## Times Less

In Do the math – The Boston Globe columnist Jan Freeman dismisses objections to the common usage of “three times less than” to mean equal to one third of.

But the Merriam-Webster editors (per JF) are completely off base if they claim that “times less” has never been misunderstood.

Any math teacher regularly sees examples of such misunderstanding – all of which are perfectly excusable given the ambiguity of the expression. eg What are “3 times less than -1” or “0.75 times less than 1”? Or consider the following exercise: When is the statement “B is less than A but C is three times less” not an ambiguous definition of C?

Yes, the “times less” expression *is* used quite often, and usually that is done in consideration of a single relationship with a nominal multiplier greater than one applied to a positive multiplicand, and in those cases its intent is generally correctly understood as meaning to invert the multiplier, but that doesn’t make it the best usage. My main objections even in the restricted context are that it doesn’t extend well to more general cases, and displaces the use of equally simple mathematically appropriate language. In blunter terms, it is “ugly” and “ignorant” in that it lacks the elegance of extensibility and ignores both the operational and terminological distinctions between additive and multiplicative relations.

Many will say “but what do these things matter when most people correctly understand what is intended most of the time?” The problem is that habits of language can lead to habits of thought and when people who have been exposed to this usage try to grapple with concepts like the scaling of graphs they are less quickly successful than they would have been had the culture around them not encouraged such sloppiness.

Of course I am not talking here about those who will eventually be technical experts. They probably have the capacity to recognize, resolve, and learn from the language issue. But those whose grasp of mathematics is more marginal are more likely to become confused and end up less able to interface with technical information when they do need it.

Unfortunately, in our “advanced” technology-dependent culture, the exhibition or affectation of innumeracy is often paradoxically considered a prerequisite for social acceptance. This sometimes leads teachers themselves to use mathematically “ugly” or “ignorant” language in the hope that it will seem more familiar to students. But in the end this does the students no service as they later end up without the language skills necessary to express (and so to understand?) concepts that really they may need in future studies.