In this respect, these classical schools sound an awful lot like the public school education I received. We learned by phonics. We diagrammed sentences. My mother actually had flash cards to drill me on math (I still have trouble remembering that 9 + 5 = 14 and not 15. ) I had Algebra I and II in junior high. There is a mention, too, of an emphasis on civics; in junior high I had a class that was actually called Civics. (I vaguely remember we did things like study the Constitution, the branches of government, how the government works--things like that.)
Weird, because you and I must have attended school at about the same time. (I graduated in 1975.) I think we learned by phonics. I remember flashcards. I had algebra I and II in junior high. We memorized math tables and occasionally poetry -- I memorized "Jabberwocky" for extra credit and can still recite it today at the speed of an auctioneer.
But we did not have a class on physics -- we had American history, Asian history, history of religions and plain old social studies. (I remember the only time I'd get school lunches was when, in the time period after my Asian history class, the day's menu was chow mein.) We must have learned some of the same material that would have been in a civics class, maybe in the all-purpose social studies classes. And I've never known what diagramming a sentence even requires. (I've seen the diagrams but could not make one myself.) I can identify things like adjectives and adverbs, subject and object, passive and active tense -- stuff like that, if that's what it's about.
I think our district might have been a bit "progressive" -- I remember an elective English class in junior high that focused on the poetry of Bob Dylan.
And in high school I was in a poetry class and was falling behind because I hadn't turned any poetry in. So I wrote a long poem about my drug-fueled experiences at a Foghat concert.
Got an A+. (I didn't mention drugs explicitly but the language in the poem suggested them.)