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. 


Monday, December 11, 2017

Property Values

     One of the first things I do when I walk into a classroom is look for vertical whiteboard space.  Ever since being introduced to the work of Peter Liljedahl, I've counted myself as a member of the #VNPS movement.  Liljedahl's research, along with the efforts of chief practitioner Alex Overwijk, combined with experience watching students use them in my own practice, has convinced me that every available inch within student reach is precious.

     However this prime real estate, which should be preserved as open space for student work, is often taken up by some dubious development.

SMART Boards are one culprit.  Think of all the wasted space underneath!

Big Ideas, Essential Questions, Objectives

What's the opportunity cost of all that lost whiteboard space?

     In her book Becoming the Math Teacher You Wish You'd Had, Tracy Zager  summarizes Liljedahl's work, as well as her reaction.  She writes:

     "When I first read Liljedahl's research, a whole series of narratives and images from the history of mathematics buzzed through my brain.  Mathematicians frequently talk about standing around blackboards or whiteboards together, thinking and talking.  This particular kind of collaboration--standing, talking, thinking, and writing--is so inherent to doing mathematics that many math buildings are designed around it.  Given that mathematicians work this way, and that educational research has revealed there are tremendous benefits to vertical, non-permanent surfaces in classroom settings, it seems we have ample reasons to set them up." (pg 322-323)  

     She offers options for teachers in classrooms with limited wall space, including hanging whiteboards on cabinets, closets, and bathroom doors, and allowing students to write on windows with vis-a-vis markers.

Mirrors work, too.  just ask Will Hunting.

     But before we resort to those measures, let's take stock of what's on our existing, classroom-wall  mounted whiteboards.  What's there?  What purpose does it serve?  Who benefits?  Who's it for?  If it's necessary information, can it be moved someplace else?  When I first started teaching back in the mid-1980's, objectives went in our plan books.  We didn't plaster them all over our chalkboards. (No one had a whiteboard back then.)  We didn't know from Big Ideas and Essential Questions.   Is there research similar to Liljedahl's that shows how advertising them promotes a thinking classroom, enough to sacrifice empty whiteboard space?
     "We all have real constraints on the size and layout of our teaching spaces,"  Zager writes.  "Nevertheless, it's worth thinking about how we can work within those constraints to provide students workspaces that promote thinking partnerships." 
     After all, it's their room too.


Tuesday, September 5, 2017

Anything's Possible

     In an attempt to make peace with the Exeter problem set, I followed Jasmine Walker's advice:

   Middle school math?  Yikes.  Anyway, here's page one:

     Problems 7-11 came pretty easy.  (Hopefully I didn't get any wrong!)  For #11 I didn't feel like writing an explanation using words and complete sentences, so I just drew a picture.

   Problems 1-6 were another story:

Problem #1
     I didn't think my answer made sense.  I had the feeling that it took a lot longer than 500 seconds for light from the Sun to reach the Earth.  So I googled it.  I was right!

Problem #2
    I knew it would take a really long time, and that I would have to consider that you have to breathe, eat, sleep, and go to the bathroom, and that different numbers take different amounts of time to say out loud.  It all seemed kind of overwhelming!  There's some interesting stuff about it on the internet, though.

Problem #3
    I used what I knew about the speed of light from problem #1 and got an answer of 232,500 miles.  Maybe I was wrong; according to google, the moon is actually 238,900 miles away, but what's 6,400 miles when we're talking about the solar system?  Close enough.

Problem #4
   Nope.  Couldn't do this one.  And I like baseball!

Problem #5
   This one felt a lot like problem #2.  However I buckled down and gave it a go.  I figured on a step being about 2 feet.

Problem #6
    Offshore pipelines?  Cylindrical mechanisms?  I nearly gave up on this one, but at the last moment before publishing this post I figured I'd give it another try.  But 1.36 miles per hour seems kind of slow, so I think I'm wrong.

To sum up:
  •  The math was a barrier in problems 4 and 6, although I did understand what was being asked  of me.  I don't feel real good about it, but that's on me.  
  • Problem 2 just didn't seem worth the effort.  
  •  I had the math skills for problems 1, 3, 5, 7, 8, 9, 10, and 11.  I can't say I enjoyed solving them, although I'll admit to a feeling of accomplishment when google confirmed my answer to problem #1.  And, in all fairness, I didn't get the emotional rewards of sitting down with a peer group and discussing my answers and attempts.  That's also a part of the Exeter experience.     
     Will I print out page 2 and start working on those problems?  Probably not.  If I'm going to solve problems like these, I need a little romance.  I need some help building intellectual need.  I can't always just gin it up on the spot.   I did my homework, but it felt like just that--homework.
     This could be the heart of the MTBoS project: how do we (as teachers) take problems like these and turn them into something that a student might want to solve?  Not because it's assigned for homework, or because it's going to be on the test, but because he's curious to know the answer.  Is that putting a premium on answer-getting over sense-making?  Does this mean that deep down I'm an answer-getter, not a sense-maker? Do I have to be one or the other?  Or can I have some of both in me?  If needing to know the answer is the motivation for learning the math, does that mean the answer-getting impulse is something that should be cultivated?
     Michael Pershan says that just like there are different genres of literature, there are different genres of math problems.  OK, so Exeter problems are not my genre.  But what genres do I like?  And what does that say about me?  The answer may lie in an experience I had at TMC '17.
     Since the host site was not within walking distance of lunch options, on several days it was arranged that food trucks be brought to campus.  Feeding 200 people out of 2 food trucks was quite an operation; the lines were long and the Atlanta heat was unforgiving.  My hunger got the better of me, and I ignored the advice from my friend (and Georgia resident) Graham Fletcher to wait inside in the air conditioning until the line dwindled down:

Big mistake.

    For the first few minutes it actually felt good to be outside after spending all morning indoors.  I got a chance to talk to Annie Perkins, who was standing behind me in line, about her experience in the Playing With Exeter Math morning session.  She explained that it was a good opportunity to just work on problems, something she had trouble finding the time to do during the school year.   As we spoke, I started to feel the famed southern heat and humidity start to kick in, and I realized that the line had barely moved.   I began to have second thoughts.  Should I go back in?  Or have I reached the point of no return?  How much longer am I going to have to wait?  Will I get sunburn?  Heatstroke?  Maybe if I started to keep track of how long it took someone to receive his lunch, I would get an idea of how much longer I'd have to stand in line.  So, in part to pass the time and in part because I was just plain curious, I asked John Stevens, who had just placed his order, to guess how much time it would take for him to get his food.
     "4 minutes and 29 seconds," he ventured.
      As I looked down at my watch to check the time, I heard a voice behind me.  It was Annie.
     "No, it won't take that long.  I've been standing here for twenty minutes, and..."
     And here Annie launched into a very cogent mathematical explanation about why it wasn't going to take John 4 minutes and 29 seconds to get his food.  My brain, baking in the heat, couldn't quite follow, but it made perfect sense.  I forgot to look back at my watch, and by the time John had his food I was back inside, in the air-conditioning with Graham, who was kind enough to refrain from saying, "I told you so."
      Should I have waited outside with Annie?  How long would it have taken me to get lunch?  How did Annie know?  It would have to depend on how long it took a person to get his food.  Would that depend on what he ordered?  Whether he paid with cash or a credit card?  If I could find the average time, and then multiply that by the number of people ahead of me in line, I could get a good idea.  It was nothing if not a math problem, but a problem that, unlike the Exeters, I was curious about.  I imagined what the problem might look like, written out and embedded inside a page of the Exeter problem set.  Would I have cared about it then?  Probably not, because it didn't come from inside of me.
     Maybe one day I'll feel like Annie, and jump on the opportunity to solve problems like those in the Exeter set.  Right now I kind of doubt it, perhaps because there's too much scar tissue.   I'd like to be able to have the choice, and the measure of how bad I want it will be my willingness to suck it up, buckle down, and learn the math.  I won't rule it out, because if there's one thing I've learned it's to never say never.  Given where I've come from, who'd have ever thought that I'd be standing outside in the hot Georgia sun, in line to get lunch from a food truck, in the middle of the summer,  at something called Twitter Math Camp, on my own dime, and turning the experience into a math problem?  Believe me when I tell you: If that's possible, then anything's possible.

Friday, August 18, 2017

It's Not You, It's Me

 Twitter Math Camp, 2017:

     In the end there were two: Mathematical Yarns with David Butler and Megan Schmidt or Math Coaches Huddle with Chris Shore.  Veteran Twitter Math Campers know that your AM session choice is all-important, because the one you select is the one you commit to for all three mornings.  That's a total of six hours, so best choose wisely.  As much as I would have loved to spend that time exploring mathematics through the medium of hyperbolic crochet with David and Megan, it was Math Coaches Huddle that ultimately got the nod.  I'm going to be doing quite a lot of coaching in my new role, and who better to help me up my game than Chris Shore.
     One thing for sure; I wasn't going anywhere near Room 711.  That's where Wendy Menard, Danielle Reycer, and Jasmine Walker would be running a session called Playing With Exeter Math. What I knew about Exeter Math I learned from Joel Bezaire: people sat around something called a Harkness Table and solved problems created at the very exclusive Phillips Exeter Academy that looked like this:


     Who in the world could possibly want to solve these? And voluntarily too?!  Playing with Exeter Math?  Really?  To me it looked about as much fun as a root canal.  You want playing?  Playing was what I did in David Butler's Friday afternoon session One Hundred Factorial: Playful and Joyful Maths; a smorgasbord of puzzles and activities we could, according to the session description, "Follow down any rabbit hole that looked interesting."  And before we all made our way down to the cafeteria, which David had set up to look like a math amusement park, he explained that the goals of the session were: learning something, making or seeing something beautiful, understanding someone's thinking, and last, but definitely not least, sharing joy together.  Now there were some SWBATs I could get behind!  We weren't disappointed:

Megan Schmidt and Steve Weimar 

Taylor Belcher and Doug McKenzie

Jasmine Walker, Jim Doherty, and Maureen Ferger

     The contrast, in my mind, was stark.  One the one hand, Exeter Math problem sets: walls of text, a whiff of elitism, for the smart kids in the honors class.  On the other, David Butler's One Hundred Factorial: visually engaging, democratic, no experience necessary.  They were diametrically opposed.  I felt the two sitting on my right and left shoulders like an angel and a devil.
    But if it's nothing else, Twitter Math Camp is a place where one can work out his or her math issues with no fear of judgment.  Which is how dinner on Saturday night turned into something of a therapy session.  Out with a bunch of TMCers at the Cowfish Sushi Burger Bar, I found myself seated next to Jasmine Walker.  I had worked with Jasmine on a Skyscrapers puzzle in the One Hundred Factorial session.  Jasmine had also helped facilitate the Playing With Exeter Math session.  Perfect! Looking for confirmation and affirmation, I asked her to compare the two.
   "They're completely different experiences, right?" I asked.
   "Not at all!" she exclaimed.  "I find the same kind of emotional happiness working on the Exeter problems as I do when I work on problems like Skyscrapers."
    I was stunned.  It sounded impossible to me, but Jasmine went on to explain that, despite their outward dissimilarities, they actually had much in common.  Like the activities in One Hundred Factorial, the Exeter Math problems could lead to unforeseen and exciting places, and engender the same amount of joyful, raucous back and forth that we had experienced together in David's session. In fact, according to Jasmine, all of the goals David had set for his session got checked off in the Exeter Math session.  From his seat across the table, Jim Doherty, another One Hundred Factorial participant and himself an Exeter problem set veteran, did his best to come to my aid, but in the end pretty much echoed Jasmine's reaction to my question.
    I've thought a lot about the conversation I had that night with Jasmine and Jim.  And what the experience has caused me to do is take a good look in the mirror.  I was angry with the Exeter problem sets.  Angry because they looked like all those word problems I had so much trouble with when I was in high school.  Angry because they reminded me of how inadequate they made me feel back then, and how inadequate they make me feel in the here and now.  Angry because they seemed so intractable, so cold, so bloodless.  I wanted them to change.  I wanted them to lower their barrier to entry, to turn themselves into something I could access, to be something I could share with Jasmine and Jim and Joel.  I wanted to be able to walk into Room 711, and I was angry because they were blocking my way.
     But what I've come to realize is something I've known all along but been unwilling to truly admit: Exeter Math isn't to blame.  The one deserving of my anger is me.  I have to take responsibility for my own learning.  If I want to play with the big kids in Room 711 I have to learn how to do the math.  And to do that, I mean to really do it, I'd have to start back in middle school and go all the way up through Algebra 2.  Not quite Billy Madison, but pretty darn close. After I did that, I could decide for myself if solving Exeter Math problems was a rewarding activity or a waste of my time.  And I don't think I'd care either way, because the decision would be mine.  All mine.