HIERARCHICAL AND NON-HIERARCHICAL CLUSTERING METHODS TO ANALYSE AN OPEN-ENDED QUESTIONNAIRE ON ALGEBRAIC THINKING

In recent years, some papers have tried to develop detailed models of the reasoning competences of the student populations tested, or to subdivide a sample of students into intellectually similar subgroups, by using quantitative or qualitative analysis methods. It is worth noting that research papers using quantitative analysis methods to study student responses to open-ended questionnaire can be found in Science and Physics education (Springuel et al., 2007), but the same cannot be said for research in Mathematics education. In this paper we focus on the application of hierarchical and non-hierarchical clustering methods referred to dendrograms and k-means approaches (Everitt, et al., 2011), trying to make sense to answers given by 118 Tenth Grade Italian students to six open-ended questions on algebraic thinking. In particular we discuss the results on the study of typical students behaviour in tackling the algebraic resolution of word problems and, at the same time, at understanding how the student semantically and syntactically control questions containing symbolic algebraic expressions (Radford & Puig, 2007). The two methods (K-means and dendrograms) both allowed us to partition and characterize our student sample, without making any a priori assumptions and giving as output student’s behaviour interesting for the researcher in Education. The first method identified 3 groups of students, the second one 5. The results we found are largely coherent with the ones already reported in the literature obtained by means of qualitative methods. For this reason, we can consider the use of both hierarchical and non-hierarchical clustering a valid tool to complement the use of qualitative analysis to study a large number of students with respect to the way they give answer to the questionnaire.