The role of proof in mathematics teaching and The Plateau Principle
AbstractOne of the most difficult learning thresholds for students of mathematics is the concept of proof. The difficulty manifests itself in several ways: (1) appreciating why proofs are important; (2) the tension between verification and understanding; (3) proof construction. The first case study describes a spurious but ‘convincing’ proof and a correct but ‘unconvincing’ proof of a deep result in linear algebra. A brief discussion introduces The Plateau Principle, an unspoken credo for successful research in mathematics, which says simply: ‘look for and be prepared to use a variety of plateaus as starting points for a mathematical investigation.’ The second case study illustrates an underlying ‘proof template’ that assists in the development of proof technique, in much the same way as a sense of perspective is essential for the ability to draw well.