Contributions to the TWG4 - CERME4

Argumentation and proof in examples taken from french and german textbooks. Richard Cabassut, IUFM d'Alsace, Didirem Paris 7
Abstract: A study of French and German curricula of secondary schools have shown that arguments of plausibility and arguments of necessity are both encouraged in mathematic teaching. We show that the mathematics textbooks used in to these curricula give examples of argumentations and proofs involving both kinds of arguments. These examples illustrate the theoretical frame presented in this paper and chosen to explain the use and the combination of arguments which can be understood through the concept of function of validation.

Pupils' awareness of structure on two number/algebra questions Dietmar Küchemann and Celia Hoyles Institute of Education, University of London
Abstract: A large sample of high attaining pupils were given a written proof test in Yr 8 (age 13.5 years) and similar tests in Yrs 9 and 10. We look at their responses to two number/algebra questions which were designed to assess whether pupils used empirical or structural reasoning. We found that the use of structural reasoning increased over the years, albeit at a modest rate, but that the use of empirical reasoning, in the form of inappropriate number pattern spotting or through the desire to perform rather than analyse a calculation, was still widespread.

The meaning of proof in mathematics educationDavid A Reid Acadia University
Abstract: The issue of what mathematics education researchers mean by "proof" and "proving" has been the topic of three recent papers. The discussion in those papers are analysed in terms of a common terminology for identifying characteristics of the meanings of proof current in research.

Natural deduction in Predicate Calculus A tool for analysing proof in a didactic perspective Viviane Durand-Guerrier IUFM de Lyon & LIRDHIST, University Lyon 1
Abstract: In this paper, we intend to provide theoretical arguments for the importance of taking account in quantification matters while analysing proofs in a didactic perspective, not only at tertiary level, where various research are still available, but also at secondary level and we argue that natural deduction in predicate calculus is a relevant logical reference for this purpose. Following Quine, we emphasize on an example the interest of formalizing mathematical statements in Predicate Calculus in a purpose of conceptual clarification. In a second part of the paper, we give some short insights about the theory of quantification before exposing the system of Copi for natural deduction. The last section is devoted to analysing a proof using the logical tools offered by natural deduction in predicate calculus.

Visualising and Conjecturing Solutions for Heron's Problem Lourdes Figueiras and Jordi Deulofeu Departament de Dida`ctica de la Matema`tica i de les Cie`ncies Experimentals Universitat Auto`noma de Barcelona
Abstract: This paper analyzes some important aspects of using visual constructions during problem solving processes. In particular, we analyze some geometric constructions made to conjecture the solution to Heron's problem, and obtain two different categories of visual representations.

Proof in swedish upper secondary school mathematics textbooks - the issue of transparency Kirsti Nordström and Clas Löfwall Stockholm University, Department of Mathematics
Abstract: We have investigated proof in two sets of commonly used Swedish upper secondary school mathematics textbooks. The frequency of proof items is low in each mathematical topic, even in the domain of geometry where pupils traditionally have learned proof. We explore the proof items with respect to different aspects of proof and discuss how they relate to students' access to proof. We show with some examples how proof often exists invisible in the textbooks and discuss the difficulty of giving a correct definition of proof at upper secondary school level.

Inductive reasoning in the justification of the result of adding two even numbers Consuelo Cañadas Santiago Universidad de Zaragoza, Spain; Encarnación Castro Martínez Universidad de Granada, Spain
Abstract: In this paper we present an analysis of the inductive reasoning of twelve secondary students in a mathematical problem-solving context. Students were proposed to justify what is the result of adding two even numbers. Starting from the theoretical framework, which is based on Pólya's stages of inductive reasoning, and our empirical work, we created a category system that allowed us to make a qualitative data analysis. We show in this paper some of the results obtained in a previous study.

A genetic approach to proof Hans Niels Jahnke, Universität Duisburg-Essen
Abstract: The metaphor of a 'Theoretical Physicist' is used to explain the thinking of students in proof situations. The paper proposes to treat geometry in an introductory period as an empirical theory and to distinguish between a formative and an established phase in the (personal) development of a theory.

About a constructivist approach for stimulating students' thinking to produce conjectures and their proving in active learning of geometry Oleksiy Yevdokimov Kharkov State Pedagogical University, Ukraine
Abstract: The paper describes processes that might lead secondary school students to produce conjectures in a plane geometry. It highlights relationship between conjecturing and proving. The author attempts to construct a teaching-learning environment proposing activities of observation and exploration of key concepts in geometry favouring the production of conjectures and providing motivation for the successive phase of validation, through refutations and proofs. Supporting didactic materials are built up in a way to introduce production of conjectures as a meaningful activity to students.