"How big of a problem" vs. "how big a problem"

Solution 1:

What about "not that high (of) a fence"? "not that red (of) a heart" "not that smart (of) a person? not that big (of) a problem?

I would argue that if you use the word that to qualify the adjective, the of conveys the meaning of comparison of a specific entity to the class of general entities to which it belongs.

I'm sure the usage can be regional, as well. There is no hard and fast rule.

EDIT

Hey, I did some more research. Dictionary.com has the following usage note for "of" :

Of is sometimes added to phrases beginning with the adverb how or too followed by a descriptive adjective: How long of a drive will it be? It's too hot of a day for tennis. This construction is probably modeled on that in which how or too is followed by much, an unquestionably standard use in all varieties of speech and writing: How much of a problem will that cause the government? There was too much of an uproar for the speaker to be heard. The use of of with descriptive adjectives after how or too is largely restricted to informal speech. It occurs occasionally in informal writing and written representations of speech.

So, I suppose that's the reason why adjectives other than "much" combined with "of" sound odd to your ear. I believe "that" can be included with "how" or "too" in this synopsis. Replacing "much" with another adjective occurs occasionally in informal writing and in speech, but isn't unquestionably standard.

When I say these constructions out loud, to me, I often want to insert the of but perhaps that has something more to say about the informality of my speech rather than the correctness of the construction. :)

Solution 2:

What nobody has mentioned is that "much" can be either an adjective or a noun (as well as an adverb). The improper use of "of" following an adjective is the result of confusing the noun "much" (How much of a problem is it?) with the adjective "much" (How much space do we need?).

So it's not that big a jump (see what I did there?) to see how someone could mistakenly apply the noun construction in a place where an adjective is involved: "How much of a scandal did it cause?" (correct) easily slides into "How big of a scandal did it cause?" (incorrect), and thence can spread to other adjectives like "How hot of a day was it?" (incorrect), etc. And in non-interrogative sentences, "Not that much of a scandal" (correct) becomes "Not that big of a scandal" (incorrect), and thence spreads to other adjectives like "Too loud of a noise" (incorrect).

Solution 3:

How is the wh-word that questions manner, means, and adjective or adverb degree in direct wh-questions, with subject-auxiliary inversion (and do-support where necessary)

• How did he get through the barbed wire?
• How does she manage to keep so trim?
• How long is that pole?
• How rapidly should I breathe?

and in embedded wh-questions, with no inversion (and therefore no do-support needed)

• They asked me how he got through the barbed wire.
• I wonder how she manages to keep so trim.
• He knows how long that pole is.
• Nobody will tell me how rapidly I should breathe.

When dealing with degree in nouns, instead of adjectives or adverbs, there are problems with how to slot the noun into the phrase. The basic noun phrase with a comparable adjective is something like a good cook, the comparative is a better cook, and the superlative is the best cook. But there are grades inbetween, as there always are.

• How good is the cook?

is the basic sentence, with all the pieces in place, but if you want them in a noun phrase suitable for questioning, you have two choices. Either you put them together in something like apposition, with an indefinite article between them to keep the phrases from running together or something

• How good a cook is he?

or, if you don't like the sound of that, you can hang more tinsel on the noun by providing it with a preposition to introduce it. The traditional one is of.

• How good of a cook is he?

Or, to process the sentences from this question, instead of this question,

• How big a problem is it?
• How big of a problem is it?

are both correct and both mean the same thing. Choices between them are entirely up to the habits, tastes, and preferences of the speaker, which will vary.