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Abstract

The National Council of Teachers of Mathematics, a US based teachers association, strongly encourages teachers to make proof and reasoning an integral part of student mathematics. However, the literature shows that, far from being integral, proving remains compartmentalized within the North American school curriculum and is restricted to a specific mathematical domain. Consequently, students suffer in their understandings and are ill prepared for the rigorous mathematical proving that many of them will encounter later at the postsecondary level. The literature suggests that compartmentalization is mainly due to teacher’s lack of experience in proof and proving and their subsequent inability to guide students through the various stages of mathematical justifications. In this article, a brief overview of the history of proof is provided. Also, the nature of proof is explained and its importance in school mathematics is argued for. Teachers can help students better if they are aware of the stages that students experience over time as they become progressively more sophisticated at proof and proving tasks. By using a developmental model of proving, teachers are more likely to guide students effectively as they move from one stage to the other. Balacheff (1988) provides one of the most commonly used hierarchies to categorize student proof schemes. Few articles within the literature, however, apply this taxonomy in explicit ways by aligning examples of possible efforts to solve proof tasks with Balacheff’s stages of proof. This article illustrates in accordance with Balacheff’s taxonomy and by using examples, how a student might tackle a proof task. It is also argued that even though explanatory proofs are relevant to school mathematics, overemphasis on verbal proofs may result in watered down mathematical proofs. With an example I also demonstrate that students can transition fairly easily from explanatory proof to formal two –column proof.

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