# Insights In High School and Want to Do Advanced Mathematics? - Comments

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1. Apr 7, 2016

### micromass

Staff Emeritus
Last edited by a moderator: May 7, 2017
2. Apr 7, 2016

### ProfuselyQuarky

Micromass, this insight was written like it was just for me. Thanks so much . . .

3. Apr 7, 2016

### Yashbhatt

Great post.

I have a question : How does one gain intuition about matrices, determinants? I can find inverses and solve determinants using properties but I don't think I truly understand them. Does linear algebra provide an insight into them?

4. Apr 7, 2016

### RubinLicht

That IS linear algebra, get a good book and work through it and see for yourself. I assume you just learned how to do it in your algebra class and the teacher didn't explain much. If so, then I was in the same situation until I took linear algebra.

5. Apr 7, 2016

### RubinLicht

Sorry strange format

6. Apr 7, 2016

### micromass

Staff Emeritus
You gain intuition about matrices and determinants only by studying the underlying geometry. This geometry is very naturally that of vector spaces and linear transformations. A matrix is then simply a very easy way to represent linear transformations, and the determinant is simply how the linear transformation acts on the volume of the unit cube. The annoying point is that it is very recommended to be able to compute with matrices and determinants before you should handle vector spaces. The effect is then that you compute with matrices without seeing what they really are. You'll have to get through this, I fear.

7. Apr 8, 2016

### adjacent

Wow. This insight is awesome. Thanks a lot micromass.

8. Apr 8, 2016

### IGU

Good stuff, but it might as well have been written for the student of forty years ago. Nowadays the world is full of students in high school doing advanced mathematics. And most of them use materials from Art of Problem Solving. Their books are excellent and fairly priced. Their classes are also excellent, but kind of expensive and not suitable for all. But for kids who love math and are good at it there's nothing better out there. Also check out their Alcumus online problem system. Free and useful. They also support social interaction and problem solving with online forums, also free and useful.

My math kid used several of their books a number of years ago, both when he was in public school and needed more and better materials, and later when he was home schooled. It was a revelation. AoPS was started by a bunch of math guys who set out to create the things they wished were available when they were in high school. I think they've succeeded admirably. You would be doing yourself a favor to become familiar with it.

9. Apr 12, 2016

### hamideh

thanks alot
could you please offer some books about Lie algebra (An, Bn,...)and exceptional like G2 algebra in physics with easy language?
thanks again

10. Apr 30, 2016

### acegikmoqsuwy

I agree one hundred percent. I am currently a high schooler and AoPS is my regular go-to for anything to do with math, whether it be studying for competitions, learning higher level math, or perhaps if I'm just bored and want someone to talk to about math. The part which really caught my eye the first time I ever went on AoPS was the following quote on their page: "Is math class too easy for you? You've come to the right place!"

Anyway, in my opinion, I agree that the article seems "outdated" in a sense, since the majority of high schoolers I know that are interested math already know topics including Projective Geometry, Graph Theory, and Abstract Algebra and are able to use them with moderate success on proof contests. The biggest problem with higher level math that's come about in my high school, as well as for many other high schoolers I know, is that even the local college courses are not enough; some kids complete Calculus I-III, Linear Algebra, and Differential Equations by their 9th grade year and then are stuck because they've exhausted all the classes from their community colleges and cannot afford the higher level classes at the actual colleges (since the school only pays a small fraction of the cost). As for me, I was slowed a tad since I was only placed in Geometry in 8th grade, but even as I have worked up to Differential Equations in 10th grade, I find myself stuck in the exact same issue.

Another thing I would like to point out: in my opinion, the issue with all the high school and college classes is not that the material is easy (since the material being taught is not going to change no matter how "hard" you make the class), but rather in the problems given. Most problems that I have seen consist of trivial observations from the definition, or "plug-and-chug." This is what makes (in my opinion), mathematics competitions far more interesting; you are not supposed to know how to solve the problem, rather you are supposed to figure out on your own what the key ingredients are that are needed to solve the problem. That is the art of problem solving.

11. Apr 30, 2016

### IGU

The solution to that problem for my math kid was to just ignore getting credit for anything. He audited a large variety of classes at local universities while in middle school and high school. Never officially, always by just asking permission of the professor who taught the class. Mostly grad classes after the first couple. It worked wonderfully and cost nothing (except for books). However I did pull him out of school to homeschool him, which made it possible to attend college classes in the middle of the day -- that's difficult to do if you are still going to high school.

He also learned things on his own. For calculus he worked his was through Apostol I & II, doing all the problems (not all the practice exercises though). Similarly he self-studied various other topics from the best books I could find for him (e.g. he learned real analysis from Tao's notes for An Epsilon of Room).
Yeah, it's pretty much impossible to get courses meant for mathematicians at community colleges -- it's not their audience. Actually it's hard at most universities (just count the ones using Stewart for calculus rather than Apostol). So with pretty much everything you have to take responsibility upon yourself for learning the material properly. Good training for the rest of life.
Yeah, that's the good part. But the problem with competition problems is that they are known to be solvable in a pretty short time -- this is completely unlike the sorts of problems mathematicians work on. And you have to be a fast thinker to do them, which discourages the slow, deep thinkers who are the ones most likely to succeed as research mathematicians. So it's a mixed bag. Works for some people, not others.

12. Apr 30, 2016

### Calaver

(Bold added by me for emphasis)

Where are you going to school that allows you to proceed so quickly to college-level math?

It's not that I don't believe you or that I mean to demean your frustrations, but my experience is not at all similar to yours (I'm currently a sophomore in high school, by the way).

The majority of high school students that I know that are "interested in math" (not quite sure what you mean by that - do you mean people who enjoy math or people who are seriously thinking about math/physics as a career?) do not know Projective Geometry, Graph Theory, and Abstract Algebra. Although, your latter statement about proof contests makes me think that you and I have different definitions of "know." Being involved in contest math myself, one does not necessarily have to understand the topic fully to get the right answer. So do they actually understand the ins and outs of it, or are they just comfortable with the basics (neither of which are bad, but that part of your post seems a bit ambiguous to me)?

In addition, I have never met anyone who has completed Calculus I-III, Linear Algebra, and Diff EQ by their 9th grade year. Is this the norm for the very advanced students in your area?

Also, with your statement:
I am a bit in awe since I have no opportunity to attend a Diff EQ class right now (as a sophomore) and being in Geometry in 9th grade is the highest you can normally do in my school (and other schools around my area).

So, in my opinion, micromass' post does not seem meant to the student of forty years ago (unless my area is just full of bums who are complete underachievers - which is possible but unlikely). He gives excellent suggestions, and as someone currently trying to do some advanced mathematics, I can say that I find the directions in which he points to be very helpful. The only reason why I didn't get more out of the Insight is because I've spent some time on these forums and seen his other posts where he mentions similar things and already started on one of the areas he mentions (namely, linear algebra - and I see he's got a recent post on that, too!).

EDIT: Sorry if weirdness happened with the quoted section of my comment on the actual Insight page. I posted it in the regular forums and for some reason the quote brackets aren't showing up on the other page.

13. May 1, 2016

### micromass

Staff Emeritus
High school kids knowing abstract algebra? I'm not saying that it doesn't happen, but I've honestly never talked to such a student before. And I've talked to hundreds of students before on PF. acegikmoqsuwy, you're clearly too advanced to get something useful out of this guide. But it's not fair to say there are many high school students with a good grasp on projective geometry and abstract algebra. I really don't know where you get this from.

Also, abstract algebra is a huge field. It takes years before you really get the basics of it. Do you really mean to say that high school students know the Sylow theorems? The fundamental theorem of Galois correspondence? The Nullstellensatz? The chinese remainder theorem? etc. Somehow I find this very very very very hard to believe...

14. May 1, 2016

### IGU

You should check out the forums on AoPS if you want to find students like that. They're generally not here.
Here's the Art of Problem Solving course on Group Theory, which has been around for only a couple of years. So you can be assured there are quite a few high school kids learning at least to this level. I'm sure some go deeper.

My kid first started studying on his own out of Herstein's https://www.amazon.com/Topics-Algebra-2nd-Edition-Herstein/dp/0471010901 when he was 13 (he got through only a couple of chapters). Later he used Jacobson's https://www.amazon.com/Basic-Algebra-Second-Dover-Mathematics/dp/0486471896 & https://www.amazon.com/Basic-Algebra-II-Second-Mathematics/dp/048647187X. By the time he went off to college he had spent five years studying modern algebra with varying degrees of intensity. One of the last algebra related things he did before going off to Cambridge was a graduate course on algebraic number theory followed by some readings on ramification from Serre's https://www.amazon.com/Local-Fields-Graduate-Texts-Mathematics/dp/0387904247. His experience is very uncommon, but not unknown. It's going to be very interesting to see what the level of the kids coming out of Proof School will be in a couple of years.

Last edited by a moderator: May 7, 2017
15. May 1, 2016

### micromass

Staff Emeritus
Exactly. And this recommendation insight is for the students who are here.

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