Maria G. Bartolini Bussi

Early Approach to Theoretical Thinking: Gears in Primary School.

Gears are part of everyday experience from very early childhood. This paper analyses a teaching experiment conducted with 4th graders in the field of experience of gears. The aim is to identify the characteristics which, given a suitable sequence of tasks and proper teacher guidance, have enabled the pupils to approach theoretical thinking, and in particular mathematical theorems. We have focused on the relationships between the epistemological analysis of some pieces of mathematical knowledge brought into play in tasks concerning gears, cognitive analysis of pupil construction of those pieces of mathematical knowledge, and didactic analysis of the teachers role in designing tasks and in offering cultural mediation. This paper presents the early findings of the teaching experiments, both at the external level of interpersonal classroom processes and at the inner level of individual mental processes.

Historical formation and student understanding of mathematics

The use of history of mathematies in the teaching and learning of mathematics requires didactical reflection. A crucial area to explore and analyse is the relation between how students achieve under standing in mathematics and the historical construction of mathematical thinking.

Research, practice and theory in didactics of mathematics: Towards dialogue between different fields

Acknowledging the complex relationships which the field of didactics of mathematics has with other research fields (e.g. mathematics, educational sciences, epistemology, history,psychology, semiotics, sociology, cognitive science), the authors analyze in this paper some cases of fruitful and some of failed dialogue between experts of the different fields. They discuss the results of these dialogues, drawing on research studies carried out by the authors, within the paradigm of the Italian research in Mathematics Education.

Experimental Approaches to Theoretical Thinking: Artefacts and Proofs

This chapter discusses some strands of experimental mathematics from both an epistemological and a didactical point of view. We introduce some ancient and recent historical examples in Western and Eastern cultures in order to illustrate how the use of mathematical tools has driven the genesis of many abstract mathematical concepts. We show how the interaction between concrete tools and abstract ideas introduces an “experimental” dimension in mathematics and a dynamic tension between the empirical nature of the activities with the tools and the deductive nature of the discipline. We then discuss how the heavy use of the new technology in mathematics teaching gives new dynamism to this dialectic, specifically through students’ proving activities in digital electronic environments. Finally, we introduce some theoretical frameworks to examine and interpret students’ thoughts and actions whilst the students work in such environments to explore problematic situations, formulate conjectures and logically prove them. The chapter is followed by a response by Jonathan Borwein and Judy-anne Osborn.

Italian Trends in Research in Mathematical Education: A National Case Study from an International Perspective

Every discussion about research in mathematics education (referred to as ‘RME’ from now on) that is carried on at the international level emphasizes contrasting and even competing approaches: the increasing number of monographs that review national contributions to RME in a self-contained way (e.g. Barra et al. 1992; Blum et al. 1992; Douady & Mercier 1992; Kieran & Dawson 1992) seems to emphasize the existence of different research traditions that are developed locally with their own store of epistemological debates, institutional constraints, research questions, methods, results and criteria, leading even to the birth of paradigms (e.g. the French didactique des mathematiques). Despite this increasing volume of information at the international level, the discussion of research questions related to local projects seems to be difficult (see Silver 1994). Two different yet related levels of problems are involved: 1. communication: How is it possible to convey to the international professional community of researchers in mathematics education meaningful information about the context of a local research project? 2. relevance: (a) What are the individual aspects of context-bound research that are supposed to be relevant to and influential for the international community, and (b) What are the general aspects of international research that are supposed to be relevant to and influential for any local research project?

When Classroom Situation is the Unit of Analysis: The Potential Impact on Research in Mathematics Education

In this commentary, I will critically elaborate on the potential impact of the coordinated papers of this volume on further development of research in mathematics education. The papers, which share common theoretical frameworks, will be categorized into three different classes: ‘demolishers of illusions’, ‘economizers of thought’ and ‘energizers of practice’. I will analyze the role played by psychology and related sciences as a possible enrichment of the frameworks, especially where technologies are concerned. Finally, I will discuss the possible conflict between the need to consider the phenomena elicited in this kind of studies and the sophistication required by the theoretical constructs, which makes the results of these studies very difficult to communicate to the international community.

Mathematical Machines: From History to Mathematics Classroom

The aim of this chapter is to present some issues concerning secondary teacher education, drawing on the activity of the Laboratory of Mathematical Machines at the Department of Mathematics of the University of Modena and Reggio Emilia (MMLab: http://www.mmlab.unimore.it). The name comes from the most important collection of the Laboratory, containing more than two hundred working reconstructions (based on the original sources) of mathematical artefacts taken from the history of geometry. In this chapter we intend to discuss, in the setting of teacher education and within a suitable theoretical framework, a single case, i.e., an ellipse drawing device, from different perspectives (historic-epistemological, manipulative and virtual), to develop expertise in selecting and adjusting appropriate tools for the mathematics classroom.

Reasoning in Geometry

Orly’s doctoral research was devoted to studying the processes of justification and proving in geometry, used by the pupils she taught in the 9th and 10th grade. After two years of research her findings lead her in a new direction.

Mathematics in Context: Focusing on Students

This chapter presents nine case studies in which school students engage in challenging mathematics outside their immediate classroom environment. In each case, students are encouraged to collaborat ...

Building a Strong Foundation Concerning Whole Number Arithmetic in Primary Grades: Editorial Introduction

This volume is the outcome of ICMI Study 23 (Primary Mathematics Study on Whole Numbers). This editorial introduction outlines the rationale, the launch, the discussion document, the study conference, the content of the study volume, the main implications of the study, merits and limits of the study, concluding remarks, and acknowledgments.