What does it mean to be good at math ?

[aerospacedout]

What does it mean to be "good at math"?

I have recently began preparing for college by picking up a calculus textbook and doing it from the beginning. Although I'm not particularly talented at math, I am a little above average when it comes to understanding physics and mathematics concepts. I never put in the time to study however so my skills have deteriorated and I'm trying to get them to an acceptable level so I would be a little more prepared for aero/mech engineering. My conceptual understanding is fine but the calculations are off and I'm missing quite a lot of fundamentals, mostly due to lack of practice. This lack of practice is also the reason I'm failing my calculus class, also very discouraging for someone who seemed to be good at math.
The problem is, even though I'm well versed(or so I thought!) in derivatives and integrals as they apply to many types of physics problems, it seems that the more math I learn, the more i realize that I know barely anything. I try to go back, relearn my fundamentals, but even if I am able to learn stuff, I am discouraged by the fact that there is just so much of it that I didn't even know existed, and that it is impossible to learn all of it. And honestly, I'm already in calc, i have absolutely no clue of how I will be able to catch up and manage to keep my other class grades up.
I read that there are around 60 disciplines in math. That's insane! How much of it do math majors learn? Engineers? How is it possible to learn (and retain) that many concepts and calculations? It seems as if, like numbers, math is infinite and extremely broad, and I never thought of it that way. I never expected there to be this much, and I honestly have no idea what being "good" at it would mean, even if I manage to get through this 650 page textbook(ambitious goal, kinda stupid and unrealistic though...). Being good at calc is completely different from being good at topology or geometry or linear algebra... I never really expected that. They're all different, just like the differences between the various liberal arts.


Is it worth it for me to keep going even if I won't learn as much as I want to, as quickly as I want to? I love math now, even if I'm not amazing at it. I've always loved physics even though my ap physics class is brutal at the moment. I was wondering if anyone could give some advice, and answer there two questions:

a)What does it mean to be "good at math", if it even means anything?
b)can you share an experience where you struggled with math and physics and managed to somehow climb your way out?

If this is the wrong section, please let me know.. I'm not sure where else to post this. And sorry for the disorganized thinking in my question, I'm just a little overwhelmed by the realization that math is actually an umbrella term for a huge, varied, never ending universe of concepts and calculations, rather than some specific class or concept.

 

[heatengine516]

You just have to take things one step at a time. It also takes a lot of time to learn and have working knowledge of a wide variety of topics. It took me basically advanced high school algebra, college algebra, trigonometry, calculus 1, calculus 2, calculus 3, differential equations, and linear algebra before I felt like I had a decent working knowledge of math, at least enough to tutor others.

You're overwhelming yourself trying think of ways to master dozens of topics in a short time. It doesn't work like that. If you're in calculus, focus only on calculus. That stuff you've never heard of, don't worry about it. You'll get there eventually.

 

[aerospacedout]

You just have to take things one step at a time. It also takes a lot of time to learn and have working knowledge of a wide variety of topics. It took me basically advanced high school algebra, college algebra, trigonometry, calculus 1, calculus 2, calculus 3, differential equations, and linear algebra before I felt like I had a decent working knowledge of math, at least enough to tutor others.

You're overwhelming yourself trying think of ways to master dozens of topics in a short time. It doesn't work like that. If you're in calculus, focus only on calculus. That stuff you've never heard of, don't worry about it. You'll get there eventually.

Thanks for the advice! So basically, i should just focus on whatever topic is at hand, until i eventually reach other next level ones? In calc right now we're doing integrals of trig functions, so I should just forget what i don't know and focus on what i should know for the class?

 

[micromass]

It seems like you finally found out how incredibly broad that mathematics is. You have yet to discover that physics (or any science) is also that large.

Sure, I believe that there are 60 (or more) disciplines in math. But you shouldn't deduce from that that there are mathematicians who are experts in all 60? Of course not. Mathematicians usually pick 1 or 2 disciplines to specialize in and they become experts in that. Somebody specializing in harmonic analysis might not know anything about number theory, for example!

A math major is actually meant as an introduction to mathematics. There you learn several important topics such as analysis, topology, abstract algebra, geometry, etc. in some level of detail. But an undergrad math major who took a class in analysis is for sure not an expert in analysis yet! It just means he knows the very basics. Even grad classes won't make you an expert in the field. Actually doing research and reading research articles will.

You haven't indicated whether you're a math major or physics. But as a physics major, you don't even need to know most areas of mathematics! You'll be fine with calculus, linear algebra and differential equations. Unless you're going into very theoretical physics, you won't need more.

So, don't be intimidated by the huge amount of mathematics out there. You are currently doing calculus, well then start by mastering your calculus book. Make sure you understand the book and make sure you work enough exercises. The only way to get good at math is to do a lot of exercises!

 

[PeroK]

I was like a boy playing on the sea-shore, and diverting myself now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

Isaac Newton

 

[stardust]

What's interesting about your story is that it reminds me of myself. When I was kid I sort of breezed through most of my classes, especially math/science classes. Then when I got to the first semester of college, I was suddenly hit with mathematics that was much more difficult than before. I mistakenly believed I could breeze through it like before. I couldn't. The irony of it was that the students who had lagged behind me had stronger study skills at this point, because I was always "breezing".

Honestly, I've become convinced that there is no such thing as being "good at something" naturally. Even Mozart had to practice playing the piano. Once I started applying myself and working very hard at it, I was able to master the concepts. This isn't to imply that this "mastering" process is ever done. What's awesome about it is that feeling when suddenly understand something you didn't before. Myself, I always feel victorious, awed, and somewhat stupid at the same time. Victorious because I almost feel like a conqueror, awed because the universe has just yielded one of her secrets to me, and stupid because it seems so ridiculously obvious to me after the fact.

The truth is my whole long diatribe could be reduced to: work hard.

 

[micromass]

Myself, I always feel victorious, awed, and somewhat stupid at the same time. Victorious because I almost feel like a conqueror, awed because the universe has just yielded one of her secrets to me, and stupid because it seems so ridiculous obvious to me after the fact.

Reminds me of the following:
A math student often totally obsessed over a problem. Can't sleep, can't relax until he figured it out. When he finally figured it out, he calls himself stupid because it was obvious. Overall, the student finds the process rather enjoyable.

 

[homeomorphic]

You haven't indicated whether you're a math major or physics.

He said engineering. So, calculus (three semesters), basic prob/stat, differential equations at the very least, and that could be it. Maybe linear algebra, too (recommended). Possibly some numerical methods, partial differential equations, complex analysis, more prob/stat, or Fourier analysis, depending on what you end up doing.

How is it possible to learn (and retain) that many concepts and calculations?

No one can learn all of math. I have a PhD in topology and one of the key reasons I quit after finishing it is that I like to understand things for myself very clearly and topology is so complicated that people usually end up taking a lot of it on faith because there's too much to study to get to the frontiers. On the flip side, I've heard a mathematician say that the desire to know the "why" of things was a good predictor of success in research. That seemed not to hold very well in my case, although who knows what would happen if I continued in topology. The other reason I quit was that it didn't seem very useful, especially considering the amount of effort that goes into it. When I was just taking classes, it often didn't seem like it was work. It was just fun. But then, when I had to do research, it really felt a lot more like work, especially when it came to writing it all down and editing and all the nit-picky details, and I didn't like it at all. Typically, even the "experts" feel like they never know enough. I think I will probably be haunted by this "never knowing enough" problem for the rest of my life to some degree, after studying math so much, but it's good to have left math research, so that at least my career doesn't depend on it so much. So, anyway, I never really figured out how to do it at the research level, despite having excellent retention of previous material. One theory I have is that I would have done better if I had gotten people to explain more stuff to me, instead of trying to learn it from the sometimes cryptic textbooks, although it wouldn't have been a panacea.

I think a lot of mathematicians don't retain a lot of the previous stuff they learned. There were grad students I knew who had forgotten a lot of calculus or trigonometry and had to relearn it when they taught those classes (they retained the most important parts that are used later on, but not the level of knowledge needed for teaching, without a little extra review). In some ways, it doesn't matter that much. If you are doing topology, it's probably fairly unlikely that you would have to do all the fancy integrals from calculus. If you did, you could just look it up. So, you can question how much you actually do need to retain. But this is all coming from someone who tries to retain a lot more than is conventional.

Being good at calc is completely different from being good at topology or geometry or linear algebra... I never really expected that. They're all different, just like the differences between the various liberal arts.

Yes and no. I find that I'm just good at math, whatever it is (as long as it's just solving textbook problems, rather than research, which I kind of suck at, despite being fairly good at solving even very difficult textbook problems). It never seems to me like I am doing something different when I'm doing a different subject. I just apply the same kinds of thought-patterns and strategies and it works out. Partly, that's due to my having diverse skill sets, but there's a kind of a similarity between all the subjects, to me.

I disagree with the idea that all there is to it is doing problems or working hard. There are a lot of tricks to making things stick in your mind. Reviewing at carefully spaced intervals, thinking about why the theorems work, visualization, and generally making things more meaningful, to name a few of the tricks. This can be hard because a lot of mathematicians and physicists (and engineers) seem to go in the opposite direction in many cases, stripping things of their meaning, making them a lot harder to understand and remember. Generally, this problem tends to be stronger with written materials, like textbooks, a little less strong in lectures, and the least strong if you go talk to them in office hours and ask how they actually think about things. Even then, it can still be a problem.

 

[aerospacedout]

Thank you for the responses! I have come to the conclusion that I just had a very simplified, romanticized view of math and mathematicians. I now see that even exceptional mathematicians(PhD in topology...thats epic) are human too and forget stuff/dont enjoy their work sometimes. I am growing fond of math due to the fact that I'm doing it on my own accord, due to my own decision, not because I was forced to by a school system. I will take your advice and continue on my (rather slow) pace. I picked up a geometry textbook recently so I'll do problems in it for fun when calc makes me rage lol.

Thanks again for your input! I appreciate the reassurance.