PHILOSOPHY MATRIX

Philosophy Matrix

Philosophy Matrix

New forms of mathematical knowledge are growing in importance for mathematics and education, including tacit knowledge; knowledge of particulars, language and rhetoric in mathematics. These developments also include a recognition of the philosophical import of the social context of mathematics, and are part of the diminished domination of the field by absolutist philosophies. From an epistemological perspective, all knowledge must have a warrant and it is argued in the paper that tacit knowledge is validated by public performance and demonstration. This enables a parallel to be drawn between the justification of knowledge, and the assessment of learning. An important factor in the warranting of knowledge is the means of communicating it convincingly in written form, i.e., the rhetoric of mathematics. Skemp's concept of 'logical understanding' anticipates the significance of tacit rhetorical knowledge in school mathematics. School mathematics has a range of rhetorical styles, and when one is used appropriately it indicates to the teacher the level of a student's understanding. The paper highlights the import of attending to rhetoric and the range of rhetorical styles in school mathematics, and the need for explicit instruction in the area.

In the past decade or two, there have been a number of developments in the history, and philosophy and social studies of mathematics and science which have evoked or paralleled developments in mathematics (and science) education. I shall briefly mention three of these that have significance for the main theme of this paper, the import of rhetoric and justification in mathematics and mathematics education. Even though all of the developments I mention below are continuing sites of controversy, I merely list them rather than offer extended arguments in support of the associated claims, since this would draw me away from the main theme. Anyway such arguments can be found elsewhere.

An important background development has been the emergence of fallibilist perspectives in the philosophy of mathematics. These views assert that the status of mathematical truth is determined, to some extent, relative to its contexts and is dependent, at least in part, on historical contingency. Thus a growing number of scholars to question the universality, absoluteness and perfectibility of mathematics and mathematical knowledge (Tymoczko 1986). This is still controversial in mathematical and philosophical circles, although less so in education and in the social and human sciences. One consequence of this perspective is a re-examination of the role and purpose of proof in mathematics. Clearly proofs serve to warrant mathematical claims and theorems, but from a fallibilist perspective this warranting can no longer be taken as the provision of objective and ironclad demonstrations of absolute truth or logical validity. Mathematical proofs may be said to fulfil a variety of functions, including showing the links between different parts of knowledge (pedagogical), helping working mathematicians develop and extend knowledge (methodological), demonstrating the existence of mathematical objects (ontological), and persuading mathematicians of the validity of knowledge claims (epistemological), see, e.g. Lakatos (1976). Below I elaborate further on the persuasive, epistemological role of proofs in ...