Learning concepts through the history of algebra - 3 - Intel

Introduction The adolescent’s notion of rationality often encompasses the epistemological view of mathematics as knowledge which offers absolute certainty. He probably has heard of a geometry in which the parallel postulate does not hold, but most likely believes that Euclidian geometry is the “real one”. We can assume that he is not familiar with Gödel’s theorems and undecidability. It is further unlikely that he has been taught about the existence of inconsistent arithmetic that performs finite calculations as correct as traditional arithmetic. These findings provide strong arguments against the view that mathematics offers absolute truth. The static and unalterable mode of presentation of concepts in the mathematics curriculum, rather than lack of knowledge, contributes to this misconception. Mathematical concepts, even the most elementary ones, have changed completely and repeatedly over time. Major contributions to the development of mathematics have been possible only because of significant revisions and expansions of the scope and contents of the objects of mathematics. Yet, we do not find this reflected in class room teaching. While the room for integrating philosophy in mathematics education is very limited, an emphasis on the understanding of mathematical concepts is a necessary condition for a philosophical discourse about mathematics. The conceptual history of mathematics provides ample material for such focus and leads to a better understanding of mathematics and our knowledge of mathematics. I will argue for the integration of the history of mathematics within the mathematics curriculum, as a way to teach students about the evolution and context-dependency of human knowledge. Such a view agrees with the contextual approach to rationality as proposed by Batens (2004). As a prime example, I will treat the development of the concept of a symbolic equation before the seventeenth century. In line with Lakatos (1976) and Kitcher (1984) my example is motivated by the epistemological relevance of the history of mathematics.