Abstract
This study reports on how students can be led to make meaningful connections between such structures on a set as a partition, the set of equivalence classes determined by an equivalence relation and the fiber structure of a function on that set (i.e., the set of preimages of all sets {b} for b in the range of the function). In this paper, I first present an initial genetic decomposition, in the sense of APOS theory, for the concepts of equivalence relation and function in the context of the structures that they determine on a set. This genetic decomposition is primarily based on my own mathematical knowledge as well as on my observations of students’ learning processes. Based on this analysis, I then suggest instructional procedures that motivate the mental activities described in the genetic decomposition. I finally present empirical data from informal interviews with students at different stages of learning. My goal was to guide students to become aware of the close conceptual correspondence and connections among the aforementioned structures. One theorem that captures such connections is the following: a relation R on a set A is an equivalence relation if and only if there exists a function f defined on A such that elements related via R (and only those) have the same image under f.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asiala, M., Brown, A., De Vries, D.J., Dubinsky, E., Mathews, D. and Thomas, K.: 1996, ‘A framework for research and curricululum development in undergraduate mathematics education’, Research in Collegiate Mathematics Education 2, 1–32.
Asiala, M., Cottrill, J. and Dubinsky, E.: 1997, ‘The development of students' graphical understanding of the derivative’, Journal of Mathematical Behavior 16(4), 339–431.
Dubinsky, E.: 1989a, ‘Teaching mathematical induction II’, Journal of Mathematical Behavior 8, 285–304.
Dubinsky, E.: 1989b, ‘Teaching mathematical induction I’, Journal of Mathematical Behavior 8, 285–304.
Dubinsky, E.: 1991, ‘Reflective abstraction in advanced mathematical thinking’, in D. Tall (ed.), Advanced Mathematical Thinking, Kluwer, Dordrecht, pp. 95–126.
Dubinsky, E.: 1997, ‘On learning quantification’, Journal of Computers in Mathematics and Computer Science Teaching 16(213), 335–362.
Dubinsky, E. and Lewin, P.: 1986, ‘Reflective abstraction and mathematics education: The genetic decomposition of induction and compactness’, The Journal of Mathematical Behavior 5, 55–92.
McDonald, M.A., Mathews, D.M. and Strobel, K.H.: 2000, ‘Understanding sequences: A tale of two objects’, in E. Dubinsky, A. Schoenfeld and J. Kaput (eds.), Research in Collegiate Mathematics Education IV American Mathematical Society, Providence, pp. 77–102.
Leikin, R. and Winicky-Landman, G.: 2000, ‘On equivalent and nonequivalent definitions I and II’, For the Learning of Mathematics 20(2), 24–29.
Piaget, J.: 1977, The Development of Thought. Equilibration of Cognitive Structures (A. Rosin, Trans.) Viking, New York.
Piaget, J.: 1980, Experiments in Contradiction (Derek Coltman, Trans.) The University of Chicago Press, Chicago, London.
Piaget, J. and Garcia, R.: 1989, Psychogenesis and the History Of Science (H. Feider, Trans.), Columbia University Press, New York.
Piaget, J. Grize, J., Szeminska, A. and Bang, V.: 1977, Epistemology and Psychology of Functions D. Reidel, Dordrecht, Boston.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hamdan, M. Equivalent Structures on Sets: Equivalence Classes, Partitions and Fiber Structures of Functions. Educ Stud Math 62, 127–147 (2006). https://doi.org/10.1007/s10649-006-5798-9
Issue Date:
DOI: https://doi.org/10.1007/s10649-006-5798-9