# Developing Mathematical Language

This is aThink Math!feature or perspective

To teach mathematics, and even mathematical language, most successfully, focus on the *ideas*—what things are, how they work, how they interrelate—and not on what they are called. Children learn their native language extremely efficiently by hearing vocabulary used in context -- by *using* words to talk about things and ideas, *not* by talking about the words, themselves. Use good vocabulary, but keep the focus on the mathematics, not on the vocabulary.

### Purpose of the *Developing Mathematical Language* (DML) feature

Originally, DML was designed to be an occasional feature, appearing only where it was especially needed, but teachers liked it so much that it was added to every lesson. It has three major aims: to clarify vocabulary *for the teacher*; to give teachers useful information about *how children learn language*; and to help teachers avoid *over*emphasizing language.

Mathematics is about ideas -- relationships, quantities, processes, ways of figuring out certain kinds of things, reasoning, and so on. It *uses* words, but it is not *about* words. When we *have* ideas, we often want to talk about them; that is when we need words. But knowing “denominator” and “addend” is not math and does not make one mathematical. Words help us communicate. Period. The ideas are elsewhere.

## Vocabulary

The vocabulary listed in *Think Math!* contains more than mathematical terms. Some of the words are informal language that might be used in class but have no *formal* meaning. Some terms are not even necessary for students, but are provided as a service to teachers who may wish to use them, or may encounter them and want to know more. Many lessons do not have *any* important new vocabulary but, because the DML feature is provided for every lesson, some vocabulary will be listed anyway. The glossary entries on the *Think Math! Information Exchange* indicate which terms are mathematical and which are just casual language. Do not make vocabulary the focus of math.

Many entries also:

**Clarify**not just by defining terms, but by showing them in correct use and, where appropriate, pointing out common incorrect uses.**Show connections with “natural language”**to help teachers be better vocabulary-builders even outside of mathematics.**Give word histories**that show how children who speak Latin-based languages (Spanish, Portuguese, Italian...) can be*resources*in class, rather than simply students who (might) need more help with English. For example, “quadrilateral,” to children who know only English, is just another complicated word. Hearing “cuatro” at the beginning of this word, though, tells a child who speaks Spanish, Italian, or Portuguese that “4” is important to the meaning. “Lateral” means side, even outside of mathematics. So "quadrilateral" has something to do with "4 sides."

**S**ee developing mathematical vocabulary to** help your students learn mathematical vocabulary.**

**Also see **glossary** for information about specific mathematical terms.**

## Written symbols

The advantage of mathematical notation, both symbolic and graphical, is that it is highly *compact* (conveying a lot of information and ideas in very little space) and *focused* (conveying the important information for the current situation and omitting the rest). For learners, this can also be a disadvantage, because one little symbol can refer to so many ideas.

fraction examples here: 2/3 being both question and answer; 2/8 + 3/8 in kindergarten and then in 4th grade

In general, the written notation is best learned *after* the concepts are solid and the child can respond to and use the *spoken language* correctly.

## Teaching mathematical *reading*

### Two-dimensional reading: charts, tables, graphs

Reading and recording in mathematics is different from reading and recording in English. English prose text is "one-dimensional"—it simply reads left to right along a line, and the top-to-bottom is merely a convenience, breaking that long line into segments that we can stack on a page.

Mathematical reading is "two-dimensional." Instead of scanning along a straight line, left-to-right, mathematical readers must often scan vertically *and* horizontally on the page. For example, the meaning of a cell on a chart or table depends on what row and column it is in; points on graphs are identified by their position horizontally and vertically; the meaning of a bar on a bar graph depends both on which bar (its horizontal position) and how tall it is (its vertical dimension); and so on.

### The language of word problems

A fictional story, or even a piece of science writing, provides a lot of context that helps scaffold the meaning. Children can figure out word-meanings from context because there’s enough *redundant* information to let them do that. By contrast, mathematical writing is typically so terse that there’s little or no redundancy to help one figure it out. Word problems rarely provide that redundancy. That is one reason why children who squeak by on state ELA tests can still have difficulty with word problems. (It's not the only reason, but it is a real one.)

For more about how to help children *learn* the language of word problems, see headline stories.