Graph theoretic methods for the analysis of data in developing systems

A full examination of learning or developing systems requires data analysis approaches beyond the commonplace pre-/post-testing. Drawing on graph theory, three particular approaches to the analysis of data—based on adjacency matrices, affiliation networks, and edit distances—can provide additional insight into data; these methods are applied to student performance in a Calculus course. Data analysis methods based on adjacency matrices demonstrate that learning is not unidimensional, that learning progressions do not always progress monotonically toward desired understandings and also provide insight into the connection between instruction and student learning. The use of affiliation networks supports the concept development theory of Lev Vygotsky and also provides insight into how students’ prior knowledge relates to topics being studied. Careful use of edit distances indicates a likely overestimate of effect sizes in many studies, and also provides evidence that concepts are often created in an ad hoc manner. All of these have implications for curriculum and instruction, and indicate some directions for further inquiry.