Q: Is it OK to use the phrase “less than” when teaching numeracy in elementary school? Example: “What is one less than five?” I suspect that many children confuse “less than” (meaning “smaller than”) with “less” (meaning “minus”).
The word “less” has had many meanings since it showed up in Old English in the ninth century, and one of the oldest, dating back to Anglo-Saxon times, involves its use as “minus” in subtraction.
However, an even older meaning—the oldest example for “less” in the Oxford English Dictionary—is “fewer,” a usage that was acceptable for hundreds of years but is frowned on today.
The “fewer” sense of “less,” which the OED observes is “freq. found but generally regarded as incorrect,” first showed up in writing in King Ælfred’s Old English translation (circa 888) of Boethius’s De Consolatione Philosophiae:
“Swa mid læs worda swa mid ma, swæðer we hit gereccan magon” (“So we may prove it with less words as with more, whichever of the two”).
Over the years, as we’ve said, “less” took on other senses, including “to a smaller extent” (c. 900), as in “none the less”; “inferior” (c. 950), as in “no less a person”; “not so great an extent” (c. 1000), as in “less time to eat”; and “a smaller amount” (c. 1330), as in “less money.”
The use of “less” to mean “minus,” the OED explains, indicates “that the number or quantity indicated is to be subtracted from a larger one mentioned or implied.”
This sense of “less” first showed up in writing in the Anglo-Saxon Chronicle, an Old English work believed to have been updated regularly from the late 9th to the mid-12th centuries.
Here’s the citation, which the OED says was written sometime before 1160: “He rixode twa læs .xxx. geara” (“He ruled for 30 years less two”).
In early writing, “less” followed the number being subtracted (as in “twa læs” above), but it now precedes the number (as in “less two”), according to the OED.
All the modern examples in the dictionary show the unsubtracted quantity (the “minuend”) followed by “less” and then the amount to be subtracted (the “subtrahend”).
This modern example is from the March 25, 1930, issue of the Times (London): “A full year’s dividend on the Preference Shares, less tax, absorbing £16,800.”
The latest OED example of the usage is from the Sept. 2, 1972, issue of the Times (London): “Cost of paint … Less VAT input tax … £500.”
We also checked six standard dictionaries and all their examples show “less” by itself following the minuend and preceding the subtrahend. Here’s an example from The American Heritage Dictionary of the English Language (5th ed.): “Five less two is three.”
Getting back to your question, is it OK for a teacher in primary school to ask pupils, “What is one less than five?”
When we were learning subtraction in elementary school many moons ago, our teachers would have said “What is five less one?” (or “five minus one” or perhaps even “five take way one”).
We think “five less one” or “five minus one” is the simplest and clearest way of expressing “5 – 1” in words, because the words follow the order of the numerals and the minus sign. And the use of “less than” here might lead to confusion between the minus sign (–) and the “less than” sign (<).
A perfect example of such confusion can be seen in an Aug. 19, 2001, question to the Math Forum, a website sponsored by Drexel University in Philadelphia:
“Why is this expression driving me crazy when at first it seems so simple: ‘three less than a number’? I believe it is x – 3 but I am being challenged that it is 3 – x.”
The response by the Math Forum’s staff includes this comment: “Many people get confused by this sort of expression, because they expect to translate directly from English to Mathish, word for word. Then ‘three (3) less than (–) a number (x)’ would seem to be ‘3 – x.’ But it isn’t. What’s even more confusing is that ‘3 less a number’ does mean ‘3 – x’ because ‘less’ as a preposition means the same as ‘minus.’ ”
We’d be wary of using “less than” in teaching subtraction to young children. But our online searches suggest that elementary school teachers generally distinguish between the use of “less” and “less than.”
From looking at educational websites that discuss basic subtraction, our sense is that the traditional wording (“five minus one” or “five less one”) is used in speaking about actual subtraction. The “less than” wording is used to compare two numbers, rather than to subtract one from the other (“four is one less than five” or “one less than five is four”).
Despite the possible confusion between “less” and “less than” in teaching subtraction, educators have been using “less than” for comparisons for nearly two centuries, according to our searches of online databases.
Here’s an example from A Manual of Instruction for Infants’ Schools, an 1829 book by William Wilson, the vicar of Walthamstow in northeast London: “Four are one less than five; four are two less than six; four are three less than seven, &c.”
And this example is from A Manual of Elementary Instruction for Schools and Normal Classes (1862), by Edward Austin Sheldon, M. E. M. Jones, and Hermann Krüsi: “The class may repeat, ‘Five is one more than four; four is one less than five.’ ”