I know this is going to sound stupid since I'm in applied precalculus

[Raizy]

But right now all math feels like is, "Here is a bunch of numbers and funny looking symbols, now do this step, then this, now do that because it just works, no questions asked. Just do it this way, understood?" And then the word problems come at me and I'm clueless, unless it resembles the one showed in the examples. I fail to see the beauty of why it works.

Is it just me being lazy minded, or is it just the nature of learning the basics, especially in an applied way (college prep instead of university)?

 

[Tac-Tics]

It sounds like you're having an issue a lot of people have in high school.

Mathematics is extremely sensitive to details. language often fails to communicate those details clearly, so we have (several) special notation(s). And even the notations are just suggestions at what's being talked about... depending on the branch, the notation isn't always so clear either.

Mathematics is not a science. It's not about "asking questions". But equally so, it's not about "just works" either. At its core, it's just logic, taking a set of assumptions and definitions and leading up to conclusions and theorems based solely on a logical argument between them.

Here's a really simple "classic" proof to demonstrate. It's something that isn't taught in high shcool. It's not hard, but I hopefully can illustrate what kind of mathematics a lot of other people on this board enjoy:

Theorem. There are an infintie number of prime integers.

Proof. We only have two possibilities. Either the primes are finite or the primes are infinite. This proof shows that there are infinite primes by showing the alternative -- the existence of only a finite number of primes -- leads to a logical contradiction.

Suppose there is a finite number of primes. We can always find the largest of a finite number of integers, so let n be the greastest prime integer. Furthermore, let k be one greater than the product of all integers from 1 to n (k = 1 * 2 * 3 * ... * n + 1). Dividing k by any integer between 2 and n leaves a remainder of 1. Therefore, k only has two divisors, 1 and k, and thus, k is prime. But k is greater than n, which contradicts our assumption that n was the greatest prime integer. Thus, our proof stands.

It's unfortunate that math is often ridiculously useful in other areas of study. In applied mathematics, the line of reasoning isn't as useful as the final theorem, so the logical side of mathematics is thrown away. Instead, students simply memorize sets of rules which have been created and proven by others. This is especially the case in calculus, where you might use a table to calculate a strange derivative or apply a law (such as the chain rule) without really understanding why it's true.

In high school especially, the "elegant" and "beautiful" face of mathematics is almost entirely ignored. The math you take there is the most you'll need if you're going into business or something. The good stuff is reserved for scientists and engineers and the *best* stuff is exclusive to mathematicians (because it is too unapplicable or too abstract for other fields).

Try looking around these boards. You'll find a lot of concepts you've never seen before and maybe you might find something interesting.

 

[qntty]

If you're frustrated with the way math is taught in high school then don't let it turn you away from math, because all the fun is stripped away in classes like those. There is order underneath what they teach you but it's almost entirely ignored. For example, instead of memorizing formulas, it helps to understand the derivations so that you can quickly derive it when needed. If you're interested in the beauty of mathematics, you might like learning some number theory. It is one of the "purest" branches of math and a good way to learn basic proof writing.

 

[Raizy]

Instead, students simply memorize sets of rules which have been created and proven by others.

This is what I'm doing right now to get my high school pre-reqs before entering a college program.

But you see there is this one thing that concerns me. I have this assumption that, if the textbook tells you something, but in your head you have a "why" question but you can't figure it out on your own. It's like the book wants you to figure it out on your own. Is this the part where genetics come into play? Or is it just a lack of the book's resources?

I have another thing that concerns, is that I am only interested in engineering, so I am not sure which type of math I should be spending lots of time on to *really* understand why the math works. Should I just rote memorize my way through the precal. stuff or am I setting myself up for failure?

 

[chiro]

This is what I'm doing right now to get my high school pre-reqs before entering a college program.

But you see there is this one thing that concerns me. I have this assumption that, if the textbook tells you something, but in your head you have a "why" question but you can't figure it out on your own. It's like the book wants you to figure it out on your own. Is this the part where genetics come into play? Or is it just a lack of the book's resources?

I have another thing that concerns, is that I am only interested in engineering, so I am not sure which type of math I should be spending lots of time on to *really* understand why the math works. Should I just rote memorize my way through the precal. stuff or am I setting myself up for failure?

You will probably get some books that are written in mind for the student to explore the math by doing problems whereas you will get other books that will offer more insight through author commentary where there are no exercises included.

If you want to learn math you have to do it. My advice is that you try and think about what the concept means from as many different possible angles as you can. Usually this is what the exercises are meant for as they ask you to apply your knowledge in a wide variety of perspectives.

Some will be intuitive like if you are dealing with say a concept of centroids of centres or averages and others won't be as intuitive, but should come to you with practice.

Memorizing things is ok. You don't have to know everything about everything when you are just starting. If you can do that though, we may hear about your name in the future I'm sure. But its ok to memorize and have a full realization of what you're doing later than beat yourself over the head because you don't fully grasp what it means.

Usually if you can do the associated exercises then you will have demonstrated some mastery of the subject and should be confident that you are able to think analytically at least to some degree.

University is harder but you can get help when you need it. Usually the lecturer/tutor/professor will do a good job in testing your ability when it comes to knowing the material. If you do the work you should generally speaking be ok with it all.

If you're worried that you are simply memorizing things without thinking about what you're actually doing then that is a good sign that you'll be ready for university, especially for a subject like engineering, since you generally deal with concepts that have a wide scope for application and understanding. But you will get a lot of problems to work on so the teachers will guide you on how to understand it so don't worry too much.

 

[PieceOfPi]

I feel your pain. When I was taking AP calculus in my senior year, my class basically boiled down to this: Here's the list of formulas (i.e. rules for derivatives and integrals) you need to know for the AP exam; make flashcards and memorize them. While to some degree it is helpful because now I don't have to look up my calculus every time to find out what d/dx(tan x) is, but it certainly was not intellectually stimulating. Back then, I wasn't even thinking about becoming a math major, because I didn't really know what math is.

When you go to college and you're still interested in math, take an "introduction to proof" type of math. Usually, one of the following courses is a good place to start: number theory, Discrete/Finite mathematics, or simply "Introduction to Proof." You might also be able to challenge yourself and take Honors Calculus too.