[bluGill]

While people are not as logical as we like to be, we all are logical. I can teach someone the rules of math - my method of teaching might be (probably is) bad, but if the student tries he will learn those rules. Latter on when given a test the student can show his work and it will be much the same as every other student trying to explain his reasoning: operations like "complete the squares" have been well described and reasoned out.

Likewise chess masters can explain their thought process while looking at the next move and other chess masters will agree the lines of thought are good (they will probably ask why not some other equally good line...). We know this explanation is good because students can watch the experts explain their thought process and replicate it in games to a small extent and to better.