In my last post I said that it was common knowledge among mathematicians that basic algebra is best described as expression arithmetic. I'm continuing my monologue on these ideas here.
Addition, as an operation of elementary arithmetic, is a procedure that takes a pair of decimals and gives a decimal that represents the sum of the numbers denoted by the taken decimals. It is often frowned upon within some mathematical circles to describe addition in this way, as it requires an overly pedantic distinction between decimals and the numbers that they represent. I can not agree with this perception, nor can I allow it to be propagated, it is not pedantic. In fact, it cuts to the heart of the mathematical method: represent a problem in a form that you can understand and easily manipulate.
The theory of cardinals is rich and deep, it requires a lot of work to prove that the sum of a pair of cardinals is another cardinal, yet in the case of finite cardinals, we need only know that the arithmetic we've developed is an adequate representation of the theoretical backings. If we are curious, and we should be, we will eventually push past the function of arithmetic towards the form of cardinals so that we might see how and why arithmetic is done as it is.
Children, on the other hand, do not need to follow the path of logic through modern mathematical topics. If we can give them tools for thinking about mathematics efficiently then we can leave it to their individual curiosity to look for deeper meaning and form. Not every person will have an impulse to understand the way things work, though they probably could have such an impulse if they were properly educated. Some people will naturally be compelled to follow the facts past what they've been told towards the truth that they are able to demonstrate themselves.
It benefits both types of people to learn efficient methods of dealing with mathematics that might otherwise have deep underpinnings. Both will learn to use math, and the one who is interested in going deeper will have already developed an effective intuition as to what principles and problems are of central importance in the use of mathematics as a tool for thought. Finally, if we are lucky, the one who searches for truth will discover that there is more to mathematics than its utility as a tool for thought, they will be compelled, we hope, to dream of ever more beautiful mathematics that eliminates the illusion that utility and ornament are disjoint.