ABSTRACT
Mathematics is not different, in the following sense: to see the beauty of the subject – to get into the mathematician's world – we must repeat, in some way, the experience of the mathematician. School mathematics has traditionally been even more like a foreign language in the sense that the rules always seem to be changing as we learn it. In elementary school, we first learned that we can't take 8 from 4; and then later that we can take 8 from 4, we just get a negative number. An ever-increasing body of research suggests that it is very much possible to learn mathematics in ways that "repeat the experience of being a mathematician." Children can do real mathematics, in the same way that they can learn to communicate in real ways in Portuguese, or learn to appreciate and create art in an authentic way. Mathematicians use language carefully – it's part of why mathematics is often called "a language.".
