I have 2 boys, ages 8 and 9.
this is a hard one, I had the same issue with the stealing, I made her write a note to say sorry and that embarresed her i think she learned
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I'm sorry that you're going through this with your kids.
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One of my son's friends is a liar. I call him on it all the time. I have told him that since he lies so much I do not believe a word he tells me or Philip. He could tell me the sky is blue and I am going to ask someone else because he isn't trustworthy enough to believe. He is constantly backpedaling with things he tells my son because he knows I don't trust him. Philip doesn't believe him like he use to. The kid's parents do not realize how bad or how often he does it and will pass it off as stretching the truth or bragging.
If my child was lying all of the time or doing it so well that I couldn't tell when he was telling me the truth, I would take away the privileges he got for being 9. He'd lose his video games, etc. It wouldn't be for a day or two, it would be lost until he'd proven to me that I could trust him. I would be having daily email conversations with his teacher so he/she knew what I knew and didn't know about what was going on at school. He would not be going over to a friends unless I had talked to the parents. In a lot of ways I would be mean about it, letting his teacher know he lies to his mom, letting his friends' parents know they have to talk to me about things because I can't trust my son to ask me or tell me things. I've told Philip that I will be his advocate but I need to know the whole story because if I go to bat for him and find out he kept stuff from me, I will not be able to do it again even when I know he's been treated unfairly. No one is going to believe me. No one is going to believe him either.
We've talked about how it is not a good idea to lie to people because they lose their trust in you. I've brought up a class mate of his who is always in trouble. The kid lies, makes up stories to cover them and basically causes problems with the other kids. The teachers, principal and yard duty do not believe his side of things any more. The other person could be completely in the wrong and gets believed. If he is involved in anything it is automatically believed he caused the problem or made it worse.
As far as the stealing goes, he'd get one chance to stop or he would be talking to a cop about what will happen to him if someone other than mom caught him. Because he's tried this twice, one time succeeding, if it was my son, I might consider having a friend's husband who's a cop have a conversation about him, out of uniform.
Oh my gf solved the forgotten home issue by telling her kids that they have an hour of homework each night whether it comes from school or from her. She got grade level workbooks and every night they either did their own homework or they did pages from the workbook. In 3rd grade they start getting planners and have to write their homework assignments down each day. The parent signs off on it nightly while the teacher checks the next day. If you routinely don't write down anything or leave things off the teacher starts checking it before you go home. 3rd graders lose their free time on Fridays if they don't follow through with their planners or remember to return homework on time. It's all on them in 4th and 5th grade. They are the ones who get in trouble for not returning or finishing homework. Philip's teacher has a class blog where she posts homework so he couldn't get away with telling me he didn't have any because we check the blog daily. He could try to tell me he did it at lunch or snack recess but I know him well enough to know he isn't skipping either recess if he doesn't have to.