With Technologically Embodied Geometric Functions, students develop conceptual metaphors that directly relate computer-based sensory motor experiences of abstract function concepts. This approach relies on four foundations:
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Freudenthal described the value of geometric experiences this way:
Geometry is one of the best opportunities that exists to learn how to mathematize reality….[N]umbers are also a realm open to investigation…but discoveries made by one’s own eyes and hands are more convincing and surprising. (Freudenthal, Mathematics as an educational task, p. 407)
Lakoff and Nuñez assert the importance of students’ sensory-motor experiences more strongly, in modern cognitive science terms:
Metaphorical thought: For the most part, human beings conceptualize abstract concepts in concrete terms, using ideas and modes of reasoning grounded in the sensory-motor system. The mechanism by which the abstract is comprehended in terms of the concrete is called conceptual metaphor. Mathematical thought also makes use of conceptual metaphor, as when we conceptualize numbers as points on a line. (Lakoff & Nuñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, p. 5)
Many other cognitive scientists and mathematics educators have contributed to this thinking and extended these insights. Conceptual metaphors related to functions include the “function machine” and assorted metaphors involving Cartesian graphs of functions.
Still, most of students’ experiences with functions remain highly abstract and symbolic. We can do better by taking advantage of the fact that geometric transformations are functions and of the ease with which students can use technology to create, drag, and transform geometric points. Students’ ensuing direct sensory-motor experiences with geometric transformations as functions can allow them to comprehend abstract function concepts in terms of their concrete experiences with dynamic mathematics software. In other words, the Geometric Functions approach can help students develop conceptual metaphors that directly relate computer-based sensory-motor experiences to abstract function concepts.