Michela Maschietto

Didactic Situations and Didactical Engineering in university mathematics: cases from the study of Calculus and proof

This paper discusses the use of the Theory of Didactic Situations (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of Didactical Engineering (DE), particularly adapted at the tertiary level. We then illustrate the potency of the TDS-DE approach in three university level Research Cases, two related to Calculus, and one related to proof. These studies deploy constructs such as didactic contract, milieu, didactic variables, and epistemological analyses, among others, to design Situations at university level. We conclude with a few thoughts on how the TDS-DE approach relates to other approaches, most notably the Anthropological Theory of the Didactic.

Uses of technology in lower secondary mathematics education : a concise topical survey

This topical survey establishes an overview of the current state of the art in technology use in mathematics education from an international perspective, including both practice-oriented experiences and research-based evidence. Three themes are discussed: Evidence for effect; Digital assessment; Communication and collaboration. Exploiting the opportunities provided by technology for teaching and learning requires a rethinking of educational paradigms and strategies. In the context of lower secondary mathematics, mathematical knowledge has the potential to be outsourced to powerful technologies, which challenges current curricula goals and teaching and assessment practices. We predict future possible trends in technology-rich mathematics education and provide a research agenda, looking forward to a position of effective integration of technology to support mathematics teaching and learning in lower secondary.

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.

Teachers, Students and Resources in Mathematics Laboratory

This paper deals with the methodology of mathematics laboratory from two points of view: the first one concerns teacher education, the second one concerns teaching experiments in the classes. Mathematics laboratory (described in the Italian national standards for mathematics for primary and secondary schools) can be considered as a productive “place” where constructing mathematics meanings, more a methodology than a physical place. It can be associated to inquiry based learning for students. An example of mathematics laboratory with cultural artefacts such as the mathematical machines (www.mmlab.unimore.it) is discussed.

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 ...

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.

The Duo “Pascaline and e-Pascaline”: An Example of Using Material and Digital Artefacts at Primary School

The paper presents the design and the analysis of teaching experiments at primary school concerning the introduction and use of a “duo of artefacts”, constituted by the pascaline i.e., the arithmetical machine Zero+1, and its digital version e-pascaline. The idea of ‘duo of artefacts’ represents the innovative component of this research work, because the e-pascaline is constructed in a complementary way with respect to the pascaline. The duo of artefacts is proposed to support student’s conceptualization processes of numbers as sign of a quantity, number sequences and recursive addition. Computation and manipulation of base ten notation are two processes that students often consider separately. This duo enables the design of situations that required those two processes to be connected and to consider their effect on each other. With duo of artefacts, technology allows the development of learning environments in which it is possible to study the articulation between material and digital manipulatives for mathematical conceptualization.