Judith Proctor's Journal

 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.