Vocabulary Considerations for Teaching Mathematics
Monroe, Eula Ewing, Panchyshyn, Robert, Childhood Education
The importance of rich and meaningful vocabulary knowledge when developing concepts is well documented and widely accepted by classroom teachers; vocabulary provides access to concepts. Because mathematics material is so difficult to read, "with more concepts per word, per sentence, and per paragraph than any other area" (Schell, 1982, p. 544), it is particularly crucial to emphasize vocabulary instruction in this content area.
Necessary Vocabulary for Developing Mathematical Concepts
The vocabulary that teachers should teach to help students develop mathematical concepts can be classified into four categories: technical, subtechnical, general and symbolic.
Technical Vocabulary. Those words generally viewed as mathematical terminology are called technical vocabulary. Technical terms convey mathematical concepts that are difficult, if not impossible, to express in everyday language. Each technical term (e.g., integer, quadrilateral) has only one meaning, which is specific to mathematics. Because these terms are encountered only in mathematical contexts, and are themselves often defined with other technical terms, they are difficult to learn and remember; learning a technical vocabulary is comparable to learning a foreign language.
Subtechnical Vocabulary. Subtechnical terms have more than one meaning; these meanings vary from one content area to another or from a content area to everyday experience. Learners may know and be able to use one or more meanings for a subtechnical term, but may not necessarily know its specific mathematical meaning. Because of their multiple meanings (e.g., the volume of a cube, the volume control on the television set, the volume of world trade), these terms can be especially difficult to conceptualize. Some subtechnical terms have multiple meanings even within a mathematical context (e.g., degrees of temperature, degrees of an angle), thereby creating additional conceptual problems. Because of this nature, subtechnical terms may be even harder to learn and remember than technical terms.
General Vocabulary. Students encounter general vocabulary words in everyday language and in their usual reading experiences. Most elementary mathematics textbooks use a general vocabulary, although these words are not likely to be taught in reading class. One 1966 study found that even if students were taught all the words presented in seven different reading series at the primary level, they would be exposed to only about half the words included in mathematics textbooks for the same levels (Stauffer, 1966). Recent research indicates the problem still exists. Panchyshyn and Monroe (1992) found that more than half of the words included in elementary mathematics textbooks were not among those most frequently used in children's reading materials. A mandate for developing general vocabulary in mathematics becomes evident when these and similar findings are considered.
Symbolic Vocabulary. Symbolic vocabulary, viewed by some to be the real vocabulary of mathematics, presents its own special problems. Most reading material uses only alphabet symbols. In mathematics, however, the reader needs to recognize not only the alphabet, but also numerous nonalphabet symbols. Numerals, the most common math symbols, represent numbers, which are themselves so highly abstract that even mathematicians find them difficult to define! In addition, a given numeral can be used to convey many different meanings. For instance, consider the numeral 2 in the following numerical contexts:
52 23 [4.sup.2] 1/2 2/3 [m.sup.2]
The 2 conveys a different, and highly abstract, meaning in each context. Furthermore, the numerical expression itself can be read in different ways - [4.sup.2] can be read as "four squared," "four to the second power," etc. Adding to the potential for confusion, the same meaning can be conveyed by different symbols. Consider how learners must refocus their thinking when division is presented as 4 / 2, 2 x ? …