Calculations and Canonical Elements Ian Stewart and David Tall University of Warwick Part 1 1. Introduction Equivalence relations are the basis of modern approaches to many topics in school mathematics, from the first ideas of cardinal number (through matching activities and correspondence between sets) through definitions of negative numbers (using ordered pairs of natural numbers), to equivalence of fractions, modular arithmetic, vectors, and many more advanced topics. We contend that these approaches to the subject have been based on an inadequate theoretical framework, causing an unnecessary schism between traditional mathematics and “modern” approaches. The missing link is the concept of a canonical element. Reintroducing this idea gives a much more coherent relationship between the structural elegance of equivalence relations in modern mathematics and the traditional aspect of computation. We tackle this in Part 1 of this paper, which follows. This in turn gives a clearer insight, as we shall see in Part 2, into certain technical and educational problems. 2. Equivalence Relations We shall assume that the reader is familiar with the notion of an equivalence relation ~ on a set S. Being a relation means that for each ordered pair of elements a, b ∈ S, we either have a ~ b (a, b are related), or a ~ / b (a, b are not related). An equivalence relation satisfies the further properties: (E1) a ~ a for every a ∈ S (E2) if a ~ b, then b ~ a (E3) if a ~ b, b ~ c, then a ~ c. This partitions the set S into equivalence classes, where for any x ∈ S we denote the equivalence class containing x by E x = {y ∈ S | x ~ y}. We find that E x = E y if and only if x ~ y , and if x / ~ y then E x ∩E y is empty, so E x , E y are disjoint. An alternative notation for E x which we shall often use is [x], so [x] = {y ∈ S | x ~ y}.