On May 27, 2013, at 12:33 PM, "Louis Talman" <firstname.lastname@example.org> wrote:
> The real problem is misunderstanding what volume is and how it's related to linear dimensions.
I think the problem is misunderstanding "understanding" itself. It isn't about any particular concept, it appears to be about all concepts. There comes a time in an algebra class, quite early, after you "solve" your first problem and nothing is the same thereafter. You are changed forever. You may be unable to solve a particular problem later on but you very much know that you are unable to solve the problem or maybe that the problem itself is unsolvable, but you don't guess because after that change takes place, you can't guess. That ignorance is gone forever. This is the most fundamental epiphany of an algebra class.
You can't spend a year of circus acts and guessing and tell me or anyone with half a mind that you have even touched the concept of algebra.