When it comes to mathematics, words are essential. A mathematician may solve numerous equations, but they need words to explain the process and the answer. Therefore, it is vital that we, as math instructors, pay careful attention to the words we use as we teach math. Otherwise, our students could easily become confused by the mathematical meaning of words and be forced to unlearn incorrect terminology and definitions.
Words for Quantities
The names of quantities are typically the very first mathematical words a child learns. The words for quantities one through ten seem random to the young child. Hundreds of years ago, to help make sense of counting words, people began to group quantities into manageable sets rather than having a long list of seemingly unrelated words.
Romans began the process of ordering quantities by recording numbers by grouping quantities into fives, V, and tens, X. Because they did not have a symbol for the intermediate numbers, such as 2 and 3, they doubled and tripled the symbol for 1. For example, 1 was recorded as I, 2 was recorded as II, and 3 was recorded as III. They continued this pattern when working with the tens. For example, 2 tens was recorded as XX, and 3 tens as XXX. Additional symbols were used for 5 tens, L, to write larger groups of 10. For example, 5 tens was recorded as L, 6 tens as LX, and 7 tens as LXX. Using this method, only four symbols were needed to write numbers 1 through 99. When recording larger numbers, additional symbols were used. For example, C was used for 100, D for 500, and M for 1000.
The early Roman numerals used the basic system of recording numbers by representing 4 as IIII and 9 as VIIII. Over time, the Roman numeral system became more concise by writing 4 as IV, which stood for one less than five, and by recording 9 as IX, meaning one less than ten. While these numbers were more succinctly written, they made calculations more difficult. Interestingly, today, clocks with hours displayed as Roman numerals will have the early Roman numeral IIII for the hour of 4 but the later Roman numeral IX for the hour of 9.
The introduction of the Hindu numerals brought about a significant improvement in recording and calculating quantities. Each number from zero through nine had a specific symbol. These digits were repeated for values larger than ten. This system used the placement of the digit within a number to determine its value, now known as place value. Unfortunately, the Indo-European words used for some of the numbers do not include the same simplicity and clarity as the written symbols. Because of the confusing, inconsistent pattern of the number names, many children today find it difficult to associate the number name with its corresponding symbol.
The East Asian languages modified their language to make the pattern of numbers more obvious. They changed their number words to be more consistent with the Hindu number symbols. For example, eleven was named ten-one, twelve was named ten-two, twenty-three was named 2-ten 3, and forty-seven became 4-ten 7. Children who speak East Asian languages have a significant advantage in understanding place value than children who speak other languages. Studies have shown that English-speaking children can benefit from learning and using transparent number words for a brief time. For example, ninety-eight would be temporarily called 9-ten 8, the transparent number name.
Sometimes, certain words are used to help students learn a concept, but the words chosen use can create confusion and lack of understanding. Other times, completely incorrect words are used. Let’s explore some of them.
The term number sentence is a regularly used expression that bothers me. According to its definition, a sentence is a group of words that makes a complete thought. If so, how does the equation 2 + 3 = 5 satisfy that definition? The term number sentence can confuse the learner when learning math and language arts. One time, a third grader was asked to write a number sentence. So they wrote, “Two plus three equals five” because they understood a number sentence to mean words, not an equation.
On the other hand, the definition of equation means to make equal. A fundamental principle of mathematics is equality. Fortunately, the misguiding term, number sentence, is finally vanishing from textbooks and tests.
Another commonly used term that annoys me is take away. One reason this term bothers me is because it uses poor English. Let’s look at this example, “seven take away five.” To make this a proper declarative sentence, take should actually be takes, changing the statement to say, “seven takes away five,” which is quite a daring feat for the number seven. If the speaker wanted to make this an imperative sentence, there should be a comma after the seven, such as, “Seven, take away five,” making the number seven sound brash. The second reason that the term take away doesn’t sit well with me is because in England, takeaway is fast food.
Mathematically speaking, a complete understanding of subtraction is limited when the child only thinks of subtraction as take away. Subtraction is not just about taking a quantity away. It also includes comparison, addition, or finding a missing part. Fortunately, this phrase is also being removed from textbooks and tests. So, what is the correct mathematical term for take away? The correct term is minus. So instead of saying, “Seven take away five,” we should say, “Seven minus five.”
Another word that should never be used is timesing. Timesing is not a word and will never make it into the dictionary. When people use this term, they are referring to multiplication. However, timesing is a childish nonword, and yes, nonword is a real word. Instead of saying timesing, we should use the terms multiply and multiple, which are genuine, bona fide mathematical words.
For example, when saying this expression, 3 x 2, the reader should say, “three multiplied by two” or better yet, “three taken two times.” Even saying “three times two” doesn’t adequately describe what is happening in this expression. Unfortunately, saying times instead of multiplied by started in the 20th century and has caused some children to be confused because they associate time with clocks.
There is another reasonably new word used when teaching elementary arithmetic that can add to student confusion. That term is regroup. The definition of regroup, according to the dictionary, is what a military unit does after a defeat in battle. As adults, we often think of this word as to re-group or to group again. However, children learn that this word refers to a process, not how to create an equality. The term regroup is not a natural word children use. You do not see children regrouping their toys and discussing how they regrouped them.
Why can’t we use the old-fashioned words of carry and borrow? These mathematical words still work perfectly fine. They are legit terms and continue to be used by programmers. Those who disapprove of using the term borrow argue that to borrow implies that something needs to be returned. However, this is not always the case. For example, languages borrow from each other all the time. If using these terms bothers you, trade is an even better word to use in place of carry and borrow. Children understand the term trade, which also includes the idea of equality.
Words for Geometry
Many times, young children are given non-mathematical words for shapes. For example, children are taught to use the word oval instead of ellipse. However, an oval can describe the shape of an egg or a running track. Another non-mathematical word children are taught is diamond, instead of the mathematical term rhombus.
Another concept children learn that causes problems later is that rectangles are specifically wide and short. When taught this way, children do not realize that a square is also a rectangle.
Textbooks often teach how to find the area of a rectangle by naming the sides l and w, length and width, respectively. Then, when textbooks instruct on how to find the area of a triangle, the sides are now labeled b and h for base and height. It would greatly help students understand the relationship between the area of a triangle and the area of a rectangle if the sides had the same names. If the term width is used to show the distance from side to side, and height is used for the perpendicular distance to width, students would better understand area and the relationship between shapes.
While talking with a senior honors student, I mentioned that a square is a rectangle. In surprise, she said, “They are? But they have different formulas!” You see, her textbook used l and w to notate the sides of a rectangle and s to notate the sides of a square. No wonder our students struggle with finding area!
Words with Oddities
Diagonal and Similar
The words diagonal and similar have very different meanings for math and everyday use. When the word diagonal is used in everyday language, it refers to a line that is not horizontal or vertical. It could also refer to a road that does not head north and south or east and west. However, the mathematical definition of diagonal refers to a line drawn between two non-adjacent vertices in a polygon. Using the mathematical definition, a diagonal line can be horizontal or vertical, depending on the rotation of the polygon.
The everyday meaning of the word similar is also quite different from the mathematical meaning. The common meaning of similar means that it is almost the same, but not exact. However, in mathematics, similar means that the line or shape is identical but proportionately larger or smaller.
More confusing terms? In math, the word tangents has two completely different meanings. A tangent line is a line that just touches a curve. However, the tangent of an angle refers to the ratio of the opposite side and the adjacent side in a right triangle.
You have heard of right angles, but is there such a thing as left angles? An early definition of right referred to being correct or acceptable. Even in the twelfth century, angles that formed an intersection of horizontal and vertical lines were known as a right angle. The same idea is reflected in the word upright. As time progressed, culture determined that the right hand was the correct or proper hand to use, so they named it the right hand. To answer the previous question, while we do have left hands, there is no such thing as a left angle.
The word billion has always been a billion, right? Today, one billion refers to one thousand millions. But initially, one billion was one million millions. In the 1800s, the U.S. changed billion to the current value of one thousand millions. However, it wasn’t until 1974 that Britain made the same change.
In a Word: Summary
Indeed, the word summary comes from the word sum. So, let me sum up my points about mathematical words. As instructors, we should introduce new mathematical words when needed, but not before. For example, the terms numerator and denominator should only be taught after the student has learned a little about fractions. We should not teach a new concept along with new terminology. When introducing a new idea, let our students, especially the younger students, see examples of the concept before teaching the definition. Once students understand the concept, the definition makes sense.
So, watch your language!
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