Willowz said:
How to understand math conceptually? What is the best way to see the relation between numbers in math? There is no certain kind of universal 'logic' in math (or is there?). So, how to understand and correctly associate the concepts in math? Is math something like a puzzle? You fill in the little pieces until you see the whole picture?
There are many perspectives you can take on understanding math, and given the scope and depth of mathematics, there are many perspectives you can form when trying to understand math.
One with I should point out with math is that the different subfields are for a greater part generalizing the math that already exists. It happens in every area of math and also happens in physical and social sciences too.
When you actually get into mathematics and especially in the higher or more general areas, you don't really focus on relationships with numbers: you focus more on structures, decomposition and generalization.
With regards to the analogy of math being like a puzzle, I think that is a valid statement. It takes time for things to "make sense" and to get to that point you often have to practice it for a while and also consistently think about it or to also teach it.
I think the best thing you can do apart from what was mentioned above is to ask "why are they doing this?" and "how is this generalizing some other concept?"
I'll give an example: when you start learning linear algebra and vector spaces you come across inner products and learn about orthogonality. In this kind of context you think about orthogonality as being "vectors" that are independent of each other.
But then you look at things Fourier series, functional analysis and hilbert spaces, and then wavelets and you can see that the idea of orthogonality is basically a structured framework for "decomposing macro objects into micro objects": that is, its a structured way to break systems into independent atomic objects.
When you start off learning linear algebra and inner products, this is not so obvious, but as you see things applied to a wide range of phenomena, you begin to see the real concept behind the ideas.
In saying that, if you want to develop those kind of insights, then you need to read a lot of math and think about the questions I stated above. That's how it works in any field whether you are a doctor, teacher, scientist, engineer, whatever: it takes time, effort, and thinking to answer the kind of questions you're asking.
