Word origins and vocabulary development
What's in a word?
Mathematical terms are often related to vocabulary that students already know, or that we'd like them to learn. For example, connecting factor with factory, a place that makes products, can help students remember what the mathematical terms mean. Exploring word origins and related words can also build students' general vocabulary. See Etymology for more.
Typically, words have more specialized meanings in mathematics than in casual use. Students can learn a lot by associating the two, and noticing the difference. See Vocabulary for more.
As with all teaching, the teacher must choose thoughtfully, neither overlooking good opportunities, nor overloading students with unfamiliar vocabulary.
Definitions of mathematical terms
"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
When are definitions important?
Generally, only after the idea is already mostly understood from use in context, from communication, and from examples and non-examples.
For more about why definition is so important in mathematics, see Definition.
For more about teaching vocabulary, see Developing mathematical language.