When I shared my comps reading list with my committee, Bill Penuel quickly replied with the suggestion that I read this article about learning trajectories by Doug Clements and Julie Sarama. I'd seen Clements present on this topic at last year's RME conference (which focused on learning trajectories/progressions), and I recognized the paper as something I found last spring too late in the writing of a final paper to really read and process, so I am happy to return to it now.

At their most basic, learning trajectories can be thought of as sequences of tasks and activities aimed at the progressive development of mathematical thinking and skill. This appeals to me because, quite frankly, I'm not all that great at focusing on single mathematical tasks. Even with a great task, I find myself wondering, "Where in the curriculum does this task fit? What should students know and be able to do before attempting it? Once students complete this task, what new thing are they ready for?" You could say I get a bit distracted in an effort to see the big picture, a habit of mine that's not necessarily new. Learning trajectories are one way of thinking about curriculum on a larger scale, and the better I understand them, the more organized my thinking can be.

Clements finds the roots of learning trajectories in a 1995 paper by Martin Simon titled

*Reconstructing Mathematics Pedagogy from a Constructivist Perspective*(which I should also add to my comps reading list, I'm sure). While it's certainly possible to create a learning trajectory thinking only about instructional tasks, Clements & Sarama stress the interconnections between the instructional sequence and the psychological developmental progression of students.

**As teachers, sometimes we make the mistake of breaking down an instructional sequence according to the structure of the mathematics, which may or may not reflect the ways students will actually construct their mathematical knowledge.**To avoid this mistake, Clements & Sarama suggest designing learning trajectories using this three-stage process:

- Specify a research-based learning model that describes how students construct the mathematical knowledge needed for the trajectory. I think this is a tough task for teachers, both because the specific models in the research are not widely known and understood and because there are surely many areas of mathematics for which specific learning models have not been thoroughly studied.
- Select key mathematical tasks to promote learning at each level of students' psychological development. Again, it takes the help of research to judge if a task truly targets a certain level of development or not.
- Complete the hypothetical learning trajectory by sequence the tasks to match the students' developmental progression.

Of course, the completed learning trajectory should (a) take advantage of specific and relevant cultural knowledge and practices of your students and (b) be subjected to repeated revision and refinement. Clements & Sarama do not understate the potential of well-constructed learning trajectories:

The enactment of an effective, complete learning trajectory can actually alter developmental progressions or expectations previously established by psychological studies because it opens up new paths for learning and development. This, of course, reflects the traditional debate between Vygotsky (1934/1986) and Piaget and Szeminska (1952) regarding the priority of development over learning. We believe that learning trajectory research, along with other research corpi, suggests the Vygotskian position that, at least in some domains and some ways, learning and teaching tasks can change the course of development. (p. 84)

Finally, Clements & Sarama make two more recommendations regarding the creation of learning trajectories. First, be sure to think carefully about how a trajectory might work for an individual student (following a more cognitive theoretical approach) and also how it might work for a class, complete with student interactions and classroom discourse (a more sociocultural theoretical approach). Second, recognize that these trajectories are always hypothetical and will work best when teachers take the time to create and re-create them to work best with their students.

References

Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education.

*Mathematical Thinking and Learning*,*6*(2), 81–89. doi:10.1207/s15327833mtl0602_1Piaget, J. & Szeminska, A. (1952).

*The child's conception of number*. London, UK: Routledge and Kegan Paul.Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective.

*Journal for Research in Mathematics Education*,*26*(2), 114–145. doi:10.2307/749205Vygotsky, L. S. (1934/1986).

*Thought and language*. Cambridge, MA: MIT Press.
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