Definitions and definitional reasoning are central to the learning of mathematics and to the teaching of mathematical content. Definitions are more than sentences that merely describe a memorized concept. According to Berger (2005), examining how individuals make personal meaning of a mathematical object or an idea is basis for how students learn mathematics. Definitions are a fundamental part of the logical structure or nature of mathematics. Although some studies have examined how preservice and in-service teachers view definitions, little research has examined the deductive structure of mathematical definitions through the use of five principles. This study uses the concept image to examine the relationship between preservice teachers’ conceptions of mathematical definitions and their concept images. In this qualitative study, seven preservice teachers at various stages in their mathematical content preparation to teach grades K – 8, were engaged in a series of tasks to systematically engage with definitions. The five logical principles served to guide the preservice teachers as they negotiated and refined the meanings of and wrote high quality definitions for the four quadrilaterals. As the preservice teachers interacted with the tasks, their discussions were recorded and coded to determine the extent to which they used these principles. The findings indicated the strength of the concept image influence the preservice teachers abilities to write high quality definitions for the quadrilaterals. The findings also indicated that the preservice teachers hold intuitive values about that often match the five principles. Through the examination of examples of high quality definitions for quadrilaterals, the preservice teachers could rewrite definitions that demonstrated the use of the principles. However, throughout the evolution of the sequence of tasks, the interplay between the five principles and their total cognitive structure were put into conflict. The cognitive structure included prior learning experiences and their own personal reconstruction of the definitions for quadrilaterals. Changes in the use of the principles demonstrated that the nature and role of mathematics is teachable and that the five principles can support such learning.
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