A philosophical study of the value of mathematical knowledge and the place of mathematics on the curriculum

In basing the school curriculum on the view that some value can be attributed to knowledge most arguments have centred on either the contingent consequences of studying particular disciplines, the claim that knowledge can be differentiated into distinct forms and that all students should be introduced to them, or the claim that some knowledge can be valued for its own sake or for its power in developing the mind. In the case of mathematics common justifications given for teaching it are that it is useful, that it promotes intellectual development or that it is intrinsically worthwhile. But a recent view argues that some knowledge is valuable because it provides people with such an understanding that allows them to reflect on questions concerning the nature and meaning of life and to be in a position to best determine what they will do with their lives. The role that mathematics plays here is investigated by an examination of the nature and foundations of mathematical knowledge. Dominant views on mathematics have nearly all stressed its a priori nature but they all have serious objections to them. By a comparison with views on the nature of scientific change a recent view on the nature of mathematical knowledge has been articulated that describes it to be in a process of evolution. At any particular time there exists a mathematical practice which consists of a language component, a metamathematical view component, and sets of accepted reasonings, questions and statements. The mathematical practice of today has evolved from a set of beliefs about simple manipulations of physical objects and consists of idealized ways of operating on the world. It is concluded that while all students should be introduced to the minimal mathematical language that is useful to everyone they should also come to understand the cultural significances of mathematics as it has evolved through man's attempts to solve problems within his environment. This comes through a study of the influences that mathematics has had on different cultures and the way that man has looked to mathematics as providing a method of solution to problems within his culture. Unlike earlier justifications given for teaching mathematics the justification based on the cultural significances of mathematics centres on all five components of the mathematical practice of tae day and provides important considerations for the structure and presentation of mathematics courses in schools