I have been doing a fair bit of tutoring trigonometry this fall – my son has a strong mathematical mind, but his previous trig units came during the first weeks of the pandemic and then in an all-online (teacherless) Algebra 2 class, so he’s been playing major catch up, and I’ve been happy to pitch in.
What we found, and discussed tonight, is that while I don’t have all the trigonometric formulas memorized anymore (and there are a lot of them), what I do have is a strong math sense – an almost innate sense of which direction to push next. This comes from a lot of repetition, internalizing what was not innate at all. (Mathematics comes less or more naturally, but is certainly not truly innate.) My math sense is strong because of internalization work I did decades ago, and that work has stayed put, even if the mathematical tools aren’t at my fingertips.
Meanwhile, I’ve been thinking about the gestalt switch that happens with young musicians. For a while, musicians focus a lot on counting each note duration – I hold this for three, then I count to four, then two two’s, etc. We have all seen this become disastrous, of course, when a young musician fails to count a rest accurately and doesn’t adjust – singing a whole solo a beat off, for example.
The gestalt switch happens when they learn to feel the shape of the rhythm: the pulses and barlines and phrases that make a piece tick. Then, no matter how off you are on counting, you aren’t going to start a strong-beat melody on beat 4 – it wouldn’t make any sense to you!
The switch happens because you’ve internalized how music works – you’ve developed a music sense. And once developed, there’s no going back; it’s as natural to you as solving algebra is now for me, 25 years after my last calculus class. It’s a part of you.