The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, (...) as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning. (shrink)

Celem niniejszego artykułu jest wyznaczenie kierunków badań edukacyjnych w działalności Polskiej Szkoły Argumentacji oraz zainicjowanie dyskusji w polskim środowisku filozoficznym i wśród nauczycieli szkolnych. Opierając się na dotychczasowej działalności Szkoły w badaniach nad edukacją i tworzeniu standardów nauczania sztuki argumentacji oraz wątków edukacyjnych wyeksponowanych podczas XV konferencji ArgDiaP, formułujemy postulaty, które mogą stanowić część planu edukacyjnego w najbliższych latach. Realizacja tych postulatów może stanowić punkt odniesienia do realizacji długoterminowego planu tworzenia standardów nauczania sztuki argumentacji.

To adequately model mathematical arguments the analyst must be able to represent the mathematical objects under discussion and the relationships between them, as well as inferences drawn about these objects and relationships as the discourse unfolds. We introduce a framework with these properties, which has been used to analyse mathematical dialogues and expository texts. The framework can recover salient elements of discourse at, and within, the sentence level, as well as the way mathematical content connects to form larger argumentative structures. (...) We show how the framework might be used to support computational reasoning, and argue that it provides a more natural way to examine the process of proving theorems than do Lamport’s structured proofs. (shrink)