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.

>>13
But you can clearly create novel utterances.  These kids can't.
Whereas you may not think you understand it, it's clear that being drilled in grammar built a grammar in your mind, like the one for your first language.  These kids lack that totally, and thus, can only create novel utterances through fucking up.  If they never made a mistake, it would all be parroting of phrases they memorized in class.

I'm doing speaking tests.  One part involves the kid reading a card with a question on it to me, and me answering, and them asking a follow-up.  On their Q&A sheet that they've been learning from, one of the questions is "Do you enjoy swimming?", where "swimming" is underlined to show that it can be changed to other verbs.  Now, if a grammar was being built, they'd be able to change that sentence into several new ones with verbs they know.  They'd be able to read sentences in such a structure.

On one of the cards, I wrote "Do you enjoy writing?"
Every single one of the kids spews forth "Do you enjoy swimming? Ah, no, uh, writing!"  They're not learning shit, they're just memorizing.  It's a problem.