Experiencing equivalence but organizing order

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.

Equivalence: an attempt at a history of the idea

“…Euclid never expresses a phrase like, 'a is equivalent to (read it "is equal to", "is parallel to", or "is commensurable with") itself'. Hilbert has to consider the reflexive property, though he does not name it (Asghari 2009): Since congruence or equality is introduced in geometry only through these axioms, it is by no means obvious that every segment is congruent to itself. (Hilbert 1902, p. 10) The only similarity between Euclid's treatment of equivalence and Hilbert's is when an equivalence of more than two objects is involved.…”