Am I Challenging Myself Too Much in Math Learning?

[algebraicpotato]

I was browsing the math subreddit on Reddit just a moment ago, and came across someone who was asking for calculus textbooks to give to his precocious eleven year old. That got me thinking: Am I pursuing subjects that are too advanced for my level without trying to pursue rigor and depth? I quit olympiad math a few years ago because I didn't like the learning environment that I was in (which was a cram school), and now I don't know what I should be learning. Right now I'm trying to explore fields like machine learning and computational science, which inevitably require learning stuff like multivariable calculus and linear algebra. Which is fine because I don't really have trouble understanding the material. But when I try my hand at something like Real Analysis and Spivak, I find myself struggling to answer basic problems. I'd say at least some of it is partly because of lack of discipline, but still. I have also grown a sort of aversion to competitive math because of my peers who do it only to improve their chances at college admissions. What should I do to ensure that I have solid math skills? Go back to olympiad or try penetrating into rigorous higher level mathematics?

 

[Vanadium 50]

What are your goals?
What are you doing to meet them?

 

[symbolipoint]

But when I try my hand at something like Real Analysis and Spivak, I find myself struggling to answer basic problems. I'd say at least some of it is partly because of lack of discipline, but still. I have also grown a sort of aversion to competitive math because of my peers who do it only to improve their chances at college admissions. What should I do to ensure that I have solid math skills? Go back to olympiad or try penetrating into rigorous higher level mathematics?

None of that. If you're a high school student, then follow the prescribed college-preparatory path of courses for Mathematics. Learn each course as well as possible each term. Review when you have the chance. Sometimes an alternative textbook is useful, but best to not try to reach above your level.

 

[algebraicpotato]

So If I understand you correctly, it would be best for me to wait learning new material until college? I feel like I have a pretty solid grasp of high school algebra, trig, calculus and what not, because I've done lots of problems. I am also of the understanding that linear algebra/proof based single variable calculus is the natural next step. What do you think is the best way to completely master high school mathematics? Do you have any textbook suggestions? In other words, how can I get ready for those undergrad level courses?

 

[algebraicpotato]

What are your goals?
What are you doing to meet them?

I want to learn higher level mathematics because my other academic interests involve them. Basically, the reason why I'm posting if it's the right thing to do is because I'm afraid I'm trying to reach for something way above my level. But...I don't exactly understand how I would prepare for those advanced courses, specifically what part of high school mathematics I should be learning more of.

 

[malawi_glenn]

I try to convice my advanced high school students when they get bored with our standard curriculum to study basics of math, like logic, proof reading and writing, set theory, algebra. real analysis - instead of pushing further with more advanced calculus concepts like fancy integral techniques, multivariable calculus, differential equations and such.

I teach a basic linear algebra class for those students (joint with a local university), where we focus on definitions, theorems and proofs. Most of the students like that approach because our standard curricula for high school math is too focused on learning recipies, formulas and applications rather than the formal concepts and proofs.

 

[jtbell]

I see from a previous post that you're in South Korea. In the US, many or most colleges and universities offer a course called something like "Transition to Advanced Mathematics" which covers proof-writing, logic, etc. Some even require it for students who want to be math majors. A Google search turns up a couple of books with that title, or similar; and links to courses at various universities.

 

[MidgetDwarf]

https://www.people.vcu.edu/~rhammack/BookOfProof/

here you go. start reading. this serves as the foundation for higher math.

you can supplement the above book with http://discrete.openmathbooks.org/dmoi3.html.

 

[malawi_glenn]

https://www.people.vcu.edu/~rhammack/BookOfProof/

here you go. start reading. this serves as the foundation for higher math.

you can supplement the above book with http://discrete.openmathbooks.org/dmoi3.html.

Yeah that is my first recommendation to my students too.

As well as this one https://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1024&context=books

 

[algebraicpotato]

Sounds good! Thanks everyone for your advice :)

 

[Vanadium 50]

I feel like I have a pretty solid grasp of high school algebra, trig, calculus and what not, because I've done lots of problems.

Unfortunately, we are not always the best judges of what we know. We can not know something and not even know there is something out there to know.

Can you prove the following?

sin(54 degrees) - sin(18 degrees) = 1/2

 

[George Jones]

Can you prove the following?

sin(54 degrees) - sin(18 degrees) = 1/2

Something to torment my wife with! Thanks!

 

[malawi_glenn]

Something to torment my wife with! Thanks!

I usually give my students to write sin54° and sin18° in exact algebraic form on tests...
not very popular

 

[Vanadium 50]

No hints, please.

 

[algebraicpotato]

Unfortunately, we are not always the best judges of what we know. We can not know something and not even know there is something out there to know.

Definitely true

Can you prove the following?

sin(54 degrees) - sin(18 degrees) = 1/2

Hmm okay so I couldn't prove it by manipulating the trig functions so basically

If you have an isosceles with angles 36º 72º 72º and a line bisecting one of the 72º angles so that a similar isosceles is inscribed in the bigger isosceles, then you can get the length of the triangles by using ratios. Then using the second law of cosines, you can figure out cos 36º, which can then be used to figure out sin 36º, then using the formula sin(x+y) = sin x * cos y + cos x * sin y figure out what sin 54º - sin 18º is...

You know what, I agree with you. I definitely do need to brush up on my trig haha.