Extending the Frontiers of Mathematics: inquiries into proof and argumentation
The proof of the pudding may be in the eating, but what is the recipe?
Do you ever wish that you had been made to explore mathematics at A-level and university? Would you like to deepen and enrich your appreciation of mathematical reasoning? Would you like to know how advanced mathematics advances? Then this may be the book for you.
The impetus and inspiration come from a style of teaching graduate students in mathematics developed by E. F. Moore in the 1960s, but it is entirely compatible with supporting mathematical exploration and discovery at secondary school and beyond.
I was fortunate to be taught by a number of Moore’s students, but unfortunate in not experiencing the real thing. Moore’s method was to require each student to agree not to look anything up in books (now this would include the internet). He would then hand out a list of theorems. Classes consisted of presentations of proofs by students, and if you had not thought out a proof for yourself you were encouraged not to attend class. He would then encourage critique of students’ reasoning and if necessary, offer his own. Many of Moore’s students went on to be famous mathematicians, but few used his method as they experienced it: it seemed to require too much effort and dedication by students.
However, several people have adapted his methods for their own personality and situation, and Edward Burger seems to be one of those. For example, the author is not reticent in presenting commentary and instruction before setting the reader loose on some problems. This means that the book would be, as the author suggests, suitable as a resource for a ‘discovery approach’ course, but it could also be used with a combination of lectures and student presentations, or for independent study. Advice is offered on choosing chapters for courses to introduce mathematical proof, to introduce discrete mathematics, to provide a survey of mathematical foundations, to provide topics for future teachers to work on, and as a capstone seminar in mathematics at undergraduate level.
The first chapter offers ten puzzles as a ‘precursor’ to proof. These vary from the familiar problems of covering a chessboard with dominos when two corner squares are removed and handshakes, to my favourite of his collection (this is a slight variant): some coins are on a table but you cannot see them or touch them; you are told how many are showing heads; you then instruct someone in how to separate the coins into two piles while paying no attention to the heads or tails, in such a way that when you tell the person to turn over all the coins in one of their piles, lo and behold there are the same number of heads in both piles.
Subsequent chapters introduce various logical and mathematical topics with titles such as ‘bringing theorems to justice: exposing the truth through logical proof’; ‘delving into the dependable digits’; ‘going round in circles: the art of modular arithmetic’; ‘infinity: understanding the unending’ and so on. Topics include introductions to irrationals, relations and functions, recursively defined functions, permutations, probability, graph theory, geometry, the intermediate value theorem, Pythagorean triples, groups, and Cantor sets. I believe the idea is to show how simple ideas, when pursued, develop into research areas in mathematics. One nice example from late in the book goes as follows. Imagine a circle and a point O on it. For any pair of points P and Q on the circle, take the chord through O parallel to PQ (or the tangent if P = Q). Let R be the point in which that chord intersects the circle again. Specify a binary operation by P•Q = R. Then this forms a group (what is the identity element?) and the idea extends to other conics, and can be extended further in algebraic geometry.
I was very excited by the opening chapter, imagining perhaps that the same style would continue, but unfortunately he succumbs to wanting to instruct the reader, or perhaps to provide the reader with essential background. I found the instructional ‘lectures’ somewhat tedious, including the final notes on how to refine and sharpen a proof, but they would probably be very useful for someone using the book as a resource or for independent study. I very much liked the way he uses section headings such as standing back in every chapter, and frequent prompts to prove and extend or disprove and salvage. The critical feature in getting the most out of the book would be in getting participants to present their reasoning to each other and to work publicly on refining and honing that reasoning. A conjecturing atmosphere would be absolutely essential!
John Mason • Open University
Extending the Frontiers of Mathematics: inquiries into proof and argumentation
Edward Burger (2007)
Emeryville: Key College Publishing and Springer
ISBN 1-59757-042-7
