Answering various comments:
The K12Inc example I gave was particularly bad, but not representative. Usually they give decent explanation (though their K-5 math sequence was redone recently and may be better than their middle school courses). And the online lessons are designed so you can just quickly cover new material and skip revision, practice, etc. It's okay for making sure we cover all the topics in the curriculum, and for other subjects we know less about, and it's easy to accelerate, and it's a free virtual school, though independent homeschoolers can do better with more effort. For us it's just a way of getting a rudimentary covering of all the basics, and we always knew we'd have to supplement at some point with something like AoPS.
The K12Inc courses in the free virtual schools can be done any time any day, so there is complete flexibility in that sense. On the other hand you usually need to finsish a whole number of courses within a standard August-May school year, so some planning is required. That's why I was pondering the question, start Algebra I now or in August. It could make a full year's difference in the timing of the future subject sequence, so one way or another there's a decision to make. We happen to be very busy the next few months (with some fun things) so, apart from finishing the non-math subjects for the school year, it'll be Alcumus for a few months, then trying a first AoPS course in the summer, then back to the regular schoolyear in August starting K12Inc Algebra I, then after getting through that in a few months, back to some AoPS courses including ones which require Alg I knowledge.
I think we've got the pacing right. DS8 covered K12Inc math K-7 in 3 years (more like 2.5). Then he'll "slow down" and do K12Inc math Alg1/Geom/Alg2/TrigPrecalc/Calc over 4 years, while expanding into the broader range of topics offered by AoPS, whether by actually taking AoPS couses, or learning it some other way. The point is, you want your mathy kid to branch out into the broadest range of topics available (within reason), not just the typical narrower path offered by the school system. But you have to cover the prerequisites first to get to these "interesting off-path" courses. And there's so much more to do way beyond the courses we are discussing here. You still need to start "basic" topics like linear algebra and group theory before you can access most higher topics. I hear people sometimes talk about "slowing down to go sideways", but if they are not careful, what they are actually doing is "just slowing down".
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Thanks. I like the look of the AoPS books, and I'm sure we'll end up getting several, whether or not we do the corresponding courses.
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This is interesting. I generally agree with what he says in various articles. He is generally expressing how mathematicians think, and he writes well. But in this article he describes (admits?) having a kind of epiphany where he went from not really thinking like a mathematician, to thinking like one. Interesting.
He also seems to imply that one could do well at lower level competitions with a "bag of tricks", but that that does not work higher up.
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In the geometry course: by discovery (of proofs). Very Socratic. I just looked at the transcript of a class on circle theorems. It covers 24 pages, including the student comments that got posted. The typical length of the teacher's contributions is 1-3 lines. There are only a few that are longer than that, and looking at them they are typically of the form [three lines of summary of last bit]"Any questions?"[two lines of intro of next bit]. The teacher asks lots of questions - here are a few random examples from one transcript:
Let's begin with the circle; now what might we draw?
Why?
Wait, aren't all quadrilaterals cyclic?
Let's see how this helps us understand cyclic parallelograms. In a parallelogram, what do we know about opposite angles?
and puts in diagrams. He posts a selection of student answers, typically only the correct ones - no idea how many incorrect ones there are, but he does sometimes say things like "Some of you are saying... But...". Presumably he also selects the suggestions that take him in the direction he wants to go! There are TAs behind the scenes, and I'd hope they are helping the students who are not submitting answers that are what the teacher wants (whether they're wrong, or whether they're just not the desired direction) but of course I haven't seen that.
Okay, I was afraid of this. Students need to be told some things. I don't like the question "Let's begin with the circle; now what might we draw?" Why would it occur to someone that a quadrilateral is something you might like to draw in a circle? Or has some prior activity in the course, a previous class, reading a book, or problem solving, prompted the student to know that a quadrilateral is in fact something a geometer might like to draw in a circle. Of course this is just one short excerpt (and I latched onto that one question). I just worry that they might be taking it too far. Don't they sometimes lecture on some pieces of theory? How do students get to know what the definitions are, what axioms there are, etc? Do they read some theory from a book outside the class?
Human civilization has advanced because, while some people invent things, discover things, solve problems, etc, the crucial thing is that these things are communicated to others and propagated throughout society. Humanity wouldn't get far if everyone had to reivent the wheel. It's true that the typical school system is way too spoonfeedy, and there needs to be much more invention, discovery, problem solving, generally thinking creatively, in the education process, but if this is taken too far, it doesn't work. Their needs to be a certain amount of simply telling people stuff.
So, should I be concerned AoPS might have the balance wrong in their approach with their courses?
