This paper proposes a reading of the history of equivalence in mathematics. The paper has two main parts. The first part focuses on a relatively short historical period when the notion of equivalence is about to be decontextualized, but yet, has no commonly agreed-upon name. The method for this part is rather straightforward: following the clues left by the others for the 'first' modern use of equivalence. The second part focuses on a relatively long historical period when equivalence is experienced in context. The method for this part is to strip the ideas from their set-theoretic formulations and methodically examine the variations in the ways equivalence appears in some prominent historical texts. The paper reveals several critical differences in the conceptions of equivalence at different points in history that are at variance with the standard account of the mathematical notion of equivalence encompassing the concepts of equivalence relation and equivalence class.
The purpose of this paper is to propose an operational idea for developing algebraic thinking in the absence of alphanumeric symbols. The paper reports on a design experiment encouraging preschool children to use the associative property algebraically. We describe the theoretical basis of the design, the tasks used, and examples of algebraic thinking in 5-6-year-old children. Theoretically, the paper makes a critical distinction between operational and structural meanings of the notion of equality. We argue that mathematical thinking involving equality among young learners can comprise both an operational and a structural conception and that the operational conception has a side that is productively linked to the structural conception. Using carefully designed hands-on tasks, the crux of the paper is the realization of algebraic thinking (in verbal mathematics) as operationally experienced in the ability to transform one number structure, with a quantity that is subject to change, into another through equality-preserving transformations.
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.
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