Do fuzzy-logic non-linear models provide a benefit for the modelling of algebraic competency?

Authors

  • Reinhard Oldenburg Reinhard Oldenburg (Math education, Math department, Augsburg University, Germany)

DOI:

https://doi.org/10.24297/ijrem.v13i.9198

Keywords:

Competence model, Non-linear Model, Estimation technique, Algebraic competences

Abstract

Statistical models used in mathematics education are often linear and latent variables are often assumed to be normally distributed. The present paper argues that by relaxing these constraints one may use models that fit the data better than linear ones and provide more insight into the domain. It combines research on statistical methodology with research on the competence structure within algebra. The methodological innovation is that models with latent variables from the unit interval are considered which allows to model relations by means of fuzzy logic. Estimation techniques for such models are discussed to the extend necessary for the present study. To assess the benefit of this modelling technique data from an algebra test is re-analyzed. It is shown that non-linear models have greater explanatory power and give interesting didactical insights. Moreover, model comparison allows to differentiate between different theoretical constructs related to algebraic understanding. Finally, a research program is outlined that aims at the development of a universal algebra competence model that can be applied to test data from various algebra tests.

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References

Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24–35.

Bardini. C., Oldenburg, R., Pierce, R., & Stacey, K. (2013) Technology prompts new Understandings: The Case of Equality. Proceedings of MERGA 2013.

Dijkstra, T. K. and Schermelleh-Engel, K. (2014). Consistent partial least squares for non-linear structural equation models. Psychometrika, 79(4), 585–604. DOI: 10.1007/s11336-013-9370-0

Epp, S. (2011): Variables in mathematics education. In P. Blackburn (Ed.). Tools for teaching logic. TICTTL 2011. Berlin: Springer LNCS 6680, 54–61. DOI: 10.1007/978-3-642-21350-2_7

Hoyle, R. H. (Ed.) (2012). Handbook of Structural Equation Modeling. Guilford Press. https://doi.org/10.1080/10705511.2013.769397

Kelava, A., Werner, C. S., Schermelleh-Engel, K., Moosbrugger, H., Zapf, D., Ma, Y., Cham, H., Aiken, L. S., and West, S. G. (2011). Advanced non-linear latent variable modeling: Distribution analytic lms and qml estimators of interaction and quadratic effects. Structural Equation Modeling, 18, 465–491. https://doi.org/10.1080/10705511.2011.582408

Kieran C. (2004) The Core of Algebra: Reflections on its Main Activities. In: Stacey K., Chick H., Kendal M. (eds) The Future of the Teaching and Learning of Algebra. The 12thICMI Study. New ICMI Study Series, vol 8. Springer, Dordrecht. https://doi.org/10.1007/1-4020-8131-6_2

Küchemann, D. (1979). Children’s Understanding of Numerical Variables, Mathematics in School, 7, pp. 23–26.

Mason, J., & Sutherland, R. (2002). Key Aspects of Teaching Algebra in Schools. Qualifications and Curriculum Authority..

Oldenburg, R. (2009). Structure of Algebraic Competencies. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), CERME 6: Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education.

Oldenburg, R. (2019). A classification scheme for Variables. In U. T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis (Eds.). Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. Utrecht: Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.

Oldenburg, R. (2020). Structural Equation Modelling – Comparing Two Approaches. The Mathematica Journal.

Oldenburg, R. (2021). Case based error variance corrected estimation of structural models. Submitted to Statistical Modelling.

Umbach, N., Naumann, K., Brandt, H., and Kelava, A. (2017). Fitting non-linear structural equation models in r with package nlsem. Journal of Statistical Software, 77, 7.

Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford (Ed.), The Ideas of algebra, K-12 (pp. 8–19). Reston, VA: NCTM.

Zadeh, L. A. (1965). Fuzzy Sets, Information and Control, Vol. 8, pp. 338-353.

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Published

2022-03-15

How to Cite

Oldenburg, R. (2022). Do fuzzy-logic non-linear models provide a benefit for the modelling of algebraic competency?. INTERNATIONAL JOURNAL OF RESEARCH IN EDUCATION METHODOLOGY, 13, 1–10. https://doi.org/10.24297/ijrem.v13i.9198

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