 # Dr. Cotter on Language

In mathematics we can’t do much without using language. Even though the mathematician may write a mass of equations, she needs words to explain their meaning. Good math instruction requires careful attention to the terms used. We don’t want students to have to relearn the mathematical meaning of words.

Number Words

The first math words we want a child to learn are the names for quantities. For quantities one to ten, the words are arbitrary. Rather than continuing to memorize a growing number of unrelated words, people hundreds of years ago began to think of numbers grouped into manageable chunks.

The Romans grouped into fives, V, as well as tens, X, when recording numbers. They had no symbol for 2 or 3, so they doubled or tripled the symbol for 1. Thus, 2 is written II and 3 is written III. With the tens, 2 tens is written XX and 30 tens XXX. Also 5 tens is L and 6 tens is LX. Although only four different symbols are needed to record numbers from 1 to 99, larger numbers require more symbols, C for 100, D for 500, and M for 1000.

Note that the early Roman numerals represented 4 as IIII and 9 as VIIII. Only later did 4 became IV, meaning one less than five, and 9 became IX, one less than ten. However, the economy gained in writing those numerals turned out to be a major obstacle to performing calculations with the numerals. It is interesting to note that most clocks with Roman numerals use the early four, IIII, but the later nine, IX.

A monumental improvement occurred in recording numbers with the introduction of the familiar Hindu numerals. Each quantity from zero to nine is written with a distinct digit. For numbers larger than ten, the digits are reused, but the place of the digit in the number determines its value, hence the term, place value.

Unfortunately, the words for numbers in the Indo-European languages predated the Hindu numerals and lacked the simplicity and clarity of the written numerals. Children today struggle matching the irregular number words to the corresponding symbols.

Remarkably, the East Asian languages were changed to make their number words consistent with the Hindu numerals. For example, eleven became ten-1; twelve became ten-2; twenty-three, 2-ten 3; and forty-seven, 4-ten 7. Many of the children speaking these languages understand place value before they even start school, giving them a great advantage in learning arithmetic. Happily, English-speaking children can gain the same benefit by using these transparent number words for a short period of time.

Simply Incorrect Words

Probably the term that aggravates me the most is “number sentence.” A sentence is a group of words that make a complete thought. How does the equation 2 + 3 = 5 fit that definition? Using the term number sentence confuses the learner in both math and language. One third grader when asked to write a number sentence wrote: Two plus three equals five. The term equation means to make equal. This equality is a fundamental principle of mathematics. Fortunately, this ill-advised term of number sentence is disappearing from textbooks and tests.

A close second in annoyance is “take away.” First of all, it is bad English to say “seven take away five.” If this is a declarative sentence, shouldn’t there be an s after take: Seven takes away five. That’s kind of bold of seven. Or, if it’s an imperative sentence, shouldn’t there be a comma after seven: Seven, take away five. Now, seven is kind of brazen. Secondly, in England take way is fast food. Seriously, using take away limits a child’s understanding of subtraction. Often, subtraction is not about taking something away, but comparing, find a missing part, or adding up. Again, I’m happy to report this phrase is disappearing from texts and tests. Oh, what should we say? How about the correct term, minus?

The next word to censure is “timesing,” which will never make it into a math dictionary. Timesing, referring to multiplication, is a babyish nonword and yes, nonword is a word. On the other hand, multiply and multiple are authentic mathematical words. The expression, 3 × 2, is best read as “three multiplied by two” or even “three taken two times.” Saying “three times two” doesn’t really describe the situation. This wording started in the 20th century and may be confusing to some children because time is associated with clocks.

Another word, fairly new to elementary arithmetic vocabulary, is regroup. According to the dictionary, regrouping is what a military unit does after a defeat. While adults think of this word as re-group, children learn it as a word to describe a process and not as equality. After all, have you ever witnessed a child regroup their toys and then talk about regrouping them? The old-fashioned words carry and borrow work just fine; they are mathematical words and programmers still use them. The old argument that to borrow implied something needing to be returned isn’t valid: languages borrow from each all the time. However, an even better word is trade, which children do understand and it does imply equality.

Geometrical Words

Preschool children are often taught non-mathematical words for geometrical shapes. Instead of ellipse, they learn oval, but an oval can also be egg-shaped or shaped like a running track. And the mathematical name for a diamond is rhombus. They also are taught that rectangles are “long and low,” disallowing squares.

When discussing the area of a rectangle, textbooks usually name the sides as l for length and w for width. Yet, when discussing triangles, the sides are named b for base and h for height. If the sides of rectangles and triangles had corresponding names, it would greatly help students see the relationship between the areas of a triangle and a rectangle. I think the best terms are width, for the distance from side to side, and height, for the perpendicular distance from the width. One day, I casually mentioned to Kim, an honor student in her senior year of high school, that squares are rectangles. She replied, “They are? They have different formulas!” Her textbook used s for the sides of a square.

Some Oddities

Diagonal and similar are two words having a mathematical meaning at odds with everyday usage. The common meaning of diagonal is a line that is neither horizontal or vertical, or a road that does not travel north and south or east and west. Contrast that with the mathematical definition: a line in a polygon (a closed figure with straight lines) drawn between any two non-adjacent vertices. Such a line could also be horizontal or vertical. Simply rotate the polygon with its diagonal until the diagonal is horizontal or vertical.

Amazingly, the usual meaning of similar is contrary to its mathematical meaning. In everyday use, similar means not exact, but almost the same; in mathematics similar means identical, but either shrunk or enlarged proportionately.

Did you know there are two kinds of mathematical tangents not even remotely related? A line just touching a curve is called a tangent line. And in a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side.

You have heard of right angles, but what about left angles? Actually, the original meaning of right meant correct or acceptable. As far back as the twelfth century, a right angle was thought as the angle formed by the intersection of horizontal and vertical lines. The word upright also reflects this meaning. Later, the right hand was so named because it was considered the correct, or proper hand. So, no, there are no left angles.

You probably thought a billion was always a billion. Although today it represents one thousand millions; originally, one billion was equal to one million millions. The meaning changed in the U.S. in the 1800s, but Britain officially didn’t change until 1974.

Summary

Yes, the word summary is derived from sum. It means we are summing up all our points. Introduce new words when needed. For example, nobody needs the terms numerator and denominator in order to begin learning about fractions. Use examples for new concepts, rather than a definition, especially for younger children. Therefore, watch your language.