This idea of intelligence as a "choice," is quite interesting. In my head, it connects to the debate of nature versus nurture. Are we smart because we're born smart, with special natural talents, or does the environment as well as ourselves have a hand in it? And to what extent?
I believe that many people are born with certain specific talents. This talent doesn't mean that they're super duper great in something naturally, but that it may come to them easily, easier that anyone else. The popular areas of "easiness" have traditionally been maths and writing... for example.
However, as you said, people can transcend that in a way. Nurture has a really big part, not from your parents' side, or neighborhood's, or friends', but also by your own will.
Here's what I think: you couldn't have grown up stupid then turned smart. It's not genetic. You might have grown up lazy, careless, ignorant... And to become hard-working, careful and knowledgeable requires some work and earned skills. You can definitely choose that. In fact, you have a huge hand in that. And here's what: I don't think any person is utterly stupid. Those who think they are, or were, haven't found their niche, their specialty, yet... that's all..