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Arandur comments on Expecting Short Inferential Distances - Less Wrong
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Eliezer_Yudkowsky: I've seen the kinds of failures of explanation you refer to, and there's also the possibility that the explainer just isn't capable of explaining all of the inferential steps because he doesn't know them. In that case, the explainer is basically "manipulating symbols without understanding them". This is why I've formulated that principle (sort of a corollary to what you've argued here) that:
"If you can't explain your idea/job/research to a layman, given enough time, and starting from reference to things he already understands, you don't understand it yourself."
That seems so simple as to be tautological. After all, you were a layman yourself once.Ideas/jobs/researches don't spring whole-spun from the ether. You have to be led along that same path yourself - either by a teacher, or by your own mind bumping along down dark corridors.
But it's not true. Consider by analogy: if you can't explain something to a 4-year-old, you don't understand it yourself. After all, you were a 4-year-old once yourself.
No, actually, sometimes you can't explain something to someone because you don't have a good enough understanding of their mental processes. It doesn't matter if you once experienced those same mental processes; the relevant memories of that time are very likely lost to you now. Explaining math to novices is a different skill than understanding math. It requires the ability to figure out why the other person has got it wrong and what they need to hear. That isn't a mathematical skill.
A distinguished math professor is probably inferior at explaining arithmetic to 8 year olds than an experienced mathematics educator, but it doesn't mean the latter has the better understanding of math. They just have a better understanding of 8 year olds.