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...


  1. Once upon a time the rectangular prism was called a cuboid - shorter if nothing else.

    I like the general questions - what a variety of responses! Particularly the possible lack of appreciation of place value.

    1. Just did some googling on cuboid. And a cube would be called a square cuboid? Maybe we should just go with "cube-ish."

    2. Or a regular cuboid. I don't know.

      It seems to be their name in the UK. For instance the English National Curriculum says for Year 1 (5 and 6-year-olds):

      Pupils should be taught to:
       recognise and name common 2-D and 3-D shapes, including:
       2-D shapes [for example, rectangles (including squares), circles and triangles]
       3-D shapes [for example, cuboids (including cubes), pyramids and spheres].

  2. Joe, I always appreciate your thoughtful posts - thank you!

    1. Thanks Linda for your kind words and encouragement.

  3. Joe,
    This also makes me think of using the different problem types in operations. A question like this appeared on our EQAO test (our standardized test) in Ontario last spring. It had a rectangular prism which only had the length and width plus the total volume listed but was missing the height. It asked them to find the height. it changes from a basic procedure question to more of a thinking based question. Its the direct connection to the problem types from primary that sometimes holds them up. If they have never been introduced or are use to finding missing part operation questions it can seriously limit their ability to answer more thinking based problems. I use the term thinking here because in Ontario the questions on the EQAO test are categorized using 3 categories: Knowledge and Understanding, Application and Thinking. The questions are either multiple choice or open response. This question was a multiple choice question. I love these types of questions like you have shown here which help us to gather so much more info about our students. It just made me think of that missing part problem which would link in well with what you have shown us. Thanks for sharing again. Always informative.

    1. Thanks Mark. And I agree with everything you've said. Maybe a follow-up could be to show a rectangular prism with length and width and total volume but missing height and see if anyone would come up with the height from a "tell me everything you can..." prompt. And as I type this response it occurs to me that there is a nice connection between this "missing part"problem and the skill of making inferences that our counterparts in the reading world emphasize.

    2. Excellent connection Joe. As an IL I am covering math and literacy but so often end up supporting Math I sometimes lose focus on Reading. I am going to think more on that and speak to some of my teachers.

  4. As usual, really interesting, Joe!

    It's that first half of What do you notice? What do you wonder?

    Without it, we're just encouraging a kind of tunnel vision. "That is only about this." And the this is just our this, or some curriculum this that's not even ours.

    I wonder about wondering too, would that uncover any unknowns...?

    (Are there really only 12 right angles? I haven't heard anyone talk about this, but it seems to me you could measure right angles all over this solid shape. Even at the vertices by twisting round... Just wondering...)

    1. Thanks Simon. Tunnel vision is a good way to describe it; in my mind I also think of it as teaching with blinders on. I was wondering about the right angles as well. I see 3 at every corner for a total of 24, but are there interior and exterior ones? Another unknown unknown just uncovered!

    2. I don't know if this is right, but with a solid shape, we're not limited to the edges are we? So imagine you had a carpenter's square at a corner, one part of it could go along an edge and the other diagonally across the face? Or you could put the set square at some point along an edge at right angles to it...

      As you're saying Joe, these uncertain territories are the interesting and revealing places to be. "Teach the tongue to say'I don't know'", as it says in the Talmud.

    3. I don't know either! What's wonderful is that this all came from a student's observation. And I'll return your gift of Talmud with another, from R. Hanina: "I have learned much from my teachers, more from my colleagues than from my teachers, but from my students I have learned the most of all."

  5. Very interesting.

    There is something to the students comments "it is a cube:" as drawn, it is hard to tell whether it is a cube or not. Other than the marked lengths, is there a way for them to decide?

    One thing I wonder about is passing the students' cards around. Do they think about what the other kids wrote or just copy? What about passing misconceptions? Like that use of semi-colons....

    I was struck by what the kids noticed about 253831. The various rounded values was probably the deepest thinking. I wonder what they would have wondered about it? For me, the immediate question: is it prime/what is the prime factorization? To dispel the suspense: 253831 = 41 x 41 x 151.

  6. Thanks Joshua. I suppose that, without the marked lengths, students could have measured the dimensions. And now that I think of it, saying that it looks like a cube doesn't necessarily mean that it is a cube!
    When they pass their cards around, I see them selecting particular observations, not just copying. Not sure what you mean about the semi-colons though.

  7. Hi again Joe! I've just nominated this post for #bestofmtbos2016.

    Part of it is to post this:

    Hello! This post was recommended for The Best of the Math Teacher Blogs 2016: a collection of people's favorite blog posts of the year. We would like to publish an edited volume of the posts at the end of the year and use the money raised toward a scholarship for TMC. Please let us know by responding via whether or not you grant us permission to include your post. Thank you, Tina and Lani.