Fuzzy relations, Possibility theory, Measures of uncertainty, Mathematical modeling.

Document Type: Research Paper

Author

Graduate Technological Educational Institute (T.E.I.), School of Technological Applications, 263 34 Patras, Greece

Abstract

A central aim of educational research in the area of mathematical modeling and applications is to recognize the attainment level of students at defined states of the modeling process. In this paper, we introduce principles of fuzzy sets theory and possibility theory to describe the process of mathematical modeling in the classroom. The main stages of the modeling process are represented as fuzzy sets in a set of linguistic labels indicating the degree of a student's success in each of these stages. We use the total possibilistic uncertainty on the ordered possibility distribution of all student profiles as a measure of the students' modeling capacities and illustrate our results by application to a classroom experiment.

Keywords


bibitem{BaKo:2002}
M. Ajello and F. Spagnolo, {it Some experimental observations on common sense and fuzzy logic},  Proceedings of International Conference on Mathematics Education into the 21st Century, Napoli, (2002), 35-39.

%[2]
bibitem{Bo:2007}
R. Borroneo Ferri, {it Modeling problems from a cognitive perspective},  In: C. R. Haines, P. Galbraith, W. Bloom, S. Khan, eds., Mathematical Modeling: Education, Engineering and Economics, (ICTMA 12), Horwood Publishing, Chichester, (2007), 260-270.

%[3]
bibitem{Do:2007}
H. M. Doer, {it What knowledge do teachers need for teaching mathematics through applications and modeling?}, In: W. Blum, P. L. Galbraith, H. W. Henn and M. Niss, eds., Modeling and Applications in Mathematics Education, Springer, NY, (2007), 69-78.

%[4]
bibitem{EsOli:1997}
E. A. Espin and C. M. L. Oliveras, {it Introduction to the use of fuzzy logic in the assessment of mathematics teachers'}, In: A. Gagatsis, ed., Proceedings of the 1$^{st}$ Mediterranean Conference on Mathematics Education, Nicosia, Cyprus, (1997), 107-113.

%[5]
bibitem{GalSti:2001}
P. L. Galbraith and G. Stillman, {it Assumptions and context: pursuing their role in modeling activity}, In: J. F. Matos, W. Blum, K. Houston and S. P. Carreira, eds., Modeling and Mathematics Education: Applications in Science and Technology (ICTMA 9), Chichester, (2001), 300-310.


%[6]
bibitem{HaCr:2010}
 C. R. Haines and R. Crouch, {it Remarks on modeling cycle and interpretation of behaviours}, In: R. A. Lesh, P. L. Galbraith, C. R. Haines and A. Harford, eds., Modeling Students' Mathematical Modeling Competencies, (ICTMA13), London, (2010), 145-154.


%[7]
bibitem{HuShi:2009}
 H. L. Huang and G. Shi, {it Robust H1 control for T-S time-varying delay systems with norm bounded uncertainty based on LMI approach}, Iranian Journal of Fuzzy Systems, {bf 6}textbf{(1)} (2009), 1-14.

%[8]
bibitem{IlSpa:2010}
L. Iliadis and S. Spartalis, {it An intelligent information system for fuzzy additive modeling (hydrological risk application)}, Iranian Journal of Fuzzy Systems, {bf 7}textbf{(1)} (2010), 1-14.

%[9]
bibitem{JaMaMa:2009}
N. Javiadian, Y. Maali and  N. Mahadavi-Amiri, {it Fuzzy linear programming with grades of satisfaction  in constraints}, Iranian Journal of Fuzzy Systems, {bf 6}textbf{(3)} (2009), 17-35.

%[10]
bibitem{KaKa:2008}
I. Kaya and J. Kahraman, {it Fuzzy process capability analyses: an application to teaching processes}, Journal of Intelligent and Fuzzy Systems, {bf 19}textbf{(4-5)} (2008), 259-272.

%[11]
bibitem{KeKa:2008}
C. Kezi Selva Vijila and P. Kanagasalarathy, {it Intelligent technique of cancelling maternal ECG in FECG extraction}, Iranian Journal of Fuzzy Systems, {bf 5}textbf{(1)} (2008), 27-45.

%[12]
bibitem{KliFo:1988}
 G. J. Klir and T. A. Folger, {it Fuzzy sets, uncertainty and information}, Prentice Hall, London, 1988.

%[13]
bibitem{KliWi:1998}
G. J. Klir and M. J. Wierman, {it Uncertainty-based information: elements of generalized information theory}, Physika-Verlag, Heidelberg, 1998.

%[14]
bibitem{MasPAr:2005}
M. Mashinchi, A. Parchami and H. R. Maleki, {it Application of fuzzy capability indices in educational comparison}, Proceedings 3d Int. Research Convention 2005, {bf 42} (2005).

%[15]
bibitem{Me:2009}
 S. Meier, {it Identifying modeling tasks}, In: L. Paditz  and A. Rogerson, eds., Models in Developing Mathematics Education (MEC21), Dresden, University of Applied Sciences, (2009), 399-403.

%[16]
bibitem{NaPa:2010}
P. K. Nayak and M. Pal, {it Bi-matrix games with intuitionistic fuzzy goals}, In: L. Paditz  and A. Rogerson, eds., Iranian Journal of Fuzzy Systems, {bf 7}textbf{(1)} (2010), 65-79.

%[17]
bibitem{Perd:2004}
S. Perdikaris, {it The problem of transition across levels in the van Hiele theory of geometric reasoning}, Philosophy of Mathematics Education Journal, {bf 18} (2004).

%[18]
bibitem{Po:1979}
 H. O. Pollak, {it New trends in mathematics teaching}, Unesko, Paris, {bf IV} (1979).

%[19]
bibitem{Sha:1961}
G. L. S. Shackle, {it Decision, order and time in human affairs, Cambridge}, Cambridge University Press, 1961.

%[20]
bibitem{SuBaBi:2006}
I. Y. Subbotin, H. Badkoodehi and N. N. Bilotskii, {it Fuzzy logic and iterative assessment}, Didactics of Mathematics: Problems and Investigations (Ukraine), {bf 25} (2006), 221-227.

%[21]
bibitem{VoPer:1193}
M. G. Voskoglou and S. Perdikaris, {it Measuring problem-solving skills}, International Journal of Mathematical Education in Science and Technology, {bf 24}textbf{(3)} (1993), 443-447.

%[22]
bibitem{Vo:1995}
M. G. Voskoglou, {it Measuring mathematical model building abilities}, International Journal of Mathematical Education in Science and Technology, {bf 26}textbf{(1)} (1995), 29-35.

%[23]
bibitem{Vo:1996}
M. G. Voskoglou, {it An application of ergodic Markov chains to analogical problem solving}, The Mathematics Education (India), {bf XXX}textbf{(2)} (1996), 95-108.

%[24]
bibitem{Vo:1999}
M. G. Voskoglou, {it The process of learning mathematics: a fuzzy set approach}, Heuristics and Didactics of Exact Sciences (Ukraine), {bf 10} (1999), 9-13.

%[25]
bibitem{Vo:2000}
M. G. Voskoglou, {it An application of Markov chains to decision making}, Studia Kupieckie (University of Lodz, Poland), {bf 6} (2000), 69-76.

%[26]
bibitem{Vo:2006}
M. G. Voskoglou, {it The use of mathematical modeling as a tool for learning mathematics}, Quaderni di Ricerca in Didattica (Scienze Mathematiche), {bf 16} (2006), 53-60.

%[27]
bibitem{Vo:2007}
M. G. Voskoglou, {it A stochastic model for the modeling process}, In: C. R. Haines, P. Galbraith, W. Bloom, S. Khan, eds., Mathematical Modeling: Education, Engineering and Economics, (2007), 149-157.

%[28]
bibitem{Vo:2009}
M. G. Voskoglou, {it Fuzziness or probability in the process of learning? a general question illustrated by examples from teaching mathematics}, The Journal of Fuzzy Mathematics, International Fuzzy Mathematics Institute (Los Angeles), {bf 17}textbf{(3)} (2009), 697-686.

%[29]
bibitem{Vo:2009a}
M. G. Voskoglou, {it Fuzzy sets in case-based reasoning}, In: Y. Chen, H. Deng, D. Zhang and Y. Xiao, eds., Fuzzy Systems and Knowledge Discovery (FSKD 2009), {bf 6} (2009), 252-256.

%[30]
bibitem{Vo:2009b}
M. G. Voskoglou, {it A stochastic model for the process of learning}, In: L. Paditz  and A. Rogerson, eds., Models in Developing Mathematics Education (MEC21) Dresden, (2009), 565-569.

%[31]
bibitem{Vo:2010}
M. G. Voskoglou, {it Mathematizing the case-based reasoning process}, In: A. M. Leeland, ed., Case-Based Reasoning: Processes, Suitability and Applications, in press, {bf 6} (2010).