Multidimensional Scaling in Cluster Analysis: examples in Science and Mathematics Education

Abstract

Several researches in STEM education research highlight the advantages of an inte- grated approach to these disciplines that relates knowledge and know-how, design and implementation, theoretical and practical problems [5, 4, 6]. In some researches, the effectiveness of these approaches on students conceptual understanding and motivation and has been studied through the use of quantitative analysis tools such as cluster analysis (CLA) [1, 7]. Through CLA it is possible to characterize students analyzing the strategies they deploy to tackle, for example, questionnaires built so as to investigate the lines of reasoning implemented by them when they are proposed with problematic situations. In particular, it is possible to characterize the students in terms of a limited number, m, of typical ways of answering the questionnaire questions [2]. Each student is therefore identified by a binary vector (each component can be 1 or 0) with m dimensions. Cluster Analysis techniques allow students to be grouped into homogeneous groups based on com- mon characteristics and the representation of these groups is ideally referred to an m-sized space. However, for reasons of simplicity and clarity, it is often preferred to perform the representation of groups in three or two dimensional spaces. One of the techniques used for this purpose is Multidimensional Scaling [3] (MDS). It allows the researcher to move from the m-sized space to a space with a smaller number of dimensions that is a function of the initial m-dimensional representation, preserving the global distances between the group elements. However, the application of the MDS methodologies depends strongly on the typology of the initial data and a non-thorough knowledge of the mathematical details at their base can lead to obtaining results that are not reliable and / or of little significance. In this paper we will study a MDS methodology based on Principal Component Analysis, with particular reference to a set of binary data, highlighting how the results obtained through this methodology can be reliable and significant for the researcher in education.