Seriously. Math is both an art and a tool. The best artists find their art by themselves.
Absolutely. It is an art and a tool in not so different a way from how a language is an art and a tool.
And like learning a foreign language, learning Mathematics is not a straight path. We would love to build up a sense of mathematics from first principles in a perfect, coherent way, but that is no more realistic than learning French by studying the etymology of every word from the get-go. No, you learn French by listening to it, by speaking it, by making mistakes, without necessarily knowing how it evolved. Later, once you are more fluent in it, you begin to read more sophisticated literature, you begin to be interested in the development of the language, and then you say, "Ah... so that's how things come to be."
A student may fully appreciate "the transcendental nature of the trigonometric functions", but what good would that do if he cannot bother to memorize (yes, MEMORIZE) the double angle formulas. How would he understand later on a real life application of Calculus, where it is taken for granted that he is fluent in the language of trigonometry.
It's funny that Lockhart uses the practice of visual arts as a metaphor. Fact is, there is a lot of dry, uninteresting stuff that went into the training of an artist. The myth that Da Vinci started out painting eggs probably isn't too far from the truth. You think Picasso painted things in the style of Guernica when he first started? Doing the dry non-interesting stuff allowed Picasso to express his artistic vision with technical facility. So what if he had the "vision" of Guernica, if he can't even handle paint competently?
From my own experience as a pure mathematician, I can tell you that my own learning curve is far from linear. When one learns topology, one has to learn all the formal definitions of open sets, compactness, and so on. Of course, one tries to motivate these definitions with intuitive notions, but ultimately, a lot of my appreciation of "the language of topology" is obtained from seeing how it is applied. One can talk about donuts and coffee cups all they want at the beginning, but that doesn't even begin to capture the beauty of it (Try talking about cups and donuts in the context of p-adic topology on a p-adic field). It's a back and forth process. Most often the person coming up with the definitions isn't him/herself fully aware of the full implications. But that's the beauty of it.