Examining the relation of correct knowledge and misconceptions using the nominal response model

This study reports an analysis of the Force Concept Inventory (FCI) using item response curves (IRC)—the fraction of students selecting each response to an item as a function of their total score. Three large samples ($N=9606$, 4360, and 1439) of calculus-based physics students were analyzed. These were drawn from three land-grant institutions with very different average FCI post-test scores. A significant number of the response curves for incorrect responses have a maximum at some intermediate value of the total score on the FCI. To analyze the incorrect responses (rather than just the correct response as is typically done), we used Bock’s nominal response model (NRM), primarily because it can fit these intermediate maximum curves. The one-dimensional NRM latent ability dimension was strongly correlated ($r=0.99$) with the latent ability determined by two-parameter logistic (2PL) item response theory applied with correct or incorrect grading even though the correct responses were not identified for the NRM. To understand the relation of Newtonian and non-Newtonian thinking, higher dimensional models were fit. The two-dimensional NRM model produced one ability dimension which could be rotated to be highly correlated with the 2PL latent ability ($r\ge 0.98$) and a second dimension which was most strongly related to misconceptions involving Newton’s third law. Cluster analysis was applied to the two latent ability dimensions producing a three-cluster solution where the cluster centroids were very similar for each of the three institutions. The clusters represented three groups of students: Newtonian thinkers still retaining some misconceptions, non-Newtonian thinkers strongly applying Newton’s third law misconceptions, and non-Newtonian thinkers weakly applying those misconceptions. The differences between overall FCI scores at the three institutions could be explained by the relative populations of the three clusters.

Figure 1

Item response curves for all FCI items in the aggregated dataset plotted as individual symbols. For incorrect responses, (A) is blue (circle), (B) is orange (triangle), (C) is green (plus), (D) is red (cross), and (E) is purple (diamond). Item characteristic curves for the 1D NRM for all FCI items in the aggregated dataset are drawn as solid curves.

Figure 2

Density contours plotting ${\theta }_2$ vs ${\theta }_1$ for sample 1.