200 - 101 = 199. Absurd? I should hope so. Unfortunately this is not the case as often as I should like it to be. Let me explain: I tutor in various subjects, mathematics being one of them, and recently a parent came in and started complaining that one of her son's maths problems had been marked wrong, when it was correct. The problem was that given at the start of this post: we tried to show her how the correct answer ought to be 99, but she just wouldn't see it. So what was the problem?
Naturally, one may argue that the processes of vertical subtraction are frequently confusing and defy understanding, but even if that is the case, it should be plain to anyone that two hundred minus one hundred and on could not possibly equal one hundred and ninety-nine. So why couldn't this mother see it? The answer is compartmentalisation. Compartmentalisation is the process of separating everything into small, neat, distinct and independent compartments. At first this seems like wisdom, and indeed it is often useful to break things down into manageable pieces, but unless one is careful, something can be lost; the big picture. The mother could not see her answer was wrong, because she had lost sight of the actual problem as a result of her focusing on the minutiae of processing.
Again, and this will be familiar to anyone who knows me, the separation of music and mathematics is both unprecedented historically, and untenable intellectually. Until the nineteenth century music and mathematics were not just considered to be connected but music was a branch of mathematics That's right, people studied music as mathematics. Yet today music and mathematics are considered polar opposites, the one being purely subjective, all about 'how you feel,' and the other purely objective, cold, hard, scientific. Neither of these views is correct; music is not purely subjective, nor is it only about how you feel; mathematics is not just about following rules and applying stale formulae. Rather, both have their spark of individuality, creativity, and passion; both have a rigid underlying structure which compliments, rather than hinders, the creativity and beauty of what is produced. Both are seeking to represent patterns and thereby beauty, and they do it in the same way.
Today, as a result of compartmentalisation, we believe that if something is creative, it can't be 'restricted' by rules. If something is 'scientific' (whatever that means) it can't be creative, but must only follow rules. If something is 'arty,' it must ex hypothesi defy clear definition, analysis, or understanding. If something is practical, it is purely practical, and has no place for expression. The high-school musician says of the maths whizz, "Oh, that's alright for him, I just don't have a mathematical brain," while his counterpart says of him, "Oh, I just don't have the creativity to be a musician."
The two should not be opposed, and yet they are; and something as been lost thereby. Don't believe me? Beethoven loved to take geometric figures and twist them, raise or lower them, taking certain points on the figures as musical notes, and write entire motifs that way. It was a frequent habit of Baroque composers to end a piece of music about God or Religion with three semi-breves, which when read is the mathematical symbol for eternity: ... These composers recognised that beauty can be found withing order, and that knowing something about the underlying structure helps, not obstructs, one's ability to create the sublime.
Compartmentalisation is a dangerous habit, for it enables you to see only one side of a coin, when the other is drastically important. I could go on, but the point is made. Dare to see the big picture; it can be daunting, it can seem terrifying, but too much is lost if you close your eyes.If everything is broken off into separate, unconnected pieces, how are we supposed to draw from our past experiences? We must see the big picture, and see the connections between everything we have seen, and what we see now.