Gifted Issues Discussion homepage Forums Recommended Resources What makes AoPS Intro Geometry "challenging"?

What exactly is it that makes AoPS Introduction to Geometry (book and/or course) more challenging/advanced than the other AoPS Introductory books and courses?

We have bought the following books with the idea of DS8 gradually working through the books, not taking the courses (though maybe he'd take the Geometry course).

Introduction to Algebra
Introduction to Number Theory
Introduction to Counting & Probability
Introduction to Geometry

I was vaguely aware that one should get about 1/2 way through Intro Algebra, before starting Intro NT and Intro C&P, and that one should finish those 3 before starting Intro Geometry. I told DS this, so he started on Intro Algebra and is about 1/4 of the way through. But he wants more variety and is itching to start the other books. I saw no reason not to start Intro NT and Intro C&P, and told DS he can go ahead with those.

I still want DS to finish those 3 before starting Intro Geometry. But I want to understand better what are the reasons for doing it this way.

DS hasn't worked systematically through the other books and the only other course experience he has is the current AMC10 course (bit on the easy side so far but he's loving it btw). My uninvestigated guess is that it's mostly harder because it takes several ideas to solve the harder problems, where even in AoPS algebra problems are normally one idea plus follow your nose. But of course, it'll make a difference whether your nose is habituated to pointing the right way...

Concretely, I do remember an AoPS geometry question assuming children could easily factor a quadratic.

In your place, I'd let your DS do things in parallel as much as he wanted, but be prepared to intervene if it did become apparent that he needed more algebra at some point.

I've seen AoPS mention this in a number of different spots (my DS is antsy to get at geometry, so I've taken note). The message seems to be that with the first 13 chapters of Algebra, you have the basic math skills you need to do geometry. However, after the other two Intro classes, you will also have developed a much wider range of problem-solving skills and approaches which will make it much easier to deal with the more complex geometry questions.

I think there are a couple things. First, my guess is that people whose strengths are more sequential than visual-spatial will find the geometry text more difficult than the algebra text (I think my kids are the opposite, though we shall see). Second, there's the proof-writing. While the other AoPS books include proof-ish problems for which reasoning needs to be explained (disclaimer: we haven't used all of those yet), geometry requires actual proofs, which I'd guess would be on the more difficult end of things for an 8 year old, though I suppose it depends on the kid.

I have a child taking the on-line geometry class right now. I love the weekly class problem sets - very challenging, more challenging than the book. At the moment, she's a little behind in her assignments - there have been a few problems that she hasn't figured out yet because she hasn't put in the time it will take. They are hard and I haven't figured them out yet either (again, an issue of time and effort and experience).

I don't see anything wrong with starting the geometry in parallel to the algebra, anticipating a pause if additional algebra is needed as mentioned by the PP, and then if it's too difficult, it could simply be put aside for later. There's no rush, after all; enjoy the banquet of math. I admit I am jealous of my child having the opportunity to learn geometry with AoPS - it's awesome.

I've re-read through ColinsMum's thread "AoPS Intro to Geometry course review"
http://giftedissues.davidsongifted....7284/AoPS_Intro_to_Geometry_course_.html
My DS8 just took an AMC 8 Basics AoPS course that he found somewhat challenging, so on that basis, it wouldn't make sense to think he'd be ready for AoPS Intro to Geometry course yet (although one could be ready for the book sooner than for a course, since you can slow down, take a break, come back to it later etc.)

[BTW the AMC 8 Basics AoPS course, was a good one for him to take. Often he'd be newly learning about a topic, rather than applying an already learned topic to problem solving, but he got exposure to a lot of topics that he can follow up on later.]

[Also, after thinking about "AoPS: Book vs Course"
http://giftedissues.davidsongifted.org/BB/ubbthreads.php/topics/197574/AoPS_Book_vs_Course.html
we've decided to generally cover material via the books not the courses, so we're not tied to a schedule. DS8 doesn't need the courses for motivation. (He'll still probably take the contest courses, and maybe some others.)]

DS8 seems happy to just work through Intros to Alg, C&P, NT, and to wait for later for Intro Geom. He should have no problem with the level of these 1st 3 books, though I'm sure there'll be plenty of challenging problems. It gives him more variety, as I think he was getting a bit sick of doing just algebra. (C&P and NT seem like more "fun".)

But he was right to question my earlier decision that he shouldn't start C&P and NT, so I really need to figure out the optimal time to start the Intro Geom book, because I'm not sure. I'm not so much concerned about the suggested prerequisites, he's close enough on those, and I could catch him up as needed. And he'd be able to learn the basic geometry material itself (e.g. from a typical lighter course). But it may be that the problems are too complex/difficult. Now if I did let him start on the book, and he realized it was too difficult, and he couldn't do the problems, and decided to set the book aside until later, then that outcome would be fine.

But here's the potential problem. With the Intro Alg book he works on every single exercise and problem (which is good) but he almost always does so without writing anything down, just doing it in his head. That's good brain exercise, but he's not developing the math writing/reasoning/communication skills that he should be at this curricular stage. (For example he'll try to solve 3 linear equations in 3 variables, he might make a mistake, which he'll discover when he checks if his solutions satisfies the equations, but he's persistent and will try again until he get's the right solution. But he's not writing it down. He's doing it in his head. I try telling him he needs to write it down, for several reasons, but he won't.) I'd really like him to develop these math writing/reasoning/communication skills, before starting Geometry for real.

So what I'm concerned about is that if he starts Geometry too soon, before reaching a certain level of "mathematical maturity" then he might start "solving" problems in a superficial way, in his head, not writing it down with proper reasoning (and possibly convincing himself he's solved a problem when he really hasn't). In these circumstances, he might proceed through the book, not getting as much out of it (and maybe not realzing this), when waiting until later would have yielded better results.

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Coincidentally, I walked in the door yestarday, and DS8 said he wanted to solve x^2+x+1=0. This led to discussion of several interesting topics. Then he said he wanted to know how to solve ax^2+bx+c=0 in general, so I guided him to derive the formula for the solutions. (This was all done mentally/verbally with no writing down, but this time I allowed that on purpose to give his brain a workout.) There's a lot more he can learn on this topic, and he'll learn it with the Intro Alg book, but he's fine with material at this level and can learn it quickly.

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I guess I'm trying to navigate an optimal learning trajectory, not superficially rushing through material, but not overcautiously holding back either. I want DS8 to wait for a time to do the Geometry when he'll get the most out of it. On the other hand I don't want to be saying, no you're not allowed to read that book yet (or to give reasons for waiting when my reasons are wrong).

Sounds as though it might work to say "you can start geometry when you like, but you have to understand it'll be important to write careful answers, so you should wait till you're ready to do that". That is, I think just telling him the truth as you see it might work just fine. And he probably will just naturally get better at writing carefully etc, given a little practice and time. I like how many problems with DS solve themselves if I can manage to be a little patient :-)

Yes you're right, and he is old enough now for me to explain just what I'm thinking, and I try to. I had told him that to do the Geometry he would need to write solutions in complete sentences and his response was "Complete sentences?! Wow!". Anyway, the more immediate problem is that he needs to write out his Algebra solutions, so a future need to write out Geometry solutions can serve as motivation to write out Algebra solutions now.

I'm not sure about the problems in the book vs the challenge problems in the courses, but my son had done C&P, NT, Alg I and Alg II before geometry and it was very challenging and a completely different level than the other courses.

Responses to the challenge problems were not sentences but multiple paragraphs with embedded diagrams, etc. and often took many hours to work out. Courses are different now with fewer of the hard challenge problems, but they are still challenging.

I would think your DS should be ready for a lot of writing down answers before thinking about the geometry course.

What do you plan for your son to take after the 4 courses you mention have been completed - follow the AoPS suggestions?

My DS will complete Intro to Geometry soon and has done the others except for Algebra he took EPGY Pre Algebra, EPGY Intermediate Algebra and is now taking Algebra II in his middle school. I have been impressed with AoPS and would like to follow their course outline but wonder if I should purchase their Alg II book and have DS go through it to see how challenging or just buy Algebra III? ( I will have him take the pre-tests but I was not sure that approx 20 questions is enough to determine how solid his Alg II is...)

Any suggestions?

The book situation is easier than you think because there are only two algebra books (Introductory and Intermediate) even though the same material is split into three courses (Alg 1, 2, 3).