Functional Misses

     What do you do when a student in class raises his or her hand...and then proceeds to give a wrong answer?  Do you:

• Call on another child to provide the correct answer under the guise of "helping" the confused student?
• Stop and intervene with the individual student while the rest of the class sits idly by and waits for you to finish?

    These are the two most frequent responses I observe (and have used myself).   As teachers we know that our students will give wrong answers.  And as teachers we know that there is a lot more riding on our response than the misunderstanding of a concept or procedure.  There are emotional consequences as well.  If we're not careful, the way in which we handle wrong answers can effect a child's perception of himself or herself as a learner, cause embarrassment, and lead to an unwillingness to contribute to and participate in class activities.  So what can we do?

    As with many things, we can look for answers to the NBA.  Specifically Houston Rockets extraordinary two- guard, James Harden.

Harden currently leads the league in scoring, and possesses the best beard in professional sports.

      In a recent article in Grantland, Kirk Goldsberry deconstructs Harden's scoring genius.  He finds there's more to Harden's offensive game than his shooting prowess and uncanny ability to draw fouls and get to the line.   He explains:

  But even when his attempts fail, they have a chance of succeeding. The entropy from his slashing drives, which scramble defenses, enables his teammates to slip into great rebounding positions. Not all missed shots are created equal, and sometimes they function a lot like inadvertent passes or shot-clock reset buttons.

 Goldsberry calls these "functional misses".

 
    In Goldsberry's example (above), Harden drives down the lane and misses the lay-up.  A teammate, however, is in perfect position for a put-back slam.  Functional miss.
   Is it possible to put this notion into practice in the classroom?  The day after reading the Grantland piece, I was helping a guided group of struggling fifth graders work through fraction addition at Rich's back table.

Hmmm.  Unlike denominators.

   

The kids dutifully took out their multiplication grids and found common multiples for 2 and 3.

  

We settled on 6.   So far, so good!

We converted 1/2 into an equivalent fraction, 3/6.  

   
     It was at this point that a student, I'll call him Jeremy, raised his hand.  Jeremy is new to our school.  His math skills seem shaky, which is why Rich decided to have him work with me in the guided group.  I was glad he felt comfortable making a contribution, and was curious to know what he had to say.
    "That's a mixed number!" he exclaimed.
    "Where?" I asked, hoping he was referring to the sum of the two fractions.  Was it possible he was two steps ahead of us?
    "Three-sixths," he said, pointing to the fraction on his whiteboard.  "Three-sixths.  That's a mixed number."
    I could sense the unease of the rest of the group.  We had spent several weeks talking about improper fractions and mixed numbers, and I was sure they knew he was wrong.  I decided to press him a little.
   "What makes it a mixed number?" I asked
   "The numerator and the denominator got bigger," he told me.
     The moment of truth.  The time to make one of the hundreds of decisions, both big and small, a teacher has to make each day, every day.  What size decision was this one?  What would you do?
   
     I decided to highlight what was correct, ignore the mistake, and move on.  Perhaps I could turn it into a functional miss.
    "What you said about the numerator and denominator getting larger is correct, Jeremy," I responded.  "Let's work on finishing the problem."

We turned 2/3 into 4/6, and then added the fractions together to get 7/6.

     The kids jumped all over it.   "That's an improper fraction, Mr. Schwartz!"
     "Yes it is," I said.  "Can you say more about 7/6?"
     They didn't let me down.
    "6/6 is 1 whole, and there 1/6 left.  That makes 1 1/6!"

Look at that!

     "There's Jeremy's mixed number!" I said, addressing everyone at the table but looking right at him.  "1 1/6.  A whole number and a fraction.  That's what makes a mixed number."
      Did I make the right decision?  I'm sure there are others I could have made, and maybe some would have been better.  I know I have to spend some time with Jeremy, who is likely in need of some intervention.  But in that moment, Jeremy had the correct answer.  Because inside of every wrong answer, there's a correct answer just waiting to get out.  If we can find the right question, we turn it into a functional miss.
    As fate would have it, that night the Rockets destroyed my sorry Knicks.  James Harden went off for 25 points and 9 dimes in just 30 minutes.  As much as it hurt, I had to smile.