You may remember that way back in May I wrote about the language we use in maths. Back then, I was thinking mainly about the importance of using unambiguous language, but I’ve been thinking about language again, particularly words which mean something specific in mathematics but also have an English meaning in everyday conversation. (I don’t know enough about languages other than English to know whether maths-specific vocab is shared with a more general usage, but I’m interested in examples people have to share, in any language.)

The word that got me thinking about this is a very simple one, “or”. I was editing an article for secondary school students about tree diagrams, written for us by one of our NRICH summer students this year, and part of the article dealt with “the probability of A or B”. I felt the need to highlight the mathematical usage of “or” which means A or B or both, because many students on meeting such a statement might assume it to be an exclusive or which doesn’t include the “both” option. It reminded me of a sign I saw outside a cafe at the weekend:

Toasted Teacakes or Crumpets with regular Tea or Filter Coffee for just £2.25

This common English usage of the word “or” strongly hints at an exclusive or; if I ordered teacakes AND crumpets, with tea AND coffee, I’d expect the bill to come to £4.50 rather than £2.25.

There are many other examples of words which mean something unexpected when used in a mathematical context – for example the word “expect”. (In case you can’t tell, a lot of my work this month has been thinking about probability problems!) I realised I was throwing in the word “expect” without ever giving a mathematical definition of expectation. Does it make sense to someone with a layman’s understanding of the word that we “expect” to take two flips of a coin to get a Head? Or do we need to take the time to explain that half the time we get it on the first flip, a quarter of the time on the second flip, and eighth of the time on the third flip, and so on, which averages out to two flips?

Part of the problem is familiarity. Those of us who are conditioned into the world of mathematics can slip easily between our subject’s specialist vocabulary and the same words used in a different sense. But whenever we are working with those who have not yet mastered the vocabulary, we may need to be explicit in making the distinction between the words in their common usage and the specific mathematical meaning we are attaching to them.

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September 20, 2010 at 17:49 |

Just for extra fun, you can come up with meanings of “or” that match the mathematical meaning in common parlance. Consider, “You’ll be OK as long as you like crumpets or teacakes,” – you won’t exactly have a problem if you like both, unless you’re extremely indecisive.

September 20, 2010 at 17:58 |

You are so right, Alison. I often find myself qualifying language when I work with children. Perhaps that’s because they’re mostly younger pupils, although I’m sure I’m guilty of assuming a shared meaning on many occasions.

It’s good to remind me of this the evening before I spend two days in a primary school working with Y1 to Y6!

September 20, 2010 at 21:12 |

Good point, Nick, although as an extremely indecisive person it’s lucky that I have a distinct preference for crumpets.

Liz, perhaps primary teachers might find this easier than secondary, because they may not be maths specialists so may not be so accustomed to thinking of the maths meaning first – I often forget that the words I’m using have other meanings at all! But I think we all need reminding that maths has its own special vocabulary and precision in meaning helps us to avoid ambiguity and misconceptions.

September 20, 2010 at 22:04 |

When teaching A-level to a group of Chinese students, I tried to keep an eye out for words which had more general meanings outside mathematics: I remember simultaneous and similar as examples. Because they were learning the mathematical use first, there was a risk of them being confused by non-mathematical usage.

On the subject of “or”, my husband was nearly denied a railcard because he couldn’t prove his age: he could prove that he was a student, but the employee wanted him to prove either that he was under 25, or that he was 25 or over and also a student.

September 27, 2010 at 09:01 |

Well Alison I have found you “OR” have I?

I know that with the youngest of our pupils its always good to get some instant feedback as to what they think I mean, so that I can clarify the sentences for them. I have a feeling that these situations are probably more common in English than in other languages ??

October 23, 2010 at 21:19 |

Often when using “or” in programming, you find that the implementation uses “shortcut evaluation”: what that means is that if you have a OR b OR c, and you find that A=TRUE, then you don’t bother to even LOOK at conditions B or C, as it doesn’t matter whether they are TRUE or FALSE.

So, definitely a non-exclusive OR. If you want exclusive or, you use XOR.

Btw, short-cut eval works with AND too, as soon as you find a FALSE…

Enjoying the blog, keep up the good work!