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Experiential models of students' geometry thinking: Case studies of two secondary mathematics teachers
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occurred in students’ mathematical understandings during the lesson. The learning trajectory began with students’ starting points, or their current mathematical understandings. Through some mathematical activity, students progressed through the learning process to conclude at some ending point. There are large learning trajectories, which span multiple concepts, and there are smaller learning trajectories, which cover individual concepts (e.g., a learning trajectory for area of polygons or a learning trajectory for area of regular polygons). In this study, I was interested in the learning trajectory that teachers described for their students during individual lessons, which spanned one to three days and covered anywhere from one to five learning goals. I also want to mention that the learning trajectory can represent learning for one student, a subgroup of students, or a class of students. Figure 1 represents what a learning trajectory might look like. I’ve chosen to create a simple figure to oversimplify learning trajectories in order to later describe and highlight results. I want to emphasize that learning is not necessarily continuous nor does it fit the curve as shown in the figure.
Figure 1. A Learning Trajectory
Let’s consider an example from the data. During working session three the teachers chose a
task that fit both of their learning goals. The teachers modified the task appropriately for their class and created lessons that incorporated the task. Figure 2 is the task and the diagram that went along with the task. Working session four was spent with the two teachers describing what happened during their classes and analyzing student work.
Lamberg, T., & Wiest, L. R. (Eds.). (2007). Proceedings of the 29
th
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.
Students’ beginning mathematical understandings
Students’ ending mathematical understandings
Mathematical activity and mathematical learning
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120
occurred in students’ mathematical understandings during the lesson. The learning trajectory began
with students’ starting points, or their current mathematical understandings. Through some
mathematical activity, students progressed through the learning process to conclude at some
ending point. There are large learning trajectories, which span multiple concepts, and there are
smaller learning trajectories, which cover individual concepts (e.g., a learning trajectory for area of
polygons or a learning trajectory for area of regular polygons). In this study, I was interested in the
learning trajectory that teachers described for their students during individual lessons, which
spanned one to three days and covered anywhere from one to five learning goals. I also want to
mention that the learning trajectory can represent learning for one student, a subgroup of students,
or a class of students. Figure 1 represents what a learning trajectory might look like. I’ve chosen to
create a simple figure to oversimplify learning trajectories in order to later describe and highlight
results. I want to emphasize that learning is not necessarily continuous nor does it fit the curve as
shown in the figure.
Figure 1. A Learning Trajectory
Let’s consider an example from the data. During working session three the teachers chose a
task that fit both of their learning goals. The teachers modified the task appropriately for their class
and created lessons that incorporated the task. Figure 2 is the task and the diagram that went along
with the task. Working session four was spent with the two teachers describing what happened
during their classes and analyzing student work.
Lamberg, T., & Wiest, L. R. (Eds.). (2007). Proceedings of the 29
th
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.
Students’
beginning
mathematical
understandings
Students’ ending mathematical
understandings
Mathematical activity
and mathematical
learning
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