In today's class, we will take time to read and discuss excerpts from two classic and influential works about the nature of
proof in mathematics & math education.
In the first, from
Thinking Mathematically, John Mason discusses justifying conjectures -- that is, proving them. His 'big idea' suggestion (convince yourself, convince a friend, convince an enemy -- and then learn to be your own enemy) gives a very interesting and often quoted plain-language idea of mathematical proof. Mason refers to puzzles from elsewhere in the book, and I have included some of these at the end of the scanned
excerpt so that you can try one or two of them.
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The second, from
Proofs and Refutations, is part of Imre Lakatos' influential doctoral thesis in the history of mathematics. It is heavier reading than the first, but is worth the effort for the insights it gives. Lakatos reconstructs a bit of the history of mathematics as if it were all happening in a classroom, with a Teacher and students Alpha, Beta, Gamma, Delta, etc.
The topic is Euler's Polyhedral Theorem, a lovely little bit of theory that says that, if you count the vertices, edges and faces of any polyhedron, the sum of the number of vertices is two more than the sum of the number of edges and faces: V-E+F = 2. To make sense of this theorem, it is very helpful to play around with virtual and/or actual models of polyhedra -- see previous posting. You can look at the simple, regular polyhedra (cube, tetrahedron, octahedron, dodecahedron and icosahedron -- the Dungeons & Dragons dice shapes!) and count the vertices, faces and edges to see if they work.
Lakatos shows that this seemingly-simple geometric theorem actually encountered a lot of opposition through counterexamples historically. Several kinds of responses to counterexamples are shown in this excerpt, including "the method of surrender" and "monster-barring". The main point for us reading this excerpt from Lakatos is to see that, historically, the task of creating mathematical definitions and theorems, proving theorems, and offering counterexamples was anything but clear, simple and straightforward! The idea of monster-barring (by changing our mathematical definitions to exclude "monstrous" cases) is something I have used in my secondary school math classes.
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Instructions for Keri, Vanessa, Shan and Alex (and optionally, Jubilee): Take a bit of time to read and make sense of these excerpts, then post four substantive comments each over the course of the next week in an online discussion amongst yourselves, considering the idea of
proof in mathematics learning and whether it is significant, important and/or understandable for learners at a variety of levels. Is there something about
proof as a way of thinking logically and critically that is important for all learners -- or not?
Hi everyone. Lucie De Blois, our guest speaker from the Université Laval, sent these two articles as a follow-up to her talk about the 'false reality' of word problems, Guy Brousseau's important concept of the didactic contract, and ways that geometry can be a way to help students develop arithmetic problem solving skills.
