Tuesday, March 14, 2017

"It Depends on the Meaning of Almost."

     My favorite statement so far this year came from a student via a recent tell me everything you can about... prompt.  This time it was:

Tell me everything you can about
4 2/5 and 3 1/2 

  The cards poured in...

...and provided many options for agree/disagree/not sure statements.  One statement in particular caught my eye:

The only thing this student could think of to say about the two mixed numbers.

I set a limit of five, but that one had to make the cut!

The final five.
     After giving the kids some time to work independently, Rich and I put them in groups with instructions to hash things out.  We listened in, and I settled down with one group and asked them to explain what they had decided about 4 2/5 and 3 1/2 being almost the same.  Agree?  Disagree?  Why?  Not sure?  Why not?
     The first student to speak up avowed that yes, they were almost the same.  Her explanation:

After converting both mixed numbers into improper fractions, she found they were 9/10 apart.  "That's pretty close."
   The next student said that she believed they weren't almost the same.  She started drawing on the whiteboard:

"They're almost a whole away from each other.  That's not almost the same."

     A third student chimed in:

"They're not almost the same.  They're 9/10 away from each other."

A fourth student stood up in support of the first student:

"I rounded 4 2/5 to 4.  And 3 1/2 is right in between 3 and 4.  If I round it up than they will both be 4.  That makes them almost the same."

     The debate lasted for several minutes, until the first student, in an exasperated voice, asked,
     "So Mr. Schwartz, are they almost the same or not?"
     Before I had time to even formulate a response, a student in the group, silent the whole discussion, piped up.  "It depends on the meaning of almost."
     Couldn't have said it better myself.
     Taking a look at the whiteboard as the kids left the room to go to their special, I felt gratified by the different ways they had thought to compare the mixed numbers.

But there was one key representation missing.

Back in my room, I played around with some number lines:

Are they almost the same?
How about now?

     Would these help, or just add fuel to the fire?  Talking about it the next morning, Rich and I realized that, in the end, it comes down to context and units.  There are situations where the difference between 3 1/2 and 4 2/5 seems insignificant (if I arrive at school 3 1/2 seconds before the bell rings, and you arrive 4 2/5 seconds before the bell rings, we've arrived at school at almost the same time), others where it's not (if I run 3 1/2 miles, and you run 4 2/5 miles, we haven't run almost the same amount of distance), and still others that seem debatable (if I have $3.50 and you have $4.40, do we have almost the same amount of money?)  In some cases it can mean the difference between winning and losing (as in a race), a delicious or inedible dessert (measuring ingredients), or even life and death (medicinal doses.)  In other cases the difference makes no difference at all.
     So I'm left wondering: What experiences and contexts could our students bring to the table?   Have we tried hard enough to free the numbers we work with from lives of lonely isolation?  From a dreary existence in the land of pseudo-context?  How can we make them come alive?  For a brief moment in time,  3 1/2 and 4 2/5 ran wild in the classroom.  The next day they went back to the black and white of the journal page and the worksheet, but man, they had fun while it lasted!


  1. Great initial prompt and awesome investigations afterward! Thanks for sharing Joe!!! I see that your students here are thinking, listening to each others' ideas, making sense of things, reasoning & proving, representing their thoughts....... So much math in such a simple idea!

  2. Thanks for sharing Joe. Such a wonderful example of the SMPs in action. Love it when students get opportunities to reason and prove, share their thinking with others, represent their thinking, make connections between ideas and concepts, and have opportunities to have misconceptions discussed and explored.
    So much great math here!

    1. Thanks Mark! I'm constantly amazed by how many great opportunities arise out of the simplest ideas. Even better when they come from students, as this one did. It makes me realize that the more open the question or prompt, the more likely we're to get something rich and thought-provoking. We just have to be willing to follow things wherever they may lead.

  3. This is such a great way to get fights going in classrooms where using math precisely is the only way to win. In particular MP3 and 6 come to mind but also MP1 and others. So simple and brilliant. Thanks!

    1. Thanks Robert! Trying to start arguments is something I'm always thinking about, but often they're arguments that have a clear winning side. This one was different. When I read the original statement I knew I had to use it with the class...it was brilliant! I never would have thought of it myself. As I wrote above in response to Mark, that's what can happen when we open things up by asking things like, "Tell me everything you can about..." We still get all the math we want (kids subtracting mixed numbers) but also much more, like the SMPs both you and Mark cite.

  4. Thanks for posting, Joe. It was great to read about students working through an open-ended prompt. "It depends" is so much more important than finding exact differences. Hope you'll keep us posted on ways they continue to consider context in their responses.

  5. Thanks Linda! What I really liked about the prompt was that most students did find the exact difference, except here finding the difference wasn't an end in itself, it was a means to an end. Thinking about this now, I realize it's an example of having the math (here computation) serve an intellectual need, rather than just doing it because it's a content standard or because it's going to be on a test.

  6. Hi Joe. I like how you realized at the end that the key to understanding the difference between the two numbers was placing them in context. When numbers are quantities that can be sensed and felt emotionally (seconds late for class, amount of medicine in a dose) they take on valuable meaning.

    1. Thanks! I'm not sure why it took me that long. Maybe it was something I understood subconsciously, but when I thought of the numbers it was as numbers with no context. That made me think about how the kids think of them, and how sometimes numbers come with contexts and sometimes without.