Learning and understanding Maths

[JayJohn85]

Do you have to learn maths piece by piece starting from the fundamental basics onto the more advanced stuff. Besides this obvious point my real question is will you understand it? I mean does understanding come eventually after you have learned the various pieces or is it possible you may never achieve true understanding? I say this because I probably am trying to run before I can walk well with my thoughts and understanding but find that I simply just don't get certain stuff well I actually don't understand all of it. Currently working on the basics I get that stuff but what I mean is I can't see the big picture.

Some dumb questions that come of the top of my head.

- How is a circle connected to a sine wave I see this on Wikipedia but don't grasp why they are connected.
- Quadratics and polynomials are equations it seems to me that you got to find the roots for and something called a discriminant is involved. You can graph these too it seems and they are also connected to waves or maybe I am wrong on this account I dunno.
- This square business I don't get using the square method. I mean has this got anything to do with square numbers yet it doesn't seem the roots follow that pattern I dunno.

I had other dumb questions I can't think of right now. Pretty much a moron right now hopefully that changes.

 

[Simon Bridge]

Do you have to learn maths piece by piece starting from the fundamental basics onto the more advanced stuff.

Pretty much - though you can jump around a bit and back-fill.
It can be like learning a language - you've got to get the basics down before you will get anywhere but you can go a long way on very little. You will notice the difference between having a working knowledge of a language, being fluent, and being natively familiar.

It does come in the end - but after lots of practice and immersion.

How is a circle connected to a sine wave I see this on Wikipedia but don't grasp why they are connected.

The sine function is defined in terms of a circle - so the two are intimately connected.
If you draw a chord in a circle, then half the chord makes an angle A with the center.
If the radius of the circle is 1 unit, then the length of the half chord is the sine of the angle A.

If you put a shotglass on a turntable, and set the turntable rotating, then crouch down to see it edge on, you see the shot-glass go back and forth with time. The plot of the position of the shotglass against time is the sine function.

Quadratics and polynomials are equations it seems to me that you got to find the roots for and something called a discriminant is involved. You can graph these too it seems and they are also connected to waves or maybe I am wrong on this account I dunno.

The roots of an equation y=f(x) are the values of x where y=0.
i.e. anything whose graph crosses the x-axis has roots.
Knowing the roots of polynomials can be quite useful - and, for beginning students, finding them provides exercise in algebra.

This square business I don't get using the square method. I mean has this got anything to do with square numbers yet it doesn't seem the roots follow that pattern I dunno.

Don't know what you mean by "this square business".

A number x, when squared, gives the area of the square that has sides of length x.

 

[micromass]

Do you have to learn maths piece by piece starting from the fundamental basics onto the more advanced stuff. Besides this obvious point my real question is will you understand it? I mean does understanding come eventually after you have learned the various pieces or is it possible you may never achieve true understanding? I say this because I probably am trying to run before I can walk well with my thoughts and understanding but find that I simply just don't get certain stuff well I actually don't understand all of it. Currently working on the basics I get that stuff but what I mean is I can't see the big picture.

This depends on the person. Many people will tell you that you can learn the math whenever you need it and that you don't need to know a math topic perfectly before you apply it to physics. Other people (including me) feel very uncomfortable using this approach and want to know the math very well before they apply it.

Since you are apparently still learning HS math, I really do recommend that you study that very well first. It's true that HS math makes it difficult to see the big picture since it's usually just a big mess of mathematical facts. You'll see the big picture eventually though.

Some dumb questions that come of the top of my head.

- How is a circle connected to a sine wave I see this on Wikipedia but don't grasp why they are connected.

If you graph $(\cos(t), \sin(t))$ for each $t\in \mathbb{R}$, then you'll get the unit circle. For example, $(\cos(24),\sin(24))$ lies on the unit circle and all other such points too. And conversely, every point on the unit circle has this form.

- Quadratics and polynomials are equations it seems to me that you got to find the roots for and something called a discriminant is involved. You can graph these too it seems and they are also connected to waves or maybe I am wrong on this account I dunno.

I don't really see how waves come in here. Can you clarify what you mean or give an example?

- This square business I don't get using the square method. I mean has this got anything to do with square numbers yet it doesn't seem the roots follow that pattern I dunno.

What square business?? Can you give an example of this too?

 

[StaceyPurcher]

Well, all that it will really cause is that most of the time the things you will learn from your lecturer or teacher will be mainly based on previous knowledge things or other basis' of understandings

If you are going to "run before you can walk" you will fall quite a bit, but that just means you'll have to work that much harder than everyone else to even just keep up...

HAHAHA, sorry for the stupid link

But yeah, it will most likely require a lot of pre-reading, so just ask your lecturer or teacher what it is you will need to know for the upcoming class or what might help to understand what you just learnt.
It will mainly just require a lot of hard work and initiative on your part...

 

[JayJohn85]

I meant finding the square or something, a method for solving a quadratic. The sine wave thing is probably me just reading stuff I don't understand too well like seeing stuff similar to Euler's identity which is moving in a circle or something I picked up elsewhere on a site aimed at helping you understand maths intuitively.

And watching a calculus video where it had the great function of calculus and this E term popped up again the one that has the same graphed curve in both functions I think. Once again is that euler's work and yea just grasping at straws that a sine wave is connected to quadratics.

lol I probably appear really stupid right now I am only doing high school level stuff and ain't on the quadratics section yet I just been obsessing though and watching lots of videos on physics and probably melting my head with too much advanced stuff.

PS> By high school I mean GCSE level advance to me at the moment is A-level. But when you look up this stuff it seems pretty heavy and at college level some of it.

 

[StaceyPurcher]

lol I probably appear really stupid right now I am only doing high school level stuff and ain't on the quadratics section yet I just been obsessing though and watching lots of videos on physics and probably melting my head with too much advanced stuff.

Well then, uh yeah, maybe focus a bit more on your course outlines and problems rather than unrelated stuff

 

[JayJohn85]

Well then, uh yeah, maybe focus a bit more on your course outlines and problems rather than unrelated stuff

True I shall do so but I get curious about the bigger picture and probably impatient.

 

[StaceyPurcher]

True I shall do so but I get curious about the bigger picture and probably impatient.

HAHAHA well done then...

 

[Stephen Tashi]

- This square business I don't get using the square method. I mean has this got anything to do with square numbers yet it doesn't seem the roots follow that pattern I dunno.

Some people won't ever learn certain kinds of mathematics. My guess is that you are at risk in that regard because the you are tyring to analyze situations in terms of words (such as "square") and trying to make simple analogies between all mathematical concepts and some visual concept, such as a circle or square. Many mathematical concepts can't be understood this way. And you cannot analyze mathematical ideas correctly by analyzing individual words. Liberal arts students are taught to analyze the content of literature by analyzing the meaning of individual words. This often doesn't work in mathematics. Words such as "square", "infinity", "converge","independent" have different meanings in different mathematical contexts. In math, only the mathematical version of "a complete sentence" has a specific meaning. What it says in its entirety can't always be understood by focusing on the individual words in the sentence. If you don't understand this, you should begin by studying mathematical logic.

 

[JayJohn85]

Some people won't ever learn certain kinds of mathematics. My guess is that you are at risk in that regard because the you are tyring to analyze situations in terms of words (such as "square") and trying to make simple analogies between all mathematical concepts and some visual concept, such as a circle or square. Many mathematical concepts can't be understood this way. And you cannot analyze mathematical ideas correctly by analyzing individual words. Liberal arts students are taught to analyze the content of literature by analyzing the meaning of individual words. This often doesn't work in mathematics. Words such as "square", "infinity", "converge","independent" have different meanings in different mathematical contexts. In math, only the mathematical version of "a complete sentence" has a specific meaning. What it says in its entirety can't always be understood by focusing on the individual words in the sentence. If you don't understand this, you should begin by studying mathematical logic.

I'll take your advice on the mathematical logic once I do my GCSE higher tier I am repeating because I want an A and then A-level. Hoping after that I will have absorbed enough that once I look at the logic stuff I grasp it. I wasn't sure what you meant by this perfectly honest until I looked up wikipedia and well to be frank that is a bit hardcore for me right now.

With cantor's set theory and well logic which I remember a friend of mine saying he did as part of his college philosophy, the logic stuff. I also have small inkling due to doing computers and we did a tiny bit on boolean. I understand what cantor was trying achieve, the poor guy no one took him seriously on that whole infinity stuff drove him to the mental. And you can't list the numbers nice wee stanford video on that too.

I know it probably encompasses a whole lot more but I think I'll stick to my basics then worry about it. Two things I want to become highly proficient at within the next couple of weeks are algebra and geometry. Ordered Euclid's elements on amazon can't wait.

 

[Simon Bridge]

I meant finding the square or something, a method for solving a quadratic.

Urrr - you mean "completing the square"?
It is related to square numbers etc - i.e. a quadratic that can be written as $(x+a)^2$ is often called "a perfect square" - but you are probably best to think of it as just a name.
Names don't have to describe the thing they label - i.e. "JayJohn" means "Blue crested man" - do you have a blue crest?

And watching a calculus video ... and yea just grasping at straws ... I just been obsessing though and watching lots of videos on physics and probably melting my head with too much advanced stuff.

Now you know why school courses go step-by-step the way they do.
You won't learn much by just watching videos that take your fancy, you learn best by structuring your education.

All maths are related to each other so you can easily get yourself messed up by drawing connections by intuition. You should concentrate on what things are in themselves. The real relationships will emerge on their own.

If you want an advanced look at A-levels - get hold of the curriculum and past exams, try to figure out which parts follow from the GCSE courses you are doing. See if you can fill in the gaps.
Bear in mind - best practice is to concentrate on the studies in front of you.

 

[mathsman1963]

Try studying Countdown to Mathematics volumes 1 and 2 published by the Open University. Quite pricey but excellent books. They bridge the gap between GCSE and A Level beautifully.

 

[tfr000]

I wasn't sure what you meant by this perfectly honest until I looked up wikipedia and well to be frank that is a bit hardcore for me right now.

A lot of Wikipedia math articles have been invaded by mathematicians. They are, unfortunately, more concerned with mathematical rigor and correctness (and perhaps throwing around lots of jargon) than in explaining in simple terms. Not usually very helpful.