"I Never Really Worried About Why."

 Just another day in grade 5, toiling away in the fields of 5.NF.A.1:

Students learned how generate equivalent fractions in grade 4, and are doing just what their teacher has told them to do:

"Whatever you do to the top, you have to do to the bottom."  Whenever I hear this, I think of The Golden Rule.  Do unto the numerator what you would do unto the denominator.  Something like that.

   
     There's a page to complete.  It encourages the students to apply The Multiplication Rule for Equivalent Fractions, which, in case they forget, is written in the middle of the worksheet:

When the numerator and the denominator of a fraction are multiplied by the same number,  the result is a fraction which is equivalent to the original fraction.

     There are lots of opportunities to practice, and there are fraction circles available so students can model what they've created.

     I approach a student and start talking to him:

Me: Hi!  What are you doing?
Student: Making equivalent fractions.
Me: Great!  How do you do that?
Student: Whatever you do to the top, you have to do to the bottom.
Me: Say more about that.  What do you mean exactly?
Student: So if I have 1/2, and I want to make an equivalent fraction, I have to multiply the numerator by 3 and the denominator by 3 and that will make 3/6.
Me: Sounds like fun!  What would happen if I multiplied the numerator and the denominator by different numbers?
Student: You'd get the wrong answer.
Me: (As I write out  1/2 X 3/5 = 3/10 on a piece of paper) So if I multiply 1 x 3 and 2 x 5 and get 3/10, that would be wrong?
Student: (politely, blithely, but somewhat exasperated) All I know is that the teacher said, "Whatever you do to the top, you have to do to the bottom."  I never really worried about why.

    Later, the math coach and I talked about the interaction.  We filled up a whiteboard with our own equations and visual models, explaining to each other what we understood, or thought we understood, about what was going on in that grade 5 class.  There's lots happening underneath the deceptively simple, oft-repeated phrase Whatever you do to the top, you have to do to the bottom, just as there is underneath the student's reflection that, "I never really worried about why."
    More than the math, it's the I never really worried about why that's had me thinking.  Here's what I've been asking myself:

• Is there a compelling reason that the student should have to worry about why?  A reason not that we think is important, but that the student thinks is important?
• Is there a difference between being worried about why and wondering about why?  What exactly did the student mean?  
• We already know what might make a student worry about why: It's going to be on the test!  You'll have trouble next year if you don't know!  But what has to happen in a classroom to make a student wonder why?  
• Is it always bad just to follow a rote procedure without understanding, wondering, or worrying about why?  Maybe that needs time to develop.  Maybe it will come later.  
• What routines or rote procedures do I follow without worrying about why?  Should I be worried about them?  Should I be more curious about them?   

     Many students I encounter are more than happy to share their thinking, their work, their questions, and even, on occasion, their life stories with me.  I'm constantly amazed by this, because I'm often a total stranger to them; some random guy who just happened to stop by their room that day during math class.  This particular 10 year old didn't have much use for me.  He probably had other, more pressing things on his mind, like getting through the assignment as quickly and painlessly as possible.  He was following the teacher's instructions and the directions on the worksheet, and was going to get all the answers on the page correct.  Who was I to add this element of stress into his life?  1/2 x 3/5?  What was that all about?  He had the how, and, in that moment, it was all he needed.  And he seemed pretty happy, maybe because he wasn't going to worry about things that, in his mind, weren't worth worrying about.  But that's OK. Sometimes you just have to damn the torpedoes and do to the bottom whatever you did to the top, and trust that later someone who knows why will help you figure it out.  Who knows?  Maybe you'll figure it out for yourself. 
     On your timetable, not 5.NF.A.1's.