Sure they should.

But that still doesn't detail for me the importance of memorizing the aforementioned parts (score, time left, foul problems-yeesh) of the game in the third quarter.

Why do schools buy those fancy scoreboards if I have to memorize the score...if I have to memorize the score, fouls, foul troubled youths, time left, time outs left, #34 favorite after game meal, I WANT A RAISE!!! :)

I love it!

Do you think it's important to know the exact score, as opposed to having a general idea of the size of the lead?

Usually I'd be hard-pressed to know the exact score of a game, but I generally try keep a good idea of the relative score: "Home team is up by a bunch (10+)", "Visitors are leading by a few (4-10)", "Score is pretty much tied (within three points either way)"

My wife when I get home: "who won?"
Me: "I did"

when you start caring too much about the exact score/ player fouls etc you simply have stopped doing the job you were contracted to do. Let the scorer do their job and you do yours!

Oh I don't know. The score matters. Last night's game is an example. With under 10 seconds left in over time, white was up 5 and blue had the ball in the FC coming out of a TO. It was worth knowing the score so we would be aware of Blue's likely tactic; score quickly and foul quickly.

Knowing the score doesn't equate to caring who wins.

That said, in the third quarter, I'm likely to just have a general idea of the score and clock.

Same here.

It's not that I care who wins, it's that I want to know if one team or the other might try using different tactics then they had been previously... running a delay offense instead of going for quick transition baskets, pressing, fouling quickly to stop the clock, etc.

Last night, last HS game of the year for me. Fourth quarter and we have a violation of some sort. I see the timer waving at me. I look at the scoreboards and they are both out. I head over to the table while asking my partner, "2:05 left, right?" He shrugs.

Plug finds its way back into the socket or something and the board comes on. 2:04. :D Visiting coach ribs me about not knowing the right time. If the scoreboard would've reset, it would've been a pretty good thing that I didn't wait until the last two minutes of the fourth to start with my game awareness.