Non-Hierarchical Clustering as a Method to Analyse an Open-Ended Questionnaire on Algebraic Thinking

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 this paper, we discuss an example of application of a quantitative, 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 use of non-hierarchical analysis allows us to interpret the reasoning of students solving different mathematical problems using Algebra, and to separate them into different groups, that can be recognised and characterised by common traits in their answers, without any prior knowledge on the part of the researcher of what form those groups would take (unbiased classification).