Sounds like a false word to me.
When young I believed that there was something called “The Smorning”, this I learned from hearing other people use it in conversation.
When I went to school I found out that it was actually"This morning".
So you’re the one that messed up “norange”, “napron”, and “ekename”. 
Doctor Seuss uses it “wrong” in One Fish Two Fish -
Oh dear, oh dear, I can not hear
Will you please come over near?
Clearly with your definitions, he should have said “cannot”. Maybe it was still not quite considered kosher in the 50’s?
I always remember this example because I have read that book a lot
I’m in my 40s (so went grade and middle school in the 1970s, early 80s) and remember my teacher making a big deal about “no one” being two words and “a lot” being two words. In my education “alot” was never correct.
Most likely, since Dr. Geisel’s ouevre is most widely appreciated by persons engaged in the quest for literacy (and their parents), the good doctor echewed the connotative difference between the one-word and two-word renderings of “can[_]not” in favor of presenting his new readers and readers-to-be with two short and common words that are easily sounded out.
Tso simple words a child can see
Would make much ore sense to me.
Distinguish croc from alligator
If you insist, but do it later.
A child needs simple easy stuff
That still is fun; writing that’s tough.
Theo. Geisel had it all filed
What would cause giggles from a child.
Much brighter than Bullwinkle Moose
Is that rhyme genius Dr. Seuss.
For what it’s worth, I am 35 years old and grew up in central Illinois. I do remember being taught, in passing, that “alot” could double for “a lot” in colloquialism (probably in 9th or 10th grade) and even at the time I remembered thinking, “Why would I choose to spell it that way when ‘a lot’ is the more accepted version?” So I don’t think the people that are saying they were taught that are crazy, but perhaps they weren’t given any guidelines on usage.
It bears mentioning that my 10th grade English teacher, Mrs. Klosterman, also vehemently fought with me over the word “horrific,” which she swore was not a word. This was my first indication that perhaps she was an idiot. And yes, for the sake of other children she may one day teach, I DO hope you’re reading this, Mrs. Klosterman. You idiot.
You’re sure that the narrator was not deaf? Perhaps he was compelling the other to come nearer so he could read their lips.
So you don’t see him alot?
On a similar note, we need to make “nother” a word. As in: That’s a whole nother thing. 
I started first grade in 1949. I struggle remembering some details of early elementary school. I cannot remember when I first saw cannot in print, but I know it was in my text books in the early 50’s. I am sure the text books were more scrupulous about formal, correct usage than the good Doctor Geisel. I do follow the argument he may have done it for ease of reading.
As for ‘‘alot’’ as in ‘‘a bunch’’, I never remember it as a single word before this thread. If I saw it, I may have dismissed it as a bit of sloppy writing.
Shakespeare used “cannot.” Charles Dickens used “cannot.” That word is as old as the hills.
A doper after my own heart. I read it every day, and I have a shirt with the title and the eponymous fish. In fact, I posted the entire text in this thread, and the mods chastised me. 
Can you imagine that?
And in that spirit, Wil Wheaton’s got it covered.
Plus, he turned me into an ewt.
(Thanks – I wasn’t aware of that example!)
I hope you got better.
I remember a teacher in the 80’s teaching us that “cannot” was the correct spelling of “can not”. I don’t recall there ever being any distinction between the two other than one is always right and the other is always wrong.
However, while I would obey the teacher’s wishes while in her class, I never saw the word “cannot” outside of her class, so I have ignored her lesson and have continued to use “can not” always and “cannot” never.
But I always remember that lesson, even if I ignore it.
Seriously? I have never seen a professional publication use “can not.” “Cannot” is the standard spelling.
I had an English teacher in HS who would rake you over the coals if you split an infinitive. Other than that nonsense, she was probably the best English teacher I ever had.
awhile or a while is another tough one to keep straight.
I’ll be at the library for awhile.
Actually, that’s incorrect. It should be either “…library awhile” (adverb), or “..library for a while” (object of preposition – i.e., a noun).