# Go slow or fast when teaching yourself math?

I don't know if this is the right subforum to put this in, but it seemed like the best.

Some background: I'm 14, in 9th grade, and go to a good public school. About 5 months ago, for whatever reason, I really got into physics and math, but mostly physics. I read Brian Greene's Fabric of the Cosmos. Meh. No math or rigor. Then I got Feynman's Six Not So Easy Pieces, and it is not so easy (although I new most of the things inSix Easy Pieces). I'm kind of trying to barrel though it, and take my time and do month a chapter (work on it during free periods), although I'm taking a break for a little while. It prompted me to learn some basic trig, so that's good. Another book I've been using since the beginning of this year was the Thompson book Calculus Made Easy. This book is quite good at explaining things, and I'm having a good time going through a few pages a week. In between, I've challenged myself with problems I've given myself, such as deriving an equation for the distance that a projectile will travel if you know the launch velocity and angle. (I felt very proud when I checked online and got it right.)

However, I'm getting an A- in math class, and got a B+ last quarter... not so great, especially since I want to be doing this stuff for a while. I have a hard time thinking on my feet, and can frequently make stupid arithmetic errors. I took the amc10, and got two easy problems wrong at the last step. I want to be able to take the aime next year, so I have to get my act together.

Sorry, that's a lot to read, and here's my question: with all this math I'm trying to teach myself, should I be taking a slower approach where I carefully go over stuff in the curriculum, SAT, etc, OR is it good to be doing stuff above your level and drag yourself up to that level, even if you don't COMPLETELY understand what you're doing?

Thanks

slow at first, and then faster.

slow at first, and then faster.

What do suggest learning first? Should I go out and buy a trig textbook?

I would suggest getting a handle on introductory:

-Geometry (Areas, Volumes, etc.)
-Trigonometry (to the extent that you understand the Unit Circle, the trig ratios, graphs, and functions)
-Permutations and Combinations (not necessary, but will be later on in University)
-Functional Analysis and Analytical Geometry (Be able to calculate slope and anything involving y=mx+b and co-ordinate geometry in your sleep)

Once you have a general understanding of these topics and a great hold on Algebra (FRACTION WORK), you should be able to start into calculus. While limits may see esoteric at first, you will in due time understand derivatives and integrals no problem. It's only the applications that get hard!

By the end of calculus you should be able to open a business where you make optimal profit for the goods you produce, find the volume of a 3-dimensional object that is produced by rotating a function around a specific axis or line, optimize the dimensions of a 3-dimensional object, understand the mathematical relationship between distance, velocity, and acceleration, and so much more!

I would suggest getting a handle on introductory:

-Geometry (Areas, Volumes, etc.)
-Trigonometry (to the extent that you understand the Unit Circle, the trig ratios, graphs, and functions)
-Permutations and Combinations (not necessary, but will be later on in University)
-Functional Analysis and Analytical Geometry (Be able to calculate slope and anything involving y=mx+b and co-ordinate geometry in your sleep)

Once you have a general understanding of these topics and a great hold on Algebra (FRACTION WORK), you should be able to start into calculus. While limits may see esoteric at first, you will in due time understand derivatives and integrals no problem. It's only the applications that get hard!

By the end of calculus you should be able to open a business where you make optimal profit for the goods you produce, find the volume of a 3-dimensional object that is produced by rotating a function around a specific axis or line, optimize the dimensions of a 3-dimensional object, understand the mathematical relationship between distance, velocity, and acceleration, and so much more!

This actually makes me very happy, as I know almost all of that stuff.

I know the basic geometry area volume equations, like pi*r2 and stuff.
I know the soh cah toa ratios, and sec, csc, and cot. I know the unit circle (so much deep understanding came into my head when I learned it, like why sin2x + cos2x = 1 and why a sin curve is the way it is.) and I know what their graphs/functions mean, along with their inverses. (arcsine, etc.)
I know some about permutations/combinations/probability, although I should probably fresh up on it. (n choose r, and the such.)
I've learned about lines in school, and how to get the slope with y = mx+b (hint: it's m) but i also know a few others like point slope and standard form.
I also believe you are talking about linear programming when you mention buisness? I know a little about that, with setting up restrictions and stuff. Also, with the relationship with distance/velocity/acceleration, I was so exhilarated when someone showed me their connection with the derivative functions. I know a little about finding the the areas under curves.

So it seems like I'm in good shape to be learning calculus at this point.
Thank you!

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That's great! Glad to hear that you are so ahead on math and are keen on learning more. :D

Please keep up the good work!

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This actually makes me very happy, as I know almost all of that stuff.

I know the basic geometry area volume equations, like pi*r2 and stuff.
I know the soh cah toa ratios, and sec, csc, and cot. I know the unit circle (so much deep understanding came into my head when I learned it, like why sin2x + cos2x = 1 and why a sin curve is the way it is.) and I know what their graphs/functions mean, along with their inverses. (arcsine, etc.)
I know some about permutations/combinations/probability, although I should probably fresh up on it. (n choose r, and the such.)
I've learned about lines in school, and how to get the slope with y = mx+b (hint: it's m) but i also know a few others like point slope and standard form.
I also believe you are talking about linear programming when you mention buisness? I know a little about that, with setting up restrictions and stuff. Also, with the relationship with distance/velocity/acceleration, I was so exhilarated when someone showed me their connection with the derivative functions. I know a little about finding the the areas under curves.

So it seems like I'm in good shape to be learning calculus at this point.
Thank you!

Why don't you try to pick up a rigourous calculus book. try the book from Spivak. Be warned however, it will not be easy reading. But if you manage to get through it, you will have a very sound idea of what calculus is. And you will be able to do rigourous proofs.
And if you don't manage to get through it, then you will know what kind of math you need to refresh.

Fine you could do calculus without doing the proofs or without being rigourous. But that's not what calculus is about. Try to learn it at a mathematical decent level... You'll learn far more like that...

Also:
-Functional Analysis and Analytical Geometry (Be able to calculate slope and anything involving y=mx+b and co-ordinate geometry in your sleep)

Functional analysis???? Please don't let it be the functional analysis that I'm thinking of...

A. Neumaier
Some background: I'm 14, in 9th grade, and go to a good public school. [...]
However, I'm getting an A- in math class, and got a B+ last quarter... not so great, especially since I want to be doing this stuff for a while. I have a hard time thinking on my feet, and can frequently make stupid arithmetic errors. [...]

with all this math I'm trying to teach myself, should I be taking a slower approach where I carefully go over stuff in the curriculum, SAT, etc, OR is it good to be doing stuff above your level and drag yourself up to that level, even if you don't COMPLETELY understand what you're doing?

You'd be doing both in parallel:
1. Make sure that you understand (and practice to do reliably) everything COMPLETELY what you need in class.
2. Learn what is beyond - whatever interests you, keeping note of the stuff that you only partially understand while you explore the rims of your knowledge.

The latter usually means that you need to better learn or practice the tools used in the partially understood context - so you need to find out how to get this missing knowledge or practice. Knowledge can be read in many places (many things are in wikipedia, which is mostly reliable on standard curriculum topics), but practice can be gotten only by actually doing it - solving exercises, trying to prove things yourself, etc..

And you'd learn one more thing:
3. Learn how to recognize at which level you understand something, and how to move from one level to a deeper one.

Understanding comes on many levels:
- having the name of a concept heard or read often enough so that it has some meaning
- having understood the definition of a concept
- being able to follow an argument containing a concept
- being able to solve exercises containing a concept
- being able to solve exercises not containing the concept but whose solution requires it
- being able to answer standard questions about the use of a concept
- being able to explain a concept to someone else
- being able to ask meaningful questions that you have about the use of a concept
- being able to understand why something is presented the way it is
- being able to improve someone's presentation of something
- being able to teach something
- being able to do research on something

Different but related things may well be understood at different levels. Moreover, understanding on each level can be
- basic (have done it once, just managed),
- moderate (can do it without mistake, if I concentrate on it and check things)
- advanced (can do it routinely, but some things are still difficult, error prone, or surprising)
- perfect (hardly need to think to do it correctly)

My advice is not to rush into calculus, especially at a rigorous level, without having a decent grasp of the basics. Although school mathematics typically does not count for much, there is no good reason why you should not be pulling A's. Furthermore, you should definitely know the basics well enough to make it to the AIME. Here, my only advice is to work on problems that are interesting and just beyond your current level. Work on problems on the AMC12 (some of the problems overlap with the AMC10). Strive to be more organized both in thought and on paper to eliminate careless mistakes.

Remember the point of these competitions is to improve your problem solving abilities. Make a commitment to learn the math and you will improve.

Why don't you try to pick up a rigourous calculus book. try the book from Spivak. Be warned however, it will not be easy reading. But if you manage to get through it, you will have a very sound idea of what calculus is. And you will be able to do rigourous proofs.
And if you don't manage to get through it, then you will know what kind of math you need to refresh.

Fine you could do calculus without doing the proofs or without being rigourous. But that's not what calculus is about. Try to learn it at a mathematical decent level... You'll learn far more like that...

Calculus Made Easy is kind of a deceiving title- it's still time consuming. But I'll have to check out the Spivak book when I finish this one (I'm in no rush.)

What I'm trying to figure out is why you would take on calculus if you haven't even learned all the stuff you should really know, i.e. trig identities. You should learn the basic things before jumping into higher math classes. If you're getting an A- and B+ in (Algebra 1 is 9th grade right?) then how can you be doing calculus lol. I can only see you being able to do basic limits and maybe some basic derivatives..

Maybe you're a math god? Idk, but I think you should stick with your math class you're in and get better grades in there :)

Good luck.

What I'm trying to figure out is why you would take on calculus if you haven't even learned all the stuff you should really know, i.e. trig identities. You should learn the basic things before jumping into higher math classes. If you're getting an A- and B+ in (Algebra 1 is 9th grade right?) then how can you be doing calculus lol. I can only see you being able to do basic limits and maybe some basic derivatives..

Maybe you're a math god? Idk, but I think you should stick with your math class you're in and get better grades in there :)

Good luck.

Maybe I didn't explain my situation well enough- I do know and understand everything in my class. It's just that my teacher's tests are notoriously hard (and even she admits that) and I'm very slow at math. I've talked with my teacher about my grades, and she says that I should be getting an A+ and would if I would make fewer mistakes. I know the trig identities (spent a week, looked 'em up). The reason I don't want to limit myself to only getting good at math CLASS is because it's a little unexciting, and also now seems like a good time to learn this stuff so that hopefully by the time I'm a junior I will be able to submit something very interesting and worthwhile to intel.

That's really all I'm up to now so far in my calculus book- limits, derivatives, integrals, and recently trig substitution. Also, a few weeks ago I joined a math circle that meets on saturday mornings, and there we do interesting contest math. I'm also trying to get good at SAT math, which I think is good because the math on the SAT is so easy, as long as you make no mistakes.

I'm definitely no math god, though.

A. Neumaier
Maybe I didn't explain my situation well enough- I do know and understand everything in my class. It's just that my teacher's tests are notoriously hard (and even she admits that) and I'm very slow at math. I've talked with my teacher about my grades, and she says that I should be getting an A+ and would if I would make fewer mistakes.

This simply means that you lack practice. So you'd read whatever your curiosity leads you to, but you'd practice diligence in applying what you know about the stuff taught in class. Check while or after you write something whether it indeed conforms to your knowledge. Diligence is _essential_ in using math - so this discipline is not a waste of time, although it may cost you quite some effort now.

Go as fast as you can but be warned if you are anything like me this won't be very fast.

It really does sound like you need to work on your current classes. It may seem boring right now, but if you can't do that kind of stuff in your sleep, you'll have a lot of trouble as you get further along in math.

Why not use these 'notoriously hard tests' to push yourself to a new level? Challenge yourself to destroy those tests. When you get to university, time pressure on tests is going to skyrocket. As much as understanding the concepts is important, being able to work quickly and accurately is really important too. I mean, if you're not at the point where you can do algebra and trig quickly and accurately without really thinking about it, calculus is going to be painful. Sometimes you'll fill an entire page with algebra just to answer one question, and trust me, hunting down those little algebra mistakes is painful.

I know it's more exciting to go beyond where you're supposed to be, but if you master what you're doing right now, you'll be able to spend time later actually learning calculus and other maths without having to play catch up on your lower level math skills.

whoa slow down tiger, if maths was a long corridor with lots of side doors you're just walking down it with out trying many of the doors, zzzzz...
Here are some random things off the top my head you could think about:

Proofs! See if you can prove the stuff you learn in class by yourself (without calculus):
Good ones are: Area of a circle, pythag., How many primes are there?, Is the sqrt of 2 irrational?

Investigate number patterns like continuous fractions, series, fibonacci numbers, the factorials, binomial coefficients

Derive things! Practice manipulating algebra, working with fractions, but take the time to look at the view.

Try to extend things you learn for more general cases or extra dimensions.

On the other hand if you don't think that is for you, start here: http://en.wikipedia.org/wiki/Galois_theory

The tempo isn't crucial to the process of obtaining this kind of knowledge.
Just make sure you actually, really, totally understand what you're studying. I cannot emphasize how important it is to understand every detail of the material you are learning about.
If you aren't sure about even the tiniest and/or seemingly unimportant part of it, take the time to find out what it's about.

This way, you will obtain a much more complete knowledge (compared to the knowledge you would acquire ignoring incomprehension at certain times). Besides that, full understanding of learnt material should lower the chances of forgetting main concepts. Of course, you'll forget the details (all the proofs, etc), but you'll be able to see the big picture, which is what really matters - that way, you'll be able to catch up on the stuff you will have forgotten after many years, much easier than if you had to learn it without having understood it the first time you had studied it.

P.S.
Even if you ignore this piece of advice, I ensure you that you will really find it hard to adopt complex concepts without having very fine elementary knowledge.