How were you taught negative numbers?

I am still extremely confused by what your are trying to say in the OP


However ... in the spirit of trying to help ... one of the things I do when adding or subtracting is to use hand gestures

moving my hand up or down to indicate + or -

making sure that I ensure my hand is very low when starting or ending on a negative value

Counting. For 4+2, start at 4 and count 2 more. Similarly for subtraction.

But when subtraction on negative numbers come along, everything just flies out the window.

That worked for the very basic stuff and quite well indeed. Until I get to things like $5 - (-3)$. Their minds go completely blank.

That worked for the very basic stuff and quite well indeed. Until I get to things like $5 - (-3)$. Their minds go completely blank.

That worked for the very basic stuff and quite well indeed. Until I get to things like $5 - (-3)$. Their minds go completely blank.

I find using temperatures can help.

For 7 - 12: "It is 7 degrees and it has become 12 degrees colder. What is the temperature now?"

For -3 + 6: "It is -3 degrees and it warms up by 6 degrees."

For -10 - 4: "It is -10 degrees and it gets 4 degrees colder."

It's a number line in disguise but the context can help.

For 2 - - 3: "What is the difference between a temperature of 2 degrees and -3 degrees?"

"Subtracting a negative is adding because you can take the two sticks and make a plus out of them"

Just kill me now.

For that one really you just need to drill in that two negatives make a positive, it's not particularly mathematical but kids won't understand if you try to explain why that's the case.

I learnt it using a number line at first - my teacher in year 8 wasn't very good so he just gave us a set of rules straight away which confused everyone.

Pretty much every student I know can accept that "subtracting a negative number means add"

I never say the "2 negatives make a positive as that causes confusion"

Maybe a sort of 'vector' representation would help? Like, represent numbers as arrows from 0 to where they are on a number line, then they can see how adding a negative number compares to adding a positive number and such like.

I properly got a hang of them when I realised [-3 - 2] was actually [(0 - 3) - 2].

Depending on how algebraically competent your students are, a good way of teaching two negatives is as follows:

$(-1)(2-2) = 0$

$(-1)(2) + (-1)(-2) = 0$

$(-1)(-2) = (1)(2)$

Intuitively I remember examples being used of debt when I learned it.

Astonishing. Do you really think that someone who struggles with 2 - 3 would have the slightest chance of comprehending this?