Making Sense of Mathematical Reasoning and Proof
This chapter charts the growth of proof from early childhood through practical generic proof based on examples, theoretical proof based on definitions of observed phenomena, and on to formal proof based on set theoretic definitions. It grows from human fo
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Making Sense of Mathematical Reasoning and Proof David Tall
Abstract This chapter charts the growth of proof from early childhood through practical generic proof based on examples, theoretical proof based on definitions of observed phenomena, and on to formal proof based on set theoretic definitions. It grows from human foundations of perception, operation and reason, based on human embodiment and symbolism that may lead, at the highest level, to formal structure theorems that give new forms of embodiment and symbolism. Increasing sophistication in mathematical thinking and proof is related to earlier experiences, called ‘met-befores’ where supportive met-befores encourage generalisation and problematic met-befores impede progress, causing a bifurcation in the perceived nature of mathematics and proof at successive levels of development and in different communities of practice. The general framework of cognitive development is offered here to encourage a sensitive appreciation and communication of the aims and needs of different communities.
Keywords Mathematical proof · Crystalline concepts · Met-before · Generic proof · Van Hiele theory · Structure theorems
This article is a product of personal experience, working with colleagues such as Shlomo Vinner who gave me the insight into the notion of concept image, Eddie Gray, whose experience with young children led me to grasp the essential ways in which children develop ideas of arithmetic and to build a theoretical framework for the different ways in which mathematical concepts are conceived, Michael Thomas who helped me understand more about how older children learn algebra, the advanced mathematical thinking group of PME who broadened my ideas about the different ways that undergraduates come to understand more formal mathematics, many colleagues and doctoral students who I celebrate in Tall (2008) and, more recently, the working group of ICMI 19 who focused on the cognitive development of mathematical proof (Tall et al. 2012). D. Tall (B) University of Warwick, Coventry, UK e-mail: [email protected] M.N. Fried, T. Dreyfus (eds.), Mathematics & Mathematics Education: Searching for Common Ground, Advances in Mathematics Education, DOI 10.1007/978-94-007-7473-5_13, © Springer Science+Business Media Dordrecht 2014
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Fig. 13.1 Outline of long-term development of proof
Mathematical Thinking in Terms of Human Perception, Operation and Reason The cognitive development of mathematical thinking and proof is based on fundamental human aspects that we all share: human perception, action and the use of language and symbolism that enables us to develop increasingly sophisticated thinkable concepts within increasingly sophisticated knowledge structures. It is based on what I term the sensori-motor language of mathematics, blending together perception, operation and reason (Tall 2013). Mathematical thinking develops in the child as perceptions are recognised and described using language and as actions become coherent operations to achieve a specific mathem
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