Ferdinando Arzarello

Meta-Didactical Transposition: A Theoretical Model for Teacher Education Programmes

We propose a new model for framing teacher education projects that takes both the research and the institutional dimensions into account. The model, which we call Meta-didactical Transposition, is based on Chevallard’s anthropological theory and is complemented by relevant elements that focus on the specificity of both researchers’ and teachers’ roles, while enabling a description of the evolution of their praxeologies over time. The model is illustrated with examples from different Italian projects, and it is discussed in light of current major research studies in mathematics teacher 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?

A method for quantifying focused versus overview behavior in AOI sequences

We present a new measure for evaluating focused versus overview eye movement behavior in a stimulus divided by areas of interest. The measure can be used for overall data, as well as data over time. Using data from an ongoing project with mathematical problem solving, we describe how to calculate the measure and how to carry out a statistical evaluation of the results.

The Impact of PISA Studies on the Italian National Assessment System

In this chapter we sketch how the discussion that started in Italy with the disappointing results of the first PISA surveys was the origin of a national assessment program that possibly led to some improvement in the outcomes of mathematics learning. We will also underline similarities and differences between PISA studies and the Italian program of assessment.

Starting Points for Dealing with the Diversity of Theories

This chapter presents the main ideas and constructs of the book and uses the triplet (system of principles, methodologies, set of paradigmatic questions) for describing the theories involved. In Part II (Chaps. 3, 4, 5, 6, and 7), the diversity of five theoretical approaches is presented; these approaches are compared and systematically put into a dialogue throughout the book. In Part III (Chaps. 9, 10, 11, and 12), four case studies of networking practices between these approaches show how this dialogue can take place. Chapter 8 and Part IV (Chaps. 13, 14, 15, 16, and 17) provide methodological discussions and reflections on the presented networking practices.

Tradition in Whole Number Arithmetic

The main topics discussed by the panel and the resulting questions to be answered are introduced along with some bibliographic references. The main topics of discussion concern the relationships between tradition and the verbal and non-verbal representations of numbers, numbers and artefacts of arithmetic and the role of technological devices in emulating traditional abaci and allowing direct interaction with the screens of multitouch devices in counting activities. Another crucial issue concerns the different languages that can be present in a classroom for historical and cultural reasons. This represents a challenge for teachers, who must cope with the ways in which words can shape the specific connotations of the meanings of numbers. Although all of these facets of numbers need to be coordinated with the standard mathematical concepts, they also appear in the multimodal representations that are used to teach them, such as words, textbooks, notes and teachers’ and students’ gestures. All of these factors intertwine and sometimes conflict with the richness of the representations and practices that children encounter outside school in their everyday lives.

Aspects that Affect Whole Number Learning: Cultural Artefacts and Mathematical Tasks

The core of this chapter is the notion of artefact, starting from the discussion of the meaning of the word in the literature and offering a gallery of cultural artefacts from the participants’ reports and the literature. The idea of artefacts is considered in a broad sense, to include also language and texts. The use of cultural artefacts as teaching aids is addressed. A special section is devoted to the artefacts (teaching aids) from technologies (including virtual manipulatives). The issue of tasks is simply skimmed, but it is not possible to discuss about artefacts without considering the way of using artefacts with suitable tasks. Some examples of tasks are reported to elaborate about aspects that may foster learning whole number arithmetic (WNA). Artefacts and tasks appear as an inseparable pair, to be considered within a cultural and institutional context. Some future challenges are outlined concerning the issue of teacher education, in order to cope with this complex map.

Approaching Secondary School Geometry Through the Logic of Inquiry Within Technological Environments

The chapter illustrates a pedagogically innovative way of using mobile technology to support the transition from an empirical to a more theoretical and logical approach to geometry. We propose an approach that shows the possibility of discovering geometric theorem statements and appreciating their universal truth using a suitable pedagogical design that draws on the work of the Finnish logician J. Hintikka as well as on Dick and Zbiek’s notions of pedagogical, mathematical and cognitive fidelities. We implement it through game-based activities, namely group activities in which, first, students play a game in a dynamic geometry environment (DGE) and then, guided by the questions contained in a worksheet task, investigate the geometric property on which the game is designed. In the worksheet task, students are asked to act as detectives using the game to investigate, formulate and check conjectures. In order to analyse the students’ productions we use a cognitive model elaborated from Saada-Robert’s psychological model, which properly describes the cognitive modalities and the empirical versus theoretical and logical approaches to geometry.

MOOC: repository di strategie e metodologie didattiche in Matematica

Abstract – This paper focuses on MOOC (Massive Open Online Courses) as a new paradigm in e-learning educational projects: it presents MOOCs for lower and upper secondary school mathematics teachers. On the one hand, a MOOC, with rich materials in innovative didactic methodologies and specific technological tools, can be understood as a repository from which teachers can draw inspiration. On the other hand - by allowing teachers from different geographic locations an opportunity for communication – a MOOC is also the place where these teachers - in a community of peers, supported by a vigilant but non-intrusive presence of trainers – have the rare opportunity to reflect together to organise strategies, processes and materials that are linked not only to age, type of school and social/individual situation, but also to the content of the discipline that has specific cognitive obstacles. The aim of this contribution is, therefore, to present our teacher education experiences through MOOCs, with a focus on the methodologies of mathematics teaching that were shared and debated with the teachers who attended the course, using as theoretical lens a new elaborated framework to analyse these new online training environments. Riassunto – Il focus di questo articolo verte su MOOC (Massive Open Online Courses) per docenti di matematica di scuola secondaria (I e II grado), da considerarsi come nuovo paradigma nei progetti educativi in e-learning. Da un lato, un MOOC, con materiali ricchi di metodologie didattiche innovative e specifici strumenti tecnologici, si puo intendere come un archivio informatico (repository) da cui gli insegnanti possono trarre spunto. Dall’altro lato, consentendo un’opportunita di comunicazione a vari insegnanti, provenienti da diverse realta geografiche, un MOOC e anche il luogo in cui tali insegnanti – in una comunita unicamente di pari, sostenuta dalla vigile, ma non invadente presenza dei formatori – si ritrovano ad avere la rara occasione di poter riflettere insieme per orientare opportunamente strategie, processi e materiali, legati non solo a fascia di eta, tipologia di scuola e situazione sociale/individuale, ma anche ai contenuti stessi della disciplina che presentano ostacoli cognitivi specifici. L’obiettivo di questo contributo e dunque quello di esporre le nostre esperienze di formazione insegnanti tramite MOOC, con un focus sulle metodologie di didattica della matematica che sono state condivise e approfondite con i corsisti, utilizzando come lente d’analisi un nuovo quadro elaborato proprio per analizzare questi nuovi ambienti di formazione insegnanti online. Keywords – MOOCs, teacher education, professional development, peer collaboration, community of practice, repository Parole chiave – MOOC, formazione insegnanti, sviluppo professionale, collaborazione tra pari, comunita di pratica