Visuo-spatial abilities and geometry: a first proposal of a theoretical framework for interpreting processes of visualization.

We propose a theoretical interpretation of visuo-spatial abilities, as classified in the field of Cognitive Psychology, in the domain of Euclidean Geometry. In this interpretation we make use of Fischbein’s theory of figural concepts and of Duval’s cognitive apprehensions. Our interpretation lays the foundations for a new theoretical framework that we propose as a tool for qualitative analysis of students’ processes of visualization as they carry out geometrical activities. In particular, we present analyses of excerpts from a set of activities designed and proposed in a didactical intervention aimed at strengthening visuo-spatial abilities of a group of students identified as the weakest from a selected 9th grade class of an Italian high school

Whole number thinking, learning and development: neuro-cognitive, cognitive and developmental approaches

The participants of working group 2 presented a broad range of studies, 11 papers in total, related to whole number learning representing research groups from 11 countries as follows. Two large cross-sectional studies focused on developmental aspects of young children’s number learning provide a lens for re-examining ‘traditional’ features of number acquisition. van den Heuvel-Panhuizen (the Netherlands) presented a co-authored paper with Elia (Cyprus; Elia and van den Heuvel-Panhuizen 2015) on a cross-cultural study of kindergartners’ number competence focused on counting, additive and multiplicative thinking. Second, Milinković (2015) examined the development of young Serbian children’s initial understanding of representations of whole numbers and counting strategies in a large study of 3- to 7-year-olds. Children’s invented (formal) representations such as set representation and the number line were found to be limited in their recordings. In a South African study focused on early counting and addition, Roberts (2015) directs attention to the role of teachers by providing a framework to support teachers’ interpretation of young disadvantaged learners’ representations of number when engaging with whole number additive tasks. Some papers reflected the increasing role of neuroscientific concepts and methodologies utilised in research on WNA learning and development. Sinclair and Coles (2015) drew upon neuroscientific research to highlight the significant role of symbol-to-symbol connections and the use of fingers and touch counting exempli- fied by the TouchCounts iPad app. Gould (2015) reported aspects of a large Australian large study of children in the first years of schooling aimed at improving numeracy and literacy in disadvantaged communities. A case study exemplified how numerals were identified by relying on a mental number line by using location to retrieve number names. This raised the question addressed in the neuroscientific work of Dehaene and other papers focused on individual differences in how the brain processes numbers. The Italian PerContare1 project (Baccaglini-Frank 2015) built upon the collaboration between cognitive psychologists and mathematics educators, aimed at developing teaching strategies for preventing and addressing early low achievement in arithmetic. It takes an innovative approach to the development of number sense that is grounded upon a kinaesthetic and visual-spatial approach to part-whole relationships. Mulligan and Woolcott (2015) provided a discussion paper on the underlying nature of number. They presented a broader view of mathematics learning (including WNA) as linked to spatial interaction with the environment; the concept of connectivity across concepts and the development of underlying pattern and structural relationships are central to their approach

Approaching Proof in the Classroom Through the Logic of Inquiry

The paper analyses a basic gap, highlighted by most of the literature concerning the teaching of proofs, namely, the distance between students' argumentative and proving processes. The analysis is developed from both epistemological and cognitive standpoints: it critiques the Toulmin model of reasoning and introduces a new model, the Logic of Inquiry of Hintikka, more suitable for bridging this gap. An example of didactical activity within Dynamic Geometry Environments is sketched in order to present a concrete illustration of this approach and to show the pedagogical effectiveness of the model

La discalculia: riflessioni intorno al tema.

In questo intervento analizzo aspetti dell'apprendimento della matematica, nel caso di difficoltà cognitive, come la discalculia, inquadrandole nel contesto culturale italiano

Micromondi e "Mathematical Habits of Mind"

In questo contributo su invito presento la nozione di micromondo discutendo come sia possibile esercitare diversi modi di pensare matematici in questo contesto. Presento esempi di tali modi di pensare i diversi micromondi studiati nell'ambito della didattica della matematica

Conjecturing and Proving in a Dynamic Geometry Environment

During this talk I will speak about the processes of conjecturing and proving in a dynamic geometry environment and discuss various examples

Contributi ai corsi base Verde, Rosso, Bianco, Azzurro di Matematica (Maths Talk, lezioni alla LIM)

Contributi ai corsi di Matematica per la scuola secondaria di secondo grado. I contributi sono esercizi in inglese (Maths Talk) e video-lezioni alla LIM

Interazioni con la Tecnologia per Sviluppare il Senso del Numero/Interacting with technology to develop number sense

Questo testo si presenta in forma di appunti organizzati e considerazioni emerse da studi pilota, condotti con bambini nelle scuole di Modena e Reggio Emilia e in quelle del progetto "PerContare " (nelle province di Bologna, Reggio Emilia e Torino), con lo scopo di fornire delle basi con le quali guardare la tecnologia a servizio della didattica per lo sviluppo del senso del numero, in modo da poter attuare scelte informate in caso si adozione di queste tecnologie in ambiti educativi. Nella prima sezione vengono illustrati alcuni aspetti dello sviluppo di abilità numeriche, in particolare il ruolo importantissimo che sembrano giocare le dita. Viene poi, introdotta l'idea della linea numerica mentale, utile anch'essa per descrivere le potenzialità offerte dalla tecnologia per il suo sviluppo e, a seguire, vengono sottolineati alcuni aspetti cognitivi che la tecnologia ha la potenzialità di mediare con successo.The purpose of this manuscript, in the form of organized notes and considerations that emerged from pilot studies conducted with children in the schools of Modena and Reggio Emilia and in those of the project "PerContare " (in the Italian provinces of Bologna, Reggio Emilia and Turin), is to lay the foundations to look at roles educational technology might play in the development of number sense, so that educators may become able to make informed choices when they wish to adopt some of these technologies in educational contexts. The first section will discuss aspects of the development of numerical skills, and in particular the important role that using fingers seems to play. Then I will introduce the idea of the mental number line, and how technology has a great potential for its correct development, and finally I will emphasize some cognitive aspects that technology has the potential to mediate successfully, adding an analysis of several existing applications in the light of theoretical considerations that emerged in the first three sections