The Mathematics of Schooling

The mathematics of schooling is my answer to the question of how mathematics education (the institutionalized social practice to try to promote some common good by teaching of mathematics) is connected to mathematics (the enigmatic phenomenon which is simultaneously an object of teaching, the name of a science and something which appears to be "used" in a number of situations).

What in an educational context is called mathematics appears to refer to something beyond education itself (science, which seems to be almost everywhere useful, almost everywhere present, etc). In an educational context, mathematics appears as something essentially different from the school itself, as something given for schooling to take as a point of departure. It seems self evident that it is impossible for school to affect what mathematics is.

The firs part of my thesis is that the way mathematics is apprehended in school is determined by school. It has, in the context of schooling, emerged ideals for teaching of mathematics, theories of how children learn mathematics, it has emerged conceptions of what mathematics is, what role it plays in society and science, and why it is important that everybody takes part in mathematics education during their adolescence.

In school, these "conceptions" - to simplify somewhat - are presented as sprung from mathematics, i.e. they are presented as an adjustment to mathemaitics, a necessary consequence of how mathematics in itself is - conceived as a science, a professional and everyday practice, something given and everywhere present, etc.

To the contrary, these conceptions have emerged in the very context of schooling. They have not been "influenced by" and "adjusted to" a beforehand given mathematics, or a mathematics developed in parallel to the development of the school system. The ideals of teaching and the theories of learning are immanent to schooling and must be understood historically and sociologically. This goes for theories regarding the formation of mathematical concepts as well as claims for a common need of mathematics education.

The first part of my thesis can be put thus: schooling has a high degree of autonomy in relation to other spheres in society - including science. Mathematics education presents itself as subordinate science and society, as its servant. To the contrary, mathematics education (as a part of schooling in general) is an autonomous "positive" force. Mathematics education - e.g. more specifically the people involved in or related to teaching of mathematics - has, to a large extent its own agenda.

This does not mean, though, that those active in the area of mathematics education are aware of the fact that they are handing down and reproducing "their own" tradition. This fact leads to the second part of my thesis.

The second part of my thesis regarding the mathematics of schooling is that the "conceptions" of mathematics which have emerged and prevail in mathematics education have been "materialized", or with another word, "reified" in mathematics. What is in fact the result of a complex historical process, and which in practice is reproduced through complicated social mechanisms, appears as caused by mathematics conceived as a universal, eternal object.

The mathematics that mathematics education circles around does thus, in some sense, exist "objectively" - so far as it appears as objectively given and thus impossible to change, particularly for individuals. From the point of view of schooling, mathematics appears as what it is, and hence as something one has to adapt to.

What this mathematics "is" is though just mathematics education itself "in another form". It is therefore, you could say, the mathematics of schooling.

Mathematics education is de facto the result of a complex historical process; it has many features which could seem enigmatic. By appearing as sprung from mathematics, i.e. as rationally derived from mathematics, i.e. as rationally derived from the inherent properties of mathematics, the enigma is transferred away from school, to mathematics. What appears enigmatic is thence not school, but mathematics.

The unfathomableness that mathematics education sees in mathematics is thus only the unfathomableness of itself in another form.

Mathematics education has, put shortly, not rational foundation in mathematics The mathematics it seems to circle around it the result of a kind of internal split.

As a third part of my thesis you could say at this internal split is an absolutely necessary condition for the proper functioning of mathematics education. At the very moment it becomes clear that the mathematics that school circles around is "its own", the institution of mathematics education will "fall" (whatever that would mean).

Here it is important to note though, that it is far from clear what would amount to such a "becoming clear". A part of the theory which have inspired my analysis is a set of ideas related to the structure of belief. Pertinent to the case of mathematics education is a phenomenon which can be described as a "fetishist split", meaning that people can "know" something but none the less act "as if" they did not know. This is normal for people in the context of mathematics education. It is normal to "know" that mathematics is in fact not the way it is officially presented by mathematics education. None the less it is also normal to act in accordance with this disavowed truth. One acts in a way which supports and reproduces the social institution of mathematics education.