This Math, Maths and Mathematics is quite simple they all mean the same thing.
With Math the American abbreviation and Maths the British abbreviation of Mathematics.
The confusion is maths is used differently by the informed to its common usage.
When those with mathematical training mean finding patterns and relationships when they talk of math(s/ematics). Which are often expressed with weird symbols.
They refer to different branches by different names such as Arithmetic, Geometry, Algebra, Set Theory, Calculus, etc. All the branches are part of mathematics.
Some mathematicians dislike how most people use terms and try to distinguish between their usage and common usage by making a distinction between the mathematics and math(s) or just math.
Regarding teaching maths so the relationships are discovered. Their are a few methods:
Montessori uses materials which make relationships clear so they can be discovered.
My favorite teacher would set projects such as investigating the lengths of right angled triangles, at some point he would suggest we squared the lengths at some point someone would go ah! Followed by the rest of the class.
My class used cards to learn when we were small. Which showed new ways of doing things and we would answer questions to test we understood it. This was not true mathematical inquiry but concepts have to be introduced some how. We were the ones that had ah moments.
Times tables are fundamentally a bad idea, I was fortunate that I could calculate the answers fast enough to fool me teachers into thinking I could remember them. Better techniques than those that I used can be found by search for:
Trachtenberg Arithmetic
I think every child should be introduced to these techniques rather than using times tables. Beads on a board would be a good tool to try and develop ah moments.
Teachers that say you can't do something such take a bigger number from a smaller or square root a negative number do a disservice. But their is fundamentally a problem that we for the most part use vectors yet some relations have scalar definitions. Complex/imaginary (what awful names as well; aren't all numbers imaginary) numbers are a result of this ugly usage confusing many children and adults for no reason. Understood in terms of set theory it is quite simple but most university science students are not aware of it. Children are suppose to have excellent intuitive understanding of sets yet we demonstrate daft asymmetric functions on vector quantities.
: The result is confusion regarding what is happening to begin with followed with confusion as to why this is taught subsequently once what is happening is understood.
Another thing which is not clear is the difference between transitive and intransitive operations.
I argued with my teachers at school about these things (they felt wrong) but only now do I understand explicitly why I was right. Many branches of maths were developed independently and are in-fact subsets of more comprehensive branches of maths. Some of the definitions are rather arbitrary and hence poor. Roll on a more sensible approach to maths education.
Charles