It's like trying to programming without being able to type. The ideas can't flow when you are constantly held up by not having mastered an underlying skill.
I almost love this analogy. There's certainly an aspect of "menial mathematics" which is like typing in that it directly translates into fluency of thought. There's also an aspect of discovery, though, that's missing from the metaphor.
There is a large difference between inferring that some equality holds based on abstract principles and actually performing the evaluation and directly tracing out why that equality holds (even very non-generally). I liken it to statistical modeling sometimes: models allow you to talk about and comprehend data on a high level, but only by directly plotting all of the data at high resolution can you let your brain's natural pattern seeking tendencies reach out for further insight.
That's a common theme there: granularity versus generality.
I almost love this analogy. There's certainly an aspect of "menial mathematics" which is like typing in that it directly translates into fluency of thought. There's also an aspect of discovery, though, that's missing from the metaphor.
There is a large difference between inferring that some equality holds based on abstract principles and actually performing the evaluation and directly tracing out why that equality holds (even very non-generally). I liken it to statistical modeling sometimes: models allow you to talk about and comprehend data on a high level, but only by directly plotting all of the data at high resolution can you let your brain's natural pattern seeking tendencies reach out for further insight.
That's a common theme there: granularity versus generality.