In the English language, context helps us decide which of the possible meanings of common words is meant. As representations of ideas, words stated free of some meaningful context often do not communicate very well. The definition of a word in the dictionary usually includes several possible meanings. The word used in context helps the listener or reader decide which among the various meanings the speaker or writer intends. In the language of mathematics, we also face the same dilemmas.
Many mathematical words have different shades of meaning.
In learning to understand how both to communicate in, and to decipher the language of, mathematics, students have to determine meaning from contextual use. Faced with a mathematics problem, students have to find ways to use mathematics to represent the situation, manipulate the representations to find solutions, interpret the solution in the original context, and look for ways to generalize the solution to a whole class of problems. Variables play a key role in the process of mathematizing a situation. Consider this problem recently given to a sixth-grade class I observed. A local contractor, Ms. Phillips, builds swimming pools.