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: