To Each According To His Need

   One class.  Three groups.  Three different classwork assignments.

This group gets two questions.

This group gets three questions.

This group gets five plus a challenge.

Here's what I'm wondering:

• Is this differentiation?
• What does it mean to differentiate?
• As a general rule, is it bad practice to put kids in ability-based groups or give them ability-based assignments?
• Are there situations where ability-based groups or assignments are appropriate?

     I believe there are times when an ability-based skills group is appropriate.  If there are five students in the class having difficulty multiplying multi-digit numbers, collecting them in one place for some further instruction makes sense to me, at least more sense than going back over the skill with the entire class.  Homework assignments might also look different for different students.  Those same five students shouldn't be working on a page of multi-digit multiplication problems at home until the skill is secure.
     In the above example, the three worksheets certainly are different, but I believe there's a better way to achieve a differentiation objective.  Here's my suggestion:

1.  Start everyone off on equal footing with just the table and a notice and wonder prompt.  This will provide time for everyone to take a breath and process the information.

What do you notice?  What are you wondering?

2.  Have students write their own questions.  Stealing vocabulary from our ILA teachers, we can ask the kids to come up with both thin questions and thick questions.  Or, from Vacca's work on Question-Answer Relationships, questions that are right there and think-and-search.

This was from a different prompt, but you get the idea.  And here's a catch: you have to be able to solve the questions you write!  

3.  Post an assortment of questions.  Let the kids decide which ones they want to solve.  But be sure to vet them first!

Again, these questions were generated from a different prompt.  They vary in difficulty.

    This model has many benefits.  For one thing, it eliminates the stigma of being in the group that got only two questions.  It also transfers ownership of the question-asking from the teacher (or teacher's manual) to the students.  Of course you can learn a lot about students from the way they solve problems, but you can also learn a lot about them from the questions they write as well as from the questions they elect to solve.  Will a bright student take the easy way out?  Will a struggler try to punch above his weight?  Yes, and these choices are very telling.
     The reality is that students in any given classroom will have a wide range of abilities and needs.  We want to preserve a sense of whole class community, and we also want to make sure each individual student is receiving what they need and working at tasks that engage them in an appropriate productive struggle.  It's a very difficult balance to maintain.  Enlisting the help of our students may make it just a little easier.

"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!