Applying clustering to statistical analysis of student reasoning about two-dimensional kinematics

We use clustering, an analysis method not presently common to the physics education research community, to group and characterize student responses to written questions about two-dimensional kinematics. Previously, clustering has been used to analyze multiple-choice data; we analyze free-response data that includes both sketches of vectors and written elements. The primary goal of this paper is to describe the methodology itself; we include a brief overview of relevant results.

Figure 1

A typical 2D kinematics question. In a horizontal plane, a dog starts at point $A$, speeds up until a point between $B$ and $C$ (not indicated on this diagram, but explicit on some question versions), and runs at constant speed past points $C$ and $D$. Students must draw a vector in the direction of the velocity at all points $A–D$. On a separate, identical diagram, they are asked to draw vectors in the direction of the acceleration. Based on similar questions asked at the University of Washington.

Figure 2

A 2D example using an arbitrary coordinate system of the pairwise distances that would be averaged to find the distance between two clusters. In this example, the three points in the lower left are one cluster and the three in the upper right are the other. The distance between the clusters is the average length of the lines.

Figure 3

Cluster tree of student results.

Figure 4

The result of removing all clusters with fewer than ten students from the cluster tree. In this diagram, the number at each intersection is the number of students in the cluster created by that intersection. The roman numerals correspond to the groups in Table .