Last year, I totally revamped my first unit on linear
relationships. My goal was to have the students develop procedural fluency from
conceptual understanding and also to work from concrete to abstract problem situations.
I designed the unit so that it was extremely heavy on tasks, and had students
notice patterns/generalize rather than a lot of direct instruction. I
anticipated that students would build their conceptual understanding at
different paces and tried to choose low-floor, high-ceiling tasks where all
students could make progress wherever they were in their understanding. I also
anticipated that it might be hard to measure and track the development of
understanding. I had seen students experience learning trajectories where it seemed like nothing was building in
exposure after exposure and then all of the sudden something clicked about the
concept.
In many ways, my revamp of this unit totally failed. While
there were probably many factors in this failure, I think that one of the
biggest ones was lack of student buy-in. I did not do a good job of
communicating my vision to students. I didn’t tell them where we were headed in
the long run because I felt like that would give away the end and defeat the
purpose of their own building and discovery of knowledge. Because my main goal
was for them to slowly make connections and I expected them to do it at
different paces, I didn’t have a neat, tidy goal for every task. This required
my students to put their trust in a vision they couldn’t understand or see,
which was an unfair expectation for them. Although students did engage in many
of the problems and start to get at the big ideas, the overall result was that
students who came in confident in math felt as though the class lacked rigor
and students who came in less confident in math felt like they still weren’t
making any progress.
So I am going to revise again. I still believe in the
premise of using a series of tasks for repeated exposure to build conceptual understanding
over time. I think most of the tasks
that I used do a good job with that. What I am going to change is my messaging
around the tasks. I need to do a much better job in getting my students to
believe that they are learning math through this process, even if it doesn’t
seem like it. I also need to do a better job in building a common understanding
of how students (and I) can tell if they have used their time in a worthwhile
manner.
I want to join this idea with my desire to figure out a way to have students explicitly reference and use the math practices. When I explained my math team’s focus on tasks, one of
my other colleagues asked, “If the goal isn’t to get the answer, what is the
goal?” For me, the process is the goal and I think the math
practices could help both my students and me assess their process of trying to solve
a problem.
I am imagining having students take some time to reflect at
the end of their work time (or even in the middle if work time extends more
than one day) on each task. They would identify one of the math practices they
had used and give an explanation or illustration of how they had used that
practice in their work. I would introduce the practices one at a time and spend
several weeks choosing tasks that particularly lent themselves to that
practice, so that we could develop a common understanding of what that practice
was and what it might look like as you are working on a problem. As the year
went on, kids would be able to choose from an increasing number of math
practices as the focus for their evidence. I would probably start with math
practice 1 and then 3, as I see those as the most over-arching, and then
sequence the rest based on which practices are the best fit for the tasks we
are working on at the time.
It would be my hope that this focus and reflection time
would give my students a much better illustration of what to value during the
time we spend working on tasks. Whether or not this is something that I will
grade (or have the students self-evaluate), I have not decided yet. I also hope that the explicit teaching and
re-visiting of the math practices in this manner will help students develop
those skills so they can really leverage them to access new content. It doesn’t
address how I will share the end-goal and overall trajectory, but I will keep
thinking on this.
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