So today as a starter I thought I’d do a recap on negative numbers. Students have starter books (see: lazy teacher) choc full of puzzles and basics revision. The one in question asked students to circle the correct answer for a set of about 20 negative number problems. Each one had four choices, and each choice was essentially calculated as so:

So above, the student should circle the bottom left box, etc.

I gave them around five minutes. During that time, I expected the occasional “aah I can’t remember how to do these…” to which I would reply “that’s why we’re doing them”. Sadly, I also frequently heard students desperately trying to recall the ‘rule’ they’d memorised (or rather, not memorised). “oh what is it again? A minus and a minus makes a plus. No wait, that’s for multiplying, a minus and a plus make a plus” “But what if it’s a minus minus a minus? That must be a minus then”.

This was going on all around the class. I’d love to say these were young students. They weren’t. These were high ability students at the twilight of their schooling. We had a brief discussion, and all sorts of random answers came flying in at me, with baffling explanations that weren’t really explanations at all.

So we started again. They were really annoyed at me for trying to wipe the slate clean. They had a method and they remembered the method and it WORKED.

I pointed out they didn’t have a method, they had a kind of rhyme, that didn’t work, nor had it been correctly remembered. They were still annoyed with me.

First I asked that they stopped using the word minus in any way shape or form. It was too confusing. If it’s a negative number, then say negative, not minus. Second, if it’s a subtraction, say subtract. That way at least we’re all talking about the same thing.

“But a minus and a minus”

I snapped a little. “Look, the reason you can’t do this has nothing to do with your maths ability. It’s because you’re trying to remember a phrase and that’s all you’re focusing on. Not one of you is even TRYING to think about it, you’re just trying to recall that phrase. So let’s please just stop, and think about what’s going on in the question.”

And it was absolutely true. They are such talented mathematicians, yet they became utterly fixated on a phrase, and lost all sight of trying to rationalise what they were doing or whether it made sense. Never has the detrimental effect of memorising a ‘trick’ been so crystal clear in front of me before.

So we drew a number line with just one negative on it, a zero, and one positive. We talked through which direction you go when subtracting a positive number, and therefore logically which direction would you go if you were subtracting a negative number?, then again for addition.

Sadly I’m utterly unconvinced that they were happy with this new ‘rationalising’ approach, and I fear I’ve confused them even more for challenging and dismissing their trick, which none of them got to work anyway. So, next lesson we’ll relent. We will get there. But what annoys me is that I’m the bad guy. I’m the guy who in trying to get them to see why minus minus blurg doesn’t work ( – 3 – 4 is not 1, yet it is often incorrectly recalled as ,a minus and a minus make a plus)Â is NOT that memorable, and negates all thinking and creates nonsense answers that nobody is even questioning.

I’m the bad guy!

I’m also faced with the horrible task of undoing bad learning, and trying to re-do it, in the face of students who are convinced they had it right.

And worse still, I stop myself and think “maybe I should just go through “a minus and a minus makes a plus” again, then they’ll remember it… for now. And we can move on.

I’ve posted stuff on this problem, but here is the simple version:

There are not “numbers, some of which are negative”.

There are numbers for describing position and for describing changes. These are the signed numbers. Simple two number operations (binary operations) are better seen as

a) -4 + 3 is “where do I get to if I start at -4 and move 3 steps to the right (or in the +ve direction)

b) 5 – 8, or better, +5 – 8, is “where do I get to if i start at +5 and move 8 steps to the left (or in the negative direction)

If they can be persuaded that this is what they are doing (it may be too late) then they should be able to create their own correct rules.

But how does that work with subtracting a negative? I really like it. Do you talk about inverses?

Yes, but indirectly. Moving 8 steps to the right is 8 steps to the right of my starting point (call it P) and shows as P + 8.

Moving 8 steps to the left is 8 steps to the left of my starting point (call it P) and shows as P – 8.

So for completeness’ sake we have to ascribe a meaning to -8 steps. There are only two directions, so the only really satisfactory meaning is -8 steps to the left is the same as 8 steps to the right, and -8 steps to the right is the same as 8 steps to the left.

Consequently P + (-8) = P – 8, which for some bizarre reason people find acceptable,

and P – (-8) = P + 8, which is where they feel that reality has gone out of the window.

There is another definition of P – Q, which is “how many steps is it from Q to P?”

So 8 – 3 gives 5 steps to the right (answer +5)

and 8 – (-3) gives 11 steps to the right (answer +11)

and -8 -(-3) gives 5 steps to the left (answer -5)

One of the problems with moving from numbers (those for measuring quantity) to signed numbers, which are those for measuring position, is that subtraction is a completely different operation physically.

And here is another method, the algebraic one:

You want 8 – (-3)

call it A (for answer !!!!!)

so 8 – (-3) = A

Add -3 to both sides and get

8 = A – 3

Add 3 to both sides and get

8 + 3 = A

so A = answer = 11

I figured this one out the other day, and kicked myself for not having done so half a century ago !

I teach the following three rules for adding and subtracting negatives:

1) Use a number line (and I emphasise that this is all you need if the second number in the question is positive).

2) Adding a negative makes a number smaller.

3) Subtracting a negative makes a number bigger.

(And I emphasise that rules 2 & 3 are the opposite of what normally happens when you add and subtract because negative numbers are in some ways the opposite of ordinary numbers)

Still huge difficulty remembering though, as they always expect a rule will tell them if the answer is positive or negative, not bigger or smaller. It’s always going to be hard stopping them from confusing addition/subtraction with multiplication/division but you are absolutely right to say that “minus and a minus” type rules make it worse.

The difficulty with signed numbers is partly due to continuing to use “bigger” and “smaller”, when “higher” and “lower” are better. Consideration of temperature, voltage, height above sea level etc shows this. “The temperature was quite a lot bigger this afternoon”.

Fair enough, but those examples are the ones they grasp best and I want them to understand these rules apply across the board. I want them to see 1 as bigger than -5, not just higher. I don’t want them to see it as higher and lower on a vertical number line, but as a property of the numbers themselves in all contexts. Too many kids can tell you 2 is higher than -3 as a temperature but, when you ask what is 5 more than -3, will then say “-8”.

I think that’s true. A friend of mine teaches addition and subtraction of negatives through a bath water metaphor. Add hot (+ ve) and you get warmer, take away cold and you also get warmer etc. I don’t like relying on metaphors before they’ve grasped the concept mathematically though. It’s a pickle…

” I want them to see 1 as bigger than -5,”

But it isn’t bigger.

Surely, the answer to the question “which is the larger sum of money, a credit balance of ten pounds or an overdraft of a million pounds?” isn’t ten pounds?

Check this stuff out. This is from A. N. Whitehead, collaborator with Bertrand Russell on Principis Mathematica, and was written in 1911.

https://howardat58.files.wordpress.com/2014/08/whitehead-intro-to-math-negative-nos.doc

Here’s another way, which I figured out a while ago:

How about this for “two minuses make a plus” :

step 1: 0 = 3 – 3 = 3 + (-3)

step 2: -(-3) = 0 – (-3) = 3 + (-3) – (-3)

step 3: take the second two terms together and get 3

since a – a = 0

This only uses the “subtract 3” means “add -3” definition, and the associative law of addition.