# Mathematics Words and Expressions

Transmission of knowledge requires language or symbolic representations. Most fields of knowledge require language for its communication. Each domain uses words, expressions, phrases and collections of words that have specific and contextual meanings. Mathematics, too, has special symbols, words and phrases, as well as everyday words, which are used in particular ways. It has its own vocabulary, syntax, and its structure.

At times, major gaps exist between a child’s language outside the class and the abstract symbolism and language of school mathematics. Because of the complexity of the mathematics language and because mathematics is a second language for most children, many experience difficulty in learning it.

The size of a child’s mathematics vocabulary is highly correlated with mathematics achievement. Mathematics language and concrete models help construct the corresponding conceptual schemas in the child’s mind. Through the linguistic component (lexical entries—language containers) the child understands, uses, and retains the mathematics concepts. Most words used for learning mathematics are derived from the native language of the learner (e.g., add) and many are carried over into mathematics from other languages (e.g., trigonometry). Mathematics vocabulary consists of five kinds of words or expressions:

• Words that have the same meaning in mathematics and outside of mathematics. For example, circle, add, cylinder, etc. For these words, students already have conceptual schemas and experiences. In other words, they bring some ideas related to these words. When these words are used in mathematics, students can expand the existing schemas and meanings to accommodate the use of these words in mathematics. For this reason, these words and related mathematics are easily accessible to students.
• Words that have different meanings in mathematics and outside. For example, distribute in common language means division, but distributive property over multiplication a(b + c) = ab + ac means multiplication by a to the sum of b and c. There are many words of this type: and, or, table, fraction, tangent, sum, product, slope, mean, mode, median, etc. Because these words have specific meanings in mathematics, the teacher needs to create new conceptual schemas either by setting patterns, concrete modeling, analogies, or reason. Relying on the meaning and usage of the word from the native language may create difficulties and misconceptions.
• Words that are unique to mathematics and do not exist outside of mathematics. For example, exponent, quotient, numerator, denominator, trigonometric ratios (such as sine, cosine), conic sections (such as hyperbola), differentiation, integration, quadratic equation, scale, figure, etc. These words are new to most students, so a teacher must take great care in their introduction to students. Concrete modeling and practice are the best ways to introduce them. For example, terms such as conic sections (circle, ellipse, parabola, hyperbola) are best understood when students actually make sections from the cone.
• Compound words that are formed using competing and even opposite meaning. Words and expressions in this category include: least common multiple, greatest common factor, simultaneous linear equations, etc. For example, the term Least Common Multiple of numbers of 4 and 6 is best understood when we compare this expression with the statement: John is an intelligent, handsome, tall boy. The words intelligent, handsome, tall are qualifiers of the word boy and are to be seen in a certain order. Similarly, Least Common Multiple of 4 and 6 must be seen in a certain order. The order of the activity is: first we find the multiples of 4 and 6. Then we mark (the numbers in bold) the common multiples of 4 and 6. Lastly, we select the least of these common multiples (12). Therefore, the Least Common Multiple of 4 and 6 is 12. It is important to emphasize the interaction of the words in the expression Least Common Multiple. The same process applies in the case of Greatest Common Factor of 20 and 24.

4     8   12   16   20   24   28   32   36    40   44   48, …

6     12      18       24      30        36        42       48, …

• Compound words formed by combining different concepts. These include: miles per hour, scale factor, density, etc. Before understanding these types of words (compound secondary concepts), students need to have a clear understanding of the corresponding component primary concepts involved in the secondary concept.

Just like in the native language, every word may have different meanings according to the context. For example, a word in the English language may be understood and its meaning found by examining its (a) entomological roots, (b) historical usage, (c) popular meaning, (d) contextual, and (e) meaning received by the listener or reader. Mathematics words, most of the time, are more consistent in meaning than they are in the native language. However, the same approach to examining the roots and history of the word may help us with the meaning of the word or expression in the context of mathematics.