Application of the Triangular Fuzzy Model to Assessment of Analogical Reasoning Skills

Michael Gr. Voskoglou^1, and Igor Ya. Subbotin²

¹School of Technological Applications, Graduate Technological Educational Institute (T. E. I.) of Western Greece, Patras, Greece

²College of Letters and Sciences, National University of Los Angeles, California, USA

Cite this paper

Michael Gr. Voskoglou and Igor Ya. Subbotin. Application of the Triangular Fuzzy Model to Assessment of Analogical Reasoning Skills. American Journal of Applied Mathematics and Statistics. 2015; 3(1):1-6. doi: 10.12691/AJAMS-3-1-1

Abstract

We apply the Triangular Fuzzy Assessment Model (TFAM) for analogical reasoning skills assessment. The TFAM is a new original model based on the Centre of Gravity (COG) defuzzification technique, which we have properly adapted and used in earlier papers as an assessment method of several human activities. The main idea of the TFAM is the replacement of the rectangles appearing in the graph of the COG technique by isosceles triangles sharing common parts. In this way, we can treat better the ambiguous cases of scores being at the boundaries between two successive linguistic characterizations (grades) of the level of the individuals’ performance. An application is also presented illustrating our results in practice, where the TFAM is compared with two traditional assessment methods (calculation of the means and of the GPA index) based on principles of the bivalent logic (yes-no).

Keywords

Analogical Reasoning, Fuzzy Logic, Centre of Gravity (COG) defuzzification technique, Triangular Fuzzy Assessment model (TFAM), GPA index

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Gentner, D., Toupin, C., “Systematicity and surface similarity in development of analogy”, Cognitive Science, 10, 277-300, 1986.
[2]	Gentner, D., Markman, A. B., “Structure mapping in analogy and similarity”, American Psychologist 52, 45-56, 1997.
[3]	Gick, M. L., Holyoak, K. J., “Analogical problem solving”, Cognitive Psychology, 12, 306-355, 1980.
[4]	Gick, M. L., Holyoak, K. J., “Schema induction and analogical transfer”, Cognitive Psychology, 15, 1-38, 1983.
[5]	“Grade Point Average Assessment”, available in the Web at: http://www.swinburne.edu.au/student-administration/assessment/gpa.html, visited October 15, 2014.
[6]	Holyoak K. J., “The pragmatics of analogical transfer”. In G. H. Bower (Ed.), The psychology of learning and motivation, Vol. 19, 59-87, New York: Academic Press., 1985
[7]	Klir, G. J. & Folger, T. A.., Fuzzy Sets, Uncertainty and Information, Prentice-Hall, London, 1988.
[8]	Novick, L. R., “Analogical transfer, problem similarity and expertise”, Journal of Educational Psychology: Learning, Memory and Cognition, 14, 510-520, 1988.
[9]	Subbotin, I. Ya., Badkoobehi, H., Bilotckii, N. N., “Application of fuzzy logic to learning assessment”, Didactics of Mathematics: Problems and Investigations, 22, 38-41, 2004.
[10]	Subbotin, I. Ya., Voskoglou, M. Gr., “Applications of fuzzy logic to Case-Based Reasoning”, International Journal of Applications of Fuzzy Sets and Artificial Intelligence, 1, 7-18, 2011.
[11]	Subbotin, I., Mossovar-Rahmani, F., Bilotskii, N., “Fuzzy logic and the concept of the Zone of Proximate Development”, Contemporary trends in the development of mathematics and its application aspects (Proceedings of the 1st International Scientific Internet-Conference), pp. 301-302, Donetsk, 2012.
[12]	Subbotin, I.Ya., Bilotskii, N. N., “Triangular fuzzy logic model for learning assessment”, Didactics of Mathematics: Problems and Investigations, 41, 84-88, 2014.
[13]	Subbotin, I. Ya., Voskoglou, M. Gr., “Language, mathematics and critical thinking: The cross influence and cross enrichment”, Didactics of Mathematics: Problems and Investigations, 41 89-94, 2014.
[14]	Subbotin, I. Ya., Voskoglou M. Gr., “A Triangular Fuzzy Model for Assessing Students’ Critical Thinking Skills”, International Journal of Applications of Fuzzy Sets and Artificial Intelligence, 4, 173-186, 2014.
[15]	Voskoglou, M. Gr., “The process of learning mathematics: A fuzzy set approach”, Heuristic and Didactics of Exact Sciences,, 10, 9-13, 1999.
[16]	Voskoglou, M. Gr., “Analogical Problem Solving and Transfer”, Proceedings of the 3d Mediterranean Conference on Mathematical Education, 295-303, Athens, 2003.
[17]	Voskoglou, M. Gr., “Transition across levels in the process of learning: A fuzzy model”, International Journal of Modelling and Application, Univ. Blumenau, Brazil, 1, 37-44, 2009.
[18]	Voskoglou, M. Gr., Stochastic and fuzzy models in Mathematics Education, Artificial Intelligence and Management, Lambert Academic Publishing, Saarbrucken, Germany, 2011.
[19]	Voskoglou, M. Gr., “Case-Based Reasoning: History, Methodology and Development Trends”, International Journal of Psychology Research, 6 (5), 505-530, Nova Publishers, N. Y., 2011.
[20]	Voskoglou, M. Gr, “Fuzzy measures for students’ mathematical modelling skills”, International Journal of Fuzzy Logic Systems, 2 (2), 13-26, 2012.
[21]	Voskoglou, M. Gr., Subbotin, I. Ya., “Fuzzy models for analogical reasoning”, International Journal of Applications of Fuzzy Sets and Artificial Intelligence, 2, 19-38, 2012.
[22]	Voskoglou, M. Gr., “Assessing the Players’ Performance in the Game of Bridge: A fuzzy Logic Approach”, American Journal of Applied Mathematics and Statistics, 2 (3), 115-120, 2014.
[23]	Voskoglou, M. Gr., Salem, A.-B. M., “Analogy-Based and Case-Based Reasoning: Two Sides of the Same Coin”, International Journal of Applications of Fuzzy Sets and Artificial Intelligence, 4, 5-51, 2014.
[24]	Wikipedia, “Center of mass: A system of particles”, available in the Web at: http://en.wikipedia.org/wiki/Center_of_mass#A_system_of_particles, visited on October 10, 2014.