Epistemic Justification and the Possibility of Computer Proof

Some mathematical theorems can be proven only with the help of computer programs. Does this reliance on computers introduce empirics into math, and thereby change the nature of proof? I argue no. We must distinguish between the warrant the proof gives for its conclusion, and our knowledge of that warrant. A proof is a priori if and only if the conclusion follows deductively from the premises without empirical justification. I start by defending this definition, and proceed to demonstrate that computer-generated proofs meet its criterion

Epistemic Justification and the Possibility of Computer Proof

Some mathematical theorems can be proven only with the help of computer programs. Does this reliance on computers introduce empirics into math, and thereby change the nature of proof? I argue no. We must distinguish between the warrant the proof gives for its conclusion, and our knowledge of that warrant. A proof is a priori if and only if the conclusion follows deductively from the premises without empirical justification. I start by defending this definition, and proceed to demonstrate that computer-generated proofs meet its criterion

Improving The Quality Of The Mathematics Education: The Malaysian Experience

Improving the quality of teaching and learning of mathematics has always been a major concern of mathematics educators. The four recurring and inter-related issues often raised in the development of a mathematics curriculum are: “What type of mathematics ought to be taught?”, “Why do we need to teach mathematics?”, “How should mathematics curriculum be planned and arranged?” and “ How can teacher ensure that what is transmitted to the pupils is as planned in the curriculum?”.The relatively brief history of mathematics education in Malaysia can be said to have developed in three distinct phases. In the first phase, the traditional approach, which emphasized mainly on basic skills (predominantly computational), was the focus of the national syllabus. In the late 70’s, in consonance with the world-wide educational reform, the modern mathematics program (MMP) was introduced in schools. Understanding of basic concepts rather than attaining computational efficiency was the underlying theme of the syllabus. Finally, in the late 80’s the mathematics curriculum was further revised. It is part of the national educational reform that saw the introduction of the national integrated curriculum (KBSM) both at the primary and secondary levels. This mathematics curriculum, which has undergone several minor changes periodically, is presently implemented in schools. The curriculum also emphasizes on the importance of context in problem solving. These three syllabi, as in any other curricular development, can be seen to have evolved from changing perspectives on the content, psychological and pedagogical considerations in teaching and learning of mathematics. In this paper, I will trace the development of the Malaysian mathematics curriculum from the psychological, content and pedagogical perspectives in relation to the recurring issues. I will argue that the development has in many ways attempted to make mathematics more meaningful and thus friendlier for students both at the primary and secondary levels. There has been also a marked improvement on the quality of mathematics education in Malaysi

Proof in dynamic geometry contexts

Proof lies at the heart of mathematics yet we know from research in mathematics education that proof is an elusive concept for many mathematics students. The question that this paper raises is whether the introduction of dynamic geometry software will improve the situation – or whether it make the transition from informal to formal proof in mathematics even harder. Through discussion of research into innovative teaching approaches with computers the paper examines whether such approaches can assist pupils in developing a conceptual framework for proof, and in appropriating proof as a means to illuminate geometrical ideas