Friday, March 16, 2018


     Grade 4.  A table, a line plot, some questions.

   You could decide to have your students complete the page as is:

Put an x in a box.  Put a number in the blank.

Or you could decide to let your students make their own line plots:

Things may get a little messy.  

Rulers will help.

He started at 0 and ran out of room on his paper!  Now what?
   There were many false starts.  Some first drafts got crumpled up and thrown away.  But it was a productive struggle with many benefits, the majority coming from the decisions that the students had to make:
  • Which way should I hold the paper?  
  • Where should I start? 
  • What if I don't have enough room?
  • How many intervals do I need?  How far apart should they be?
  • Which fractions go where? 
     Did everyone get to all the questions?  No.  Is the opportunity cost worth it?  Yes.  We can't continue to bemoan the fact that students have trouble persevering and solving problems when what they mostly do in class is put x's in boxes and fill in blanks with numbers.  And in case you're wondering, rather than starting all over, the student who started at zero and ran out of room simply taped another piece of paper onto the end of his original sheet.  Lesson learned.

Tuesday, February 20, 2018

"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.    

Sunday, January 28, 2018

Equations I Have Known

The Triple-Header Run-On

The Vertical Right Side Equal Sign  

The Upside Down Double Flip

The Order Of Operations Parenthetical 

The Multi-Operational Spectacular With Arrow

The It Works For Every Other Operation So Why Not Division?

The Double Division Combo Special

The Fraction Run-On, Whiteboard Edition

The Wait a Minute, I Get To Write On the Table?

     "Mathematics is the language with which God has written the universe," said Galileo.  These first attempts at using that language, while not always perfect in their grammar or usage, deserve to be celebrated.  If we look only for what's wrong, we'll miss the creativity, ingenuity, and inventiveness our students gift us as they themselves try to make sense of the universe around them.

Friday, January 12, 2018

Play On

     At the conclusion of her book Exploring Mathematics Through Play in the Early Childhood Classroom,  Amy Noelle Parks makes an extraordinary statement:

     There is a great deal of evidence supporting the incorporation of play into the classroom, and that evidence can be particularly useful in getting support from administrators and parents.  However research on what works alone cannot guide our actions in the classroom.  For example, we could imagine a research study that demonstrated that administering electric shocks to children led to higher test scores.  And yet, even in the face of this "evidence", no one would advocate such a practice.  We are responsible for asking not just whether a pedagogy works, but also whether it is ethical to use with children.  (pg. 130)

     Parks cites Russian philosopher Mikhail Bakhtin and his work on a theory of ethical behavior called "answerability."  She interprets Bakhtin to mean:

     As a teacher I cannot simply turn to guidelines-even developmentally appropriate ones-to decide what it is okay to do in my classroom.  If my children are miserable, it is not enough to say that they have had the appropriate amount of play and so must return to seatwork.  "I myself-as the one who is actually thinking" must decide what it is ethical to do based on what I see happening with the children in front of me.  (pg 131)

     Reading Parks reminded me of questions I have often asked myself:  What if teachers were required to take some kind of educational Hippocratic Oath, an oath that bound us to act in a moral, ethical manner towards the children whose education and care we're entrusted with, an oath that, if broken, would result in the forfeiture our licenses?  In what ways does the institutionalized system in which we work rob us of autonomy and make us complicit in harming the very students we mean to help?  How do we advocate for change without losing our jobs and, with them, our livelihoods?  These are questions I don't like to ask, because when I answer them truthfully I know that, had I taken a Teacher's Hippocratic Oath, even one that said simply "First, do no harm," it would have been violated many times over.
     I'm no longer in the classroom.  No longer subject to the pressures, demands, and restrictions that come along with employment in a school system.  In my new role as a consultant and a coach, I have lots more freedom to do and say what I want.  Currently I'm working with some kindergarten teachers, trying to figure out how to make their math block more student-centered, engaging, and, well, fun.  Trying to figure out how to negotiate their district's expectation that 5 year-olds slog through relentless testing, and torturous lessons from an industrial, mass-produced curriculum, with its attendant rigor and relevance, its common core college and career-ready connections, its mind-numbing, one-size-fits-all mediocrity, while still leaving some time for their kids to explore and play around with math.  Which is what led me to Amy Noelle Parks.

      We cannot justify practices that we identify as harmful because they are required in standards, by the district, or in order for children to be successful in later grade levels.  In fact, Bakhtin refers to these outside requirements as "alibis", and argues that we cannot use them to justify behavior we know to be unethical. (pg. 132)

     I've used those alibis, every one of them, to rationalize and excuse behavior I've known to be unethical.  Reflecting now, I realize that much of the work I've done since I've left the classroom has been an attempt to find some expiation.  So what now?  Parks encourages us to look around our classrooms and ask ourselves the following questions:

  • When do children seem joyful?
  • When do they laugh?
  • When are they most engaged?
  • When do students cry?
  • When do they get angry?
  • When do I feel happiest and most relaxed? (pg. 132)
   These are the questions I want to ask myself, and want the teachers and administrators I work with to ask themselves,  when visiting classrooms; not just kindergarten classrooms, but all classrooms.  As Parks concludes:

     Attending to those questions pushes us toward the creation of a humane as well as educative classroom environment, and almost certainly toward a classroom that includes time for play.  Literacy scholar Deborah Hicks, in discussing Bakhtin's ethics, wrote that the commitment required by answerability was "more similar to faithfulness, even love, than to adherence to a set of norms."  As the adults who are responsible for small children for large parts of their lives, we need to bring that faithfulness to our work with them, just as much as our concern for standards or testing outcomes.  (pg 132)

And to that I can only add, Amen.