OP - I have been in your shoes, criticized by my mother for my lack of ambition, and for not using the gifts I was born with. What she didn't realize is that my ambition wasn't to make gobs of money or be a world travelling woman of means. My ambition was to get married and stay married and raise happy, healthy kids. I wanted family and security and a peaceful home, and time to write and paint - none of which my mother was ever able to achieve despite all her success and ambition.
My mother is retiring now and looking back on her life, and she's started to see the value of what I've actually achieved.
My point is - do what's right for your daughter. If she
enjoys the extra-curricular work, then by all means let her do it. It won't hurt her. If she doesn't like it, or if she's dragging her feet and complaining, then don't make her. Keep her close by if you think that will make both you and her happier. Growing up is not a race. It's a time to be enjoyed and savored, both by your daughter and you. If she grows up secure and happy, she'll do very well in life, no matter what.
As for the other folks, talking about what makes a gifted kid...
* You can't do complex algebra if you've never been taught - so a kindergartner who isn't doing algebra isn't necessarily NOT gifted.
* Conversely, you can teach almost any toddler to read. So the fact that your toddler reads does NOT make them gifted.
* Being gifted isn't about performance, it's about potential, about the ability to synthesize data, and about the speed at which a child learns.
* You can be gifted in one academic area and not in another. A gifted mathematician is not necessarily a gifted reader.
* Not every gifted child loves to read. Or loves math. Or loves whatever it is you think they should love.
* A child can be both gifted AND learning disabled. (And boy, I'll tell you, that's no fun for parents and teachers!)
* Not every gifted child is ambitious or driven.
* Social skills are important, but you shouldn't try to force a child to be something they aren't. If your child is a total nerd who just wants to read The Lord of the Rings during Kindergarten recess, and couldn't care less about Dora the Explorer - let her be! And maybe think about putting her in a congregated gifted class, so she doesn't feel entirely alone.
I spent entirely too many years trying to change my daughter, and I'm pleased to say I utterly failed. Thank goodness! Because she's still a happy, nerdy, content-to-float-through-school kid, who has far more good sense than I ever did at her age.
* Watch out for cultural issues in GT classes, too. My daughter worries a lot about her classmates, because while she puts in just as much work as she likes, and consistently pulls As and Bs, many of her classmates are very, very stressed and feeling the pressure. Some cry. Some throw up. Most of the class is Asian, and many take extra prep classes. The kids in my daughter's school describe an A minus grade as an "Asian Fail" - because those are the kids who freak out over A minuses.
A few times my daughter has come home and hugged me, thanking me for not putting that kind of pressure on her.
Good luck!