Chapter devoted to the theme of proof in the Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future

"Proof and proving in mathematics education"

Maria Alessandra Mariotti

It is our pleasure to announce the release of the Handbook of Research on the Psychology of Mathematics Education (Rotterdam, The Netherlands: Sense Publishers).
This volume has been prepared as a celebration for the 30th anniversary of the PME International Group (1976-2006). It is a compilation of the research produced by PME since its creation,  and has been written to become an essential reference for Mathematics Education research in coming years.
The chapters offer summaries and synthesis of the research produced by PME, presented to let the readers grasp the evolution of paradigms, questions, methodologies and most relevant research results during last 30 years. They also include extensive lists of references. The chapters raise also the main current research questions and suggest directions for future research.
You can get more information and download a nonprintable PDF version of the book from Sense Publishers' web page at
http://www.sensepublishers.com/books/otherbooks/90-77874-19-4.htm.

One of the chapters is devoted to the theme of proof "Proof and proving in mathematics education" Maria Alessandra Mariotti

Introduction
Nowadays, differently to ten years ago, there seems to be a general consensus on the fact that the development of a sense of proof constitutes an important objective of mathematical education, so that there seems to be a general trend towards including the theme of proof in the curriculum. Take, for instance, the following quotation.
Reasoning and proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions, no matter what topic is being studied. (NCTM, 2000, p. 342)
I wonder whether these words would have been possible only a few years ago, and still now the idea of “proof for all” claimed in this quotation is not a view that most teachers hold, even in countries where there is a longstanding tradition of including proof in the curriculum. I’m thinking of my country, Italy, but also, as far as I know, France or Japan. In fact, the main difficulties encountered by most students have lead many teachers to abandon this practice and prompted passionate debate amongst math educators, which has produced a great number of studies. Proof has also been a constant theme of discussion in the PME community and at PME conferences, which has given rise to a large number of Research Reports, although not at the same rate every year (the reader can find a useful, although not yet complete, collection of references at the site: http://www.lettredelapreuve.it/).
The debate is far from being closed and has generated a number of research questions which have evolved in the decades, also in accordance with an evolution of the general trend of Mathematics education research. A quick overview of PME contributions on the theme of proof - probably not dissimilarly from what happened for other themes - shows a move away from early studies, focussed on students’ (and more rarely teachers’) conceptions of proof, and generally speaking on difficulties that pupils face in coping with proof and proving, towards more recent studies where researchers present and discuss opinions on whether and how is it possible to overcome such difficulties through appropriate teaching interventions. As a general trend, it is possible to observe a change in the methodology: reports on teaching experiments have increased while reports discussing quantitative analysis based on questionnaires have decreased, though questionnaires and quantitative analysis remain the main methods in large scale investigations, when nationwide investigations or cross cultural comparisons are carried out. We will return to this point in later.
An exemplar of earlier studies is the classic research report presented by Fischbein & Kedem (1982). Repeatedly quoted and subsequently re-discussed, this study focused on the crucial tension between the empirical and formal approach to proof. Exemplars of recent studies centred on teaching experiments can be found among the number of reports concerning the use of Dynamic Geometry Environments: a specific section will be devoted to discussing these.The variety and the complexity of the PME contributions, differently related to proof, required a drastic selection in order to give a reasonable account in the space of a chapter and within the limits of my capacity: I therefore apologize for the unavoidable incompleteness.
The discussion is organized according to three main streams of research, identified by three main categories of research questions, which I summarize as follows:
Proof at school. What is the status of proof at school? This quite general question is formulated differently in different studies, but the general characteristic aim consists in searching for a global view that captures widespread phenomena and possible correlations between them.
Students’ Difficulties. The general issue concerns the study of students’ difficulties, and it refers to two main questions, roughly corresponding to describing and to interpreting students’ behaviours in proving tasks. What are the main difficulties that students face in relation to proof? Which might be the origin of such difficulties?
Teaching Interventions. Is it possible to overcome the difficulties that students meet in relation to proof? How can teaching interventions be designed? What general suggestions can be given?Before starting the discussion I’d like to share with the reader some
introductive reflections. As clearly pointed out by Balacheff (2002/04), the epistemological perspective taken by the researcher is not always made explicit, and this can be considered one of the main reasons for the failure of communication: instead of correctly fuelling the debate, contributions risk becoming blocked in the impasse of misunderstanding.
Epistemological issues are not often directly addressed in the Research Reports presented at PME Conferences. Nonetheless the centrality of these issues in the debate was clearly discussed by Gila Hanna at 20th PME Conference (Hanna, 1996) and two papers have explicitly dealt with them in the recent past (Godino & Recio, 1997) and (Reid, 2001).
Both contributions focus, in different ways, on the differences in the meaning of the term proof as it appears in people’s use of this word. The first paper takes a wide perspective, describing some of the meanings of the term proof in different contexts, such as Mathematics and mathematical foundations research, sciences, and Mathematics class.
The second paper focuses on the domain of Mathematics education research, where different usages of the term proof are identified. While I refer to these papers, together with more recent contributions (Balacheff, 2002/04; Reid, 2005), for an explicit comparison between different perspectives, I will try to make my position explicit through a short introduction that might facilitate understanding of what I am going to present in the following sections.