Interesting article, and it does describe things that we should strive for---an intuitive sense is a key sign of mastery of a subject. However, the idea that w

Interesting article, and it does describe things that we should strive for---an intuitive sense is a key sign of mastery of a subject. However, the idea that we should learn math the way we learn languages has a flawed premise. Human beings have been equipped by evolution with the capacity to acquire language by exposure. Cognitively normal children aquire language, including the complexities of grammar, through simple exposure to people speaking that language during a particular period in their development. It's unfortunate that this ability doesn't persist through life, or apply to more than one language for most people. But for children, language acquisition is completely effortless. This is not the case for other cognitive skills, such as reading and (to our point) mathematics. My son cannot learn calculus simply by hanging out in my office. While our end goal may be the sort of effortless intuition in mathematics that we experience in parsing complicated sentences, the process of acquiring those two skills is of necessity widely divergent.