Geometry - What is the purpose?

[lewelma]

My son is definitely a mathy. He started AoPS Intro Algebra at 9 1/2 and will finish the book in 2 1/2 years at the end of 6th grade . I am starting to think about Geometry. I would like him to take Algebra 2 in 9th grade, so I am curious what math he should take in 7th and 8th grade.

 

The problem: AoPS puts out Geometry, but also Probability and Number theory. A 1 year AoPS course so far is more than a year course for my son. He is young, so I assume he will speed up, but I am considering a simpler Geometry (MUS or Jacobs) so that he will have time to do AoPS Probability and Number theory. If he does AoPS geometry, I assume there won't be time for the other 2 courses.

 

So the question is: My son has limited time, which mathematical fields do I have him spend time learning? I was a statistician in a previous life, so of course I would put probability ahead of Geometry, but I also know that the SATs have a lot of geometry in them. Should he do MUS/Jacobs geometry + AoPS probability and number theory, or should he do AoPS geometry and just 1 other AoPS course if he has time?

 

Second question: Proofs teach logic, but so do most of the AoPS books. And I also plan for him to learn tradtional logic. So what is the true purpose of in-depth proof-based geometry? I was a High school math teacher in a previous life, so I do know the importance of geometry to more advance math -- but long, advanced proofs are of the ancient era, and I am not convinced of their importance today.

 

Thanks for your advice,

 

Ruth in NZ

[In The Great White North]

I was under the impression that AoPS Number Theory and Counting and Probability were shorter books that did not take a whole year, so he may well have time to do them all. I would choose more on learning style. AoPS is more deductive; the others are more plug and chug. Do you have to explain each lesson to him or does he get joy out of figuring it out on his own?

 

but long, advanced proofs are of the ancient era, and I am not convinced of their importance today

 

Not necessarily. Ds is a freshman in college this year and has to explain, in words with reasons, each step of his calculus homework and tests. Sounds rather like a formal proof to me. And he is not a math major. This is the required freshman calculus class that everyone has to take.

[lewelma]

My son likes to figure it out on his own -- he self teaches AoPS. He also just said that "it goes into your brain better if you figure it out, rather than being told." AoPS is definitely his learning style.

 

I was just thinking that given how long the intro algebra book is taking, that perhaps an easier geometry book would allow him to dabble in the other mathematical fields before Algebra II. I supposed there are 2 more questions: How hard and time consuming is AoPS geometry compared to AoPS intro Algebra? How much faster do kids work as they get older?

 

Ruth in NZ

[regentrude]

The AoPS Intro to geometry book does NOT take as long as the Intro to algebra book.

The algebra book has a lot of material, is way more than a traditional algebra 1 course, and my DD was not able to finish the complete book within the school year. The geometry book is shorter and she will finish within the school year (even though she did some algebra as well.) .

If you need to shorten the course: There are a few chapters which can be omitted if the student is only interested in the "normal" geometry and not in competitions (13 and 19). Ch. 18 is an intro to trigonometry which could be covered in precalculus.

 

Formal proofs are not ancient. Formal proofs give the student an insight into what mathematicians do - even today: they come up with theorems and prove them. I would definitely include proofs in the geometry course - but as I have a math loving, proof hating, DD, we choose judiciously and did not make proofs the main focus. She did enough of them to know how.

 

This said, I see no problem in choosing an easier geometry text if you want to fit geometry into one semester and spend the other one on probability.

[Jane in NC]

-- but long, advanced proofs are of the ancient era, and I am not convinced of their importance today.

 

 

 

Gulp. Apparently I am of the ancient era!

 

The two column proofs of traditional high school geometry (cannot speak for AoPS) are an odd beast, but proof is the substance of mathematics, not algorithms!

 

Perhaps Kathy in Richmond can answer this question for you as she has used all of the AoPS books, I believe. If she does not weigh in, you could send her a PM.

 

Jane

[Myra]

Here's what my son did (or is still doing!)

7th Algebra I - Jacob's

8th Geometry - Jacob's

9th - Algebra II - Lial's

10th Intro to Counting & Probability, Intro to Number Theory - AoPS

11th - Pre-calc - Chalk dust

12th - Finish Pre-calc as Chalk dust is a 3 semester program then do economics and/or statistics

 

 

Myra

[kiana]

Gulp. Apparently I am of the ancient era!

 

The two column proofs of traditional high school geometry (cannot speak for AoPS) are an odd beast, but proof is the substance of mathematics, not algorithms!

 

Perhaps Kathy in Richmond can answer this question for you as she has used all of the AoPS books, I believe. If she does not weigh in, you could send her a PM.

 

Jane

 

 

IIRC, aops uses paragraph proofs rather than two-column. I like this as it conforms far better to more advanced proof-based classes.

[Jane in NC]

IIRC, aops uses paragraph proofs rather than two-column. I like this as it conforms far better to more advanced proof-based classes.

 

Thanks for the heads up.

 

I really need to buy some AoPS books for my own entertainment.

 

Jane

[Kathy in Richmond]

Hi lewelma,

 

I don't find it unusual that a very young student would take a little over two years to complete AoPS intro algebra, since it encompasses what is typically called algebra 1 along with much of what is called algebra 2. Good for you for not rushing him through it.:) There are so many gems in the challenging problems to puzzle over - and taking that time to puzzle over them is a terrific way to build mathematical thinking.

 

As regentrude said, the geometry book includes a year's worth of material. As per AoPS, the level of challenge is still high. I think that a seventh grader who's highly talented and interested in math could complete it in one year. Though I would include chapter 13 (power of a point) - that's the sort of nonroutine math that could really ignite further interest (speaking as a total math nerd who had a teacher in her rural PA school who introduced her to power of a point among other things back in the early 70's and set her off on a lifetime math path...but I digress :tongue_smilie:).

 

My kids never used the AoPS introductory books, since they were all published after they completed those courses. We used Jacobs 2nd Edition for geometry, but we also used the geometry chapters in the original AoPS problem-solving texts. In particular, the geom chapters (especially the chapter on triangle geometry) in volume one of that series are excellent for introducing topics like inradius, circumcircles, etc, that aren't treated in typical US geometry books. He'll have to really think though the problems! You wouldn't go wrong do use the Jacobs/ AoPS problem-solving book combination, either. Just make sure to get the second edition of Jacobs.

 

The AoPS introduction to number theory and counting/probability books are meant to be done in a semester course each, so you could use those two texts for grade eight. Sounds like a fun year, too.

 

I'm not sure if I understand your second question. Are you asking whether the two-column proofs of say, Jacobs, are important? I think that proofs are extremely important, but whether they're introduced in two-column format (a la Jacobs) or paragraph format (a la AoPS) is immaterial. Most kids need to see the two-column format first. But of course mathematicians who make their living proving stuff don't write proofs up in two columns. So the paragraph format actually resembles more what mathematicians really do.

 

But I might have misinterpreted your second question?!

 

Got to run now to math tutoring :)

[LoriM]

Jane--I have thought the same thing! I have even "shopped" them on Amazon; is that the best resource for AoPS texts?

[In The Great White North]

Jane--I have thought the same thing! I have even "shopped" them on Amazon; is that the best resource for AoPS texts?

 

Me too. You have to be careful with Amazon because the AoPS books are often grossly overpriced there compared to the AoPS website.

[lewelma]

Wow. Thanks so much for all your advice. This definitely give me something to chew on.

 

My view on geometry is definitely biased towards using math for engineering or statistics, rather than mathematical research. In the eight classes that I took in university, there were no proofs. I also have been told that the SATs do not have long proofs on them. I definitely think that dabbling in that style of geometry is valuable, but if AoPS geometry was going to take my son as long as AoPS intro algebra, I was not sure that this would be the best uses of his time. In New Zealand, they do not do ANY proof-based geometry (I have taught here), so perhaps I am also being swayed by another's countries view of what should be taught to high schoolers. But all of you have given me much to chew on, and I really appreciate it!

 

Ruth in NZ