She shouldn't have reported you to the school authority, thats not proper at all.i lobe the way you reacted. Extra lesson is by choice, personally, I don't do that because of the stress and drama attached, you will work extra and some parents won't even pay.

For the child, I feel sorry for her, you know every kid has their learning pace, I was also very dull while in primary school but then, God helped me out. Not for the mother but for love sake, draw the child to yourself and just help her in the way you can, I just have a feeling that she would be transformed if she is shown love.