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I have exactly the same problem with my Mum - 84 and no dementia - who thinks that at 60 I shouldn't be wanting to still grow my hair to my waist (it's almost there, but not quite - thanks to a scissor-happy hairdresser
a couple of months ago).
I told Mum bluntly it's my hair and I'll wear it how I like and it's none of her business what length I have my hair. Had I not been this blunt with her she'd have gone on and on every time I see her about the length of my hair, so we now agree not to discuss this subject.
And you're quite right, cobden - it is none of her business.
My daughter would tell me the same if I tried to insist she had her hair cut. Being a mother does not give you a right to bully your children, especially once they are adults. And adult children should not allow themselves to be bullied.
However, if dementia is in the mix, it is harder. I agree with the others that you should adopt the least confrontational stance and say you will 'think about' making an appointment next week. That may stop her continuing to argue. Keep it vague so you're not actually lying, as I know you're not comfortable with that. *
Chances are she won't remember the conversation and then you say the same next time you visit.
Of course, it may be she is going through an argumentative phase, as my mum did, and she'll simply pick up on something else. There were times when the simplest conversation with Mum turned in to her getting in a real strop with me over the mildest comments and then I used to just walk away until she calmed down.
Arguing back simply stokes the fire, is upsetting for both parties and is ultimately pointless.
*This is similar to advice given to new mothers who are advised to smile sweetly and say thank you when their mother in laws tell them how they should be bringing up the new baby. Nod, don't commit and do it your own way.