Hierarchical and Non-Hierarchical clustering methods to study students algebraic thinking in solving an open-ended questionnaire

Abstract

The problem of taking a data set and separating it into subgroups, where the members of each subgroup are more similar to each other than they are to members outside the subgroup, has been extensively studied in science and mathematics education research. Student responses to written questions and multiple-choice tests have been characterised and studied using several qualitative and/or quantitative analysis methods. However, there are inherent difficulties in the categorisation of student responses in the case of open-ended questionnaires. Very often, researcher bias means that the categories picked out tend to find the groups of students that the researcher is seeking out. In our contribution, we discuss an example of application of hierarchical and non-hierarchical analysis method, to interpret the answers given by 118 Tenth Grade students in Palermo (Italy), to six open-ended questions about algebraic thinking. We show that the parallel use of the two quantitative analyses allows us to interpret in deep way the reasoning of students solving different mathematical problems using Algebra. These clustering methods also allow us to highlight different students groups, that can be recognized and characterized by common traits in their answers, without any prior knowledge on the part of the researcher.