By Lyn D. English
How we cause with mathematical rules is still a desirable and not easy subject of research--particularly with the speedy and various advancements within the box of cognitive technological know-how that experience taken position in recent times. since it attracts on a number of disciplines, together with psychology, philosophy, laptop technology, linguistics, and anthropology, cognitive technological know-how offers wealthy scope for addressing matters which are on the middle of mathematical studying.
Drawing upon the interdisciplinary nature of cognitive technology, this ebook provides a broadened viewpoint on arithmetic and mathematical reasoning. It represents a circulate clear of the normal suggestion of reasoning as "abstract" and "disembodied", to the modern view that it really is "embodied" and "imaginative." From this angle, mathematical reasoning contains reasoning with constructions that emerge from our physically reports as we have interaction with the surroundings; those constructions expand past finitary propositional representations. Mathematical reasoning is inventive within the experience that it makes use of a couple of strong, illuminating units that constitution those concrete reports and remodel them into types for summary suggestion. those "thinking tools"--analogy, metaphor, metonymy, and imagery--play a major function in mathematical reasoning, because the chapters during this ebook display, but their capability for reinforcing studying within the area has bought little popularity.
This publication is an try and fill this void. Drawing upon backgrounds in arithmetic schooling, academic psychology, philosophy, linguistics, and cognitive technology, the bankruptcy authors offer a wealthy and accomplished research of mathematical reasoning. New and interesting views are provided at the nature of arithmetic (e.g., "mind-based mathematics"), at the array of strong cognitive instruments for reasoning (e.g., "analogy and metaphor"), and at the alternative ways those instruments can facilitate mathematical reasoning. Examples are drawn from the reasoning of the preschool baby to that of the grownup learner.
Read or Download Mathematical Reasoning: Analogies, Metaphors, and Images (Studies in Mathematical Thinking and Learning Series) PDF
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Extra info for Mathematical Reasoning: Analogies, Metaphors, and Images (Studies in Mathematical Thinking and Learning Series)
Given our ordinary concept of 'Just as many as," Cantor proved no such thing. He only proved that the sets were equipollent. But if you use Cantor's metaphor, then he did prove that, metaphorically, there are just as many even numbers as natural numbers. The same comment holds for other proofs of Cantor's. Literally, there are more rational numbers than natural numbers, since if you take the natural numbers away from the rational numbers, there will be lots left over. But Cantor did prove that two sets are equipollent, and hence they metaphorically can be said (via Cantor's metaphor) to have the same number of elements.
But there are many aspects of naive set theory in mathematics that are not consequences of this metaphor and cannot be defined using it. Container-sche- 2. METAPHORICAL STRUCfURE OF MATHEMATICS 41 mas are just cognitive mechanisms that impose conceptual groups. Though you can have sets of objects further grouped into subsets by additional container schemas inside an outer one, those internal container schemas are not themselves made objects by the Sets-As-Container-Schemas Metaphor, and since they are not objects, they cannot be members of the set.
Would we deny that there are now four chairs in the room? Of course not. But doesn't that mean that 2 + 2 = 4 is in the world? Of course not. The reason should be clear. The category chair is a human category. Suppose the four chairs in the room are different kinds of chairs-a desk chair, an armchair, a patio chair, and a recliner. Those are four very different objects. It is our human conceptual system that categorizes them all as chairs. Counting presupposes the grouping of things to be counted.