I've thought of a good example of 2 different ways of thinking.

I was at an early-music workshop where the teacher handed out a spagnoletta, which is a stately renaissance dance. All spagnolettas are much alike.

The teacher said something like, "In a spagnoletta, the second derivative of the fifth beat is tangential to the congruent triangle on the hypotenuse of the next measure." Well, that's not actually what she said, but whatever she said, none of us understood it. I could see complete bewilderment on many faces.

I made a buddy at that workshop, a woman named Maggie who played the most lovely recorder music but didn't know how to count. She asked me about the spagnoletta, and I said, "Pretend we are singing words to this. The pick-up note is an unimportant word, something like 'when, it's, oh or if.' The sentence goes for not-quite two measures. The 12th beat is the next unimportant word, so put a tinny pause or diminuendo before it. Then start a new sentence."

She got it right away. It's how I do it, too. For me, counting beats works down to the quarter-note level, but smaller than that, I keep the beat but use language, not numbers.