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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Notes

  1. 1.

    FCI test can be downloaded at following websites: https://www.physport.org/assessments/assessemnt.cfm?A=FCI, http:// modelling.asu.edu/R&E/Research.html

  2. 2.

    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).

  3. 3.

    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.

References

  1. Bao, L., Hogg, K., & Zollman, D. (2002). Model analysis of fine structures of students models: An example with Newton’ s third law. American Journal of Physics, 70, 766–778.

    Article  Google Scholar 

  2. Bao, L., & Redish, E. F. (2006). Model analysis: Representing and assessing the dynamics of student learning. Physical Review Special Topics Physics Education Research, 2, 010103.

    Article  Google Scholar 

  3. Battaglia, O. R., Di Paola, B., & Fazio, C. (2018). An unsupervised quantitative method to analyse students’ answering strategies to a questionnaire. In S. Magazu (Ed.), New trends in physics education research (pp. 19–46). New York, NY: Nova Science Publishers Inc.

  4. Borg, & Groenen, P. (1997). Modern multidimensional scaling. New York, NY: Springer Verlag.

    Google Scholar 

  5. Brewe, E., Bruun, J., & Bearden, I. G. (2016). Using module analysis for multiple choice responses: A new method applied to Force Concept Inventory data. Physical Review Physics Education Research, 12, 020131.

    Article  Google Scholar 

  6. Brookes, D. T., & Etkina, E. (2009). “Force”, ontology, and language. Physical Review Special Topics Physics Education Research, 5, 010110.

    Article  Google Scholar 

  7. Calinski, T., & Harabasz, J. (1974). A dendrite method for cluster analysis. Communications in Statistics, 3(1), 1–27.

    Google Scholar 

  8. DiCiccio, T. J., & Efron, B. (1996). Bootstrap confidence intervals. Statistical Science, 11(3), 189–228.

  9. Ding, L., & Beichner, R. (2009). Approaches to data analysis of multiple-choice questions. Physical Review Special Topics Physics Education Research, 5, 020103.

    Article  Google Scholar 

  10. Ding, L., & Caballero, M. D. (2014). Uncovering the hidden meaning of cross-curriculum comparison results on the Force Concept Inventory. Physical Review Special Topics - Physics Education Research, 10(2), 020125. https://doi.org/10.1103/PhysRevSTPER.10.020125.

    Article  Google Scholar 

  11. Di Paola, B., Battaglia, O. R., & Fazio, C. (2016). Non-Hierarchical Clustering to analyse an open-ended questionnaire on algebraic thinking. South African Journal of Education, 36(1), 1142.

  12. Everitt, B. S., Landau, S., Leese, M., & Stahl, D. (2011). Cluster analysis. Chichester, England: John Wiley & Sons, Ltd.

    Google Scholar 

  13. Fazio, C., Battaglia, O. R., & Di Paola, B. (2013). Investigating the quality of mental models deployed by undergraduate engineering students in creating explanations: The case of thermally activated phenomena. Physical Review Special Topics - Physics Education Research, 9(2), 020101.

  14. Fulmer, G. W. (2015). Validating proposed learning progressions on force and motion using the Force Concept Inventory: Finding from Singapore secondary schools. International Journal of Science and Mathematics Education, 13(6), 1235–1254. https://doi.org/10.1007/s10763-014-9553-x.

    Article  Google Scholar 

  15. Gilbert, J. K., & Boulter, C. J. (1998). Learning science through models and modelling. In B. Frazer & K. Tobin (Eds.), The International handbook of science education (pp. 53–66). Dordrecht, The Netherlands: Kluwer.

    Google Scholar 

  16. Gower, J. C. (1966). Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika Trust, 53, 3–4.

    Article  Google Scholar 

  17. Grunspan, D. Z., Wiggins, B. L., & Goodreau, S. M. (2014). Understanding classrooms through social network analysis: A primer for social network analysis in education research. Cell Biology Education, 13(2), 167–179.

    Google Scholar 

  18. Hake, R. R. (1998). Interactive engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses. American Journal of Physics, 66, 64–74.

    Article  Google Scholar 

  19. Hestenes, D., & Halloun, I. (1995). Interpreting the force concept inventory: A response to March 1995 critique by Huffman and Heller. Physics Teacher, 33, 502–506.

    Article  Google Scholar 

  20. Hestenes, D., & Jackson, J. (2007). Revised Table II for the Force Concept Inventory (Unpublished). Retrieved from http://modeling.asu.edu/R&E/Research.html. Accessed March 2016.

  21. Hestenes, D., Wells, M., & Swackhammer, G. (1992). Force concept inventory. Physics Teacher, 30, 141–151.

    Article  Google Scholar 

  22. Huffman, D., & Heller, P. (1995). What does the Force Concept Inventory actually measure? Physics Teacher, 33, 138–143.

    Article  Google Scholar 

  23. Jammer, M. (1957). Concepts of force. Cambridge, England: Harvard University Press.

    Google Scholar 

  24. Lasry, N., Rosenfield, S., Dedic, H., Dahan, A., & Reshef, O. (2011). The puzzling reliability of the Force Concept Inventory. American Journal of Physics, 79, 909–912.

    Article  Google Scholar 

  25. MacQueen, J. (1967). Some methods for classification and analysis of multivariate observations. In L. M. LeCam & J. Neyman (Eds.), Proc. 5th Berkely Symp. Math. Statist. Probab. 1965/66 (Vol. I, pp. 281–297). Berkely, CA: University of California Press.

    Google Scholar 

  26. Mantegna, R. N. (1999). Hierarchical structure in financial markets. The European Physical Journal B, 11, 193–197.

    Article  Google Scholar 

  27. Morris, G. A., Harshman, N., Branum-Martin, L., Mazur, E., Mzoughi, T., & Baker, S. D. (2012). An item response curves analysis of the Force Concept Inventory. American Journal of Physics, 80, 825–831. https://doi.org/10.1119/1.4731618.

    Article  Google Scholar 

  28. Nieminen, P., Savinainen, A., & Viiri, J. (2013). Gender differences in learning of the concept of force, representational consistency, and scientific reasoning. International Journal of Science and Mathematics Education, 11, 1137–1156. https://doi.org/10.1007/s10763-012-9363-y.

    Article  Google Scholar 

  29. Planinic, M., Ivanjek, L., & Susac, A. (2010). Rasch model based analysis of the Force Concept Inventory. Physical Review Special Topics Physics Education Research, 6, 010103.

    Article  Google Scholar 

  30. Rouseeuw, P. J. (1987). Silhouttes: A graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics, 20, 53–65.

    Article  Google Scholar 

  31. Savinainen, A., & Viiri, J. (2008). The Force Concept Inventory as a measure of students conceptual coherence. International Journal of Science and Mathematics Education, 6(4), 719–740. https://doi.org/10.1007/s10763-007-9103.

    Article  Google Scholar 

  32. Saxena, P., Singh, V., & Lehri, S. (2013). Evolving efficient clustering patterns in liver patient data through data mining techniques. International Journal of Computer Applications, 66(16), 23–28.

  33. Scott, T. F., & Schumayer, D. (2017). Conceptual coherence of non-Newtonian worldviews in Force Concept Inventory data. Physical Review Physics Education Research, 13, 010126. https://doi.org/10.1103/PhysRevPhysEducRes.13.010126.

    Article  Google Scholar 

  34. Scott, T. F., Schumayer, D., & Gray, A. R. (2012). Exploratory factor analysis of a Force Concept Inventory data set. Physical Review Special Topics Physics Education Research, 8, 020105.

  35. Semak, M. R., Dietz, R. D., Pearson, R. H., & Willis, C. W. (2017). Examining evolving performance on the Force Concept Inventory using factor analysis. Physical Review Physics Education Research, 13, 019903. https://doi.org/10.1103/PhysRevPhysEducRes.13.010103.

    Article  Google Scholar 

  36. Springuel, R. P., Wittmann, M. C., & Thompson, J. R. (2007). Applying clustering to statistical analysis of student reasoning about two-dimensional kinematics. Physical Review Special Topics Physics Education Research, 3, 020107.

    Article  Google Scholar 

  37. Steif, P. S., & Hansen, M. A. (2007). New practices for administering and analyzing the results of concept inventories. Journal of Engineering Education, 96, 205–212. https://doi.org/10.1002/j.2168-9830.2007.tb00930.x.

    Article  Google Scholar 

  38. Stewart, J., Miller, M., Audo, C., & Stewart, G. (2012). Using cluster analysis to identify patterns in students’ responses to contextually different conceptual problems. Physical Review Special Topics Physics Education Research, 8, 020112.

  39. Struyf, A., Hubert, M., & Rousseeuw, P. J. (1997). Clustering in an object-oriented environment. Journal of Statistical Software, 1(4), 1–30.

    Google Scholar 

  40. Wang, J., & Bao, L. (2010). Analyzing Force Concept Inventory with item response theory. American Journal of Physics, 78, 1064–1070.

    Article  Google Scholar 

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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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Fazio, C., Battaglia, O.R. Conceptual Understanding of Newtonian Mechanics Through Cluster Analysis of FCI Student Answers. Int J of Sci and Math Educ 17, 1497–1517 (2019). https://doi.org/10.1007/s10763-018-09944-1

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Keywords

  • Assessment
  • Cluster analysis
  • Engineering freshmen
  • Force Concept Inventory