Mathematics has its roots in ancient cultures, both in practical matters such as how to build a pyramid, and in more philosophical and mystical matters.

Mathematics is one of the keys to understanding the modern world. Arithmetic is central to both our money systems and our computers. Computers themselves emerged out of the foundations of mathematics and questions as to what problems were effectively solvable.

In my own experience there are many aspects of reality that are hard to understand without a grasp of the related mathematical concepts. The problem is that it is relatively poorly understood, despite its place of prominence in modern school syllabi, and it is hard to teach well.

PopularMaths

Explaining advanced mathematical topics to the average reader, or to point out interesting patterns that are not commonly known yet most people can see and appreciate is the domain of PopularMaths, which like PopularScience aims to explain the wonders of what modern mathematicians and scientists explore in ways that the average layman without a decade or so university experience has a chance of grasping.

IanStewart wrote a book called FromHereToInfinity which I read during my first year of an abandoned TheoreticalPhysics degree (from which I switched to mathematics, mainly studying PureMathematics). This gave me a fantastic intuitive overview of the wonders I was to explore, and I have to say that it inspired my learning much more than the actual teaching: I wanted to learn from lectures so that I could see what this book describe in full glorious detail.

LearningBasicMaths

Whilst I many not recommend the method itself as one to be studied, the book TheTrachtenbergSystemOfSpeedMathematics, which shows methods of doing MentalArithmetic more quickly, has various interesting ideas, along with what does and does not need to be trained by repeated practice.

My own view as to LearningBasicMaths is that we should learn from lessons learned in ComputerScience over the past few decades: first and foremost, to master a few basic generic techniques and learn how to apply them, rather than trying to cover every practical application as its own area of study.

I would advocate learning passages of a non-mathematical nature by RoteMemory first, as is done with PassageMeditation, so that you are able to learn formal sequences of words with reasonable ease and reliability. For example, the first verse of PsalmFiftyOne reads:

Have mercy on me, O God,
according to your unfailing love;
according to your great compassion,
blot out my transgressions.
It is a nice meditative exercise to quietly and calmly repeat the words above, of longer fragments of this psalm, up to and including the whole psalm, to a slow and gentle rhythm, focussing as much on each word as possible.

When this practice is mastered, consider learning the following 'passage' in a similar formal way (that is, just learn what word comes after what without any interest in actual meaning: meaning and suchlike comes later):

One Two Three Four Five
Six Seven Eight Nine Ten
This is something we all learn as a child. Now consider:
Seven Fourteen Twentyone Twentyeight Thirtyfive
Fortytwo Fortynine Fiftysix Sixtythree Seventy
Which many of us would recognise as the seven times table. Next consider:
Five Twelve Nineteen Twentysix Thirtythree
Forty Fortyseven Fiftyfour Sixtyone Sixtyeight
This time we are taking the seven times table and subtracting two from each number. If you try to do this by actually working out, say, 'Fiftysix minus Two' in your head each time, you'll find that after repeating the sequence over for a few minutes your head will have had to work more than you might expect.

Eliminating mental effort is an undervalued key to doing maths well: it is only when we can do things effortlessly that we really start to see applications of what we have learned in reality. The reason is that if something takes effort, we must consider whether or not to do it, and in most cases this won't seem justified. As the effort required goes down it is easier to justify to yourself the idea of trying just because you can and it's easy; when it becomes as easy as reading and breathing you just get a feel for what will happen if you try to do something, and often you'll see things immediately.

This familiarity with the basics only comes with practice, and really only comes properly if things are done in a way that is relaxed, fun and joyful: if you dislike maths it just makes it harder for you.

If you think that this is too basic to be worth spending time on, and that there are more practical things to be learning, bear in mind that modern computers effectively use arithmetic and basic logic to repeatedly rearrange lists of numbers: and nothing else. You computer screen, so far as the computer is concerned, is just a list of numbers, or a few lists of numbers. An image in photoshop is just a few lists of numbers. A document in your wordprocessor is just a few lists of numbers. It is by doing a well chosen small set of simple things well, and doing them very quickly that the magic of the modern computer emerges.