# Conceptual Understanding of Newtonian Mechanics Through Cluster Analysis of FCI Student Answers

## Abstract

The Force Concept Inventory is a multiple-choice test and is one of the most popular and most analyzed concept inventories. It is used to investigate student understanding of Newtonian mechanics. A structured approach to data analysis can transform it in a “diagnostic” instrument that can validate inferences about student thinking. In this paper, we show how cluster analysis methods can be used to investigate patterns of student conceptual understanding and supply useful details about the relationships among student concepts and misconceptions. The answers given to the FCI questionnaire by a sample of freshman engineering have been analyzed. The analysis takes into account the decomposition of the force concept into the conceptual dimensions suggested by the FCI authors and successive studies. Our approach identifies latent structures within the student response patterns and groups students characterized by similar correct answers, as well as by non-correct answers. These response patterns give us new insights into the relationships between the student force concepts and their ability to analyze motions. Our results show that cluster analysis proved to be a useful tool to identify latent structures within the student conceptual understanding. Such structures can supply diagnostic insights for classroom pedagogy and teaching approaches.

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We included questions involving Newton’s first or second law in the same sub-questionnaire since our previous experiences, involving interviews of high school students, made evident that very often they see the first and the second Newtonian laws as undifferentiated. Moreover, the first law is often seen as a consequence of the second one in the cases where the resultant force, and consequently the acceleration, are equal to zero. Such a view can be also pointed out from the analysis of other interview studies (Brokes, & Etkina, 2009).

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The metric choice is often complex and depends on many factors. If we assume that two students, represented by arrays ai and aj and negatively correlated, have to be more dissimilar than two uncorrelated (Rbin(ai, aj) = 0), a possible definition of the distance between ai and aj, is: $$\sqrt{2\cdot \left(1-{\mathrm{R}}_{\mathrm{bin}}\right)}$$. A distance dij between two students equal to zero means that they are completely similar (Rbin = 1), while a distance dij = 2 shows that the students are completely dissimilar (Rbin = −1). When the correlation between two students is 0 their distance is $$\sqrt{2}{d}_{ij}=\sqrt{2}$$. A new NxN symmetrical matrix, the distance matrix, containing all the mutual distances between the students is then defined.

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## Acknowledgments

We wish to express our thanks to Prof. Rosa Maria Sperandeo-Mineo for her continuous advice and support during the development of this study.

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Correspondence to Onofrio R. Battaglia.

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