I'm going to try my best to summarize a lot of what I have been thinking about lately in terms of curriculum design and pedagogy that supports students in "doing mathematics." Although there is more to it than what I write here, the following three pillars seem to be a good place to start:

1. Each lesson should revolve around a single task, prompt, or question

Students can't "do mathematics" if we don't offer them an opportunity. I think it is most helpful to think of this as a shift from the traditional lesson format of lecture, guided practice, individual practice to a more fitting lesson format of launch, explore, summarize. The traditional lesson format is designed precisely to have students mimic the teacher's way of knowing and doing mathematics. The second lesson format shifts the responsibility of doing, inventing, and creating to the student. By offering students this responsibility, we also give them the opportunity to develop mathematical habits of mind

2. Each task, prompt, or question should be student-driven

Many people interpret student-driven curriculum as the notion that students decide what to study. While I think that might be a possibility (by having students generate and pursue extensions of the task, prompt, or question), I prefer to think of this as curriculum that is designed to respond to a student's current way of knowing by always pushing on their equilibrium point. In this way, it becomes difficult for a curriculum to stand entirely on its own, independent of the student community. While curriculum can/should serve as a reference, it is ultimately up to the teacher to craft a lesson that, in a response to an assessment of a student's current mental model, puts the student in a position of cognitive conflict. This iterative process results in a connected mathematical trajectory that is student-driven.

3. The "resolution" of each task, prompt, or question must come from the students

Mostly, I mean here that no body other than the students should provide resolution (ex. textbook, teacher, video, etc.) When students see that the validity of their thinking will be measured by something/somebody other than themselves, they will always end up judging their thinking against a "greater" authority. In these situations, the "doing" becomes solely about figuring out how to arrive at someone else's understanding. When a task is open, the "doing" becomes more about arriving at a resolution that makes sense to the individual student/group. There may be differences in opinion and reasoning among students, which is good. In order for a pair, group, or class to reach a resolution, the authority must come from the logic and mathematics of the students. I would even suggest that any task that prompts students to uncover a pre-existing solution will inhibit their ability to engage in "doing mathematics."

Even though much of this sounds dramatic, I truly don't think it is too far removed from what is possible in even the most standards-driven schools. Thomas Romberg writes of a problem-based approach similar to this in which he sees students "studying much of the same mathematics currently taught, but with quite a different emphasis." I think this is a fitting description of what I propose here. I would push for much of this type of design to occur within a problem-based environment, but I don't think it is necessary. One could just as easily work with students to study, abstractly, a unit on exponential growth, trigonometric ratios, or any other traditional standards-based unit. This type of design is difficult and requires patience, but the impact on teaching, learning, and doing mathematics has meaningful impact.


04/09/2012 7:58am

I'm intrigued by the Resolution description. It sounds more and more like a science class, where the goal is to blow something up or inflate a balloon or turn the test tube purple. A math classroom where the resolution "feels right" enough for the student to be satisfied would need to be like a science class, where they can both...

a) see or experience the solution
b) validate it against possible sources of error.

For this to happen, the goal has to be clear ahead of time (a la Dan Meyer's 3 Acts). I'm thinking of the catapult lab that our math and science classes did a few weeks ago. The only goal was to measure the horizontal distance of each differently-weighted cube, then graph them.

If our goal was to make a hole-in-one with the catapult, we would have had much more buy-in.

04/09/2012 11:35am

I love your analogy to a science class. Some might argue that we could define mathematics as the science of patterns...exploring and looking for regularity. We often think of science as a hands-on, experimental process of conjecturing and testing. We would benefit from a similar vision for mathematics, I think.

I would agree that the "task" must be clear to begin with. I think students play a vital role in making this so. For instance, they need to make sense of problems and, collectively, identify what the task at hand is. Depending on your definition of "goal," I would be less certain to make this explicit. I would only disagree if we define "goal," as a specific learning outcome (way of knowing and thinking). If we, or students, start with a rich task/question that responds to students current ways of knowing and invest in them the mathematical authority to sort things out for themselves, meaningful mathematics is sure to follow. Whether or not their final "understanding" mirrors mine/ours is not quite as important to me.

I also particularly liked your last comparison about the catapult lab you did. The goal to make a hole in one sounds like a more engaging, and potentially meaningful inquiry, than measuring specific distances. It is more open for the students and, I would imagine, leads to more interesting investigations.

What are your thoughts?


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