Unsupervised quantitative methods to analyze student reasoning lines: Theoretical aspects and examples

This paper is part of the Focused Collection on Quantitative Methods in PER: A Critical Examination.] A relevant aim of research in education is to find and study the reasoning lines that students deploy when dealing with problematic situations. This can be done through an analysis of the answers students give to a questionnaire. In this paper, we discuss some methodological aspects involved in the quantitative analysis of a questionnaire by means of two different clustering methods, a hierarchical one and a nonhierarchical one. We start from the coding procedures needed to obtain analyzable data from the questionnaire and from a definition of a correlation coefficient suitable for measuring student similarity in the case of binary coding of student answers. Then, criteria to choose the optimal number of clusters are discussed, and for the same purpose a new coefficient is introduced that measures the total amount of information we can obtain from a clustering solution. We show that each cluster can be characterized by its centroid that summarizes the answers most frequently given by the cluster students to the questionnaire. Finally, an example of the application of these procedures to a student sample is given, and a comparison between the two clustering methods is discussed.