend.

     We weren't disappointed.  The question sparked some good conversation.  There was debate about which size cake to buy, what kind of drinks to serve (one group decided that bottled water was the way to go because kids might be allergic to apple juice and soda, another group agreed but decided to save money by just serving tap water), whether it was necessary to have potato chips and pretzels, whether or not it was important to have money left over, and the difference between 1 pint of chocolate, vanilla, and strawberry ice-cream and 1 pint of chocolate, vanilla, or strawberry ice-cream.
     The AM class worked in groups of three.  Not so great.  One (or sometimes two) tended to dominate while the other(s) were left with nothing to contribute.  You'd think I would have learned this by now.
     We switched it up in the PM.  Rich had them work in groups of two.  As they started in on the problem, I remembered something that I heard this past summer from a teacher at a workshop.  When she has kids collaborating this way, she makes each use a different colored marker.  Her kids know that she is expecting to see a balance of colors.  It helps promote accountability.

I had never tried it before, but it seemed to work!

       Here's what I like about this approach:

• The kids generate the questions.
• The kids get to select which questions they'd like to solve.
• The questions vary in level of complexity and necessitate the use of a variety of skills.
• The kids can work on these at centers, independently, or in guided groups as the unit progresses.

The day the kids were working on their solutions, this tweet came across my feed:

It's like teacher christmas! Except that we buy all the gifts ourselves and they are greeted with subdued acceptance! pic.twitter.com/4Md59dIOW7
— Justin (@JustinAion) September 10, 2014

   I'd like to experiment with big whiteboards for this type of problem solving activity.  Using whiteboards would allow the kids to erase when they make a mistake or want to revise their work instead of crossing out or just tossing the paper and getting another one.  Justin, who knows a lot about whiteboards, tells me that I can get them at Home Depot, and that they'll even cut them to size if I tell them I'm a teacher!
    So school's in.   We are excited about building on all the good work that the fourth grade teachers did with this group last year, and I think we are off to a good start.

My Confession, Part 2: I Am Saved by the MTBoS

    This poster is displayed prominently in my room.

     I tell my students that it's there to provide them with encouragement and hope, but really it's there for me.  It gives me comfort, even though I highly doubt Einstein had the type of dysfunctional relationship that I have had with math.

    Several years ago I was plucked out of out of my elementary classroom, where for over 20 years I taught reading, writing, science, social studies, health, and, yes, math to 7 and 8 year olds.  I put up bulletin boards, made sure everybody got cupcakes and juice on birthdays, planned and took field trips to zoos and museums, drew smiley faces on papers, filled out report cards, did the million things, both large and small, that all elementary school teachers do.  And then one September it was over.
    I was given the title "math specialist", a title that made me cringe, because my mathematical ability is by no means special.   But as long as the math didn't get too difficult, I had no doubts about my ability to help struggling learners.  In fact, I felt I had a small advantage; I could empathize with their struggle and perhaps be able to re-teach and explain concepts in ways that might make sense to them. For inspiration I looked to a familiar world, the world of sports.  I could be like Charlie Lau!

A mediocre hitter himself, Lau (right) is considered the most influential hitting coach in the history of Major League Baseball.  His disciples included Hall-of-Fame third baseman George Brett, a lifetime .305 hitter.

      I spent the first several years in my new position mostly pulling kids, both individually and in small groups, out of their classrooms and back to my room for one-to-one and small group instruction.  We call this "basic skills".  I was patient. I was sympathetic.  I took out manipulatives.  We did journal pages together, maybe got a start on the homework.  We studied for unit assessments.  I did what I could to patch them through a curriculum that raced relentlessly forward and never slowed down long enough for them to catch up, then sent them back into the inferno.
    Two years ago, something happened that changed everything.
   One day,while surfing the internet for math resources, I followed a link to Dan Meyer's TED Talk: Math Class Needs A Make-Over.  There's a powerful sequence when he shows a page from a textbook...

Just looking at it made my eyes glaze over and gave me a familiar sinking feeling in the pit of my stomach.

 ...and then strips everything away until he's left with just the visual of the chairs going up the lift.

   "Which section do you think is the steepest?" he asks.  That was a question I could answer.  It was a question anyone could answer.  You could just eyeball it and make an intuitive guess; you didn't need any "math".  I was hooked: how would we find out?  If there was math that would help answer that question, then that was math I wanted to learn.  Very powerful stuff for a kid who just wanted to crawl under his desk during math class.  I replayed the video over and over, and from there went straight to his blog, started at the beginning, and began reading.  I became convinced that if he had been my math teacher, things would have been much different, and realized that what I had been doing wasn't really teaching.

   Next came Paul Lockhart's  A Mathematician's Lament.

 I remember reading it and thinking, "This must what it would've been like to read a samizdat in the post-Stalin USSR." I imagined math teachers passing worn and dog-eared copies to each other, one step ahead of supervisors waiting to confiscate the manifesto and denounce them as heretics.  I just couldn't believe that a real, honest-to-goodness math teacher would write something that was so damning of his profession and that so accurately captured my learning experience.   I wanted to cry when I read this:

...if I had to design a mechanism for the express purpose of destroying a child's natural curiosity and love of pattern-making, I couldn't possibly do as good a job as is currently being done-I simply wouldn't have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education. 

    It would be hard to overstate what this meant to me.  It meant that I had, somewhere buried deep inside, an ability to do math.  Maybe it was small, but it was something that could be nurtured and, given the right conditions, it could grow.   It had been crushed out of me all those years ago, but with some help it could be found again.
 
   I suppose I followed a well-worn path: from Dan, who taught me about 3-Acts and intellectual need; to Andrew, whose work at estimation180 has had the biggest impact on my practice; to Fawn, whose humor,  humanity, and creativity  has helped me keep my eye on the ball; to Michael,  whose relentless and passionate search for meaning inspire me to dig deep; to Graham, an elementary compadre who keeps me company in a middle- and high school world.  And there are many others.  I took Jo Boaler's course, and learned about Carol Dweck's growth mindset research.  So it was true.  I could learn, not just how to do math, but maybe even to like math.  And if it was true for me, it could be true for all those other strugglers out there: the finger counters and the red x'ers, the fraction flunkies and the long division losers, the times table fist bangers; the confused, the lost, the drowning, and the already drowned.
   So I joined a wild and wonderful community called the MTBoS. I lurked.  I started commenting on other people's blogs.  I started my own.  And I've grown more as a professional in the past two years than in the previous 25 combined.
   There is another, smaller picture hanging in my room.

   This one has been with me since my first years teaching.  But it has taken me all this time to realize that the words apply just as much to me as they do to my students.  We no longer have to use our imaginations to envision what engaging, exciting, and nurturing math classes can look like.  The teachers who are embracing and exploring new ways to make math meaningful in their classrooms are taking no small amount of risk.  But they are doing no less than what they expect of their students.  I am proud of them, and proud to be counted in their number.
    School's in.  It's time to get back to work.

My Confession, Part 1: I Undergo Mathematical Trauma

   I was never good at math.  My struggle, my inability to get it, has colored my feelings toward the subject, feelings which remain even today.  It started early on:

This is from my first grade report card.  My parents saved lots of things.

      Towards the end of my second grade school year we moved, and I enrolled in the neighborhood elementary school for the final few weeks.  The teacher figured me out real fast:

I eventually learned how to tell time.

     Things never got better.  I never caught up, never caught on, and I suppose this is when my confusion turned to feelings of inadequacy, fear, anxiety, and hostility.  Here's  a work sample from grade 4:

4 out of the 7 problems are marked incorrect with a red "x".
How's that for meaningful feedback, Michael Pershan?

(It's funny what you remember.  It must have been in this class that the teacher asked us to put long division problems on the board for our classmates to solve.  When it was my turn, I wrote something I supposed would be really difficult, with 99 as the divisor.  99 seemed like a "hard" number to me.  The girl who was chosen to solve the problem laughed, "99 is easy to divide.  It's close to 100."  I didn't get it.)
 Again, my math warranted a report card comment:

The "of course" really stung.  This was doubly devastating because I had a crush on Mrs. Hughes.
Despite her hopefulness, it didn't improve.

  Things degenerated in middle school, and I am thankful those report cards have gone missing.  I suppose things culminated in my algebra 2 class in high school:

I was fortunate to get Ds; Mr. Momberg felt sorry for me.  I know I took geometry and trigonometry in high school, but that's as far as I got.  I planned to get as far away from math as I possible could.

  The idea to explore my personal relationship with math comes from an assignment in a course I should have signed up for:  Justin Lanier's smOOC Math is Personal.   It also arises from feelings I have about becoming a more active member of the MTBoS, and connecting with people who come from mathematical backgrounds very different from my own.  But what actually got me digging up my old report cards was reading a recent post from Fawn Nguyen, who shared excerpts from a book called The Number Sense, by Stanislas Dehaene.  This one jumped right off the page:

… most children enter preschool with a well-developed understanding of approximation and counting. In most math courses, this informal baggage is treated as a handicap rather than as an asset. Finger counting is considered a childish activity that a good education will quickly do away with. How many children try to hide when they count on their fingers because “the teacher said not to”?
Despising children’s precocious abilities can have a disastrous effect on their subsequent opinion of mathematics.
… It seems more likely that many of these “mathematically disabled” children are normally abled pupils who got off to a false start in mathematics. Their initial experience unfortunately convinces them that arithmetic is a purely scholastic affair, with no practical goal and no obvious meaning. They rapidly decide that they will never be able to understand a word about it. The already considerable difficulties posed by arithmetic to any normally constituted brain are thus compounded by an emotional component, a growing anxiety or phobia about mathematics.

     That was me, the kid hiding his fingers behind his back.  The kid with his head buried in his book, pretending he knew what he was doing and praying not to get called on.
     So it is ironic that I find myself in my current position, which I suppose goes to show that you really never know where life will lead.  And I am now part of a community where sometimes people talk like this...

New on my blog: if m≠1 is rational, can y=mx be tangent to y=sinx? http://t.co/blu5hOQPZl
— Dave Richeson (@divbyzero) August 27, 2014

...and I haven't much of a clue what they're driving at.
    But it is that same MTBoS that has made me see that it just doesn't have to be that way, something I describe in My Confession, Part 2.

 
   
   


 

Don't Worry So Much

       Reading Chris Lehmann's post about encountering former students as adults has inspired me to share some thoughts.  He's absolutely correct: one of the great things about teaching is that, if we hang around long enough, we sometimes get to see the adults our former students become.  "The perspective of seeing students become adults," he writes, "Can powerfully inform the way we teach."
     True story:  One of my first years teaching, over 25 years ago now, I had a student I’ll call Jennifer.  Jennifer was a very sweet second grader, but she struggled academically.  She was reading below grade level, her writing was poor, and she lacked many basic math skills.  As the year progressed she fell further and further behind.  I was really worried about her.  She may have been the first student I brought before our Student Assistance Committee, and she was eventually referred to our Child Study Team, who recommended she be evaluated.  When her father came in to sign off on the eval plan, he turned to me with the pen poised over the dotted line and asked,
     “Mr. Schwartz, if she was your daughter, what would you do?”
What did I know?  I didn't have kids.  I was in my mid-20’s, single, only few years out of school.  His question left me flustered.  I felt this weight of responsibility, as if her entire future was riding on my response.  I don’t recall exactly what I said; I but know I fumbled around uncomfortably.  And I think he sensed my worry and uncertainty, because after he signed, he looked back at me and said,
     “It’s OK.  She’s going to be fine.”   I was relieved, but couldn't help thinking, “What does he know that I don’t know?”
    Flash forward: Just a few years ago I was standing outside school one afternoon on bus duty.  As the last bus pulled away, a black Mustang rolled up to the curb.  It was Jennifer’s brother, who I had also taught when he was in second grade.  After catching up with him, I asked about Jennifer, thinking back to that day when her father asked me that very important question.
 “She doing great,” he told me.  “She’s a nursing student at Rutgers.”  I could only smile.

   Around the same time that I learned about Jennifer, I found myself sitting with my wife across the table from a different Child Study Team, in a school not my own, signing off on an eval plan for our daughter.  She had struggled for much of her elementary school years, and her fourth grade teacher was very worried.  Her test scores were bad.  She was reading below grade level.  Her writing was poor.  And don’t even ask about her math.  I told her teacher over and over again, “Don’t worry so much.  She’s going to be fine.”  I knew this because I knew some things about my daughter that her teacher didn't know: that she had nursed a grandmother dying of cancer and never flinched; that she could command a room full of three year olds better than many teachers at my wife’s pre-school; that she conquered a fear of animals and learned how to ride a horse and jump over a fence.  I knew my daughter was an amazing child, with talents and abilities that her teachers had no idea she had.  They were worried because her test scores were low.  I knew she had attributes that no standardized test could ever measure.  She had courage, persistence, and empathy.  I could see into the future.  I knew what Jennifer's father had known.  I could see the adult she was destined to become, and I thought about Jennifer, and knew I was right.

      The perspective of seeing students become adults can powerfully inform the way we teach.
      Can we also say that the ability to imagine our current students as the adults they will become is equally as powerful?  I think so.
      How does it inform the way I teach?  Because I only see my students in school, I need to resist the temptation to define them by their “school selves”; their school behaviors and the work they produce.  With the intense focus on testing and its high-stakes implications, it's often hard to do.  But these are narrow definitions, and the kids I teach are much more than the sum total of their test scores.  They have strengths and abilities that I never see.  So instead of trying to cram all the math I can into them in the limited time I have with them, I try to set aside time to talk: in the hallway on the way from their classroom to my room, over a game we play once we arrive, on the playground when I’m on recess duty, in the all-purpose room during line-up when I’m on morning duty, or just sitting at a table.  Just to show them I’m interested in what they’re interested in.  Even the ones who don’t deserve it.  And when I help teachers plan projects and activities, I try to allow for means of expression not normally associated with a math class: writing, art, and sports, something I want to do a better job of this year.
     Anyway, it's not exactly news that the nature of the student-teacher relationship has profound implications for learning.  David Kirp put it well in his op-ed piece last Sunday.  The process of teaching and learning is, "an intimate act... (full of) complicated and messy human relationships."
   I am blessed to be able to have taught as long as I have, because it has given me the opportunity to see former students grow into adulthood.  And I am thankful for Chris Lehmann's post.  As I contemplate the beginning of a new school year, he reminded me that I must first connect to my students as one human being to another.  If some math sneaks in, that’s just icing on the cake.

Making a Big Thing Out of it Would've Been a Good Idea: Practice Standard #6

   Is there a better example of the importance of  Math Practice Standard 6, Attend to Precision, than the Stonehenge fiasco in Rob Reiner's classic "rockumentary" This Is Spinal Tap?
     As the band's tour starts to disintegrate, lead guitarist Nigel Tufnel (Christopher Guest) suggests
resurrecting their famous "Stonehenge" stage act.  Lead singer David St. Hubbins (Michael McKean) is skeptical; they no longer have the required scenery.  Nigel insists, and begins sketching on a napkin as Ian (Tony Hendra), the band's manager, looks on.

"So we build a new one.  And this is it, look!"

Ian:   Consider...consider it done.
David:   So you're just going to take care of it like that.  You're going
         to find someone to design it...using that as a plan?
Ian:     Let's try.  Let's try.
David:   If you can do it, I'll do the number.

When Ian receives delivery, he is quite surprised:

Ian:    This looks actually perfect. I mean it's, uh, the right
        proportions.  It'll be this color right?
Artist: Yeah. Yeah.
Ian:    Yeah.  That's...that's...that's just terrific. It almost looks
        like the real thing.
Artist: Well good.
Ian:    When we get the actual, uh, set, when we get the piece,
        it'll...it'll follow exactly these specifications. I mean even
        these contours and everything?
Artist: Um, I'm not understanding it. What do you mean "the actual piece?"
Ian:    Well I mean...I mean when you build the actual piece.
Artist: But this is what you asked for, isn't it?
Ian:    What?
Artist: Well this is the piece.
Ian:    This is the piece?
Artist: Yes.
Ian:    Are you telling me that this is it?  This is scenery?  Have you
        ever been to Stonehenge?
Artist: No, I haven't  been to Stonehenge.
Ian:    The triptychs are...the triptychs are twenty feet high.
        You can stand four men up them!
Artist: Ian, I was...I was...I was supposed to build it eighteen inches high.
Ian:    This is insane.  This isn't a piece of scenery.
Artist: Look, look. Look, this is what I was asked to build. Eighteen
        inches. Right here, it specified eighteen inches. I was given this
        napkin, I mean...
Ian:    Forget this!  %#@*  the napkin!!!

Unwittingly, the band goes onstage to perform.  As it is lowered to the stage at the climactic point in the song, they stare in disbelief as dwarves dressed as druids dance around the model.

After the show:
David:   I do not, for one, think that the problem was that the band was
         down.  I think that the problem may have been...that there was a
         Stonehenge monument on the stage that was in danger of being
         crushed by a dwarf.  Alright?  That tended to understate
         the hugeness of the object.
Ian:     I really think you're just making a much too big thing out of it.
Derek:   Making a big thing out of it would've been a good idea.
Ian:     Nigel gave me a drawing that said eighteen inches.  Alright?
David:   I know he did, and that's what I'm talking about.
Ian:     Now, whether he knows the difference between feet and inches is not
         my problem.  I do what I'm told.
David:   But you're not as confused as him are you?  I mean it's not your
         job to be as confused as Nigel is.

I'd like to acknowledge Matthew Petty, a systems engineer who blogged about this last September.  I came across his post while looking for some pictures for mine.  (Thanks Matthew!)  He draws from this a lesson in Requirements Management, which he describes as, "How you ensure that what you build corresponds to what the customer is asking for."  Certainly a relevant "real world" application, whether we're the customer or the designer.
  Of course, mathematically speaking, all is not lost for Nigel.  While he may not know the difference between feet and inches, he does know that eleven is one more than ten: