It seems odd if your kids have been in school for a long time and you have met a few times already. But honestly, I would be cautious, too, about another person I do not know transporting my child to a third location when I know nothing about them nor their habits.
It's a lot easier to ask to get to know you some than to ask all the questions she can likely figure out from spending a little time with you.
Do you smoke? Do you smoke in the car?
Do you drive safely or do you talk and text and put on makeup while you drive?
Do you swear? Do you talk down to your kid and call them stupid or belittle them in other ways? Tell them to "shut up"? Do you act like another one of the teen girls when the girls are together, or do you distinguish yourself as the responsible adult in charge?
Who ELSE might be around when her daughter is there?
I don't know you, nor exactly how old your daughter is. But, I will be honest. She was judging you. And good for her. She has that right and I don't blame her. She may find you perfectly acceptable and a good example, or she may find things about you that she prefers to discuss with her child before (or after) she spends time at your home or under your watch.
I actually LIKE when other parents want to meet me. I want to meet them, TOO.
My kids are 15 and 12. My daughter has a friend from 3rd grade that she met that year (her first time in public school). I did not know her family, and in public school it can be really difficult to get to know parents if they do not come to the one school party they kids have. Even then, it is tough.
The girls became pretty friendly, and we had her over to spend the night for a birthday. And the mom ended up hosting my daughter for a lunch out and a movie. But she never invited daughter to spend the night. And I was glad. She was not married, but they lived with her boyfriend. I had no issues with the woman, nor her child, but would not have wanted my daughter to spend the night at her home. Judgey? Sure. That's my right, to judge who will be watching over my vulnerable impressionable young daughter.
Now.. fast forward.
The girls are in 7th grade now. Few weeks ago, daughter was invited to a sleep over party for her. The girls had not been in close touch outside of school much for the last year... they did things, but at other venues. Come to find out, since we'd last had any real conversation (mom and me) she had moved out, begun dating someone else, and gotten MARRIED to another guy completely! I had never met the man. Never been to where they now lived. Nothing.
You can bet, we chatted for a good hour before I agreed to let daughter go to the sleepover. And, when I dropped her off, I went in, met new husband and stayed and chatted with him as well (along with mom) for a good 30 minutes. Frankly, I liked him a lot more than her previous boyfriend. I had an instant "this is a good guy" vibe from him, which I didn't get from the previous one.
Saw him (and her) next morning when I picked her up, too. And last night, actually, at a school function. Stepping up being there for his new family.
Would I let her go with just him? No. But I would be okay with him picking up the two girls together or running them to the movie or something. The other guy... no. I wouldn't have.
Various reasons. But that is MY prerogative to make that decision.
It's ok for parents to judge other parents. I'm so tired of this idea that seems to be fomented that it is NOT ok. Why not? That's, you know, my JOB as a parent... to make judgments about what my child is taught, exposed to, etc.
Does that give me the right to criticize you and your choices? No. And I wouldn't. But I would most certainly tell my child I do not agree with whatever I find inappropriate and allow or disallow her interaction based upon whatever my criteria are.
So, all that said,
Unless she says something totally rude to you, then I would be GLAD that she is interested in the kind of person you are. She is a caring parent, who likely would be the kind of parent you could be comfortable letting YOUR kid spend time around. Just another way to look at it.
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By the way, wanted to make a note, that the mom in question is a teacher. So it may be that she meets a TON of parents. And her emphasis on "really" getting to know you might have been as simple as a distinction between the parental meet and greet on Open House night and an actual getting to know something about you.
I'm not a teacher myself, but I've been to plenty of open houses over the years and met almost every single one of both of my kids' teachers over the years (it gets harder in high school, b/c sometimes the classes change after open house and you there isn't a 2nd open house for 2nd semester classes).
But under no circumstances would I consider that "meeting" of the teacher adequate to gauge whether I would trust my child in their household or in their car, or under their supervision elsewhere. And I am pretty confident that the teacher would have the same feeling about that "meeting" being an inadequate (and frankly inappropriate) getting to know you time. So maybe it is just a distinction she was making from that perspective.
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eta (again) ... you've edited your original post. Just wanted to point that out, in case some chiming in late are wondering about any peculiarities mentioned in early responses... like the "really getting to know you" comment. And that the fact that you are new to a private school wasn't specified. And I don't recall, "She told me she has been asking about us and what people said." being in the original post, either.
hmmm... also just noticed that you changed what you said originally.. first time out, you said that you offered to take her daughter to the skating rink for the girls to skate. NOT that you offered to meet them there and stay and supervise.
Sorry... but all these little changes affect how this woman comes off. NOW she sounds a little kooky. When you were asking to TAKE her daughter, that is totally different than meeting them there. Do you not see that?