Words and Mathematics, by Dr. Joan A. Cotter
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. Roman Numerals 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. Hindu Numerals 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. Wrong Words 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. Number Sentence 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. Take Away 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.” Timesing 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
