Théorie des situations didactiques

par
Guy Brousseau

Journey into mathematics an introduction to proofs

 "Le passage de la  pensée naturelle à  l'usage d'une pensée  logique comme celle  qui régit les raisonne- ments mathématiques  s'accompagne de la
 construction, du rejet,  de la reprise de  différents moyens de  preuve : rhétoriques,  pragmatiques, sémantiques ou syntaxiques.
   L'examen d'une preuve est une attitude réflexive. Il faut que la preuve soit formulée et présente tout au long de l'examen donc le plus souvent écrite. Qu'elle puisse être confrontée à d'autres preuves écrites elles aussi, à la situation à laquelle elle renvoie.
   En général, la preuve ne pourra être formulée qu'après avoir été utilisée et éprouvée en tant que règle implicite soit dans l'action soit dans les discussions."

Commande aux
Editions La Pensée Sauvage

One of the main purposes of this book is to help the (novice) reader at the undergraduate level how to read and write proofs. The author points out at the beginning that a major function of proof is that of explanation; ie. explaining why a result is true. On the other hand, he also uses some very good examples early on to illustrate the limitations of inductive reasoning. For example, he gives a spectacular example involving a special case of Pell's equation for which the first n for which it is false has 1115 digits! The author subsequently introduces the verification function of proof by introducing mathematical induction as one method for checking the validity of a mathematical statement for all n.
  The book discusses binomial coefficients, polygonal areas, the irrationality of square root 2, the Pythagorean theorem, Pythagorean triples, the Diophantine method of finding Pythagorean triples, the area and circumference formulas of disks by Eudoxus and Archimedes, the quadratic formula, complex numbers, the cubic formulas of Cardona and Viete' respectively, the quartic formula, and lastly proofs of the irrationality of e and pi, which is essential reading for every high school mathematics teacher. The author presents all these topics by interweaving historical information which makes it very entertaining; in fact, this is a strategy used effectively throughout the book. In so doing, geometry, algebra, number theory, and analysis are all intertwined in this entertaining journey.
  A major feature of the book which makes it particularly relevant to novices, is the way in which the historical background has been interwoven into the discussion. It shows how techniques and understanding developed over time, with several people building on the work of their predecessors.
  Several exercises are given in the book where the reader can apply and extend the ideas discussed in a chapter. This clearly forms an integral part of the book, and the author correctly points out that merely reading about mathematics is no substitute for doing mathematics. In conclusion, this book is a valuable contribution to the teaching of proof at the undergraduate level, and strongly recommended.

M. de Villiers

Proofs from The Book

 "We have no definition  or characterization  of  what constitutes a proof  from The  Book: all we  offer here is the examples  that we have selected,  hoping that our  readers  will share our  enthousiasm about  brilliant ideas, clever insights and wonderful  observations. We also hope that our readers  will enjoy this despite the imperfections of  our exposition."
  "To a large extent this book reflects the views of Paul Erdös as to what should be considered a proof from The Book"

The editors

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The NCTM Curriculum Standards (National Council of Teachers of Mathematics, 1989) call for decreased attention to two-column proofs and increased attention to alternative expressions of mathematical argument, such as "deductive arguments expressed orally and in sentence or paragraph form". Teachers are to promote or increase students' opportunities "to make and provide arguments for conjectures", "formulate counterexamples; follow logical arguments; judge the validity of arguments; [and] construct simple valid arguments". The Standards suggest that mathematical argumentation be contrasted with other forms of argumentation (such as political or commercial advertisements), in some of which logical errors can be detected.
   Similarly, the Professional Standards for Teaching Mathematics (NCTM, 1991) indicate the teacher's responsibility to choose tasks that "require students to speculate,… to face decisions about whether or not their approaches are valid". Teachers are encouraged to play a role of orchestrators of classroom discourse in the direction of mathematical reasoning, asking students to explain and justify their ideas. The Professional Standards expect that as a result of those efforts from the teacher, students will become competent in the use of mathematical argument to support the validity of their conjectures.
   A group of five mathematics educators will discuss the role of the teacher in fostering mathematical argumentation with respect to the epistemological opposition between argumentation and proof and the educational opposition between mathematical reasoning and two-column proofs. (The discussion will focus on the American curriculum but will be enhanced by the experiences of the contributors, as they come from four different countries.) The aim is to provide a multidimensional conceptualization that identifies the conditions and constraints of the teacher's work and to illustrate the dilemmas that the teacher may encounter.
   The symposium will have five sections. Each contributor will approach the opposition between argumentation and proof from a different angle. The symposium wil be concluded by a panel discussion in which will be raised the key questions, it will allow the five participants to react to the other presentations, and open the floor for comments and questions from the audience. The following sections describe the parts into which the symposium will be divided...

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Pupils need for conviction and explanation within the context of dynamic geometry

This Master Thesis (University of Durban-Westville, South Africa) aims at evaluating the feasibility of introducing "proof" as a means of explanation rather than only verification, within the context of dynamic geometry. Pupils, who had not been exposed to proof, were interviewed and their responses were analyzed.

The research attempted to determine whether pupils were convinced about explored geometric statements and their level of conviction. It also attempted to establish whether pupils exhibited an independent desire for why the result, they obtained, is true and if they did, could they construct an explanation, albeit a guided one, on their own.

What's in a Proof?

Contact the author...

 The reader will discover a rather quick answer to a complex  question, but many links to follow and pages to explore.

 This special issue of the Math Forum's weekly newsletter highlights interesting conversations taking place  during December of 1998 on Internet math discussion groups. The readers of the Proof Newsletter may be interested in following the discussion of CALC-REFORM -- a mailing list hosted by e-MATH of the American Mathematical Society (AMS) -- focussing on the "Pedagogy of "Big Theorems" (5 Dec. 1998). Hereafter are reproduced some quotations :

"...proving 'big theorems' is not pedagogically effective. Proofs are simply too long and too overwhelming for students. It is hard to learn 5 or more steps at once, the brain gets overloaded...." - Kazimierz Wiesak

"Proofs are not intended to build intuitive understanding. But saying 'accept this!' is probably worse pedagogically, depending, of course, on goals...." - Walter Spunde

"My piano teacher always amazed me by playing complicated Bach pieces at sight. She explained the method: she recognized groups of notes (chords and chord sequences for example), and hardly had to think when playing them; her fingers knew what to do, and she had to think only for the broad sweep of the piece. Whereas when I played at sight, each note involved finger juggling and thinking. No wonder I found it difficult (and still do)! Polya wrote about the value of teaching mathematical ideas using repetition but with variation, and pointed to the composers for examples. Perhaps sight-reading is another example where we could learn from the composers." - Sanjoy Mahajan