A violinist who still worries about fingering positions cannot hope to impress with the beauty of tone or the elegance of phrasing, and an opera singer without the requisite high notes would try in vain to stir our souls with searing passion. In good art as in good mathematics, technique and conception go hand in hand. –
I don’t know if I’ve ever spent as much time thinking about mathematics as I have lately. I am an interpreter by training and aside from the calculus, trigonometry and statistics I took in undergrad, and the business type math I took in grad school, my world is a world of language. I do know about learning language, teaching language, and speaking a foreign language, however, and it occurs to me that it is much like learning math.
Math starts out much like a foreign language, something unfamiliar with it’s own symbols, and meanings, and structures. It’s hard work at first and requires repetition, memorization, and practice. You’re not really comfortable with a new language until you use it over and over, consistently and successfully. There is no resting on your laurels, because fluidity depends on practice and vocabulary depends on exposure to rich sources of language.
You don’t discover the language on your own. You don’t invent the language. The language is already there for you to master, to seize, to make your own if you’re willing to work for it. It’s a beautiful thing to move seamlessly between two or more languages but it doesn’t magically appear by placing a dictionary under your pillow and wishing it so.
You learn language by listening to others speak, by looking words up in the dictionary, by learning the rules of grammar and then memorizing the “irregular” applications that break all those rules. Finally, you are able to communicate with others in this new language and the more often you do so, and the more challenging the conversations become, the more you grow in your ability to communicate.
You can always tell who had the benefit of a good foundation through formal instruction (grammar, syntax, pronunciation, vocabulary) and who just "picked it up" informally, on their own and without academic support. The latter tend to be more limited in the range and depth of communication in that foreign language. There is a limit to what they are able to accomplish with the "tools" in the "toolbox" so to speak. Those with the strong foundation move through their discipline with ease, with confidence and with mastery. I think the same applies to many other disciplines be it ballet, basketball, the violin or dare I say, mathematics?
Yes, learning math is much like learning a language. When you master the basics through diligence, perseverance, and the benefit of good instruction, one day without realizing it, you’ve opened the door to a whole new world.