To get right down to it: in my day (growing up in the 70s and 80s), and in my world (upper-middle-class suburban Cleveland), confidence in girls was not a feature, but a bug. A bug to be swiftly and fully stamped out as soon as its little feelers began to wave. I grew up surrounded by women and girls constantly saying self-deprecating things like I'm so dumb, I can't do math, I can't (fill in the blank). Whether by overt intention or just widespread cultural norming, I'm convinced that the reason so many of us were were taught from a young age to think and speak this way was to make sure we could snag a husband. If we were too confident, too independent, too opinionated, too competent, we would not obviously need a man to help us get our helpless selves through the day. Feeling useless (and therefore offended), they would reject us, and we would live out our lives bitter and childless and unfulfilled and alone.
Clearly, this reflects a rather dim view of men as well as of women, because if one really succeeded through this particular strategy, one got to spend one's life with an insecure asshole. But I remember distinctly being told that men didn't like women who were "too smart." I remember being told that by a boy I had asked to a homecoming dance (in a perverse twist, the tradition at our school was for girls to ask boys to dances). And so, being a dutiful sort, I adopted the language and the practice of self-deprecation. And I have struggled against it ever since.
I feel reassured that this way of thinking and being has at least begun to change. But I still see evidence that the bad old ways hold sway, especially in certain parts of the country. What to do? So many things. But one easy thing we can do today is to stop talking about ourselves as stupid and incompetent--not only but especially around young women and girls.