Topics: Mathematical Language

Understand it well yourself, 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.

Only *after* a word is “sort of clear” can you then try to make its meaning more explicit with more examples, descriptions, or definitions.

Young children acquire vocabulary at an astonishing rate — a full 50% of what will be their adult vocabulary by the age of five! They do that *entirely from use in context*. New words are acquired extremely rarely from a definition and *never* solely from one. (Adults can get some words from definitions, but even adults learn new words and meanings mostly from context and usage *first*, which is why the glossary entries on this site give, along with definitions, correct (and, where appropriate, incorrect) usage. Some “Developing Mathematical Language” sections in *Think Math!* suggest teaching strategies other than “saying what the word means” — strategies for just using the new word (with *no* explanation) until students are using it, too, and only *then* clarifying the meaning formally with other words (that is, through discussion and/or definition).

In everyday conversation, definitions are of little real help. Try to think, for example, how you’d define “chair” to include all the different kinds of objects, wooden, plastic, stuffed, formal, etc., that are “chairs.” Or how to define “cat.” And look in a dictionary to see how much you must already know in order to understand the definition! For casual use, context and experience are enough for us to “get the idea” without being able to give formal definitions. And when definitions are looked up *before* a child has the general idea, we get strange results. The child looks up “extinguish,” sees it means “put out,” and writes “before I go to bed each night, I extinguish the cat.”

In mathematics, definitions are essential, because examples, alone, can’t ever nail down meanings precisely enough for the careful use mathematics makes of words. Even so — and especially in elementary school — definitions really don’t quite “work” until one *mostly* understands anyway. Only then can a definition help clean up the details, *refining* and *making precise* what one already knows in a fuzzy and approximate way. And even then, in elementary school it is generally hard to get a definition precise enough to do the job without using words and ideas that the child doesn’t know!

Definitions are problematic when they are presented first, but they eventually become important.

“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 theThink Math!project, June 26, 2002

**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 × 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 2 × 12 = 25, twelve-and-a-half is not a “factor” of 25. (See factor for a careful definition.)

We are often asked where *Think Math!* defines basic geometric terms like *point* and *line*. It doesn’t. It *uses* these terms, but does not define them, and *should not*. *Point* and *line* are, even for mathematicians, undefined terms. (If they were to be defined, they would have to be defined using other words. *Some* word would have to be taken as “understood,” or the process would either be circular or infinite! *Point* and *line* are among several words that are taken as just “understood,” the *basis* upon which other terms are defined.) School texts should handle them that way, too (but often don’t). *Point* is used explicitly in 2nd grade—with *endpoint* and *midpoint*. Such terms can appear in a glossary, but must be understood from examples and context, and not defined.

Glossary entries for such “undefined” terms *can* say a bit about what these terms do *not* mean, to clarify common misunderstandings.