“”There is no royal road…”” –Euclid to Ptolemy. Mathematics has some abstract and complex concepts to master. While Math cannot be made easy, or learned without effort, it can be made accessible. We will look at Henri Picciotto’s strategies for helping students gain access to mathematical mastery… -Themes: mathematics-rich contexts, drawn from real-world or fanciful problems, where math concepts can be introduced, explored, developed, and reviewed. Well-chosen themes can bring algebra to life, uncover connections to other parts of mathematics, and support the claim that math does indeed have applications. -Tools: objects (and electronic environments) which provide concrete and manipulable models of abstract and complex ideas, thereby making them accessible and interesting. They are the “”objects-to-think-with”” called for by Papert in Mindstorms (1980) -Problem Solving: symbol manipulation is often the main focus of the math courses. In fact, some teachers mistake good symbol manipulation for rigor. Picciotto (and lots of research since) suggests that symbol manipulation become a tool for problem solving, but that Problem Solving itself be the main operating mode throughout the course. -Spiral organization: extended exposure to concepts using the multiple representations made possible by the interplay of tools and themes. This approach allows for substantial preview and review, and helps highlight connections between concepts.
