Definitions

Share Cognitive Science Implications for Teaching Mathematics
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According to Dr. David Chard, “Vocabulary knowledge is as essential to learning mathematics as it is to learning how to read.” [Chard]
It is entirely possible for a person to possess an extensive vocabulary but be so deficient in a particular area (mathematics, chemistry, economics, medicine) to be essentially illiterate in that area. That is frequently the case in mathematics. For many mathematics students, reading a mathematics textbook is like reading the following.

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

The above “nonsense paragraph” was created by selecting a paragraph from an elementary algebra mathematics book and replacing each mathematics term with an arbitrarily chosen word from a list of non-English words.
The above “nonsense paragraph” is a faithful representation of what is seen when a student reads a math book without knowing the requisite definitions.
Clearly the “nonsense paragraph” has no instructional value. Likewise, paragraphs in a mathematics textbook have no instructional value unless the student has learned the requisite definitions. The burden is on the student, not the author.

[Chard] Dr. David Chard, Vocabulary Strategies for the Mathematics Classroom, http://www.eduplace.com/state/pdf/author/chard_hmm05.pdf

Verbal communication is a critical component of teaching, learning, and studying mathematics. Communication always involves two parties: the presenter and the receiver. In order 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.

Construction of a Simple Concept
Construction of a Simple Concept

No longer is there any doubt that “All new learning requires a foundation of prior knowledge.”
Memorization of stipulative mathematics definitions provides a form of foundational knowledge upon which to build a complete concept.
Simple mathematics concepts are constructed with a definition as a central core tightly wrapped in several layers of illuminating information.
Therefore in order for any one of teaching, learning, or studying to be effective it is necessary the both the instructor and the learner know precisely the stipulative definitions of all the words being used.

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