Metaphorical Reasoning : Origins , Uses , Development and Interactions in Mathematics *

People's innate neurological perceptions, mental simulations, intuitions, and schemas provide the familiar source entities, relationships, and actions for a metaphor. People metaphorically project this information on to the target problem to construct new concepts, relationships and actions. People also reason metaphorically to: connect mathematical ideas, improve recall, understand mathematical representations, and enhance their computational environment. Metaphorical reasoning capacity depends not on age but on people's understanding of the source which provides the potential for metaphorical reasoning. Use of a particular metaphor decreases increasing target understanding. A metaphor's source-to-target projections specify the actual metaphorical inferences. Metaphorical reasoning may serve as a permanent resource rather than a temporary scaffold as experts automate their metaphorical reasoning for routine problems. Overlapping metaphors increase coherence of mathematical understanding and compensate for each metaphor's limitations. Facing difficult mathematical relationships in a problem situation, people can create a chain of metaphors from an intuitive source to the mathematics to the problem situation. Finally, metaphorical reasoning differs from the following types of reasoning: embodiments, intersection, analogical, example-based, symbolic play, symbolic mnemonics, and distributed.