Understanding math conceptually.

[Willowz]

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?

 

[Astronuc]

Quantitative relationships.

 

[Jack21222]

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?

What part of math are you having trouble with? There is a universal logic in math. Everything in math can be proven by starting from a small set of axioms. For me, concepts in math can be most easily understood by thinking of what they mean physically.

For example, addition is understood by taking separate groups of objects and putting them together. Subtraction is taking items away from a group. Division is breaking up a group into equal parts. Multiplication is just a fancy way to add many groups. Equations are just saying that two things are the same. Algebra is just a method to say the same thing in different ways (a way to keep true statements true). Derivatives are just how things change. Integration is just a fancy way to add a bunch of small things.

What exactly are you having trouble with? I assume you aren't talking about anything beyond calculus, because you said "numbers" in math, and you'll be hard-pressed to find numbers in anything beyond what I just described.

 

[Robert1986]

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?

"We never understand math, we just get used to it."

 

[Drakkith]

"We never understand math, we just get used to it."

Whats so hard to understand about it?

 

[Willowz]

Quantitative relationships.

I did a search on "quantitative relationships". Coulden't find anything. I just need some scafolding or some bedrock to stand on.

A text about metamathematics [meant for people] without a rich understanding of mathematics, would probably solve my problem.

 

[Robert1986]

Whats so hard to understand about it?

Ask von Neumann. http://www.best-quotes-poems.com/mathematics-quotes.html

 

[lavinia]

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?

my experience is that math is understood only through insight. Insight comes unexpectedly when one focusses intensely upon something - not only in math. The axiomatic approach does not work in practice. I have never met a mathematician who thinks in terms of axioms - although I have met logicians and physicists that do.

 

[Willowz]

Another way of putting this is, 'Do mathematicians think mathematically?'

So far I've taken a look into "Conceptual Mathematics: A First Introduction to Categories". Is that a good start or appropriate for addressing my question?

 

[micromass]

Another way of putting this is, 'Do mathematicians think mathematically?'

So far I've taken a look into "Conceptual Mathematics: A First Introduction to Categories". Is that a good start or appropriate for addressing my question?

Hmm, I did not read that particular book. But I suspect it is just a categor theory book. Now, category theory is quite interesting: it gives some kind of metalanguage of mathematics. In the sense that most mathematical structures and questions can be expressed w.r.t. categories. It also provides a framework to transform certain questions in one field of study, into other questions in a complete different field.

However, when I read your OP, I didn't get the idea that this is really what you're looking for. But if the preceding paragraph interests you, go read the book!
Sadly, I don't really have an alternative for you...

 

[chiro]

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.