Understanding vs Memorization

I took one math class that gave the quadratic formula as "memorize this and plug everything in."  I took another math class where the procedure for getting to the quadratic formula was actually done step by step, answering the question "why is this formula so HUGE?"

I prefer understanding over simple memorization.  I feel students would do better in math if they were made to understand every concept.  Being "bad" at math is actually not grasping the core concepts, or not having them explained well.  If each student can individually understand concepts, they will progress through a course just fine.  This may mean learning at their own pace, even if it takes twice as long as other students, but at least they "get it." 

Why can't all math be based on "understand it and apply it"?  Sure, test scores determie how well you know it to a point, but actually making sure each student understands would help even more.  Instead of "No Child Left Behind" that asks for only test scores to be high, why not have something that requires UNDERSTANDING to be high.  It's harder to measure, and takes more time, but its worth a lot more than being able to guess correctly on a test.

Japanese kids "learn" English. But they don't understand it.  They memorize it. So instead of learning English as any other person on the planet learns any language (that is, learning a lexicon, and a grammar), they memorize everything.  So instead of having, for example, the words "dog" "run" "car", and an adequate grammar, which can produce "The dog ran after the car.", they just memorize the phrase "The dog ran after the car.", and this is true of EVERY THING THEY LEARN for the most part.  Japanese teachers just don't get it, when you point this out to them.