Do we need to teach grammar?

I am not doubting the necessity of using correct grammar. When writing or speaking, however, I never consciously think about why a sentence sounds right. It is a feeling more than anything else.

I learned English at age eleven, never studying the correct grammar. Spanish and Italian, OTOH, I was taught in school – pretty unsuccessfully. I can piece together sentences, but that is all.

Instead of teaching concepts, wouldn’t it be more efficient to expose children to correct grammar, as early and thoroughly as possible?