Visualising negative numbers

[watervole]

 I have a math pupil who came to me because she'd slipped through the net at school.  She isn't stupid, but she locks down into virtual silence if she can't understand something and stops paying attention or making any real effort.

I've made quite good progress with her in several areas, but negative numbers are a major stumbling block.

Her basic approach to maths is to learn a METHOD and ruthlessly apply it to everything.  But she doesn't try to understand what she's doing or why.
 

When she thinks about things, she can actually do them, but as soon as she's worked out a METHOD she stops trying to think about what she's doing.

Her written arithmetic is great, because she has rules to follow. Her mental arithmetic is terrible - she will default to counting on fingers  (we are gaining, but she's very resistant to new techniques and she missed out on learning how to use number bonds to ten to aid in adding numbers like 17+6  - you take three from the six to get you to 20 and then add the remaining three to get to 23 ).

But she has decided that 'minus' means 'take away' and therefore -16 - 4 has to be -12... And similarly with other related problems. In spite of all my efforts to get her to focus on a numberline and the direction of movement for positive and negative, she will work out a METHOD, say 'find the difference between the two numbers and slap a minus sign on the result' and apply it to any question with a minus sign.

 

She doesn't try to visualise anything - which means I really have to work with her as soon as a problem begins 'Abigail has six rabbits and each rabbit has 600 grams of straw' etc.  (she's improving in this area, but will still randomly multiply/divide the numbers in the question unless forced to slow down and read it carefully)

 

How can I get he to visualise negative numbers in some form that will make sense to her?  (She's just turned 10, and she loves wildlife in every form.  Stubborn but not stupid)

 

Temperature doesn't work very well.  She can accept that temperatures go negative, but if I say the temperature is -6 degrees and it drops by 3 degrees, you can't guarantee that she'll see that as -9...

We discussed negative heights below sea level, and money owed to banks, but neither seemed a model she was happy with.

The basic problem is that she cannot accept that 'minus 3' is a number in it's own right.

 

I'd appreciate any ideas that people have.

 

Comments

[eledonecirrhosa]

Debts and owing people money, perhaps?
EDIT: oh dear, I've just re-read and seen you mention banks. What about her personally owing money?

[watervole]

I tried that, but it didn't really help.

She sees it as a number but not a negative number.

[bugshaw]

Hmm, for a method, "if a sum starts with a minus sign, put a zero in front of it"?
-12 -4
is
0 -12 -4

Otherwise I'd try something like a numberline but on a y=x slope to give another sense of direction.

[watervole]

I started trying something a bit like that with a 0 at the front. Like you, I thought it might help, but I got met with the stony wall of silence....

I think she saw it as adding an illegal third number to the sum...

I quite like the idea of a sloping number line. I might be able to use that - might be able to rotate it various ways to do things like height above/below sea level.

I'll try it on Thursday.

[rpdom]

How about the concept of a lift in a tall building that also has many basement levels. Call ground floor "0" and floors below that "-1", "-2" etc. Then subtracting is going down and adding is going up.

It's really just a number line turned 90 degrees, but might be easier to visualise.

[watervole]

Ooh, I like that one. If she's ever been in a lift that has -ve floors she might get it.

Our university had levels down to -3

[elisi]

So, I've been pondering this, because I find these things interesting and I'm a very visual person... It's sort of echoing everyone else, but I thought that maybe Minecraft could be a good way of visualising the concept. Building blocks on top of each other for addition, and digging holes for subtraction.

I even drew a very basic little diagram when I was having my breakfast this morning. :)

If she could get herself a METHOD where + means going up and - means going down, that should work? So the more minus, the further down she goes.

[feng_shui_house]

I can empathize. I have the same problem. Mostly I get around it by either using a calculator, or else working the problem on paper several times to check I haven't had a OOPS moment. Really not fun when balancing a checkbook!

It's not that I don't WANT to visualize math the way most people do, it's that I can't. My brain skitters away from it and wants to 'short-cut' and refuses to concentrate.

And yes, I do count on my fingers.

Every time I come up with a rule/method, there are situations that don't apply. Wish I could help.

The best visualization I can think of would be a ladder- because I'd be ON the ladder climbing and descending... don't know if that's any use to your student.

[tictactoepony]

I've been thinking about this - how about using lifts in a department store as an analogy?

e.g. She enters at floor zero. If the lift goes down 2 floors, then down 3 more floors, how many floors down is she?

[watervole]

Probably only work if you go into the basement. It's the final negative number that is part of the issue.

[supermouse]

How does she do if she has a diagram she can literally see, labelled -10, -9... 9, 10? Does having it as a thing she can see in front of her make any difference? How is she on graphs with positive and negative values on the graph on both axes? Plotting it out and writing it down might become METHOD, but then I still multiply large numbers by drawing a table, which was the only method that ever stuck for me on that one.

Otherwise, there's repetition and memorisation maybe, to get a table of values as a mental reference? That's how I do C/F conversions... badly, but I can't ever handle remember the formula to multiply, so I have to make a rough chart and guess from memorised values.

[watervole]

We started out with a numberline, and she was fine with it, until she worked out a METHOD for doing it in her head without the numberline.

Unfortunately, because she didn't fully understand what she was doing, or what it represented, she started applying the METHOD (I capitalise, because she really loves her methods) in all the cases it didn't apply to (ie. adding two negative numbers)

I'm forcing her back to the numberline, and telling her that she has to use it for all negative number problems even if she wants to shortcut.

[supermouse]

I love the way you describe the thinking process and hope you manage to chip out the misconception.