Monday, October 10, 2016

Unknown Unknowns

     Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know.  We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns--the ones we don't know we don't know. is the latter that tend to be the difficult ones.

Scenario One
     You're a fifth grade teacher in the middle of a unit on finding the volume of right rectangular prisms.  You pose the following task to your class, maybe as a do now, maybe on an exit card:

Find the volume of the rectangular prism.

Question: What are you likely to find out?
Answer:  Which students know how to find the volume of a rectangular prism.

Scenario Two
     You're a fifth grade teacher in the middle of a unit on finding the volume of right rectangular prisms.  You pose this task to your class, maybe as a do now, maybe on an exit card:

Tell me everything you can about this figure.

Question: What are you likely to find out?
Answer: A whole lot more than in Scenario One.

Wait...the answer to what?  There's no question!


There's a trend developing.

An over-achiever hits the ceiling.
A struggler enters on the ground floor.
Some other student observations:
  • There's nothing inside.
  • I know this shape is made up of squares.
  • The perimeter is 14 units.
  • It's a cube, AKA a 3D square or rectangle.
  • It's a full cube with a top and everything else.
Here's a breakdown of the 29 respondents who elected to identify the shape:
  • cube: 13 
  • rectangular prism: 11
  • square: 4 
  • special rectangular prism: 1 
   After looking through the responses, Rich and I realized we had some work to do.  We had managed to uncover some misunderstandings and misconceptions about 3-dimensional shapes and their attributes that we didn't know existed, among both the students (What makes a cube a cube?) and ourselves (Is it correct to say a rectangular prism has sides?  Are the terms sides and faces interchangeable?)  These and other matters would need to be addressed.  But first...

We gave the kids their cards back without comment, then asked them to pass them around.  We also asked them to take some notes.  After looking at several other cards, they would get their own back and get a chance to edit their original response.
Remember the struggler?  

His revised card.

     The idea for this kind of task isn't original or new.  It comes from Steve Leinwand via Dan Meyer, and I came across it browsing through Dan's archives a few months ago.   I tried it out again last week in a grade 4 class studying place value:

Some known unknowns surfaced, including:
  • Confusion about the difference between a digit and a number.
  • Confusion between the value of a digit and its place value location.
  • Imprecise language when trying to describe a digit's location.
And an unknown unknown:
  • How do we tell if a number is odd or even?
      American psychologists Joseph Luft and Harrison Ingham first came up with the idea of unknown unknowns in 1955 as part of an analytic technique they created called the Johari Window.  It's a technique used by the intelligence community, and it may have beneficial applications to our field as well.  The questions we ask and the tasks we pose yield information about our students.  But when those questions and tasks are of a closed and narrow nature, the information we receive is limited.  It may confirm or disprove what we think we know, which is no doubt important.  But what don't we know about our students?  What don't they know about themselves?  What don't we know about ourselves?  How can we gain entry to those hidden places, where misconceptions and misunderstandings lay buried under piles of fractured definitions, half-broken algorithms, and jumbled digits and symbols?
   Tell me everything you can about...

Monday, September 26, 2016

SMP Scavenger Hunt

    Find evidence of each Standard for Mathematical Practice.  In one week, with at least one example from every grade level.  No cheating by looking at the teacher's manual.  On your mark, get set, GO!

SMP 1- Make sense of problems and persevere in solving them
Grade 2, Friday, September 16

Even after his teacher told the class to put their journals away, this student continued to work on a self-created broken calculator  problem.  I was impressed with the way he challenged himself, his determination, and the willpower he showed not to use the broken key.

SMP 2- Reason abstractly and quantitatively
Grade 4, Thursday, September 15

The fourth graders were presented with the above problem.  Because they were in the middle of a unit on estimation and rounding, all students obediently rounded each of the numbers to the nearest hundred.  They added 500 + 200 + 300, got 1,000, and answered yes.
  All, that is, except for this student:

As he later explained, there was no need to round the numbers to the nearest hundred.  He front-ended them, added 400 + 200 + 300, got 900, saw the 63 left from 463, and immediately knew they would have more than 950.  Done.  He was also able to calculate mentally how far past 950 they would be, without even being asked by the teacher. 

SMP 3- Construct viable arguments and critique the reasoning of others
Grade 1, Wednesday, September 21

Is there an easier way to incorporate this practice than Which One Doesn't Belong?  I've used this prompt countless times, and never once has a student explained that the basketball didn't belong because, "It's the only one that bounces.  If you try to bounce the other ones they'll get smushed."  Gotta love those first graders!

SMP 4- Model with mathematics
Grade 3, Tuesday, September 20

Multiplication and addition equations, pictures and number lines.  Same problem, many paths, many models:

SMP 5- Use appropriate tools strategically
Grade 5, Monday, September 19

Volume time again.  I described this project here.  Calculators and rulers are the tools of choice.

SMP 6- Attend to precision
Grade 5, Monday, September 19

Estimation on a number line.  Too low, too high, and just right.  Which student attended to precision?

SMP 7- Look for and make use of structure
Grade 3, Wednesday, September 14

This student is using the structure of the hundreds grid and what he knows about place value and patterns to help him fill in missing cells.

SMP 8- Look for and express regularity in repeated reasoning.
Grade 5, Wednesday, September 21

Deriving the formula for finding the volume of a rectangular prism.

Some reflections:
  • I'm fortunate to work in a position that gives me access to a wide range of grade levels.  The fact that I was not able to get into a kindergarten classroom doesn't mean that there isn't great math work and thinking happening there, because there is.  
  • I realized there were two practice standards I didn't fully comprehend.  My friend Graham Fletcher helped me with SMP 8, which he describes as "algebrafying."  The other was SMP 2, and I'm still not totally sure my example fits. 
  • I found many activities and tasks with overlapping standards.  For example, the volume project that I used to illustrate SMP 5 could also fit with SMPs 1, 3, 4, 6, and 8.
     Most days I find myself lost in the content standards: what they really mean, how they fit together and progress across grade levels, whether or not our curriculum really does align, how we can do a better job engaging kids and hitting them with meaning.  Pulling back for a week and viewing the math in my school through a practice standard lens was refreshing, and made me realize that I was missing the forest for the trees.  Lesson: don't miss the forest for the trees.

Wednesday, September 14, 2016

It's Never Too Late to Learn

Some teachers wonder then what their role in the classroom could be if it is not focused on demonstrating and explaining strategies to students and monitoring students' progress in using these strategies.

Susan B. Empson and Linda Levi

      I'm late to the math party.  I came to my current position as a K-5 specialist from a general elementary education background.  I taught all subjects during the 23 years I spent in my grades 2 and 3 classrooms, and math didn't stand out in any significant way.  I knew who Marilyn Burns was, but if I had ever met the luminaries that my colleagues in the MTBoS speak of so highly, people like Van de Walle and Fosnot and Kamii and Richardson, for example, it was long ago in teacher school, and I'd certainly forgotten who they were.  So I play catch up.
    Cognitively Guided Instruction (CGI) is something that, upon hitting the MTBoS, I heard referred to again and again.  Glowingly.  Reverently.  And so this summer I decided to find out what the fuss was all about.  I wanted to explore Extending Children's Mathematics: Fractions and Decimals, and put together a small, very informal book study PLC composed of our two grade 4 teachers and my former partner and fellow specialist Theresa.  We met twice in July and once in August to discuss our reading assignments and share the work we had done.

Kudos to my colleagues who gave up some of their summer evenings to participate in this project.

    It was an incredible experience, and we learned so much from the book and from each other.  We increased our own content knowledge, and are now armed with some tools that I hope we can employ to better analyze student work.  
     A few weeks ago I came across a set of photos I had taken late last school year of responses to an item on a grade 5 unit assessment.  The problem seemed pretty simple, and I was struck by the many different ways students had attempted a solution.  At the time I was ignorant of CGI, but looking at them now through a CGI lens I can better understand and describe what's happening.

The student multiplied 4 x 2 and 4 x 5/8 and added the two products together.  This was a good illustration of the distributive property of multiplication over addition.  This student was also able to use relational thinking to express 20/8 as 2 1/2.  He is able to decompose 20/8 into 8/8 + 8/8 + 4/8.  He understands that 8/8 equals 1 whole, and that 4/8 is equal to 1/2.

    The students who got it right were almost uniform in implementing the above strategy, although there were some outliers:

Convoluted, but it worked.  This student is also able to use relational thinking to express 2 5/8 as 21/8 and 4 as 32/8.   Why did he feel it necessary to have common denominators?

   But what had really caused me to stop and take some pictures was the many and varied ways that students had managed to be wrong.

This student converted 2 5/8 into an improper fraction.   Did he really understand why 2 5/8 was equal to 21/8 or was he just following a procedure: multiply the denominator by the whole number and then add the numerator?   The multiplication across the numerators is incorrect.  Careless mistake or confusion about the identity property of multiplication?

No question here.  Issues with the identity property of multiplication.

This was a common mistake: multiplying 4 x 2 then just adding the 5/8.  This student has used an equal sign to separate the expressions, rendering an incorrect equation. 4 x 2 does not equal 8 + 5/8.

This is diagrammed as if the student were multiplying 42 x 5/8.  When multiplying 5/8 x 4 the student flipped the 4 into 1/4.  Perhaps he had heard something about "invert and multiply"?  However he did not do this when multiplying 5/8 x 2.  In the top left corner he is adding his two products, but how he got 10/628 is a mystery.

With no actual work, I deduce that this student multiplied 4 x 2 to get 8, then, in the mixed number, multiplied 2 x 5 and 2 x 8 to get 10/16.  He finished by adding 8 and 10/16.

At first look it appears as if this student multiplied 4 by every digit he saw in the mixed number.  How did he get 160?  By multiplying 8 x 20?

     These fifth graders are now off to middle school.  But their work remains, and looking at it through the lens of what I've learned this summer is going to inform what I do, not only with the new batch of fifth graders, but with all math learners in our school.  If I read my CGI right, these kids are victims of an over-reliance on procedural memorization.  They have bits and pieces of algorithms that they can't put together because they have weak conceptual underpinnings.  They're easy to spot. However it's possible that many of the kids who did get the correct answer also have shaky conceptual foundations, but are just better at memorizing a procedure.  How will we find them?
     CGI argues for a decrease in the amount of time teachers spend, "demonstrating and explaining computation and problem-solving strategies to students."  Instead, teachers are encouraged to allow their students' intuitive strategies to emerge first.  "Children have some conceptually sound understanding of fractions, even before instruction," Empson and Levi write, "(but they) can learn to ignore this understanding in favor of models introduced in school that portray fractions in narrow ways."
     This is a tremendous shift away from the traditional "I do, we do, you do" model, and entails teachers taking on a different role.  "This new role," the authors of CGI explain, "Centers on helping students communicate strategies to other students, directing questions to specific students to help them draw connections between these strategies and more basic strategies, introducing equations to represent students' strategies, and highlighting the fundamental properties of operations and equality that that underpin these strategies."
     That's a lot to ask.  But I've read the book and I'm a believer.  Let's roll up our sleeves and get to work.

Monday, August 29, 2016

Talking Math With Your Kid: End of Summer Edition

    Having aged out of camp, and unable to subsist on the cash received from her intermittent babysitting gigs, my daughter faced the same prospect her brother had faced two summers before: get a job. That's how, on a warm, late June afternoon, with her dad in tow for moral support and a folder full of very skimpy resumes, she found herself pounding the pavement in downtown Princeton, New Jersey.

Nassau St.

Throughout the spring, she had deflected every conversation on the subject of summer employment.  There was school, and dance, and a busy social calendar.  She expressed ambivalence about working at a day camp, and anxiety about working retail.  Never a good math student (and that's putting it kindly), she was afraid, she explained, of working the cash register.
   "I don't really know how to make change!  And I'm afraid that there will be a long line of customers and they'll all get mad at me!"
   "You don't have to make change," I answered.  "You just punch in the numbers, and the machine figures out the change for you."
   Variations of this conversation, and others concerning the urgency of looking for work while there were still positions available for teens, continued up through the final day of her high school sophomore year.   Finally, she ran out of time and excuses.   School ended on a Thursday; on Friday we were on the hunt.  I could sense her unease, and knew that the entire project could end up in tears, raised voices, slammed doors, and recrimination.  This was going to be a delicate operation; with a ticking time bomb of a soon-to-be 16 year old, it would test all my fatherly skills and powers.
    The plan was to walk around town, looking for businesses with Help Wanted signs in the window.  But she had her own, difficult-to-verbalize litmus test: regardless of whether or not they were hiring, some stores were simply off-limits.  We fell into a rhythm: JaZams toy store: No. Thomas Sweet: Absolutely not.  Nina's Waffles: A half-hearted yes.  Hulit's Shoes: Heck no! As frustrated as I was by her refusal to jump on every opportunity, I kept my cool as we trudged on.

Here was one that passed the test.

    We stopped for lunch at PJ's Pancake House (they were looking for help; another defiant no.)  She had left her resume at four stores.  It was a start.  We were emotionally exhausted.  We would go home and regroup, but not before making one more stop on our way out of town.

Bon Appetit.  A gourmet food store in the Princeton Shopping Center.  Her older cousin had worked there when he was in high school.

  She protested, but I put my foot down.  Sensing I wouldn't budge, she walked in reluctantly, asking for the manager as I browsed the shelves.  Several minutes later she found me.
   Panic and barely concealed tears.  "Dad!  They want to interview me!  I don't want to work here!  What do I do!  You made me come in here!"
  "Calm down.  Let them interview you.  It will be good practice.  Why don't you want to work here?  Ben worked here and he liked it."
   "I just don't don't want to work here!"
  "You might not even get the job.  Sit down with the manager.  It will be a good experience either way.  Ask him what he might want you to do."
   "But I don't want to work here!"
   Back and forth, our voices rising, then lowering so as not to make a scene, until she gave in.
   I waited outside, peering in the window, pacing up and down the outdoor patio.  I knew she was mad at me for having forced her into the store, and I hadn't thought things would proceed so far so fast.  But I  hoped that all would be forgotten in the afterglow of a successful interview.  Boy, was I wrong.
   When she came out to meet me, all the pent-up emotion came flooding out.
  "Dad!" she cried, fear and anger mixing together.  "They want to hire me!  Right now!!  I don't want to work here!  What do I do?"
  "That's great, honey!" I was proud she had made such a good impression.  "Why don't you try it out?  Did you ask what they wanted you to do?"
  She unfolded a piece of paper.  On it were the words: cash register.  Of course.
  "We've been over this a million times!"  I was exasperated.  She had a job offer in hand.  She could start tomorrow.   "You don't have to make change.  If someone gives you cash, the register will tell you how much change to give the customer."
  "I know that," she quavered, voice rising through the tears.  "It's that I don't know what coins to give back!"
   There it was.
   It had been there all along, but I didn't hear it.  Her fear wasn't about figuring out how much change was due.  True, she would have a lot of trouble doing that, but she knew the register acted as a calculator.  It was about finding the right combination of coins.  Not only that, she knew that a customer would expect the combination be comprised of the least number of coins possible.  She was unsure she would be able to do that, especially under pressure, with a line of impatient customers backing up out the door and onto the street.   Her litmus test, which had seemed so arbitrary, finally made sense: the store had to have as little foot traffic as possible, and at least the potential of non-register related work.
    I told this story one evening at Twitter Math Camp, along with what I had taken away:
  • It was possible for a student to go through elementary school, middle school, and Algebra 1 and Geometry in high school without mastering the ability to make change with the least number of coins;
  • That this was less a reflection on her than it was on a system of math education that, from early on, left her feeling inadequate, disconnected, and disaffected;
  • That math anxiety is real, that it interferes with performance, and that bad things happen when we put undue pressure and time constraints on kids.
And bless Andy Gael for his response: "I love how your daughter advocated for herself."

     She went back in and told the manager she'd get back to him.
     She eventually found a job:

JerZJump: A family entertainment center.  Birthday parties, open play, camp field trips.  No cash register.

Her first paycheck.

Here's how the story ends:
     Summer is over, and the novelty of the job has worn off.  The smile you see above hasn't been seen in quite a while.  There's no air-conditioning, the hours are irregular, the pay stinks, and the kids are uncooperative and unruly.  Lately she's taken to spending time with her babysitting loot and the contents of the house change collection:

"Make 38 cents with the fewest possible coins."

Nothing like a little intellectual need.


Monday, August 15, 2016

Real Mathematicians

   First on my to-do list when I got back from TMC '16 was get a new notebook, because in Minneapolis I learned that real mathematicians go graph ruled.

My old notebook (left). My new notebook (right).

     Day 1 of Tessellation Nation. Michelle Niemi  is hard at work with a folded piece of paper and a scissors.  "I wonder if I could get a snowflake to tesselate?"
     I'm sitting next to her, deep into Christopher Danielson's turtles.  "I don't know," I tell her.  "But someone here probably does."
     Enter Dr. Edmund Harriss.  Co-author of Patterns of the Universe, researcher, professor at the University of Arkansas, and all-around great guy, Edmund would later blow my mind when he explained to me why a square can have wiggly sides and 72 degree angles.  But now he's got a problem to work out.
     I watch him sit down next to Michelle.  After listening to her for a few moments, he takes out a pad of graph ruled paper and a pencil.

     Putting the turtles aside, I turn my attention towards Edmund as he begins to explain to Michelle how he is going to accomplish this task, getting a snowflake to tessellate.  He starts drawing squares.  He's visualizing the folds.  He's marking where to cut.  He sees symmetrical designs in his mind's eye.  Mid-summer, and it's snowing in his head.   Michelle has questions.  She wants to know what he's thinking.   So do I.  Edmund is patient.  What's clear to him isn't clear to us.  He's got to go over it several times.
     And it occurs to me: I am watching a real mathematician solve a problem.  And it's thrilling!   Because I've never been this close before!  I'm not sure I really understand what he's talking about, and I don't care!  I'm just caught up in the excitement of watching him work.  And I think to myself, "This is what mathematicians do.  They solve problems.  On graph paper.  I need a graph ruled notebook.  And I need to stop thinking of myself as only a math teacher.  I need to be more of a math doer."
     Edmund finishes.  He takes a piece of paper, folds it into squares, and makes the cuts.  He's left with the pieces of a snowflake.
     "You could have every child in your class cut one of these out, and put all the pieces together to make a tessellation."    

     That evening: 

Once I started looking, I saw graph-ruled notebooks everywhere.

Henri Piccioto explained how to create a graph ruled notebook.

Megan Schmidt's spiral obsession, inspired by Edmund Harriss, began innocently during a school meeting as she kept herself a graph ruled notebook.

Of course sometimes graph ruled notebooks aren't available, and mathematicians need to improvise:

Jonathan Claydon did calculus on a napkin at dinner one night.

I love my new notebook:

I'm using it to solve problems for an informal summer book club...

...and it really came in handy for helping me understand the tessellation in my uniform tiling with curvature problem.

Up until last month, the closest I'd ever come to watching real mathematicians at work was at the movies.  Both real...

He helped defeat the Nazis!

...and otherwise.

Nice use of a vertical non-permanent surface, Will Hunting!

One of the great thrills of TMC is getting to watch real mathematicians do their thing live and unplugged.

John Golden (left) and Henri Piccioto (right).

So, nearly a month after it ended, I've finally found my #TMC1thing.  It's to put as much effort into doing math as I do into teaching math.  On a vertical non-permanent surface when I can, on a napkin if I have to, but mostly in a graph ruled notebook.  Like real mathematicians do.