I think there’s a common mistake in thinking along the lines of “social is good, outgoing is good, shy is weak and bad, introverted is inferior to extroverted.” In the USA in particular, there seems to be a heavy pro-extrovert anti-introvert bias.
Labels are never perfect, but it’s somewhat accurate to label me as somewhat introverted. I am not shy. I have very little problem standing up in front of hundreds of people and speaking. I often enjoy socializing… when it seems to have some substance and authenticity to it, and when I want to, and when I don’t feel obliged to. But I have a strong bias against empty socializing, people who talk without substance or authenticity, and social situations where people babble and mingle superficially.
I don’t know if your daughter is introverted as I am, but I think if she isn’t, then I would tend to expect she will want to do social things by herself. If she is a bit like me and other introverts I know, I think making her do social things that she doesn’t want to can easily backfire. However, it’s valuable to discover what things you really don’t want to do, and what things you actually would like to do, but feel kind of like not, but might regret not doing it. For those, it can help if the deal is that you try going but you can leave if you decide you really don’t like it, or you at least talk it through and figure out if you’re going to regret it or not.
I think one of the most important things a parent can do is communicate and listen and give a child some say in what they end up doing or not doing.
So I’d say:
Should I encourage her to be more social by pushing her to go to birthday parties and play dates and stuff like that?
Only gently, and with conversations where you talk to her about what she really wants to do or not and why. Ask her how she feels about it? Does she sort of want to go? Etc. If it’s clear she really doesn’t want to, I’d not make her.
She has a few friends. Teachers tell me other kids go over to her and then she’ll play, after being approached by the other kid.
Sounds ok, especially if she gets to be good friends with some. I had a few friends, because they were the ones who were worth being good friends with. I wouldn’t think the goal is to be friends with everyone.
She has not yet played team sports. She prefers things like crafts.
Sounds normal to me. I didn’t much like team sports, unless it was something really fun, like British Bulldogs or War or Dodge Ball. I certainly didn’t want to sign up for after-school obligations to play one! Really, I would encourage any kid to find the things they are actually really interested in, and support them in doing that. Part of supporting them in that is not adding random standard “all kids should do this” obligations/activities.
When it comes to birthday parties, she may or may not always want to attend.
Sounds entirely normal to me. Again, I’d discuss but not make her go to one she really doesn’t want to go to. It can be tricky figuring out which is which sometimes but if you can talk that out with her and be her ally in figuring it out who is trustworthy and doesn’t make her do things she really doesn’t want to, that will be a great thing.