Mathematics as an Educational Task

Description

Like preludes, prefaces are usually composed last. Putting them in the front of the book is a feeble reflection of what, in the style of mathe­ matics treatises and textbooks, I usually call thf didactical inversion: to be fit to print, the way to the result should be the inverse of the order in which it was found; in particular the key definitions, which were the finishing touch to the structure, are put at the front. For many years I have contrasted the didactical inversion with the thought-experiment. It is true that you should not communicate your mathematics to other people in the way it occurred to you, but rather as it could have occurred to you if you had known then what you know now, and as it would occur to the student if his learning process is being guided. This in fact is the gist of the lesson Socrates taught Meno's slave. The thought-experi­ ment tries to find out how a student could re-invent what he is expected to learn. I said about the preface that it is a feeble reflection of the didactical inversion. Indeed, it is not a constituent part of the book. It can even be torn out. Yet it is useful. Firstly, to the reviewer who then need not read the whole work, and secondly to the author himself, who like the composer gets an opportunity to review the Leitmotivs of the book. Front Matter....Pages I-XII The Mathematical Tradition....Pages 1-16 Mathematics Today....Pages 17-51 Tradition And Education....Pages 52-63 Use and Aim of Mathematics Instruction....Pages 64-98 The Socratic Method....Pages 99-108 Re-Invention....Pages 109-130 Organization of a Field by Mathematizing....Pages 131-146 Mathematical Rigour....Pages 147-154 Instruction....Pages 155-161 The Mathematics Teacher....Pages 162-169 The Number Concept — Objective Accesses....Pages 170-241 Developing the Number Concept from Intuitive Methods to Algorithmizing and Rationalizing....Pages 242-286 Development of the Number Concept — The Algebraic Method....Pages 287-311 Development of the Number Concept — From the Algebraic Principle to the Global Organization of Algebra....Pages 312-331 Sets and Functions....Pages 332-400 The Case of Geometry....Pages 401-511 Analysis....Pages 512-580 Probability and Statistics....Pages 581-614 Logic....Pages 615-661 Back Matter....Pages 662-680