## Tuesday 2 April 2013

### Minus versus Negative: Some Mathematical Grammar

Ooh, today you're getting a discussion of maths and grammar, aren't you lucky?

We had a "moderation day" last week. It's an INSET day where teachers moderate their coursework. As you can probably imagine, the maths department was incredibly swamped that day. NOT!

Maths doesn't have coursework, so theoretically, we didn't have anything to do. In practice, however, we had absolutely loads to do, because we are still teachers, and a teacher's work is never done. That sentence has far too many commas. Should I really be writing a post about grammar? You can always put bad grammar down to style, can't you? My style is to use too many commas. Like, this.

Anyway, the maths department decided to take a long lunch on this moderation day, and went to a popular pizza restaurant armed with multiple two-for-one codes. The joke "how many maths teachers does it take to split a restaurant bill?" comes to mind.

We spent most of the meal maths debating. This often happens to us. Luckily the restaurant was almost empty, or it could have been quite embarrassing. We were scrawling equations on napkins using board markers (the only pen we ever have on us) by the end.

The subject of the debate? Should the word "minus" only ever be used as a verb?

Read the following sentence out loud: The weather today is -2 degrees. How did you say it? Did you say "minus 2 degrees"? Or did you say "negative 2"? I would guess that if you are from the UK you probably said minus. I know that's what I say. It's definitely what the weather people say on TV.

Now read this out loud: x - 5 = 7. Did you say "minus" again? You might have said "take-away", possibly "subtract", or even "less".  I would say minus, probably because this is what all of my maths teachers used to say to me.

Third test: what rule were you told about why -5 x -2 = 10 not -10? Say this rule out loud. Did you just say something like "a minus times a minus makes a plus" or "two minuses makes a plus"?

Can you see a slight issue? We're using "minus" as an adjective, meaning negative, and we're also using it as a verb* meaning subtract. And thirdly, we're using it as a noun when we say "a minus" meaning a number less than zero.

We're all quite comfortable with this word having several meanings. But what about students? When they first learn about "directed numbers" (as they're known), does this odd quirk of English confuse them?

I can see why some might think this. I understand that the language of mathematics should be used very carefully. I've always been very interested in grammar, which was why I did A level French (which, incidentally, everyone thought was weird: most of my teachers assumed I would be studying maths and the three sciences). At uni I took some modules that were about logic, which is pretty much just another word for language. I've taught enough EAL students to know that you need to choose your words carefully. But to be honest... I'm not completely convinced.

In a "number sentence" (a wonderful expression, thank you primary school teachers), I will always pronounce a dash (or hyphen, or em dash, or en dash) as "minus". Let me tell you why: that little symbol represents two things at once; it's the operation of taking away, and it's also to indicate something that is being negated (notice that both of these things are actions: I'm not saying it represents a negative number, I'm saying it represents something that is being negated). You absolutely need this symbol to represent both at once, because you want to be able to swap between the two meanings depending on how you feel.

Take this example:

3 - 4 (x - 7) = 10

If you wanted to solve the above equation, there are a few ways you could do it. Before reading ahead, please solve it.

How did you treat the minus before the 4? Did you see it as indicating something you're taking away from 3? Or did you see it as attached to the 4, making it a negative 4?

Did you do this:
3 - [ 4x - 28] = 10 (expanding the bracket with 4 as the multiplier)
31 - 4x = 10
etc

or this:
3 [- 4x -- 28] = 10 (expanding the bracket with -4 as the multiplier)
3 - 4 x + 28 = 10
etc

Or something different?

Can you see that if I, as a teacher, had indicated in some way that the minus before the 4 was a negative symbol, the first method wouldn't really make sense? And if I had indicated it was a take away, the second method doesn't really make sense, because students aren't taught that subtraction follows the distributive law.

The duplicity of the minus is one of those mathematical things that makes sense when you are mathematically fluent. Just like in English, how we have words that look the same and sound the same but mean two different things. As a fluent speaker of English, I don't even notice these. Look, I just used one! Notice! I didn't have to think: wait, is this the verb to notice, or a kind of sign stuck on a wall? I just used the word. And guess what, when I learnt French, my professeurs didn't just remove all homophones from the syllabus so that, as a learner, I wouldn't get confused, they left them in, so that I could aim to become fluent. Why should maths teachers do this? Don't we want our students to become fluent in maths?

Yes, we should be careful with our language in maths lessons. We should make sure when we say "line" we don't mean "line segment". But we cannot protect our students from the difficult to understand bits. We need to expose them to these things.

What do you think?

Emma x x x

*Technically it is a preposition rather than a verb. But these days we use it as a verb, saying things like "minusing" and "you minus the five from both sides". I know technically these uses are wrong, but it's what we say. Just like how we say "timesing" and "timesed" because we use the word "times" as a synonym for multiply now.

1. You know my thoughts Urma... negative for a 'thing' and minus for 'something that is going to happen'.

So a number / quantity is never ever ever a minus it is a negative or positive. We are allowed to omit the positive at times.

If we were to not omit the positive 'ever' this would clear the matter up easier I believe.

Mr 'Oats so Simple' LOL

1. Gee, I wonder who you could possibly be! As if calling me "Urma" wasn't enough of a giveaway!

Interesting thing about missing off the positive. Perhaps we should always say the "positive" to avoid confusion.

2. PS:
Perhaps we should always write little + and - to instruct the direction of the number 'when students are learning any rules' once they have mastered it, they can be more slick with leaving them off where they are not really needed as the serve their mathematical apprenticeship and become more competent with rich understanding.
From early years... (+3) x (+4) = (+12) [whilst learning to master]
Only when truly mastered by students should they begin removing unnecessary signs?
I.e. it becomes 3 x 4 = 12.
We should teach direction immediately with positive sign being closed in with brackets [to help master] and not just when negatives are put on the ‘learning table’.

MR ‘O.S.S.’
Chris

1. I think this could possibly work but you're right, they'd have to be little signs, not the same as we use for adding and subtracting.

2. Yes I quite agree Urma. Again using a different style of - and + signs for minus / plus and negative / positive would help to prevent a misconception in their use.

Chris

3. I agree with everything you've said Emma - but then I usually do (after you've made me)!

I like the way you've highlighted that it's not always as simple as simply using "negative" for a number and "minus" for the action, as it's sometimes unclear what it is even to ourselves.

I'm sure even Mr. OSS can see the difficulties.

By the way - I'm with you, I (nearly) always use the word minus for both, but emphasise using "negative" in some situations when it is absolutely clear that we are dealing with a negative number.