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

DS10 finished this course today, so I thought I'd report on what it was like, for those who might consider it or another AoPS course.

Short version: very valuable, much harder than I expected.

I had thought I was playing safe and giving him a gentle introduction to AoPS by picking this course - he's done a fair amount of geometry in other settings over the years, including ALEKS high school geometry back in Feb'12. Not a bit of it, and thank goodness I didn't encourage him into anything harder!

The course (their details here) ran over 24 weeks. Each week there was a "homework" exercise sheet of roughly 10 questions, usually 8 short-answer and marked automatically and immediately, and two proofs that were marked by humans. The marking was picky, in a good way - DS didn't often lose marks, but often had it pointed out to him where he could have proved something more elegantly or explained himself more clearly. I mostly agreed with the markers, but not absolutely always :-) DS took a completist attitude and insisted on doing every question - it might have been better to be prepared to give up sometimes, but that wasn't going to happen. I unstuck him when he needed it, and I scribed for him sometimes. Proofs were written in LaTeX and always had to be accompanied by a diagram, which DS produced in Geogebra - he is now reasonably competent in LaTeX and very proud of being "the Geogebra wizard", far more familiar with it than I am. (Geogebra was a big help and he often used it even for the short answer questions - obviously this means he had help that he wouldn't have in competition, but I felt it was good for him to use it. The answers typically involved surds and so couldn't just be read off, but there was a fair bit of "hmm, geogebra says that angle is equal to that one, can I prove it?"!) Most weeks there were one or two questions that gave him serious trouble. Every few weeks there'd be one that gave me some trouble too! Here's the very last question of the course for anyone who'd like to try their hand - it's a nice example, requiring only very elementary techniques, but nevertheless reasonably challenging:


Each week except the last few, there was also an assignment of Alcumus topics to do; this tended to be light relief for DS who was already well advanced in the Alcumus geometry before the course started (another reason I hadn't expected it to be hard). There was usually a chapter of the textbook to read, on which I have to admit DS skimped.

There was an online class which DS couldn't attend in real time because it was in the middle of our night, but transcripts were posted, showing both the instructor's words and those of the students' that had been posted (it's all text based; students ask and answer questions, the instructor decides which student typing goes into the class transcript, and there are also TAs who can have side conversations with students who need it). He enjoyed reading the transcripts and would, I'm sure, have loved being able to attend class. One good thing about the format is that there's very little in the way of clues to student age.

There was also a class forum, for which I had high hopes, but it was so quiet that DS was too shy to post there except for one exchange early on. The instructor posted extra questions each week to the forum, but hardly any of them attracted responses - maybe other students also found themselves fully occupied by the homework. The occasional student question to the forum was promptly and helpfully answered.

Given that really the only aspect of the course DS used was the homework problems, and given that I could have set and marked these myself with some help from the book and competition papers, one way to look at this would be as a very expensive way to have someone else do for him what I could have done myself! But he would never have kept up the pace without the external pacing the course provided, and it was very helpful to have the proof pickiness come from someone else - it happened more than once that I criticised a proof, he decided to leave it as it was, and then the marker criticised the same thing :-)

He had a bit of a love-hate relationship with this course, and now that it's over, he definitely does not want to do another course immediately, and tbh I concur. Overall, it probably took him 5-7 hours/week, which is about what they estimate; it's just that I had assumed that would be an overestimate for him, and it wasn't. Given a 12 hour school day (including travelling) and two instruments to practise, that meant it was really a substantial chunk of his free time. On the other hand if he were being homeschooled, I have no doubt we'd be hooked. As it is, I think we'd need to be looking at a shorter course that ran mostly in school holidays. I have my eye on Intermediate Number Theory, possibly in the summer... but we'll see. For the next few months, I think just Alcumus will be fine.

Wow-- THANK you for the very thorough review. Tremendous insights into this.

Got it! (At least I've convinced myself with a picture.)
Geometry was my weak point in competitions since we didn't do it much in school. I once successfully solved a competition problem using coordinates, and formed the false hope, despite warnings to the contrary, that this was the way to go. The warnings were right. Maybe I need to take this course.

Just some key points, many details omitted, and hopefully my picture didn't fool me. No peeking til you've tried it.

I never did either competitions or (much) geometry at school, and I learned a lot from this course :-) Since DS's school maths teacher was sometimes letting him do these problems at school, I undertook to produce and send in a set of model answers each week, so that when DS was stuck the teacher didn't have to solve the problems in his head to help DS while also dealing with the rest of the class doing something completely different. It was more of an undertaking than I thought when I made it! (And it's both encouraging and probably good for school-home relations that the one time I had to confess I hadn't solved one of the problems yet, DS's teacher promptly sent me a solution!)

ETA yup, your solution is consistent with DS's (though my guess from the wording is that the method was not identical - there are several ways to approach this I think).

Yes, great review! DD has done some Alcumus and work with some of the AoPS books but it does make me wonder whether DD should ever take this course because I am not very mathy at all and it has been sooooo long since I had geometry. So it would take a significant amount of effort for me to even try to 'unstick' her, assuming I could even figure it out. Does AoPS have tutors available, or people you can ask if you get stuck with a problem and don't have a helpful parent or friend around? Otherwise I am not sure how DD would be able to deal with this course, as she is quite mathy but does get very frustrated when stuck, and I am of very little use despite my most earnest efforts.

Oh, parental unsticking is definitely optional, and not necessarily even a good thing. Arguably, it's better for them not to do every question. But DS's frustration tolerance, while excellent for his age, wasn't really up for that, and it was natural for me to help.

There are hints available for the questions. The instructor and TAs watched the forum so students ask there, and there's a mechanism for doing so completely anonymously if they like. Also, I didn't mention that the students arranged to meet "in the classroom" to work on any leftover problems just before each deadline - again, in the middle of the night for us though!

What I'd say is that a student does have to be able to ask coherently for help - I don't know what happens if someone struggles silently, but I suspect they just drop out - and may need to be able to tolerate a delay before they get help (eta but not a longer delay than someone in a b&m school might have if they get stuck on a homework question). I'm sure you could help with those skills even if you did get stuck on the maths, and modelling a good reaction to being stuck might in itself be useful!

You can get a full refund several weeks into the course - up to the start of the third class iirr - so it's possible to try a course and bail if it's not going to b work.

We have integrated maths here and I think geometry is the discipline that gets least attention. It is also the only one I didn't at least do to first year at university so I wouldn't be a lot of help. It looks quite scary!

Thanks for the details, ColinsMum! I couldn't quite tell how it would from the course descriptions I saw, and the help that is available sounds like it is set up well. And I often get chances to model not being able to get an answer to something!! although DD often moves on to something else before I am done showing her how to persevere, so I continue to work on that

If you could get a kid to actively participate in the school and the community it would help them a lot I bet.

I have been looking at Introduction and intermediate to counting and probability for my ds 10. He is pretty stretched right now but maybe for some summer work. It is pretty cool stuff that could get a kid hooked if they hang in there but might be too much review.

We tried some ALEKS stuff and thought it was horrible. How did your son like it?

My DS10, soon to be DS11, took an Intro to Python programming course from AoPS this summer and it was run very similar to the Geometry course you describe. The course was pretty tough for him and I contacted a local university and found a tutor which turned out to really help (as I was of almost no use to him). He now often likes to find difficult math problems and write code to find the answers.

He is currently working with EPGY Geometry and is definitely being challenged. It has been a long time since I took Geometry and I do not remember being presented with some of the topics in such great depth. In the past year he has completed EPGY's 5-6, PreAlgebra, Algebra and Intermed Algebra. Geometry is the first course that has really challenged him but he seems to be rising to the challenge so far.

I think we may try the AOPS Counting and the Number Theory classes after he finishes Geometry. I was thinking of trying to just get the books and go through it without the online course but not sure how that might work.

Thanks for the info on AoPS Geometry.

Does he do Alcumus, and if so, does he have a high level and all topics blue in the counting and probability there? That's the obvious thing to use as a guide. The intro c&p is reputed to be the easiest intro course, but it looks to me as though the intermediate requires a level of mathematical sophistication that makes it unsuitable as an immediate follow-on from the intro course. For example, it does induction half way through, something which (for reasons I've never fully understood!) many people find to be a conceptual roadblock.

It was useful for a few years, and he used to go in phases of liking it a lot for a while and then getting bored with it. Definitely only useful for routine problems, but doing the routine questions and working out what was going on from that beat being actually taught the material, for him. I doubt he'll use it again at this point, though.

Thanks for the great review!

You mentioned that the class was all text based. Just wondering how well they can convey the geometrical ideas without drawings.

Bad choice of words on my part - they did have diagrams, but no audio, video [in the classes - there are some video lessons I think, but they weren't assigned] etc.

I see. Thanks for the clarification.

He did some Alcumus (not much) when he was doing the Pre-Algebra but none recently. I will have to check. His teacher at school has been trying to get the students to do the Alcumus and give him their info so he can check up on them.

Thanks for that in-depth review. I also would have expected it not to have been quite so challenging for your DS. On the other hand, I did find ALEKS (as least the pre-algebra) somewhat superficial and definitely way easier than the Pre-Algebra DS subsequently studied at school.

Thank you for posting! Incidentally, here is a recent WTM thread 'AoPS--self-study vs online class':

http://forums.welltrainedmind.com/topic/497646-aops-self-study-vs-online-class

Re: *Intermediate* Number Theory:


Re: relative difficulty of classes:

Re: online class vs self-study:

ETA: also, from here:

http://forums.welltrainedmind.com/topic/495394-number-theory-vs-counting-probability-in-aops

it looks like 'intro to number theory' < 'algebra 1' < 'intro to counting and probability'.

As far as this "One good thing about the format is that there's very little in the way of clues to student age," is the point just that it's possible to never disclose it, or is it encouraged not to disclose it, or doesn't it matter either way? Is there any philosophy behind this "age-anonymity"?

As far as costs I was thinking.
$X0 for a book
$Y00 for an AoPS course
$Z000 for a Math Summer Camp (for 2-5 weeks, e.g. Epsilon, MathPath, MathCamp).
We had a thread on Math Summer Camps
http://giftedissues.davidsongifted..../all/What_elite_math_camps_are_ther.html
The $Z000 price for a Camp takes people aback, but some here have paid it, and others are thinking about it. In that context, $Y00 for an AoPS course seems more reasonable, even though you can get a book for only $X0.

Our DS7 has been taking K12Inc online courses (for free in a virtual school) and he has been going through the math courses like a hot knife through butter. They are okay for basic presentation of material, but the exercises are rudimentary and unchallenging, and I've heard similar complaints about most course options out there. I gather that AoPS is a rare case (unique? anyone know of others for math?) of very challenging courses which we desperately need to prevent all the pitfalls of whizzing through shallow courses, and never meeting real challenge.

I've been convinced that AoPS Intro to Geometry is a good one for DS to take (eventually, when the time is right), and I could possibly learn a fair bit from it myself, which would not be the case for any other intro or interm course.

We need to pick one or more AoPS courses for DS to take before Intro to Geometry, and before the lack of challenge goes too long. (I'm asking for suggestions.) We have to be selective due to time, cost, potential burnout. Though I could teach any of them myself using just the book, I gather from the review and other comments, that it is worth having the structure, timeline, feedback from non-parent grader/marker, and other potential dialogue.

Out of the following AoPS courses, does anyone have recommendations? And what is a (partial) order of these courses in terms of difficulty, and also considering prerequisites? (DS would meanwhile be taking a standard USA sequence of courses. He's doing K12Inc's PreAlgebra now.)
Algebra 1
Introduction to Counting & Probability
Introduction to Number Theory
Algebra 2
Introduction to Geometry
Algebra 3
Intermediate Counting & Probability
Intermediate Number Theory
MATHCOUNTS/AMC 8 Basics
Advanced MATHCOUNTS/AMC 8

Also does anyone have experience with younger kids (8, 9, 10) handling the live online lessons with the fast reading and typing (with a parent right there to maybe help type)and the fast thinking, of course? Does it work for young ones? How much learning happens in real time during the live lesson, rather than at other times?

DS (turned 11 last week) began EPGY in 4th grade and went through their 5th-6th course in a bit under two months, next completed Honors Pre-Algebra in about a month and a half; finally completing 4th grade by doing Honors Algebra. He worked alone almost entirely which was great as before EPGY I felt I was too involved in his math.

He took the summer off for math but took AoPS Intro to Python Programming which was difficult for him. I contacted a local university about 2/3 of the way through the course and found a student to tutor him for what turned out to be about 10 hours. The tutor really helped him understand and he enjoyed the course and now really likes to write programs. I think the AoPS course was hard in general but also he struggled to type quickly enough to be too involved. It was his first time being involved in an online chat learning format and I think it was difficult for him but all in all a great experience. I was not really involved at all except for at the very beginning to see how the course worked and the chat was being moderated.

He began 5th grade this year and started EPGY Intermediate Algebra which he completed in two months so he recently started Intro to Geometry and is definitely being challenged but seems to enjoy it. I think Intro to Geometry is the first real challenge he has faced with his EPGY courses, not so much that he did not learn in the other courses as he definitely did, but it did not seem to be overly tough for him.

I think it will take him about 4 to 5 months to complete the Geometry course. Afterwards I hope for him to do the AoPS courses on Counting/Probability and then Number Theory. I purchased the books and am hoping he can do it without the online class as the timing is not great for us.

FYI - my son is in public school but is able to work on his EPGY math course during the math period.

Firstly, kudos to ColinsMum for talking the plunge with AoPS and then doing us the service of putting together a detailed and positive review.

I have been wondering what to do with DD after the SG primary series peters out and also dread pushing ahead without a coherent plan/syllabus. AoPS seems like a good way to build on the foundation that DD already has from SG primary maths and it appears to present its material in a very coherent and incremental way.

I had DD take the 'pretest' and she went through it like a dose of salts with one transcription error from the test paper to her 'work paper' causing an incorrect answer.

I will be signing up my DD for a pre algebra I class in February.

What I meant was that I never saw anyone disclose age; DS didn't come under any pressure to do so, and I don't see that he would have done, even if he'd been more active. When I first heard about AoPS I assumed (as apparently many people do) that there'd be audio or video participation involved, and that, obviously, would reveal age clues. A typing interface does not in the same way. At the start of the class, I mailed the instructor about a couple of things, and mentioned DS's age in an "I hope this won't show but just in case it does" way, and got the response that I shouldn't worry provided he was up for it and "we have lots of younger students". I didn't clarify whether this meant "younger than your DS" or "younger than average to be learning the material" - I suspect the latter, but I saw everything that DS saw about the other students and I couldn't tell which ones might be his age or younger. Yours might not have come across this so much, being homeschooled, but DS finds it a bit tedious being known for being good at maths at school (though he sort of enjoys it too) so it was very welcome that he felt safe from that in AoPS. [Well, in practice, he was invisible so of course he was safe; but he also didn't see anything to suggest that he'd have attracted that kind of attention if he'd been more visible.]


Don't know so much about difficulty (though see the comments upthread quoted from WTM). I think the "x < y means course x contains material that is prerequisite for course y" partial order contains these and their consequences:

Forall n. Algebra n < Algebra n+1

Forall X. Introductory X < Intermediate X

(better put those in ;-)

I think probably

Forall X. Algebra 1 < Introductory X

for being able to handle basic algebra including quadratics, at least (this was certainly assumed in the geometry course).

Their material suggests:

Forall X. Algebra 3 < Intermediate X

but it's not clear to me whether this is really a question of prerequisites, or more a "mathematical sophistication required" thing. We have the Intermediate Algebra book which goes with their Algebra 3; I had in mind to ask on their classes forum whether there were specific chapters to revise, before signing DS up for Intermediate Number Theory (if he does want to do that later).

I have read that all three intros are incomparable wrt prerequisites, but geometry is harder than the others. I'm less sure that I've seen it from them in writing, but I think the same is true of Int C&P and Int NT. I'm less sure about the AMC8 courses; clearly they're at introductory level of difficulty, and I think they neither have nor are prerequisites within your course list. What I'm not sure of is whether there'd be anything to be gained from them if one already had a good understanding of the introductory courses' material.

The AMC8 ones, and Intro to Number Theory, have the advantage of being shorter than the other reasonable possibilities for a course to do before Intro to Geometry - that's probably an advantage for a toe-dipping! I think if I were you, I'd probably decide which AMC8 one would be more your DS's level, and offer him the choice of that or Intro NT.

I don't have experience, but I suspect that depends more on the individual's characteristics such as processing speed than on age, really. DS is a relatively slow processor, I think - I predict that would be his lowest WISC index by a fair way, if we had him tested - and I think I am too. I see (some, but reassuringly not all, even of the best) colleagues well able to learn in real time from seminars, something I still can't generally do! (I can get the gist, of course, from a competently given one, but if I want to fully grok what's going on I have to think about it alone later. It's rare that I can ask a penetrating question at the end of the seminar.) That said, pretty sure I personally would find a class in which there was a written record of everything the teacher said in front of me at all times much easier than an ordinary spoken one!

Thanks for clarifying that. So you can give full details of age, background, etc to AoPS "staff" to get the best feedback on course suitability, but you can be totally (age, background,name)-anonymous to the other AoPS "customers".

Thanks for analysing the courses. I've now had a good explore of the AoPS website. We are looking for a "toe-dipping" course, just to see if AoPS will work out yet for DS8. (I'm concerned he'll find the format unappealing at his age, or that he won't be able to keep up with the non-mathematical aspects of it.) It's a good idea to look at the shorter (and cheaper) courses, and for the lower level courses that means:

Introduction to Counting & Probability $265 ($307 with books) 12 lessons of 90 minutes each.
Introduction to Number Theory $265 ($312 with books) 12 lessons of 90 minutes each.
MATHCOUNTS/AMC 8 Basics $245 12 lessons of 90 minutes each.
Advanced MATHCOUNTS/AMC 8 $245 12 lessons of 90 minutes each.

It seems that MATHCOUNTS/AMC 8 Basics only needs Prealgebra, while Intros to C&P and NT need about the first half of Algebra 1, so that suggests AMC 8 Basics as a course to start with. I realize the AMC 8 courses could be redundant if you do all the "Intro" courses, but on the other hand if it falls through (if DS8 doesn't take to AoPS at the time) there's not much lost and he doesn't need to "repeat" some course to cover "irredundant" material. It would just be (difficult, compared to K12Inc) problem solving, which is what he needs.

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DS8 just finished the K12Inc Prealgebra, and found it very easy. The problem is that these courses are designed to only require a rudimentary "mastery" of the material. He could probably get through a few more of these "rudimentary" types of K12Inc courses pretty easily, but I'm worried about how that would turn out. (It's not just K12Inc. The problem is with any course aimed at typical students.) I'm torn as to how to proceed. I'm seriously thinking of not getting the next K12Inc course (Alg1) until August, and instead working through Alcumus for a while, then taking an AoPS course and generally trying to "toughen" DS8 up a bit, and also get a much broader coverage of maths topics than just what's in the standard Alg1/Geom/Alg2/TrigPrecalc/Calc path. On the other hand Alg1/Geom/Alg2/TrigPrecalc/Calc is just basic stuff a mathy person should learn before going to more advanced stuff, so why delay, why not just keep going through these courses as soon as one is ready? (DS8 is in a virtual school and can do courses at his pace. I'm trying to optimize the pace.) It's quite a dilemma. Thoughts, anyone?

\end{ramblerant}

My thought would be that exposure to finding problems hard is important, far more important than getting through any particular material, so yes, I'd notionally put that in first and then see what else there was space for. Depending on personality and what you use for that and maybe other stuff too, you might find your DS is happy to spend all the time that seems reasonable to spend on maths on the hard problem stuff, or you might find that his tolerance for that is initially limited and it's good to have some more routine stuff in the mix too.

Do you have complete flexibility about when to start classes, and can you let them go to the backburner for a bit without penalty if you like? E.g., suppose your DS started a K12Inc course now and then hadn't finished it when the AoPS AMC8 course starts in March, could he temporarily drop the K12Inc work for the duration of the AoPS course, if it wasn't convenient to keep both up? If so, I don't see a downside to starting all the hares running and just playing it by ear. If, on the other hand, once you start the next K12Inc course you need to keep at it consistently, then I'd be inclined to agree that taking a break to work on Alcumus and AMC8 stuff now might well be sensible.

The thing I find helpful to remember is "he has time". If you take a decision that turns out not to be absolutely optimal - e.g., he does something he turns out not really to be ready for, so that he forgets it and needs to do it again later - no big deal.

I think the appropriate place to start with AoPS depends on interests and mathiness of the kid and whether the kid hates or feels challenged by exposure to material he/she cannot master.

My DS did some online math courses at 8 hated them. Hated the material. Hated the math. Hated the repetition. At 9, he tried AoPS Algebra I. This was a while ago and it's much different now, but it was a sink-or-swim situation (no help from parents, pretty much mastering typing and LaTeX at the same time and no prior exposure to hard problems). It was a bit brutal, but he loved it and it restored his love of math. He subsequently took all the intro and some non-intro AoPS courses and he agrees with Colinsmum that geometry was the toughest of the intro courses (AoPS founder also says this).

If you are considering toe-dipping, I'd recommend starting with pre-algebra. If he's already had pre-algebra *and* has mad and well-developed deep problem solving skills, I'd consider algebra but AoPS pre-algebra will not be the same as a standard pre-algebra course. Perhaps the pre-algebra 2, rather than the first one if you wish, but it would be kinder to start developing perseverance with harder problems with material he has some familarity with rather than also learning a new subject. And this is really, really different than typical math.

The AMC/Mathcounts problems are easier since they all require speed so the problems can't be that deep. There are tricks learned and initial problem solving skills, but a big difference between those skills and challenge level geometry problems which DS would work on for an hour sometimes.

I don't think of doing C&P and NT as delaying, but rather enriching the typical and somewhat limited typical school sequence. Since the courses are not full year for any of them, the kid ends up further accelerated anyway. So DS did algebra I, algebra II, geometry, C&P, NT, pre-calc in far less time than alg I, II, geo would be in a regular sequence, but he much learned more math.

Some kids don't like the format. Some kids don't like math problems they can't solve. Some kids get frustrated with the difficulty. But some kids eat this stuff up and if you have such a kid, AoPS is simply fabulous.

Now that you say this, I wonder why I didn't also suggest considering it :-) DS had in principle done the geometry material a long time before he did AoPS geometry, after all, and still got challenged. One thing to note is the "full refund before third class" policy and the fact that there are Prealgebra classes starting in February. So if you're not sure whether Prealgebra will be right or will be too much repetition and AMC8 would be better, you could sign him up for a prealgebra course and bail if it does turn out to be too easy, with time to enroll in the AMC8 course instead in that case. If the geometry course is typical, you'll see three sets of challenge problems in the prealgebra course before you reach the point of commitment, which should be enough to make a good judgement of whether it's suitable.

How is your DS finding Intermediate Number Theory, solaris? In particular, how long is he needing to spend each week, roughly, and is he doing all the questions? The further the geometry course recedes into the distance, the more tempting it looks to have DS do that course in the summer... but I don't want to overload him.

I'm not Solaris, but DS experience with the classes was similar to your son's in terms of geometry. Hard but rewarding. Number theory (both versions) was not nearly in the same league. NT was easy, fun, and not much work.

Thanks both! So, different experiences there from the sound of it. Interesting... Solaris, I wonder, could you maybe share (by pm if you prefer) just one of the questions that were hard on that first sheet, to help me get an idea of what we're talking about?

ETA DS was also unwilling to use the forum, and come to that, so are my own students, so I'm no help.

Lots of great comments. I'll get back to the rest, but I was curious about this.


This makes me like AoPS even more. But it makes me wonder, how exactly are their classes structured. Do they lecture about some theory, with adequate proofs/explanations? Or do they lead you to "discover" things via a sequence of problems/exercises/questions? Or a combination of these? Or what?

For comparison, the K12Inc Prealgebra course simply stated formulas (after adequately defining n!)
P(n,r)=n!/(n-r)!
C(n,r)=n!/[(n-r)!r!]
without any explanation of why LHS=RHS. (Usually they give some kind of explanation for things, but not this time.) Then they have some "worked examples" which are easily skipped past, then a routine quiz.

How does AoPS take a student who hadn't seen n! before, and lead them to know these formulas and understand why they are true? And what other activities surround the learning of that particular piece of maths? (Or substitute any other piece of maths for the purposes of this discussion.)

How does AoPS work?

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.