Epistemological Questions About School Mathematics

Abstract

A disciplinary educational research field cannot, I think, avoid tackling general questions about the educational aims it pursues, as well as more specific questions concerning the teaching and learning of its contents, the nature of these contents and its methodology and theoretical foundations. Mathematics education is not an exception. These questions and their possible answers define a specific area of inquiry that has been termed the philosophy of mathematics education. Ernest (1991a, b, 2009), whose work has been influential in shaping this area of inquiry, suggests that the philosophy of mathematics education revolves around two axes. On the one hand, the philosophy of mathematics deals with the philosophical aspects of research in mathematics education. On the other, it deals with the aims of mathematics education (Ernest, this volume). Taking both axes together, the phi- losophy of mathematics education tackles questions such as our understanding of, and the meaning we attribute to, mathematics and its nature. It also includes questions about the purposes of teaching and learning mathematics, the meaning of learning and teaching mathematics and the relationship between mathematics and society. The answers that we can offer to the previous questions go beyond mathematics itself. In order to tackle those questions, we need, indeed, to go beyond mathe- matics and step into new territory. We need to immerse ourselves in a series of theoretical domains like history, politics, ontology, metaphysics, aesthetics, epis- temology, anthropology, ethics and critical philosophy (Ernest, this volume). Consider for instance, the question about the relationship between mathematics and society. Since ancient times, what we call today “schooling” has been related to societal needs. The education of the scribes in Mesopotamia is a case in point. Mesopotamian scribes were instrumental in the organization and administration of the City (Høyrup, 2007). The mathematics that they learned and practised was influential in the measuring and distribution of lands, the collection of taxes, the calculation of the amount of food to be distributed to the soldiers, etc. One of the three oldest known problems goes back to ca. third millennium BC. It was found in 1975 by an Italian Archeological Mission while excavating the site of the Royal archives of the city of Ebla. The problem, contained in text TM.75.G.1392, is about the amount of cereal that is required to be distributed among a large number of individuals. In Fribergs’ (1986) reconstruction, the problem reads as follows: Given that you have to count with 1 gu-bar for 33 persons, how much do you count with for 260,000 persons? (Friberg, 1986, p. 19). The mathematics that the Babylonian produced and that scribes learned in school (what they called the “House of Tablets”) was not a disinterested endeavour. It was related to the way the Babylonian administrative and political body sought to respond to societal needs.1 It does not mean, however, that all Babylonian math- ematics was about solving practical problems. This is the case, for instance, of many geometric problems at the basis of what has been called “Babylonian alge- bra.” The first problem of a tablet known as AO 8862 that goes back to ca. 1750 BC reads: 1. Length, width. Length and width I have made hold: 2. A surface have I built. 3. I turned around (it). As much as length over width 4. Went beyond, 5. To inside the surface I have appended: 6. 3` 3. I turned back. Length and width 7. I have accumulated: 27. Length, width, and surface w[h]at? (Høyrup, 2002, p. 164)2 Without