Should I teach myself pre-calculus and calculus or ?

[amsteridam]

Hello, I'm a new user, but I have been lurking around the forums for a little while here and there.

So let me get to the point. I am extremely interested in Mathematics, Physics, Cosmology, Chemistry and Biology. So much in fact that I genuinely want to learn anything and everything possible in these subjects. Problem is, I'm 15 and nearing my second year in high school. I feel like I'm not being challenged at all. I'm in an advanced math class, but won't be starting the basics of Pre-Calculus for a few months. Currently I have read many of Stephen Hawking, Neil deGrasse Tyson, Euclid, and other math/science books. I believe myself to be above the average intellect and capable of advanced math.

So basically what I'm wondering is, should I wait and be taught or just teach myself so when the time comes, I can move onto larger things.

(Btw I have the time to complete these kinds of things.)

Any Advice?

 

[Number Nine]

Reading Hawking and Tyson means absolutely nothing as far as mathematical knowledge/talent goes, unless you're reading their research articles. Obviously, you should always be teaching yourself and reading ahead, but the fact is that you are almost certainly not ready for "advanced mathematics". In fact, given that you haven't even studied pre-calc, you probably don't have any idea what "advanced math" (i.e. proof based math) actually looks like. If you think you're too smart for the class, then show it by mastering the material. Skipping ahead will not benefit you.

 

[brocks]

If you have the time, then by all means, do it yourself. See if you can borrow a copy of the book they use in your school's precalc class (if not, buy a cheap used copy of Stewart's or Swokowski's or Larson's precalculus text), and work through it yourself. If it's a typical modern textbook with a hundred problems per section, just do a fair sampling of them, maybe a dozen or so per section, so you don't get bogged down, but at least look at most of them to make sure you understand what they are asking and how you would solve them.

I normally advise people to get used copies of a couple of texts for alternative approaches to topics you find difficult, but I think maybe that advice is becoming dated because there are so many free sources on the internet, like the Khan Academy and MIT's OCW. So just use Google to find explanations for any topic that baffles you.

Once you're ready for calculus, you can do the same thing, and use the video lectures at MIT or other websites to help. Math requires no chemicals or physical apparatus to do experiments, so there's really no limit to how far you can go with self-study. But it would probably be more practical and enjoyable to simply demonstrate to your school that you are ready for calculus, and have them advance you to the regular class. If they don't offer it, maybe a local community college does.

Best of luck to you.

 

[micromass]

Self-studying pre-calc or calc is ok. But there might be reasons not to do it. First, you'll going to see the material anyway in your classes. So you'll see the material twice! This might lead that you'll be bored to dead in your classes.
Second, self-studying too advanced material might lead to misconceptions about the material. And such a misconceptions are hard to get rid off.

What I would suggest that you do is not to self-study pre-calc and calc, but rather expand on your current knowledge. There are a lot of things that a normal school curriculum doesn't cover, but that's very cool to know. Here are some suggestions:

• You know trig? Well, why not study spere-trigonometry then? This will teach you how to deal with triangles on spheres, and it will also teach you the very basics of higher geometry.
• You know geometry? Well, why not study projective geometry? This branch of geometry is very useful in designing games and optics. It is also a very important topic in pure math.
• You know geometry? Try to read the geometry book by Coxeter then? It outlines all kind of fun topics in geometry. Coxeter is one of the best mathematicians of this century, so it should be an honor to read a book by him.
• You know trig and geometry? Why not study polar and spherical coordinates? These will pop up all the time in your later studies. So get to know them well.
• You know algebra? Why not study matrices then? Things like determinants and ranks are very important, and they're not too difficult as well.
• Or read a nice proof book. Once you get the hang of proofs, you don't know how you could live without them. Proving can be difficult at first, but the sooner you start, the easier!

 

[Sankaku]

Micromass has good suggestions (as always). I might also recommend picking up a copy of "What is Mathematics?" by Courant. You may not be ready to tackle all of it right now, but it will give you an idea of how mathematicians think and provide some fun things to look up in other places. It is not expensive:

https://www.amazon.com/dp/0195105192/?tag=pfamazon01-20

...you are almost certainly not ready for "advanced mathematics".

Advanced mathematics means different things to different people. Don't sneer at him just because he is unaware of your terminology.

 

[romsofia]

Do something like micromass suggested; especially the polar/spherical coordinates since you'll need that for calculus two/three and higher math.

Good luck.

 

[Nano-Passion]

Micromass has good advice here. I've learned classical mechanics in high school, and a year later (this summer) I decided to go through all the problems of classical mechanics to master it.

Now I have to take classical mechanics for a third time basically in college... Definitely not the best choice I could have done. It could lead to boredom.

I self-taught myself calculus and found it immensely interesting, but I won't know how I will feel next fall in class. When things are easy lectures tend to be pretty boring. Especially because my community college has lectures for 4 hours.

Also, I am similar to you in the domain of interests. Except that on top of those add psychology, neuroscience, and some philosophy. But I'm 19, there is a good chance you will develop an interest to some of the other sciences as you grow older.

Your interests tell me a couple things about you. You see the big picture in things.. which enable you to expand your interest to other fields. The more connections you develop between different subjects the more you see the beauty of things. You might be a passionate person. You might very well see the beauty in little things in life while pondering for a long time about it.

Your self of identity is based around your quench for knowledge. You love knowledge.

Did I get some of these right?

 

[Sankaku]

I've learned classical mechanics in high school...

Slightly off topic, but here is another terminology note. Although what you learned in high-school was certainly 'classical', what many universities call "Classical Mechanics" is Lagrangian and Hamiltonian mechanics:

http://en.wikipedia.org/wiki/Lagrangian_mechanics
http://en.wikipedia.org/wiki/Hamiltonian_mechanics

A course will often use a text such as this:

https://www.amazon.com/dp/0201657023/?tag=pfamazon01-20

This is somewhat further along in a physicist's training.

Because this forum is global, you may find different interpretations of seemingly simple things like 'advanced' mathematics or 'classical' mechanics.

 

[Nano-Passion]

Slightly off topic, but here is another terminology note. Although what you learned in high-school was certainly 'classical', what many universities call "Classical Mechanics" is Lagrangian and Hamiltonian mechanics:

http://en.wikipedia.org/wiki/Lagrangian_mechanics
http://en.wikipedia.org/wiki/Hamiltonian_mechanics

A course will often use a text such as this:

https://www.amazon.com/dp/0201657023/?tag=pfamazon01-20

This is somewhat further along in a physicist's training.

Because this forum is global, you may find different interpretations of seemingly simple things like 'advanced' mathematics or 'classical' mechanics.

Sorry, what I should have said was pre-calculus based classical mechanics. Not even calculus based. :( But I'm taking that next fall.. about time.

 

[romsofia]

Sorry, what I should have said was pre-calculus based classical mechanics. Not even calculus based. :( But I'm taking that next fall.. about time.

Just call it Newtonian mechanics, because even without calculus you solved problems using mainly Newton's laws.

 

[amsteridam]

Micromass has good advice here. I've learned classical mechanics in high school, and a year later (this summer) I decided to go through all the problems of classical mechanics to master it.

Now I have to take classical mechanics for a third time basically in college... Definitely not the best choice I could have done. It could lead to boredom.

I self-taught myself calculus and found it immensely interesting, but I won't know how I will feel next fall in class. When things are easy lectures tend to be pretty boring. Especially because my community college has lectures for 4 hours.

Also, I am similar to you in the domain of interests. Except that on top of those add psychology, neuroscience, and some philosophy. But I'm 19, there is a good chance you will develop an interest to some of the other sciences as you grow older.

Your interests tell me a couple things about you. You see the big picture in things.. which enable you to expand your interest to other fields. The more connections you develop between different subjects the more you see the beauty of things. You might be a passionate person. You might very well see the beauty in little things in life while pondering for a long time about it.

Your self of identity is based around your quench for knowledge. You love knowledge.

Did I get some of these right?

Thanks for your college experience and I bought a college Pre-Calculus textbook and have started working through it. You actually did get most, if not all correct lol!

Self-studying pre-calc or calc is ok. But there might be reasons not to do it. First, you'll going to see the material anyway in your classes. So you'll see the material twice! This might lead that you'll be bored to dead in your classes.
Second, self-studying too advanced material might lead to misconceptions about the material. And such a misconceptions are hard to get rid off.

What I would suggest that you do is not to self-study pre-calc and calc, but rather expand on your current knowledge. There are a lot of things that a normal school curriculum doesn't cover, but that's very cool to know. Here are some suggestions:

• You know trig? Well, why not study spere-trigonometry then? This will teach you how to deal with triangles on spheres, and it will also teach you the very basics of higher geometry.
• You know geometry? Well, why not study projective geometry? This branch of geometry is very useful in designing games and optics. It is also a very important topic in pure math.
• You know geometry? Try to read the geometry book by Coxeter then? It outlines all kind of fun topics in geometry. Coxeter is one of the best mathematicians of this century, so it should be an honor to read a book by him.
• You know trig and geometry? Why not study polar and spherical coordinates? These will pop up all the time in your later studies. So get to know them well.
• You know algebra? Why not study matrices then? Things like determinants and ranks are very important, and they're not too difficult as well.
• Or read a nice proof book. Once you get the hang of proofs, you don't know how you could live without them. Proving can be difficult at first, but the sooner you start, the easier!

I love this response! I have just begun a pre-calculus study, but I will absolutely check out the things you've suggested(especially matrices and polar and spherical coordinates)--Thank you all for taking time in your responses and criticism. I will take it all into account with my education path.