We must have had the same 10th grade English teacher–I remember her using this very example!
One of the reasons for this indeterminacy is that natural speech can be very flexible when processing a particular unit for different syntactic reasons. We try to represent this in writing, (e.g., by making albeit one word in print), but the fact is that the usual conventions of writing can’t really reflect this flexibility in its totality–nor is it really meant to–nor is it really necessary for writing to achieve its purpose.
You can label it “one word” for some syntactic or semantic purposes, but that won’t necessarily stop natural speech usage from chopping it up for other syntactic purposes. Phrasal verbs are a good example of this.
This is such a weird thing to remember but I do remember my 3rd grade teacher had a few sayings on the wall (in teacher style cut-out letters) and one of them was “Not words!” with ‘alot’ and ‘gots’ as well as “Ain’t ain’t a word.” (Good luck with that in Texas!). So maybe it was an example of what not to do?
When I was starting out I had a manager that explained it this way - she walked to the white board on the left side and wrote “A” and then walked to the right side and wrote “lot”.
A __________________________ lot
She also did that for:
Data _______________________ base
Easy to remember it that way.
Easy to remember what? Database hasn’t been hyphenated for a quarter of a century.
Not hypenated. Two words. However, I will grant that the lesson was about 20 years ago and I think “database” is pretty much accepted as one word these days.
There’s a typical migration words go through from two joined words, to hyphenated, to one word. There are terms I used to consider were properly hyphenated that I now prefer to run together as a single word. E-mail is rapidly heading in that direction, to cite one example.
Two words.
As for the whole “cannot” argument, one linguistic hill that I am bound and determined to stand my ground on — and if I die upon it alone, so be it — is that “cannot” and “can not” have two different meanings.
If I cannot go to the concert, then I am incapable of attending it. If I can not go to the concert, I AM capable of NOT attending; whether I am capable of attending is unknown. A statement of the form “I can (X)” always indicates the positive presence of my ability to do X, even if X is to fail to do something, and thus can never be logically equivalent to a statement of the form “I cannot (X)”.
Since I see no similar use for making a distinction between “alot” and “a lot”, I don’t think the comparison applies. Descriptivist or not (and I am), you can still argue for the usages you think create the clearest communication, and I have to think that arbitrarily ramming two words together is counterproductive.
…and “reign in” our more boisterous debaters.
At least the ones we didn’t give “free reign” to.
Very good point! I know I’ve occasionally used “I can not” in just the way you describe. It’s basically equivalent to “I don’t *have *to…”, and is usually followed by something like “…but I will anyway, because I feel like it.”
That’s what I suspected. The managers of today have moved on to other pressing concerns.
It’s one lexical item but two words, using the most widely-accepted definition of ‘word’ in English, which interprets a space between letters as indicating that they’re separate words.
Lots of things are lexical units - phrasal verbs are the ones we encounter most often - but that doesn’t mean that, outside of a linguistics paper, it’s incorrect to describe “up there” as two words. “Word” has too vague a definition to be useful in linguistics anyway (except of course for discussions about the word word), so insisting on a proper linguistic usage of it would be nonsensical, because there isn’t one.
Also, the word from Word is that “a lot” is two words (which is not a lot). 
I think the reason alot hasn’t become widely acceptable in English (yet) is that very few words combine with articles to make one word. There’s “awhile” (which has slightly different usage to “a while”) and probably a couple of others that I can’t think of now, but they’re rare.
When you do see an “a” as a prefix, it means (with the exception of “awhile”) “not including the features of,” like in asymmetry, asocial, asexual, so the grapheme “a” already has widespread usage as a prefix.
Two words, one expression. Or, as is apparently the right term, one lexical item 
I just learned a new lexical item from the Dope!