…there was already something in his gaze beyond the capacity of any ape. In those dark, deep-set eyes was a dawning awareness—the first intimations of an intelligence that could not possibly fulfill itself for ages yet, and might soon be extinguished forever. (1.6)
So, is Moon-Watcher special because he's smarter than your average ape or is he special because he's smarter than your average man-ape? The novel goes back and forth a bit. Strength and skill are individual talents, but they're also evolutionary, or species, talents.
Part 1, Chapter 3
If he survived, those patterns would become eternal, for his genes would pass them on to future generations. (3.10)
The slab changes not just Moon-Watcher's mind, but his genetic material. It's not just teaching; it's changing evolution. The book makes Moon-Watcher the hero, but the real protagonist could be seen as the human race itself. (Which is maybe why many of the characters in the book are so bland. The human species as a whole doesn't have much of a personality.)
Part 1, Chapter 4
Then, not for the first or last time, he proved himself a genius. (4.16)
Moon-Watcher is the smartest man-ape around. Even the smartest man-ape isn't so smart, of course—but you could probably say that for humans too, if you were a super-energy alien.
Part 3, Chapter 19
The time had not yet come when Man could leave his mark upon the Solar System. (19.31)
The suggestion here is that humans at some point will leave their mark on the solar system. Humans will get ever stronger and more skillful, until they can put a big, "Joe Was Here" sign out on Pluto.
Part 4, Chapter 21
"Frank Poole, who is specially qualified for this type of work, will go outside the ship and replace the faulty unit with the backup." (21.27)
Poole is qualified to go out and space walk. The book thinks that's cool. It's into being qualified.
Part 4, Chapter 24
"I don't want to insist on it, Dave, but I am incapable of making an error." (24.23)
Hal is supposed to be incapable of making an error. Alas, he is full of them. This is why he's so much fun; all the other characters are professional and qualified and strong and skillful. Hal, with his neurotic burping brain, is much more sympathetic.
Part 5, Chapter 31
Work is the best remedy for any shock, and Bowman now had work enough for all his lost crewmates. (31.1)
Is work really the best remedy for any shock? Maybe if you're all efficiency and competence all the time, like David Bowman. He's more of a robot than Hal ever was.
Part 6, Chapter 44
If he was indeed mad, his delusions were beautifully organized. Everything was perfectly real; nothing vanished when he turned his back. (44.7-8)
This seems like it's Clarke's goal. 2001 is a delusion (or fiction) that is carefully organized; each piece of equipment fits together, and the whole clinks along. The movie is much more of a mess (which is part of why it's better).
If this is some kind of intelligence test, I've probably failed it already. Without further hesitation, he walked back into the bedroom and began to undo the clamp of his helmet. (44.30)
The novel opens with man-apes getting their intelligence tested; now towards the end, Bowman is worried about having his intelligence tested. 2001 is a world where smarts and skill are important, and need to be quantified. The future is a giant SAT. Yuck.
Part 6, Chapter 46
How obvious—how necessary—was that mathematical ratio of its sides, the quadratic sequence 1:4:9! And how naïve to have imagined that the series ended at this point, in only three dimensions! (46.8)
Being a star baby means you understand more math. Be jealous.