This thesis seeks to answer one single question: "what is an equivalence relation?" A more correct, though longer, version of this question is "what are the qualitatively different ways in which people experience an equivalence relation?" The second question is not simply a version of the first one. It has a completely different nature and consequently demands a completely different answer. The answer to the first question can be found in any textbook on the foundations of mathematics; while the second question can be answered only by conducting research where people are given a chance to reveal their conceptions of equivalence relations. These two questions embody two integrated phases of this thesis linked together with a transitory phase. The first phase starts with a definite answer to the first question, i.e. the standard definition of equivalence relations. This definition is used to design a certain situation consisting of certain tasks embodying the corresponding notion. The initial intention of the situation is to get students to define certain predetermined concepts related to the notion of interest, and the effectiveness of the situation is characterized by the extent of students' success to do so. The tasks are tried out on a smallish sample of students. To put it bluntly, the situation fails to achieve its aim. In the process of interviewing the students it becomes clear that the standard definition is just an advanced means of organizing by which the given situation {and many others} can be organized. More importantly, there is a growing realization that the initial intention of the study ignores the richness of the students' ways of organizing the situation in favour of maintaining a narrow criterion for success. Relinquishing the latter in favour of the former is the turning point from the first phase to the second. The second phase is a transitory phase in which more weight has been put on what students use to organize the given situation. Although the focus of this phase is not on the notion of equivalence relation, the students' works reveal some unexpected aspects of this notion. This suggests the possibility of using the original tasks for pursuing an unexpected purpose in the main (i.e. third) phase of this thesis. The main phase of the thesis adopts a phenomenographic approach to reveal students' conceptions of equivalence relations. These conceptions are inferred from the ways that the students tackle the tasks, regardless of the extent to which they fit into the standard account. It is shown that these conceptions correspond to certain 'historical' counterparts, where some prominent mathematicians of the past have tackled certain situations that from the vantage point of today's mathematics embody the idea of equivalence relation. These correspondences put forward a critical distinction between "equivalence" as an experience and "equivalence" as a concept. This distinction calls 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 illchosen from a pedagogical point of view, but wellcrafted from a mathematical point of view.
