Monday, September 24, 2018

The D Word

     In the course of my work with teachers, I'm often asked about a little something called differentiation.  Maybe you've heard of it.
     They might say something like:
     Admin is requiring us to have differentiation options in our lesson plans.  What's a good way to account for all the levels of learners in my class?

    More often than not they won't use the d word.  They'll say something along the lines of:

    I've got a lot of low kids in my class.  It takes them forever to solve a problem.  Some of them can't even read.  The high kids finish right away and then just sit around and wait while I try to help the others.  What can I do?

    When I hear comments like this, I set aside (for the moment) my concerns about labeling learners as high and low.  I compliment the teacher on her observation that something is wrong, that not all needs are being met, that precious class time is being wasted, and that she wants to do something about it.  But what? 
     Sometimes teachers will assign certain groups of students certain problems, or adapt assignments, like this:

    One common method I've seen employed is dividing students into ability-based groups.  Teachers might call this guided math, or math workshop.  After a whole-class lesson, the teacher will meet with the so-called struggling students and lead them through the problem set while the "high fliers" work through the problems on their own, and then move on to some game or other activity.  If there's time, the students in the teacher-led group may get to play a game.  There's rarely enough time.
   While I do believe there is a time and place for teachers to meet with small, skill-based groups, too often this model becomes the daily, default structure.  The groups calcify, and all the inherent social, emotional, and academic implications manifest themselves.  So what's the answer?
    Here's a differentiation template that I've used.  I take no credit for inventing it, only experimenting with it in some of my elementary classrooms to good effect.  I've written about this before, but as we begin the 2018-2019 school year I feel it's a good time for a revisit.  It goes like this:

Step One:
   Grab a scissors.  Take whatever problem you want the kids to solve and remove the question.  Keep just the text.  If the questions are generated based on a chart or some other graphic, keep just the graphic.
   Some examples:




Step Two:
     After a quick notice and wonder, ask the students to write their own questions, questions that can be solved using the information in the prompt.  First in their notebooks and then, after vetting, on chart paper.  Ask students to vary the question type between "thin" and "thick" questions.
   Some examples:

Gr 5

Gr 2

Step 3
    Select a variety of questions and post them around the room.  Allow students freedom of choice to solve whichever problems they want, using whatever strategies they want.
     Some examples:

     I like this model because it avoids the stigmatization of students being put in high, middle, and low groups and it allows for students to self-differentiate, both in terms of the questions they write and the questions they elect to solve.  Engagement is increased because, rather than being told which questions to answer, students have been given the opportunity to pose their own questions and decide which they'd like to tackle.  It's the definition of a low floor/high ceiling task, and it provides opportunities for small group interactions within a whole group community setting.  The teacher can elect to work with whatever students she feels need guidance without the students feeling that they've been singled out in an ability-based group.  Most important, it's fun.
     If you'd like to know more about how this has played out in classroom settings, see posts here, here, and here.  Comments are open for your ideas, observations, and reflections.



  1. Love it, Joe. By cutting it down to the situation, everyone's mathematizing is valuable, and there's more to see than just this question which I think I can or can't answer. Plus, how much richer is the discussion going to be?

  2. This challenges the idea that there's one dimension, a road that we're either way ahead on, or way back on. That there are high kids and low kids. That they can't be involved in conversation with each other.

    I've found that the kids who have the interesting questions aren't necessarily the same as those with most experience handling numbers. Same with the ones who like to explore. Or who want to explain things. Or the ones who watch what's happening. We force it all into some grotesque beauty contest when we narrow down what we're asking of our students.

    I love how your scissors work takes away the lining-up-in-order of lowness that is so common. Instantly there are more dimensions, we lose awareness of any line.

    1. Thanks Simon. As always, you express what I mean to say much better than I'm able to. That "losing awareness of any line" is really important to me. I'm not sure we can completely eliminate it, but it's nice to forget that it's there, if even for a time.

  3. Your post has me thinking about a transformation in content knowledge I've been making way too slowly over my entire life.

    My content knowledge transforms when I realize how many different ways there are to understand any given area of mathematics. Many more than just "the formal operations in that area," which was my impression for a long time. Instead, you can eg. notice & wonder mathematically, make reasonable predictions, select an appropriate representation for a question, create new questions about the math, understand connections between that area and others, etc.

    That content knowledge transformation increases our options for remediation and differentiation.

    If math is just formal operations, then the best and really the only way to remediate is to show students enough examples of those operations until they can repeat them. If math is just operations, then the only way to differentiate is to move students to the next set of formal operations.

    As my field of vision on mathematics increases so too does my imagination for ways I can help students, for ways I can enrich a task beyond its formal operations.

  4. Thanks for the comment Dan. For me, this goes to the very heart of what it means to do mathematics. In the school setting, it is very defined and rarely taken to mean things like noticing and wondering, creating new questions, etc. (This is actually a subject that Edmund Harriss and I are having a very robust e-mail exchange about.) This is due to many factors, but one big one is the nature of standardized testing. Until the quality of the questions you ask is valued more than the accuracy of the ones you answer, then teachers and students will continue to be measured by that very narrow definition. And the differentiation and remediation will continue to be more of the same, only slower and louder. Who's responsible?