Definition
From Thinkmath
"Definitions in mathematics should be precise and unambiguous. In practice, this means that a definition should tell you exactly what you need to do to determine whether any object does or doesn't fit the definition." ---Peter Braunfeld, advisor to the Think Math! project, June 26, 2002
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Why are definitions important in mathematics?
Vignette A fifth grade teacher asked her student teacher to review the ideas of factors and multiples with the class. The student teacher started by asking the children to list the factors of 25. At first, the students confused the idea, and listed 50, 75, 100... She clarified that she was looking for "numbers that could be multiplied to make 25." She was surprised when the children listed only 5, and then stopped. It is not uncommon for children to "miss the obvious" because it doesn't seem to be "interesting" enough! There doesn't seem to be enough "multiplying" going on in 1 x 25. But she asked for more and, after a long pause, one of the children said, tentatively, "two times twelve-and-a-half?" after which another child practically exploded with the answer "four times six-and-a-quarter!" The class was proudly showing their knowledge of fractions, which they'd just studied.
So, how many factors does 25 have? The student teacher was then unsure, herself. Just the number 5? Or 1, 5, and 25? Or are the zillions of other possibilities legitimate, too, because they are "numbers that could be multiplied to make 25"? It all depends on what we mean by factor. For that reason, definitions in mathematics are essential.
In casual conversation, we can tolerate a fair amount of ambiguity, but in mathematics (and law, for that matter), ambiguity causes trouble.
Mathematics builds new ideas on already established ideas. We can't build a new idea on "it depends what you mean," so we need, right at the start, to agree on what we mean. Moreover, we can't share our discoveries with others unless they agree on the same meanings for the words we use. We can't, for example, claim that "a prime number is a number that has only two factors, 1 and itself" unless we agree that even though 
, twelve-and-a-half is not a "factor" of 25. (See factor for a careful definition.)
Formal definition and and informal meaning
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How to build mathematical vocabulary
Understand it well yourself (see glossary), and use it naturally (and correctly) in communication when you need it, in a rich enough context so that students can "figure out" the meaning from context, just as you might expect them to figure out the meaning of a new word that occurs in the middle of a story.
A "rich enough context" might be one in which you are pointing to examples and non-examples, or it might contain other clues that help students get an idea of the meaning of the word.
After a word is "sort of clear," you can then try to make its meaning more explicit with more examples, descriptions, or [[definition]s.
For more, see Developing mathematical language.
