That was fascinating, thank you. To quickly find cube roots I would generally employ linear approximation and get close enough.
(29)^1/3
f(x) = x^1/3
f '(x) = (1/3)x^-2/3
27 ~= 29
f(29) ~= f(27) + 2f '(27)
= 3 + 2*(1/3)(1/3^2) = 3 + 2/27 ~= 3.074
since 29^(1/3) ~= 3.072, that's not bad for a quick and dirty.
What are your thoughts on the new Common Core math teaching system that's been adopted by most US states?
Note, if you go looking for information on it, look at the source materials from the creators. The specification only provides the concepts and the curriculum but not the course materials. Professionally produced course materials are still being created, so some teachers have created their own materials, some of which are terrible. The worst of these are then scanned and put on the Internet as if they're representative of the entire program, inciting conservatives who (falsely) believe the system is imposed by the federal government.
From what I've seen, the curriculum calls for teaching methods like what you describe (and which is generally the way I do math in my head). One blog I saw was incensed because when solving 7 + 7, instead of just memorizing '14' they wanted the students to break that down into 7 + 3 + 4, recognize 7 + 3 as a group of ten, then add 4. That to me provides for a deeper understanding and would eventually make working in other bases much easier. The only reason the second place in our numbering system represents a group of ten is because we have ten fingers. If they understand grouping it would be much easier to switch to say hexadecimal by counting groups of 16. C + 9 = (C + 4) + 5 = 15.