[a_p]

My favorite story about how Ernst Kummer [1] did arithmetic, from Hoffman's The Man Who Loved Only Numbers[2]:

"One story has him standing before a blackboard, trying to compute 7 times 9. "Ah," Kummer said to his high school class, "7 times 9 is eh, uh, is uh...." "61," one of his students volunteered. "Good," said Kummer, and wrote 61 on the board. "No," said another student, "it's 69." "Come, come, gentlemen," said Kummer, "it can't be both. It must be one or the other." (Erdos liked to tell another version of how Kummer computed 7 times 9: "Kummer said to himself, 'Hmmm, the product can't be 61 because 61 is a prime, it can't be 65 because that's a multiple of 5, 67 is a prime, 69 is too big-that leaves only 63.' ") "

[1] https://en.wikipedia.org/wiki/Ernst_Kummer

[2] http://www.amazon.com/The-Man-Loved-Only-Numbers/dp/B004R6HX...

[fyi80]

> Erdos liked to tell another version of how Kummer computed 7 times 9: "Kummer said to himself, 'Hmmm, the product can't be 61 because 61 is a prime, it can't be 65 because that's a multiple of 5, 67 is a prime, 69 is too big-that leaves only 63.'

That's how I do problems like that, too (I am not a genius mathematician), and it is exactly the sort of thinking that kids should be doing all through K-12. Estimation, intuitive reasoning, analogy, pattern matching, logic, etc.

[rimantas]

  > Estimation, intuitive reasoning, analogy, pattern
  > matching, logic, etc.

Alas, most of the things on your list require a pretty solid foundation to work. You need patterns already committed into your brain to do pattern matching and to see analogy, you need to have internalized experience for intuition to work, you need to have previous exposure for any meaningful estimation. All to often people forget foundation when they move to the upper layers and sadly sometimes this leads to thinking that foundation is not necessary. And now matter how you look at it there will always be bits of the foundation that require rote learning.