Frowny Face

   I feel Brian's pain.  There are few things worse than being undermined by a poorly designed worksheet.  This one popped up in a grade 4 class last month, in the middle of a unit on rounding:

This worksheet had 10 word problems on it.  I'm not sure why you would want to estimate; the questions appear to call for precise answers.  And why to the nearest ten?

     As I was walking around the room, looking at student work and helping those with questions, I could feel my anger rising.  "Here we go again," I thought.  Until something on one student's paper caught my eye:

A small frowny face.

     I knelt down by her desk.
     "Why the sad face?" I asked.
     "I don't like this problem," she responded.
     "Why not?"
     "Well, when I round each of those numbers to the nearest ten, I get 540 and 250.  Those are not so easy for me to subtract."
      "What would make things easier for you?"
      "I could round the 536 to 500 and the 246 to 200 and just subtract 500 - 200 and get 300.  That's easy for me because I know that 5 - 2 = 3."
     "Then you go right ahead and do that!" I told her.
     "But the directions say to round each number to the nearest tens!"  She was nervous that the teacher might mark it wrong.
     "That's OK," I assured her.  "I'll talk with her."

Her answer.  Unit issues aside, isn't it better than two question marks?

     Worksheets like these, in which problems exist for the sole purpose of having kids practice a skill, fail on several levels.  They poorly serve the concept they're designed to reinforce (rounding), and they force kids into a single way of thinking (to the nearest ten.)
     This student knew there was something wrong.  She expressed her displeasure in the only way she knew how: by drawing a frowny face.  I love that frowny face!   Keep 'em coming, kids!

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.  And...it is the latter that tend to be the difficult ones.

Secretary of Defense Donald Rumsfeld
February 12, 2002

                

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!

Hmmm.

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...