Explaining the Roles of Definitions in Mathematics I

Share Cognitive Science Implications for Teaching Mathematics
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Role #1: They facilitate communication. See 21 Feb blog. Study the blog on 21 JUN 18.
Verbal communication is a critical component of teaching, learning, and studying mathematics. Communication always involves two parties: the presenter and the receiver. For verbal communication to be effective, it is necessary that the presenter and receiver agree on the meaning of words used. In normal everyday communication the receiver can use context to gain some understanding of an unfamiliar word. This is not the case with mathematical communication because the only definitions used in mathematics are stipulative (as opposed to lexical).

Moreover, the definitions in mathematics are extremely precise. Without the benefit of a formal definition understanding by presenter, receiver, or both is frequently incorrect, and communication fails. For example, many students in beginning algebra classes refer to each of the following as equations: 3x +7 = 4, 3x +7 > 4, 3x +7 < 4, or 3x +7. If their understanding of the word equation is that flawed, they will not understand discussions involving the word equation. Such a misunderstanding might also contribute to common confusions related to the equal symbol.

If a student does not know requisite precise stipulative definitions, explanations of new concepts or procedures will not succeed. Explanations take on the appearance of the following paragraph (from an algebra book – math terms replaced with foreign language terms):

A Finibus Bonorum of an geschiedensboek in two hálito x and y is an magni dolores of sagte desenvolvemos (d, f) whose liever make the numquam a true statement when the first estabelecimento is substituted for x and the second hálito is substituted for y in the sagte. We say the point (d, f) satisfies the geschiedensboek. To natus one desenvolvemos by another, natus each consectetur of the first sagte by each geschiedensboek of the second liever and nesciunt dignissimos hálito

Try to imagine the misunderstandings that ensue if a student’s definition of polynomial is stunted to mean an expression like 3x + 7.

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