Gifted Issues Discussion homepage Forums Recommended Resources AoPS Intro to Geometry course review

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.

Pardon me, but this is just awful.

(OT: Judging just from the excerpt above, you'd be much better with at least EPGY than with k12.com. EPGY does not use so-called discovery method as much as AOPS, but at least EPGY would never just give a formula to use (and memorise), without deducing it.)

I agree with others... the K12 approach is awful.

Have you seen the AoPS promotional mega-sample at http://www.artofproblemsolving.com/Resources/Files/Excerpts.pdf

Chapter 4 in that file is on divisor counting. The introductory material is basically a special case of regular permutations. Looking through that should give you an idea of how they would approach a somewhat more advanced combinatorics topic.

I didn't see any similar topics in their preA book on a quick glance through. The chapter above is from the Number Theory book which is after Algebra.

The sample has lots of good material to get a feel for how they teach. It might be more useful to you than the shorter book samples. I have no experience with the classes.

I can not resist quoting from one of the AOPS articles (also included in the pdf above):

What is Problem Solving?
by Richard Rusczyk
http://www.artofproblemsolving.com/Resources/articles.php?page=problemsolving

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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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?

The examples I gave were random, non-consecutive questions - there's a lot of missing context! (I would just send you the whole transcript, but they are strict on not sharing their materials, reasonably enough.) The "what might we draw?" question was in the context of trying to answer the question "can we draw a cyclic trapezoid" and was eliciting the answer "a pair of parallel lines".

Yes, they read from the book outside class, as I said in my very first post. However, they don't necessarily read the chapter that goes with the class before the class. Definitions and axioms/results are introduced in the class, one at a time, results normally being derived in class.

Lecturing doesn't work so well in text form - it's kind of like writing a book :-)

I agree with your general point, but personally I don't think you need be concerned about AOPS - though obviously, you'll make up your own mind when you try it! What AOPS covers so well is the ground between unguided discovery and lecturing: what we might call guided discovery. Students may end up feeling they discovered the standard results for themselves, but if you look at what happens, it was carefully planned: they were prodded just as much as was necessary to take each step. I think this is crucial in mathematics: it gives them a chance of not simply memorising the standard results, but rather, reconfiguring their brains such that the standard results become obvious. For this, giving a proof is not sufficient (at least, not for fairly inexperienced mathematicians): it helps to be led through the proof with some discussion of why each crucial step is a reasonable step to take. For example, in geometry, knowing which lines it might be useful to add to a diagram, and why, is often the hard part. I haven't seen anywhere else do anything like such a good job of getting this kind of thing over.

Oops, I took them to be a consecutive sequence of questions. Somehow I envisioned an exchange like

AoPS teacher: It's a circle. Now, class, what do we draw in a circle?
Students, in unison: A quadrilateral!

which I took to be of the same ilk as

B&M teacher: It's Thursday tomorrow. Now, class, what color do we wear on Thursday?
Students, in unison: Purple!

(Don't ask me to explain how my brain works. I can only give examples. )


I've reread it now. With all the comments I think I have a clearer picture of how AoPS works, and I'm sure it works well with their target audience.

So we'll pick a course this summer and try it. I say we, because we'll definitely attend all sessions live and I'll be sitting right there, probably typing, absorbing everything to maybe explain later, and just observing AoPS as an interested mathematician/educator.

I thought I would comment a bit more on EPGY's Intro to Geometry course. I noted earlier in this thread that my son had taken several EPGY courses and we were very happy with them. He had then moved into Geometry and was being challenged which was great.

Unfortunately, it seems the reason we had previously been happy with EPGY is that we never needed any help. For their Geometry course, EPGY has a CD based system which was problematic.

We intially had difficulties downloading the CDs and were informed the CDs were defective so we had to wait on new CDs to arrive. Then, the progress reports function kept stating that my DS had not performed any of the homework even though he had, when we inquired about this we received no initial response. The tutor was changed and when we asked again, we learned that the previous tutor had not input any of the information and we were requested to resend in all of the homework. The new tutor is clearly overwhelmed as she could not reply timely (partially due to time difference and also she seems to have very little time to spend on EPGY) and when she did it was obvious she did not read my DS's questions. Ultimately she stated that the Proof Environment was "pretty sensitive" and "most of my students if not all struggle with the format". There was a simple proof my son was unable to complete and the tutor sent the steps to complete it, however, he still could not get the Proof Environment to accept her steps. We were ultimately told to do the proofs on paper, scan them and send them in for grading.

I felt that it was too much hassle to continue with EPGY's Geometry and I have requested a full refund. I am uncertain if they are having personnel issues or exactly what the problem is. It does seem that the more difficult part of the course is trying to understand how to get the Proof Environment to work.

DS is now doing the AoPS Intro to Number Theory and we will then try to find a Geometry course.

How is the Intro to Number Theory going?
Does your son like it? It seems like it would be a fun course.