American Journal of Educational Research
ISSN (Print): 2327-6126 ISSN (Online): 2327-6150 Website: http://www.sciepub.com/journal/education Editor-in-chief: Ratko Pavlović
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American Journal of Educational Research. 2015, 3(3), 330-339
DOI: 10.12691/education-3-3-12
Open AccessArticle

Fuzzy Logic in the APOS/ACE Instructional Treatment for Mathematics

Michael Gr. Voskoglou1,

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

Pub. Date: March 04, 2015

Cite this paper:
Michael Gr. Voskoglou. Fuzzy Logic in the APOS/ACE Instructional Treatment for Mathematics. American Journal of Educational Research. 2015; 3(3):330-339. doi: 10.12691/education-3-3-12

Abstract

In this paper principles of fuzzy logic are introduced for comparing the performance of two student groups concerning the comprehension of real numbers in general and of irrational numbers in particular. The first group was taught the subject in the traditional way (control group), while the APOS/ACE instructional treatment was applied for the second group (experimental group). The two groups are represented as fuzzy subsets of the set of the grades (from A to F) achieved by the students in a pre-instructional and a post-instructional test and the centroid defuzzification technique is applied on comparing their performances. The results of our classroom experiments show that the application of the APOS/ACE approach can effectively help students to enlist the real numbers in a powerful cognitive schema including all the basic sets of numbers.

Keywords:
Fuzzy sets centroid defuzzification technique teaching and learning the real numbers APOS/ACE theory

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References:

[1]  Asiala, M., et al. (1996), A framework for research and curriculum development in undergraduate mathematics education, Research in Collegiate Mathematics Education II, CBMS Issues in Mathematics Education, 6, 1-32.
 
[2]  Dubinsky, E. & McDonald, M. A. (2001), APOS: A constructivist theory of learning in undergraduate mathematics education research. In: D. Holton et al. (Eds), The Teaching and learning of Mathematics at University Level: An ICMI Study, 273-280, Kluwer Academic Publishers, Dordrecht, Netherlands.
 
[3]  Espin, E. A. & Oliveras, C. M. L. (1997), Introduction to the Use of Fuzzy Logic in the Assessment of Mathematics Teachers’ Professional Knowledge, Proceedings of the First Mediterranean Conference on Mathematics, Cyprus, pp. 107-113.
 
[4]  Klir, G. J. & Folger, T. A. (1988), Fuzzy Sets, Uncertainty and Information, Prentice-Hall, London
 
[5]  Maharaj, A. (2013), An APOS analysis of natural science students’ understanding of derivatives, South African Journal of Education, 33(1), 19 pages.
 
[6]  Perdikaris, S. (2002), Measuring the student group capacity for obtaining geometric information in the van Hiele development thought process: A fuzzy approach, Fuzzy Sets and Mathematics, 16 (3), 81-86.
 
[7]  Piaget, J. (1970), Genetic Epistemology, Columbia University Press, New York and London.
 
[8]  Subbotin, I. Ya. Badkoobehi, H., Bilotckii, N. N. (2004), Application of fuzzy logic to learning assessment. Didactics of Mathematics: Problems and Investigations, 22, 38-41.
 
[9]  Voskoglou, M. Gr. (1999), The process of learning mathematics: A fuzzy set approach, Heuristic and Didactics of Exact Sciences (Ukraine), 10, 9-13,
 
[10]  Voskoglou, M. Gr. (2009), The mathematics teacher in the modern society, Quaderni di Ricerca in Didattica (Scienze Mathematiche), University of Palermo, 19, 24-30.
 
[11]  Voskoglou, M. Gr. (2009), Fuzzy Sets in Case-Based Reasoning, Fuzzy Systems and Knowledge Discovery, Vol. 6, 252-256, IEEE Computer Society.
 
[12]  Voskoglou, M. Gr. (2009.), Stochastic and fuzzy models in Mathematics Education, Artificial Intelligence and Management, Lambert Academic Publishing, Saarbrucken, Germany, 2011; for more details look at )
 
[13]  Voskoglou, M. Gr. (2011), Fuzzy Logic and Uncertainty in Mathematics Education, International Journal of Applications of Fuzzy Sets and Artificial Intelligence, 1, 45-64.
 
[14]  Voskoglou, M. Gr. & Subbotin, I. Ya. (2012), Fuzzy Models for Analogical Reasoning, International Journal of Applications of Fuzzy Sets and Artificial Intelligence, 2, 19-38
 
[15]  Voskoglou, M. Gr. (2012), A study on fuzzy systems, American Journal of Computational and Applied Mathematics, 2(5), 232-240.
 
[16]  Voskoglou, M. Gr. (2012) A fuzzy model for human reasoning, International Journal of Mathematics and Engineering with Computers, 3(2), 61-71.
 
[17]  Voskoglou, M. Gr. (2013), Problem solving, fuzzy logic and computational thinking, Egyptian Computer Science Journal, 37(1), 131-145.
 
[18]  Voskoglou, M. Gr. (2013), An application of the APOS/ACE approach in teaching the irrational numbers, Journal of Mathematical Sciences and Mathematics Education, 8(1), 30-47
 
[19]  Weller, K. et al. (2003), Students performance and attitudes in courses based on APOS theory and the ACE teaching cycle. In: A. Selden et al. (Eds.), Research in collegiate mathematics education V (pp. 97-181), Providence, RI: American Mathematical Society.
 
[20]  Weller, K., Arnon, I & Dubinski, E. (2009), Pre-service Teachers’ Understanding of the Relation between a Fraction or Integer and its Decimal Expansion, Canadian Journal of Science, Mathematics and Technology Education, 9(1), 5-28.
 
[21]  Weller, K., Arnon, I & Dubinski, E. (2011), Preservice Teachers’ Understanding of the Relation Between a Fraction or Integer and Its Decimal Expansion: Strength and Stability of Belief, Canadian Journal of Science, Mathematics and Technology Education, 11(2), 129-159.
 
[22]  Zadeh, L. A., Fuzzy Sets, Information and Control, 8, 338-353, 1965.
 
[23]  Zazkis, R. & Sirotic, N. (2010), Representing and Defining Irrational Numbers: Exposing the Missing Link, CBMS Issues in Mathematics Education, 16, 1-27.