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Many times I have heard President Obama use "a" before an adjective that starts with a vowel, such as "a economic stimulus" or "a occasional thing." Am I crazy that this sounds wrong to me? Shouldn't it be "an"?
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You have either misheard him, or you have heard the pausal 'uh'.
I've heard him do that too. Yes, it is wrong and pretty jarring to the ear.

Fawning by the press notwithstanding, I think we can all agree that he's not exactly a good extemporaneous speaker. My guess is that when he does that he has (in mid-sentence) decided to change what he wants to say, thus the mismatch in the article and the following word.

I haven't heard him make this mistake when he is reading a prepared speech but speakers sometimes depart from their prepared remarks so it could happen even then.
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This often happens when people are speaking extemporaneously, without a printed text from which to read. It becomes, in effect, a sort of "articulated pause + indefinite article". I have every confidence that he knows the difference between 'a' and 'an'.
Actually, Obama's grammar is correct in the examples you provided. The rule of using "a" with an object that begins with a consedant, and "an" with objects that begin with vowel sounds is correct, but you are forgetting an important detail that our english teachers should have taught us. While it is preceding the object of the prhase, it is not* necessarily *immediately preceeding it. Adjectives do not count, only the object (noun).

"a stimulus" is correct, therefore "a economic stimulus" is also correct, because the object of this phrase is still stimulus, not economic. Economic is an adjective.

"a thing" is correct, therefore "a occasional thing" is also correct, because the object of this phrase is still thing, not occasional. Occasional is merely an adjective.

Likewise, "an economy" is correct and therefore "an dismal economy" is also correct.

The more common slang, in having molded and reshaped this rule to apply to whatever word immediately follows either "a" or "an", rather than the object of it's phrase is technically grammaticality incorrect. I know we all do it, but it is still just exactly what it is -- slang.
Anonymous"a stimulus" is correct, therefore "a economic stimulus" is also correct, because the object of this phrase is still stimulus, not economic. Economic is an adjective.

"a thing" is correct, therefore "a occasional thing" is also correct, because the object of this phrase is still thing, not occasional. Occasional is merely an adjective.

Likewise, "an economy" is correct and therefore "an dismal economy" is also correct.
No, no, a thousand times no! This is not correct. This is extrememly incorrect. "A" preceds a word beginning with a consonant sound, and "an" preceeds a word beginning with a vowel sound. The choice depends on the word immediately following the article, be it adjective, noun, interjection or sneeze.
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khoffThe choice depends on the word immediately following the article, be it adjective, noun, interjection or sneeze.
I must make a note ... Emotion: big smile [Y]
An ah-choo!