I’ve been thinking a lot recently on the language we use when we are teaching mathematics. This was partly prompted by a thread I saw on the TES maths forum discussing the use of the phrase “smaller than” to describe relationships between negative numbers. The discussion became quite heated as people argued their point, but one point that cropped up which resonated with me is that the most important people in a discussion like this are the learners – as teachers we must equip the children in our care to function in the real world and have a good grasp of mathematics.

For me, clear unambiguous language and definition is one of the most important things a mathematician does. If we all mean something different, we can never agree on our mathematics. In all mathematical communication it’s important to understand what everyone else understands by the terms we use. A well-documented problem, researched by many finer minds than me, is that we often use words that have a meaning in English as well as in mathematics. One example of this is the word “multiply” – in English usage this word has connotations of growth, which goes some way to explaining the common misconception multiplication always makes the answer bigger.

On to the less than/smaller than debate. I risk revealing my vast ignorance about primary mathematics and the way my primary colleagues work, but I would suspect that when children first meet the symbol < it is in the context of ordering positive whole numbers. If we are thinking about comparing measured quantities in the real world, “smaller than” is a very natural phrase to use. Alison is 1.70m tall, Sam is 1.87m tall. 1.70<1.87 so Alison is smaller than Sam.

However, I am uncomfortable about using the phrase “smaller than” when comparing negative quantities. Maybe my unease is unnecessary, but I worry that it may be ambiguous to some children in a way that “less than” wouldn’t be. “Less than” can clearly be defined to indicate that a number is positioned further to the left (or down if we’re working vertically) on a number line. I would rather not define “smaller than” in the same way, because to me, “small” is for talking about real world, positive things. I can imagine contrived examples about one person’s debt being smaller than another, and I often speak of behaviour near an asymptote at x=0 by talking about x being small and positive, or small and negative.

People will disagree with me, I’m sure. But my own personal opinion is that once children’s understanding of the concept of < is secure enough for them to be considering ordering negative numbers, we should be insisting on precise, clear unambiguous language and reading the symbol as “less than”.