(Page 1 of 8) - Experiential models of students' geometry thinking: Case studies of two secondary mathematics teachers authored by Rhodes, Ginger.

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LEARNING TRAJECTORIES: CASE STUDIES OF TWO SECONDARY TEACHERS

Ginger Rhodes

University of Georgia

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Teachers acknowledge that the ways that students think mathematically influences their
instructional practices. Through this study I considered how learning trajectories might provide
insights into the ways that teachers understand and make use of students’ mathematics in
classrooms. Specifically, I examined the ways that two teachers intended student learning to
progress in lessons and their reflections of how learning actually progressed. Results indicated
that the projected learning trajectory doesn’t always coincide with enacted learning trajectory.
One reason for these differences was whether the students or the teacher led the direction of the
learning trajectories during enactment.

There is a growing body of literature in mathematics education that suggests teachers make

instructional changes when they focus on student thinking (Carpenter, Fennema, Peterson, Chiang
& Loef, 1989; Lubinski & Jaberg, 1997). In addition, research also indicates that instructional
practices that supports and builds on students’ thinking will promote students’ mathematical
understanding (Fennema et al., 1993; Hiebert & Wearne, 1993). Teachers must have an
understanding of students’ mathematical thinking in order to create learning opportunities that
builds on and supports that thinking. Learning trajectories provides a framework for considering
the complex ways that teachers make sense of and use their students’ mathematical thinking. This
research paper will report on a study that examined the learning trajectories that teachers create of
students’ mathematical thinking through their teaching practice.

Theoretical Framework

In order to support and build on students’ mathematical thinking one must consider the

relationship between the learning activity and learning process. In other words, when planning
lessons one must consider how a student might engage in a mathematics task and what learning
might happen because of that engagement. In recent years, mathematics educators have noted this
importance of instructional planning when the goal is to build on and support students’ current
mathematical thinking (Gravemeijer, 2004). Learning trajectories are models that represent
children’s starting points, the changes that occur due to the mathematical activity, and the
interactions that were involved in those changes. In Simon’s (1995) discussion about learning
trajectories he noted that the “path by which learning might proceed” is hypothetical because “the
actual learning trajectory is not knowable in advance. It characterizes an expected tendency” (p.
135). Simon noted that the hypothetical learning trajectory (HLT) includes three components: the
intended direction, the learning activities, and the hypothetical learning process. Teachers can
refine and modify existing learning trajectories through interactions with children. Steffe (2004)
notes, “the construction of learning trajectories of children is one of the most daunting but urgent
problems facing mathematics education today” (p. 130). He also mentions that children and a
teacher/researcher should co-produce these learning trajectories.

Lamberg, T., & Wiest, L. R. (Eds.). (2007). Proceedings of the 29

annual meeting of the North

American Chapter of the International Group for the Psychology of Mathematics Education,
Stateline (Lake Tahoe), NV: University of Nevada, Reno.