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

Semiotic mediation: fragments from a classroom experiment on the coordination of spatial perspectives

Abstrac

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

ABSTRACT. 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 of some failed dialogue between experts of the different fields. They discuss results of these dialogues, drawing on research studies carried out by the authors, within the paradigm of the Italian research in Mathematics Educatio

L’educazione geometrica attraverso l’uso di strumenti: un esperimento didattico

Abstract. In questo lavoro ci si propone di presentare un esperimento didattico a lungo termine sull’introduzione in una classe V elementare di strumenti e modelli per la prospettiva. La fase di cui ci occuperemo, temporalmente iniziale rispetto l’intero esperimento, riguarda l’uso di particolari artefatti culturali, che hanno permesso la realizzazione della trasposizione nella classe del costrutto teorico della mediazione semiotica

Difficulties with whole number learning and respective teaching strategies: the case of Ole. Introduction and video clip (electronic supplementary material).

Rottmann T. Difficulties with whole number learning and respective teaching strategies: the case of Ole. Introduction and video clip (electronic supplementary material). In: Bartolini Bussi MG, Sun XH, eds. Building the foundations: Whole numbers in the primary grades. New ICMI Study Series. Vol 23. Cham: Springer; 2018: 512

Chapter 11. Classical and Digital Technologies for the Pythagorean Theorem

This paper aims to discuss the use of material tools, called mathematical machines, and digital tools in approaching the Pythagorean theorem. These mathematical machines are related to different proofs of the theorem. Teaching experiments with 13-year old students were carried out within the laboratory approach developed from the theoretical frameworks of the Theory of Semiotic Mediation and Instrumental approach in mathematics education. Their analysis shows that behind the kinesthetic experience with the machines, there are important cognitive processes such as the identification of invariants, relationships between the components and usage schemes. It also shows the only manipulation of the first machine does not imply the emergence of the mathematical meanings embedded in the materials tools and the crucial role of the teacher with his different instrumental orchestrations in that process

Experimental approaches to theoretical thinking: artefacts and prof

Abstract. It is well known that any sort of instruments, from straightedge and compass to a variety of computational tools created in the course of history, are deeply intertwined with the genesis and development of many abstract concepts and ideas in mathematics. As will be discussed in this chapter, their use introduces an “experimental” dimension in mathematics, and a tense dynamics between the empirical nature of the activities with them, which encompasses perceptual and operational components, and the deductive nature of the discipline, which entails a rigorous and sophisticated formalization