Thursday, March 26, 2020

A New Favorite

    Something I've been carrying around in a folder for years finally got a trial run this past winter in one of the grade 2 classrooms I visit, and it was a big hit.  Lifted from the Georgia Frameworks (you can find a detailed lesson plan, directions, and game sheets in Grade 2, Unit 1, pgs. 80-89), it's called Capture the Caterpillar, and if I can make any sort of contribution during this current crisis, it's an activity I'd highly recommend.  It will work just as well at home as it does in the classroom.
    Capture the Caterpillar combines place value, comparing numbers, addition and subtraction, number sense, and strategy as you attempt to get as close to a target number as possible.  All you need are:
  • A deck of cards (10s and face cards removed, though Queens can be 0s)
  • A pair of dice
  • A game sheet and a score sheet 
  • Counters or anything that can stand in as counters, such as beans, paper clips, etc.
Here's how it works:

  • Pick two cards and generate a target number.  In the example above either 28 or 82; I chose 28.
  • Roll two dice.  Find the sum.  Take that many counters.  I rolled a 3 and a 2 and that makes 5, so I took 5 counters.  
  • Place the counters in the caterpillar and try to get as close to 28 as possible.  You can place them anywhere you want, but you need to use them all. 

Here I made 14.  1 ten and 4 ones.  That's 14 away from 28.

23.  2 tens and 3 ones.  That's 5 away.  Closer... but can we do better?

32.  3 tens and 2 ones.  How far away from 28?  Is this the closest we can get?  How do you know?

     In the above example, 32 is the closest number to 28 we can make given the 5 counters.  We would score 4 points because the difference between 32 and 28 is 4.  Play as many rounds as you like (in the game it's 5) and try to get the lowest score possible.
   Here are two second graders playing this game.  Their target number is 22 and they have 9 counters.  Pay attention to how they collaborate; how they count, how they determine the differences between their numbers, how they talk through the rules, how they plot strategy:

Here's a completed scoresheet:

Two more versions:

Some notes:
  • The Georgia Frameworks include only the 100s and 1000s caterpillars.  Honestly, it's been so long I'm not sure where the 10s caterpillar came from.  I can't find it on either the new (2019) or older (2014) edition of the Frameworks.  Did I do a cut and paste job and make it myself?   Here's the image in a google doc.  Or you can make your own.  Better yet, have your kids make it!  (Actually you don't even need the caterpillars.  Just circles will do.)  
  • The scoresheet in the Frameworks has a column for expanded form.  We eliminated that column and replaced it with a column to indicate the difference between the two numbers. Here it is.   We were about to experiment with having the kids plot their target numbers along with their attempts on a number line, but never got the chance. 
  • An alternative to competing against another individual or team is competing against your (or the class's) best score.  I like that better anyway.  
  • For a change, try to make the number that's farthest away from the target number and try for the highest score possible.
  • Try it with 3 dice. 
  • Forget drawing cards and trying to get close to a target.  Instead roll the dice, grab counters, and see how many different numbers you can make.  Is there a systematic way to go about the task?  Is there a relationship between the number of counters and the number of different numbers you can make?  
  •  One possible assignment: take a picture of a set up and ask students to come up with the   closest/farthest number possible.  This is a good number sense routine.
    My hope is that this activity can be used by parents working with their children at home and teachers working virtually with their students.  I can only imagine what it must be like for all those trying their best to continue to provide meaningful educational experiences for their kids right now.  Know that, when all is said and done, you will be counted among the many heroes who held our society together during this most difficult and challenging time. 


Sunday, March 1, 2020

Math Class Needs A Makeover Turns Ten

     Ten years ago this month, on March 6, 2010, a tall man in a maroon sweater vest gave a Ted Talk.  It clocked in at 11 minutes and 24 seconds.   Here's how he described what he did for a living:
     "I sell a product to a market that doesn't want it but is forced by law to buy it.  I mean, it's just a losing proposition."
     He called the talk "Math Class Needs a Makeover."  His name was Dan Meyer.  He was a high school math teacher, and he was tired of losing.


     How long does it take to change your life?  I remember exactly where I was the first time I saw it.  Sitting at my computer in Room 10A, in that elementary school somewhere off the New Jersey Turnpike, I was an unlikely math specialist with a very conflicted relationship to the subject, searching for something, anything, that might make a difference in the lives of the basic skills kids I was supposed to be helping.  What I saw took my breath away.  Here was an actual math teacher giving voice to and validating everything I felt was wrong about math instruction while showing a way forward.  It was a manifesto.  It was a challenge.  And, for me, it was nothing short of revelatory.  From there it was a few mouse clicks to his blog, which I devoured, anxiously awaiting each new post, and his "Blogulty Lounge", where I discovered the work of Andrew Stadel, Fawn Nguyen, and Michael Pershan, among others.  (I'll admit here for the first time that it was years before I got the pun: of course an online Faculty Lounge was a Blogulty Lounge.)  I began to experiment with ideas I found in this new online universe in the K-5 classrooms in my school and, in emulation, started my own blog as a way to record and reflect on that work.  How long does it take to change your life?  11 minutes and 24 seconds.


     Ten years, 33 languages, and over 2,800,000 views later, "Math Class Needs A Makeover" is as powerful and important today as it was then.  Will we ever truly be able to measure its influence?  One of the very first comments on the talk came from a 17 year old named Timo Bronseth.  He wrote:
     "By the time Meyer's idea has overthrown our school system, maybe I'll be teaching it!"  
     Timo's now 27.  I'd like to imagine he's out there somewhere, doing a 3-act task with his class, trying to overthrow the system.

Monday, September 23, 2019

Houston, 1964

     If I appear to be asking too many questions and not providing any answers, I am merely trying to convey the flavor of MSC ( the Manned Spacecraft Center) in 1964, when we had a mandate to fly to the moon but few hard facts with which to work.  It was primarily a question-asking operation at this stage, defining those things which needed answers...

                                            Michael Collins
Carrying the Fire, p. 67

Michael Collins before his flight on Gemini 10, July, 1966


     For my money, the best part of a 3-Act task is Act 2.  After the focus question has been established, but before any math work has been done, Act 2 is where students need to ask the teacher for the information they'll need in order to solve the problem.  It's crucial to the entire enterprise: if the right questions aren't asked the key information will not be provided and the problem will remain unsolved.  For students used to being given all the information they need embedded within the problem (Johnny has 3 apples.  Billy has 4 apples.  How many apples do they have in all?) understanding what information might be useful, and then asking for it, isn't always easy.  But the skill is all-important.

     Michael Collins is best known as the astronaut who didn't land on the moon (he orbited while Armstrong and Aldrin made the first crewed landing on its surface).  Caught up in the excitement this past summer surrounding the 50th anniversary of the moon landing, I read his book Carrying the Fire, in which he describes his experiences as an Air Force pilot, his astronaut training, and his flights aboard Gemini 10 and Apollo 11.  It's compulsively readable and highly recommended!


    I didn't realize just what an incredible accomplishment it was to put a man on the moon until I read Collins's book.  The lives sacrificed (of the fourteen candidates chosen along with Collins in the third astronaut cohort, four would die in training accidents, as well as Grissom, White, and Chaffee in the Apollo 1 cabin fire), the money, resources, time and expertise dedicated to the mission; in 2019 it's hard to imagine our country uniting behind a project of that magnitude.  

     Like a 3-Act task, the project started with an Act 1 focus question: How do we put a man on the moon before the end of the (1960s) decade?  Then came the complicated part: What do we need to know in order to accomplish the goal?  Well, start asking questions.  And remember, one unasked question could put the entire enterprise in jeopardy, so, no pressure.


   Collins's book has questions on nearly every page.  There are way too many to list them all, but here's a very small sample:
  • Would the possible thickness of the dust layer on the moon exceed the height of the lunar module?  
  • Would the static electricity on the lunar module cause the dust to adhere to the lander, obscuring the astronauts' view?
  • How much fuel would be required for a rendezvous and docking around the moon?
  • What would the temperature be in the spacecraft during the constant sunlight on the way to the moon?
  • What would be the effect of weightlessness on body functions?
  • Could a guidance system find its way to the moon and and back again?
  • If the spacecraft experienced communications failure, would the astronauts be able to take over navigational responsibilities?  How accurately would they be able to measure the angle between a selected star and the moon's or earth's horizon?

     In 1964 in Houston a lot of answers had to be provided before any rational person could assess the chance of success.
   Those questions were answered in test chambers, in flying simulators, in labs and factories and classrooms, on computers and chalkboards and notebooks, and in the data collected from each successive space flight, starting with the first Project Mercury flight on May 5, 1961 through Project Gemini, and culminating with the Apollo 11 flight, July 16-24, 1969.  Before the end of the decade.

Collins (center) flanked by Armstrong (left) and Aldrin (right) in quarantine after Apollo 11 splashdown, July 24, 1969.


    Only a select few had the ability and the talent to be able to answer questions like the ones that needed answering on our way to the moon.  Brilliant mathematicians, engineers, physicists, geologists, and doctors crunched the numbers and solved the equations, collected and interpreted the data.  Once given the necessary information, not every student is going to be able to come up with a solution in Act 3 of a 3-Act task.  Am I interested in that?  Yes.  But I'm more interested in Act 2: What questions did you ask?  What information do you think you need to know?  And remember, one unasked question could put the entire enterprise in jeopardy. 


        If I could use only one word to describe the earth as seen from the moon, I would ignore both its size and its color and search for a more elemental quality, that of fragility.  The earth appears "fragile" above all else.  I don't know why, but it does.  As we walk its surface, it seems solid and substantial enough, almost infinite as it extends flatly in all directions.  But from space there is no hint of ruggedness to it; smooth as a billiard ball, it seems delicately poised in its circular journey around the sun, and above all it seems fragile.  Once this concept of apparent earthly fragility is introduced, one questions whether it is real or imagined, and that leads inexorably to an examination of its surface.  There we find things are very fragile indeed.  Is the sea water clean enough to pour over your head, or is there a glaze of oil on its surface?  Is the sky blue and the cloud white, or are both obscured by yellow-brown air-borne filth?  Is the riverbank a delight or an obscenity?  The difference between a blue-and-white planet and a black-and-brown one is delicate indeed.
     ....The beauty of the planet from 100,000 miles should be a goal for all of us, to help in our struggle to make it as it appears to be.

Carrying the Fire, pgs. 471-472

Michael Collins, July, 2019




Thursday, August 8, 2019

The Earth Appears Fragile Above All Else

     Inspired by the 50th anniversary of the moon landing, I read Carrying the Fire, Michael Collins's compelling, funny, honest account of his astronaut training and experiences aboard both the Gemini 10 and historic Apollo 11 space flights.  He had no ghostwriter; every word is his own.  Here are some excerpts.  I put them in verse (and added titles) because they just seemed like poetry to me.  
    Hope he doesn't mind.

Only One Word

If I could use only one word to describe the earth as seen from the moon
I would ignore both its size and its color
and search for a more elemental quality, that of
The earth appears
above all else.
I don't know why but it does.
As we walk its surface, it seems solid and substantial enough,
almost infinite 
as it extends flatly in all directions.
But from space
there is no hint of raggedness to it;
smooth as a billiard ball,
it seems delicately poised 
in its circular journey around the sun,
and above all
it seems

Carrying the Fire

I have been places and done things you simply would not believe, 
I feel like saying;
I have dangled from a cord a hundred miles up;
I have seen the earth eclipsed by the moon, and enjoyed it.
I have seen the sun's true light, unfiltered by any planet's atmosphere.
I have seen the ultimate black of infinity in a stillness undisturbed 
   by any human being.
I have been pierced by cosmic rays on their endless journey from God's place
   to the limits of the universe,
   perhaps there to circle back on themselves
   and on my descendants.  
I have no intention of spending the rest of my life looking backward
   I do have this secret, 
   this precious thing,
   that I will always carry with me.

What Any Pilot Knows Is the Most Useless Measurement

any pilot knows
from ready-room fable 
or bitter experience that
the length of the runway behind him is the most useless measurement he can take;
it's what's up ahead that matters.
We know we cannot dwell 
on those good things that have already happened,
but must keep our minds one step ahead, 
especially now,

In Memoriam: 
June 16, 1961-November 16, 2002
His humor, courage and character are indelible, 
like footprints on the moon.


Thursday, November 1, 2018

"All in All, a Pleasant and Educationally Sound Experience for the Children."

OK, not the first math lesson I taught, but pretty darn close:

December 17, 1986

     At the time this lesson took place, I had been teaching for about 3 1/2 months.  The administrator conducting the observation was Dr. Frank Gardella, the district's math supervisor.  (Frank, who would soon leave East Brunswick, is now a professor at Hunter College in New York City.  Years later we would reconnect during some summer PD at Middlesex County College.)
     Even after reading the write-up, I'm not exactly sure what happened during this lesson.  Did it come out of a teacher's manual?  If not, where did it come from?  Did I make it up?  Clearly it was aimed at developing the relationship between addition and subtraction.   Unifix cubes were handed out.  I tried to connect my students' ideas of what related meant in their lives (family relations) to what it might mean for addition and subtraction equations.  Best I can tell I led the students through some direct modeling with addition facts with sums of 14, matching them to subtraction facts with a minuend of 14, and then did the same with addition facts with sums of 13.  The unifix cubes were used.  I modeled what I wanted on a piece of chart paper and the kids followed my lead at their seats on paper of their own.  It appears that this took 22 minutes.  Then we played a game of "practice races" for 10 minutes.  Finally I collected the unifix cubes and gave a homework sheet.
Here are Frank's comments:

A kind, humane administrator is a blessing for any teacher, first year or otherwise.

         What might I tell my "rookie self"?  What might I do differently?
  • The lesson was very teacher directed.  Now, as an intro, I might throw up some related facts on the board and ask: What do you notice?  What are you wondering?  Allow the kids to do more of the mathematizing.
  • I liked that I used unifix cubes.  But now I would let them explore on their own, in pairs or groups of three.  Maybe something like: Take 13 unifix cubes.  How many different addition and subtraction equations can you make? Then I might walk around and monitor their work, and find some related equations that I could use as examples.  (How did we do that in 1986?)  After consolidating some of the learning, I would give them a choice of using any number up to 20.  
  • I'm not sure what "practice races" are, but I feel confident I wouldn't be doing those.
  • I need a better closure.  Collecting cubes and giving a homework sheet doesn't cut it.  Maybe: Tell me everything you can about: 6 + 5 = 11  and 11 - 5 = 6
     Some other thoughts:
  • As a first year teacher, I was fortunate to have, in addition to Frank Gardella, some very supportive administrators.  For example my principal, Mike LaRaus.  I'll never forget what he told me back on my first first day of school, that September of 1986.  I showed up at like 6:00 AM, after a sleepless night, nervous as anything.  He found me, near paralyzed in my classroom.  He told me it was normal to feel that way, that I would always get that feeling on the first day of school.  Then he said, "Just relax and do your thing.  No one's going to bother you.  I'm not even going to set foot in your classroom for the first two weeks of school, and neither will any other administrator.  Get your footing and then we'll talk."  I can't tell you how relieved that made me feel.  Thanks, Mike!
  • Are you surprised I have a copy of the evaluation? I have them all.  Every single one I received during my 31 years of teaching.  What strikes me is how bare bones it is.  Three pages.  The two narrative paragraphs above, the first on page 1 and the second on page 3, with a checklist of performance practices, from Exceeds Expectations through Not Observed, on page 2.  The last formal observation I received was on January 31, 2017, and it came to me via e-mail.  I printed it out.  It's 14 pages long.  No wonder Frank left.
  • It's interesting to think back to the 25 year-old, first year teacher that I was.  Yes I was nervous at first, but I was also a little cocky.  I thought I knew a lot more than I really did.  (Now I know I don't know all that much.)  Also, I was a bit stand-offish.  (If you don't believe me, ask my wife.)  In time I learned how to be a good colleague; a supportive and sharing grade-level teammate and a helpful and contributing member of the staff and the wider school community.  That is to say, I grew up.
  •  I'm spending a lot of time this year coaching first year teachers.  They're brand new, right out of college.  Many of them have wanted to be teachers since they were kids, when they'd spend hours in their rooms "playing school".  Now their dreams have become realities.  They're nervous and excited, overwhelmed and overworked, and stressed out.  I love them.  I want so much to help them, to make their lives a little less stressful.  To let them know that they're doing a good job.  They're not much older than my own kids, and when I sit with them and talk to them I think about how I'd want someone in a position of authority to treat my son and daughter when they are just starting out in their first jobs.  Frank could've torn the lesson apart, but he didn't.  (Maybe he did think it was a "good lesson!"  Maybe he saved his real criticism for our post-observation meeting.  I don't recall.)  But I didn't yell at anyone, didn't make anyone feel stupid; I wasn't sarcastic or intolerant.  He recognized that.  My issues were with pedagogy and instruction, and those things can be improved with time, patience, and a desire to work at getting better at the craft.  I'm still trying to get better.

My first class.


Thursday, October 18, 2018

In Memory of the Overhead Projector

 In schools all over America there are overhead projector graveyards.

Quick, save one to put in the Museum of Outdated EdTech!

When I first started teaching, these projectors were invaluable.

Having one in your room was a status symbol.

     If you wanted to show something to the class, you had to have a transparency.  Some programs came with transparencies already made.  Awesome!  If they didn't, or if you wanted to show a sample of a student's work to the class, you had to make one yourself.  Not awesome!   You had to put a blank transparency in the paper tray of the copy machine and pray the copier didn't get jammed.  It always got jammed.  And it was always in the morning, five minutes before the opening bell, with a line of teachers waiting to make copies for their classes, moaning and groaning and giving you the hairy eyeball.  Then spreading the news all over the school that it was you who jammed the copy machine, so you'd have to skulk around the hallways all day and maybe even avoid eating lunch in the faculty lounge.  Also the bulbs always burned out, often right before you were getting observed doing a lesson that required the overhead projector.  Then you had to beg a colleague to borrow theirs, because there was always a shortage of overhead projector bulbs because they cost like hundreds of dollars and didn't get put in the budget.  And also because teachers hoarded them.  I know this to be true because once a teacher in my building retired and when the new hire came in and looked in the desk there were three brand new unused overhead projector bulbs in the bottom drawer.
     Happily, those days are behind us.  Behold one of the greatest advances in education in the last half century:

The document camera.

      We can show anything we want to the class without the hassles of the old overhead projector.  A picture in a book, the directions to a game, and, most importantly, student work:

     No longer do we have to wait.  We can share the work immediately, in the moment.  Imagine implementing the 5 Practices for Orchestrating Productive Math Discussions.  Now imagine having to make transparencies of any student work you wanted to share.   Document camera for the win.
     Of course many classrooms don't have document cameras.   Not to worry; take a picture with your phone's camera:

Everybody's got a phone.  Email it to yourself, or AirDrop it.

     The overhead has gone the way of the ditto machine, and the opaque and filmstrip projectors.  Most teachers today have probably never used one, and if they do remember them, it's from their days as students.  And while I will admit to some nostalgia for the calming whirr of the cooling fan and the intimate mood lighting, as well as a touch of sadness at the thought of those once proud machines piled forlornly and haphazardly in the back of a forgotten storage closet, their time has come and gone.

A document camera graveyard??


Sunday, October 14, 2018


     The days had become cooler and shorter.  The leaves on the trees began to yellow and I saw birds flying in flocks--probably on their way to warmer climates.  The nights were colder and longer.  I could not sleep and I went outside for a breath of fresh air.  There were no more lights coming from the bungalows and the sky was full of stars.  God, or whoever He is, was still there, observing his creation.  A new theater?  A new man?  The old idolatry was here again.  The stone and clay idols had been exchanged for a Gertrude Stein, a Picasso, a Bernard Shaw, an Ezra Pound.  Everybody worshipped culture and progress.  I myself had tried to become a priest of this idolatry, although I was aware of its falsehood.  At its best, art could be nothing more than a means of forgetting the human disaster for a while.  I walked over to the colony.  Most of those whose names the bungalows bore had departed this world, with its illusions, forever.  Those who worshipped them would soon follow.  I lifted up my eyes to the starry sky again and again as if in hope that some revelation might descend upon me from above.  I inhaled the cold air and shivered.

Isaac Bashevis Singer
Lost in America