Welcome to the site.
I am Dr. Kaushik and i believe i can help you with your problem.
Well first of all it is very disheartening to see that you are being subjected to bullying and rude behaviour from your own grandchild and I can understand your frustration and dejection over this issue as no grandparent would feel comfortable when a 7 yr old shows such sort of contempt towards them.
However having said that I have a different take on this whole issue as I believe that at her age of 7 years she is not capable of differentiating between good and bad behaviour, reason being she was probably never taught about it by her own parents, especially her mother to whom she looks up to for guidance in many ways than one can imagine. You see for any girl of your granddaughter's age her mother is her role model and she tries to imbibe all those acts and traits which her mother either displays or asks her to carry out so much so that a girl child tends to ape her mother to greater extents and even harbors a dream to grow up and become like her mother.
So what I am trying to imply here is that in this situation the girl is not to be blamed as she being a child does not know any better and she plainly follows what her mother asks her to do or even if the child has inherent selfish and mean streak towards others but unfortunately her behavior is not being curtailed by her mother or father, instead she is probably being supported albeit silently behind the curtains by her mother. Well this very well can be a possibility as perhaps the child's rude behavior is just a way of her mother to get back to you and make you feel miserable for reasons best known to her or to you.
You see young children are like clay and it is up to the parents to give them appropriate shape and form meaning that if the girl has been showing irreverent behavior towards elders including you it is up to the parents to stop her and lead her by example and instill in her the right values and ethics of life and try to mould her personality into a pleasant one. Unfortunately this effort from her parents is missing in fact it seems like her mother has a silent role to play in her mis-demeaner .
Having said that you seem to be a balance headed, dignified, reasonable and self respecting lady, so thinking that you will see the abstract here in this situation I will thereby urge you to understand that it is futile to loose your cool when this seems more like a ploy to create a sense of unrest and apathy in your life through the bad behavior of the young child , so all I can say that you can just try to ignore her behavior because by feeling upset and dejected you are giving more stimulus to the child and her mother to seek pleasure by creating more misery in your life.Also talking about this to her parents seems to be of no use as they have turned a deaf ear to your requests, besides this should have come from them by their own will as the child is their responsibility and as long as the child is dependent on them her actions and behavior are reflected upon their upbringing and rearing tactics.
I hope this helps you to see the bigger picture, so try not to get worked up and feel upset and angry about this , as by doing so you will only make her behavior more profound as she seems to be doing this under the influence of someone else who either has some ulterior motive behind it or that person does not care at all what becomes of her own child.
Wish you all the best.
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I just would like to ask for one more point of assistance. Her mother is not rude herself, and though she may have an ulterior motive she occasionally reprimands the child, though weakly. The child is quite nasty to her also when she doesn't get her way. I understand that it is not the child's fault as she hasn't been taught. I will try not to "lose my cool" but I think I should tell her when her behavior toward me is rude (without screaming at her). I don't think I have to put up with this silently, and I am in dread of her being a teenager. I think her mother is treating her like an uncontrollable baby rather than a growing girl.
Yes I agree with you that the girl is not being reared by her mother properly as quite possibly she is treating her like a baby and not understanding that she needs to teach her proper etiquettes and social behaviour towards others as this is the right age to start on this because once the child grows then it is too late to resurrect the inappropriate behaviour.
So you may continue to point out her rude behavior but try to strike a conversation with her mother and tell her that it is about time for her to take a more stricter role as a mother and apply positive and negative reinforcements and punishments such as taking away her liberties or perks which she enjoys when she does not behave properly, grounding her or giving her time outs, appreciating her when she behaves well so as to provide her some thought on how to differentiate between good and bad behaviour. These tactics and strategies need to be applied by her mother on a regular basis so as to shape her personality and behaviour and make them pleasant and comfortable for others.
I hope this helps.