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.

Learning what something actually means rather than something = something else makes everything a hell of a lot more enjoyable. I'm currently studying Chemistry and Biology and both teachers have completely different teaching methods.

The chemistry teacher will simply put premade sheets on a projector, tell us to copy them out and then give us questions, which is such a fucking dull way to teach things that I find it hard to remember everything.

The biology teacher, however, is completely different in that he will spend the majority of the lessons telling us not only about the stuff we are required to know, but also the things that are only slightly related to it, giving us plenty of things to link together with each other, making the course a lot more enjoyable and a lot easier to do well in, purely because you actually understand what's going on rather than memorizing things that, for all you know, might not mean a thing.

In a perfect world, all teachers would teach like that, but the thing is that not everyone can keep up with teaching like that. While some of us are excelling in biology, some people are struggling because they prefer to learn by drilling the same thing into their minds over and over until it sticks.

Because these people can't cope the learning way, but we can cope with the memorizing way, a lot of teachers will use the memorizing way over actually teaching why everything means what it means.

Of course, this could go all the way back to "what if people are taught like this throughout all their lives, so they never get used to memorizing things and will understand things a lot better?" It'd make chemistry a hell of a lot more interesting, that's for sure.