Why teach math? Clearly how we teach math has a great deal to do with how it has always been done, and the “why” was therefore not often questioned. Arithmetic has been part of the school curriculum since ancient times. Land has to be surveyed, taxes assessed, and accounts figured. These were done by the educated elites, while ordinary laborers had little use for numeric skills. With the rise of market economies, almost all daily needs were met through the market, so everyone needed some number sense, and benefited from some arithmetic skills. So far as daily living and workplace needs go, manipulating money and quantities of goods to be bought or sold, profits, interest and so on are the main practical needs.
At one time, pencil and paper arithmetic was clearly a useful skill, while today this is something that is hardly done outside of school. It is still important to be able to understand practical arithmetic problems so that you know which key or spreadsheet symbol to use. Traditional elementary math was moderately effective in conveying this sort of understanding, but it isn't really known how important practice with manual arithmetic is for conveying this sort of understanding or for developing “number sense”.
Cultural institutions evolve, presumably meeting some cultural need. Education is an interesting case because the overt function of school (teaching of useful subjects) does obviously have some validity, but doesn't seem sufficient to explain the curriculum. Math education being in excess of obvious need is not a new thing. Since classical times, geometry and logic were part of the curriculum, often with various odd numerological accretions, such as (during the middle ages) extensive classification of special kinds of numbers, going well beyond square, prime, … .
Education has often been highly circular, with its function being to prepare you for further education, then at some point to qualify you for entry into the elite. The connection between educational goals (curriculum) and fitness for privilege has never been entirely clear. This may have worked in ancient China or in the industrial west, both as a way of sorting for intelligence, diligence, etc., and as legitimizing the social structure through opportunity for upward mobility creating an argument that privilege was deserved. But the educational enterprise has been shaped by deep tradition as a gateway to privilege, and that this conflicts with the goals of universal education.
The disconnect that often exists between curriculum and what we use in life (what we remember) is one of the most obvious puzzles. Some possible reasons (several may apply):
When you consider the social function of school as a whole you appreciate that it can have value largely independent of the usefulness of the subjects taught: