Developing mathematical language
From Thinkmath
This is a Think 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.
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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 overemphasizing 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. Do not make vocabulary the focus of math.
The glossary on the Think Math! Information Exchange indicates which terms are mathematical and which are just casual language. 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."
The role and risks of definitions
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!
Teaching vocabulary by using words in context
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 they mostly also rely on meanings from context and usage, which is why the glossary entries on this site give correct and incorrect usage to the teachers, where appropriate, along with definitions.) Some DMLs give teachers strategies other than “saying what the word means,” strategies for just using the word (literally without explaining it) until students are using it, too, and only then clarifying the meaning formally with other words (that is, through discussion and/or definition).
Words we never quite teach
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. The glossary on the Think Math! Information Exchange takes that approach.
Written symbols
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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.
