Experiencing equivalence but organizing order Amir H. Asghari Published online: 23 December 2008 # Springer Science + Business Media B.V. 2008 Abstract The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our own conceptions of it. This paper considers various conceptions of equivalence, in history, in mathematics today, and in mathematics education. It reveals critical differences in the notion of equivalence at different points in history and a meaning for equivalence proposed by mathematicians and mathematics educators that is at variance with the ways that learners may think. These differences call into question the most popular view of the subject: that the mathematical notion of equivalence relation is the result of spelling out our experience of equivalence. Moreover, the findings of this study suggest that the standard definition of an equivalence relation is ill-chosen from a pedagogical point of view but well-crafted from a mathematical point of view. Keywords Equivalence relation . Equivalence . Experience . Organizing . Historical variations 1 Introduction The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. In fact, Halmos (1982) contended that “[it] is one of the basic building blocks out of which all mathematical thought is constructed” (p. 246). It also plays a central role in mathematics education, either simply as a subject to be taught explicitly or as a foundation, disguised or not, for teaching other mathematical concepts. Yet, in whatever way it has been used, it has recently manifested itself in a single form: “it is well known and standard” (Halmos, 1982, p. 245); it is a relation that has three properties, namely that it is reflexive, symmetric, and transitive. Educ Stud Math (2009) 71:219–234 DOI 10.1007/s10649-008-9173-x A. H. Asghari (*) Shahid Beheshti University, Tehran, Iran e-mail: asghari.amir@gmail.com