Metaphor and Mathematics:

The Development of Meaning©

Tim Whiteford PhD.

Language plays a central role in the way we learn and use mathematics in our lives. The mathematical meaning we construct are encoded and communicated through our use of language. So too are the mathematical misconceptions we all develop at one time or another.

We use many everyday words in math but we tend to use them in a different way than we do in everyday life. Words like 'product', 'square', 'prime' and 'number' can have quite specifically mathematical meanings compared with their everyday use. 

Sometimes, words are used in a metaphorical sense such as in "how high can you count?", "the face of a clock" or " reducing a fraction". The metaphorical use of language in math can be both positive as in the "face' of a clock, or negative as in "reducing' a fraction. Here the term "reduce" can lead students to believe that 1/2 is smaller than 2/4 because  2/4 has been 'reduced' to 1/2.

The development of a student's mathematical vocabulary is a constant and ongoing process. A student may first discover the word "circle" in kindergarten but will develop her understanding of the meaning associated with that word throughout her school experience and beyond. A student's understanding of a particular piece of mathematical vocabulary grows through her participation in developmentally appropriate experiences. A kindergarten student might develop her understanding of the word circle by tracing around different circular objects while a fifth grade student might develop her understanding of the relationship between the circumference and diameter of a circle through activities with a piece of string and a calculator.

The way a student comes to develop an understanding of a particular word is not through the memorization of a specific definition but through the use of the word in appropriate contexts. This involves a series of negotiations through which the student may use the word inappropriately and not get the response or effect she was looking for. As the word is used more and more appropriately so the level of understanding of the meaning of that word is developed. Formal definitions have a valuable role to play when we are trying to distinguish between terms or define a word more specifically when the need arises. 

The mathematical vocabulary a student experiences throughout her school life is also characterized by increasing levels of specificity. While "top number" and "bottom number"  would be an appropriate term for a first grade student to use when first experiencing fraction notation, numerator and denominator would be far more appropriate terms for fourth grade students to use.

Here are some interesting sources of mathematical terminology and vocabulary. Although some provide terms associated with particular grade levels, the best sources for determining when to introduce new terms is the nature of the content  being taught and the developmental level of the students involved.

The Harcourt Brace Math Glossary - An alphabetically organized list of terms and their formal definitions.

A Math Dictionary for Kids - A well presented and nicely animated site, but it's Australian so some of the definitions are interestingly different.

A Mathematical Quotation for every occasion -  a neat collection of neat things said about mathematics.

The Everyday Mathematics Glossary of terms - clear and precise definitions