Cristina Sabena

Università degli Studi di Torino | UNITO · Dipartimento di Filosofia e Scienze dell'Educazione

Recognizing the omnipresence of mathematics across various contexts, this paper addresses the untapped potential of museums as rich venues for informal mathematics learning beyond traditional educational settings like classrooms. This paper presents the InformalMath program, designed for the professional development of primary and middle school tea...

Questo lavoro si ispira alla prospettiva dell'educazione matematica informale, nella qua-le l'attività matematica è libera dai vincoli imposti dalle tradizioni curricolari, dai libri di testo e dalle prove di valutazione. In tale prospettiva, è stato progettato un percorso didattico che integra delle attività di insegnamento-apprendimento della mat...

Informal Mathematics Education is an emergent field of research in which out of school spaces become protagonist of intentional learning designs. In our research we exploit cultural spaces such as art and history museums to engage teachers in a challenging teacher education programme called "InformalMath". In InformalMath teachers work in communiti...

L’intreccio tra componenti concettuali e figurali è una caratteristica fondamentale del pensiero geometrico e costituisce un importante obbiettivo didattico a tutti i livelli scolastici. In questo articolo si considerano le altezze dei triangoli inquadrando il problema da un punto di vista teorico e analizzando un’attività didattica sperimentata in...

In our study we exploit digital tools in order to foster a relational approach to mathematics undergraduate students. Specifically, we designed formative assessment workshops based on peer feedback, with the aim to promote self-assessment and reflection on one’s own mathematical activity. In the paper we describe the design of an educational activi...

In questo lavoro si presentano e analizzano due esperienze didattiche rivolte alle classi IV e V primaria, incentrate sull’interpretazione di dati e sulle loro rappresentazioni grafiche. Queste esperienze si inseriscono in un percorso più ampio che mira a costruire le basi per educare cittadini consapevoli, capaci di utilizzare strumenti scientific...

Literature has already shown that gestures play a relevant role in classroom interactions between students and teacher. We integrate the perspective of Theory of Semiotic Mediation with the notion of Semiotic Bundle to illustrate how gestures can be used as “pivot signs” in semiotic chains. This means that gestures can be performed by the teacher t...

We discuss how strategy games may be exploited to develop mathematical reasoning and argumentation competencies in kindergarten. According to a design-based methodology, 5-year-old children have been involved in a teaching experiment based on a game called “Thirteen buttons”. We have conducted a qualitative data analysis, based on video recording a...

This paper examines mathematical meaning-making from a phenomenological perspective and considers how a specific dynamic digital tool can prompt students to disclose the relationships between a function and its antiderivatives. Drawing on case study methodology, we focus on a pair of grade 11 students and analyze how the tool’s affordances and the...

This chapter considers the application of augmented reality (AR) technology for developing covariational reasoning and aims also to show general potentialities of AR technology and to give some hint of the theoretical principles behind them. Specifically, the chapter discusses the first design-cycle from a design-based research study that aims at b...

This contribution addresses the theme of technology for formative assessment in the mathematics classroom and in particular the ways connected classroom technology may support formative assessment strategies in whole class activities. Design experiments have been developed through the use of a connected classroom technology by which students may sh...

The paper situates mathematics teaching-learning processes within a multimodal perspective and discusses a semiotic approach apt to seize this dimension, namely the Semiotic Bundle lens. This analytical tool considers the great variety of semiotic resources through which mathematical meanings emerge and evolve in the classroom, ranging from embodie...

This study aims to identify types of teachers’ interventions that can foster or hinder the processes of inquiry in the classroom because of the teacher’s behavior. The Method of Variation Inquiry developed by Arzarello guided this study, which took place in a learning setting where students learnt pre-calculus concepts through exploring real-world...

We focus on formative assessment processes carried out, by the teacher and the students, through the use of digital technologies. The research is situated within the European Project FaSMEd, in which a new model connecting the role of technology to classical views on formative assessment is proposed. Through data analysis from teaching experiments...

A multimodal perspective on mathematics thinking processes is addressed through the semiotic bundle lens and considering a wide notion of sign drawing from Vygotsky’s works. Within this frame, the paper focuses on the role of gestures in their interaction with the other signs (speech, in particular) and investigates the support they can provide to...

This paper is based on a design-based research project investigating how to use digital resources to help activate formative assessment processes in the classroom. Performed as part of FaSMEd, a European Union project, our own project adopts a comprehensive theoretical framework, including the different functionalities of technology, formative asse...

In this contribution we analyse data coming from the research project FaSMEd, which aims at investigating the role of technologically enhanced formative assessment methods in raising the attainment levels of low-achieving students. Our working hypothesis is that low attainment is also linked to affective factors and that, consequently, these factor...

This contribution addresses young children development of spatial competences, and investigates the didactic potentialities offered by a programmable robot. The theoretical framework addresses the delicate relationship between space as lived in everyday experience versus space as a mathematical notion, and takes a multimodal perspective on mathemat...

In this work, we present the analysis of the ways in which formative assessment processes can be developed, by the teacher and the students, thanks to the support given by technology. The analysis is carried out focusing on two case studies developed in France and Italy within the European Project FaSMEd, and is developed with two main aims: (1) hi...

It is widely shown in literature that gestures take part in classroom interactions. Adopting the perspective of Theory of Semiotic Mediation, we illustrate how they can act as “pivot signs” in semiotic chains, namely they can link signs referring to a situated context (such as the activity with an artifact) to the mathematical domain. More specific...

The paper analyses how connected classroom technologies can be exploited to foster formative assessment practices in the mathematics classroom. Referring to a three-dimensional framework developed within the European project FaSMEd and to Hattie and Temperley's levels of feedback (2007), an excerpts from a classroom discussion in grade V is analyse...

This paper investigates the introduction of the derivative notion and, specifically, the introduction of the derivative function, as a significant moment in the development of mathematical work on functions. In particular, we analyse the process of genericization that two Italian teachers conducted with their grade 13 students, in order to make the...

In this chapter we present the main ideas of an educational Vygotskian semiotic approach, emphasizing in particular some crucial questions about its methods of inquiry. We resort, on the one hand, to Leont’ev’s (1978) work on activity, and, on the other hand, to Vygotsky’s cultural psychology. Considering a theory as an interrelated triplet of “com...

Our contribution to this Round Table on assessment stems from our joint participation to the FaSMEd Project. FaSMEd (“Improving Progress through Formative Assessment in Science and Mathematics Education”) is a European Project (FPVII, 2013-15) aiming at investigating the use of technology in formative assessment classroom practices in ways that al...

This workshop is focused on the role technology may play in supporting the formative assessment process. Different examples from the case studies developed in France and Italy within the European Project FaSMEd will be analysed and discussed. In order to highlight the choices we made in relation to the aim of the project, before discussing the exam...

Relationship with mathematics is a crucial variable in the professional development of pre-service primary teachers: it can largely affect pre-service teachers’ reactions to educational prompts and also their future didactical choices in classroom. Many researches showed that pre-service primary teachers have often developed a negative relationship...

On the one hand, recent results in neuroscience and communication bring the role of multimodal resources in cognitive and communicative processes to the fore. On the other hand, Bruner highlights the different role of narrative and logical thinking in human understanding. Before this background, we investigate the teaching and learning processes wh...

In our research on gestures in mathematical understanding, we start by assuming that gestures, along with other body aspects (e.g., gazes), are an important component of emotional and cognitive manifestations and an inseparable component of speech discourse, as research in psychological and related domains have shown (e.g. Goldin-Meadow, 2003; McNe...

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’ ro...

This paper refers to the concept of semiotic and theoretic control describing resources to conduct decisions in epistemic processes. We consider an argumentation process from a complex problem-solving activity involving different conceptual frames related to parabolas. Using a micro-analytical interpretative lens, we will show that, in order to car...

In this study we use Habermas construct to analyse argumentation processes related to strategic interaction problems. These problems provide suitable environments to develop and analyse students’ planning and control processes, of a paramount importance in mathematical problem-solving. Some theoretical tools to study planning and control processes...

This paper reports on a study from a wider project of thesis, whose main focus is investigating, in the secondary school context, the transition between two important mathematics domains: elementary Algebra and Analysis. Following Kuhn (1969), we detect some fundamental paradigm shifts between the two. They represent a change in the way of thinking...

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...

The case study of the epistemological gap involves two theoretical approaches, APC and IDS. It describes a networking case that starts from a situation of seemingly contradictory analyses, develops a common methodology, and leads finally to conceptualizing and locally integrating the new concept of the epistemological gap into both theories.

By referring to the data presented in Chap. 2, the chapter introduces the theoretical approach of Action, Production, and Communication (APC) and the related tool of the semiotic bundle. APC provides a frame for investigating semiotic resources in the classroom. It addresses the use of semiotic resources from a multimodal perspective including the...

In this case study, the epistemic role of gestures is considered empirically. The analysis of gestures is included into the AiC analysis of a small excerpt of the data from Chap. 2 by means of the notion of semiotic bundle, which forms a crucial component of the APC-space. For this purpose, APC and AiC are coordinated and then locally integrated in...

The chapter provides the basic information on the set of data that is used throughout the book. Data from a video recording show two students, Carlos and Giovanni, when investigating the exponential function in a dynamic geometry environment. An interview with the teacher gives background information.

The relationship with mathematics of future primary teachers is very often built on negative experiences with mathematics as students and characterized by strong negative emotions towards mathematics. This phenomenon is alarming because of its consequences on teachers’ development and practice. Nevertheless, many future primary teachers reveal a de...

Mathematics education is strongly interested in defining “what is necessary for teaching mathematics effectively”. The main directions of research emphasize the cognitive side of the answer to this question, trying to describe what kind of knowledge is needed in order to teach mathematics effectively. Starting from the point that teachers’ affect p...

As researches in mathematics education and also as teacher educators we are strongly interested in facing the question “what is necessary for teaching mathematics effectively”. The main directions of research emphasize the cognitive side of the answer to this question. In our view, attention from a purely cognitive dimension has to shift to a wider...

We use eye tracking as a method to examine how different mathematical representations of the same mathematical object are attended to by students. The results of this study show that there is a meaningful difference in the eye movements between formulas and graphs. This difference can be understood in terms of the cultural and social shaping of hum...

In many countries elementary school teachers are not specialized in mathematics and low mathematics self-perception and negative emotions towards mathematics often prevail among the pre-service teachers. These negative beliefs and emotions are an allarming phenomenon because they can interfere with becoming a good mathematics teacher. In this paper...

We present a model to analyze the students’ activities of argumentation and proof in the graphical context of Elementary Calculus. The theoretical background is provided by the integration of Toulmin’s structural description of arguments, Peirce’s notions of sign, diagrammatic reasoning and abduction, and Habermas’ model for rational behavior. Base...

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 evalu...

Our research addresses the role of examples to foster the students’ development of the mathematical concepts, and of their mathematical ways of thinking. We consider the notion of example space introduced by Watson and Mason (Mathematics as a constructive activity: learners generating examples, 2005), particularly when it is not formed by a simple...

Research on teachers’ beliefs is based on the shared assumption that teachers’ mathematical beliefs are closely related to teachers’ intention of practice (and then to teachers’ practice). But if many studies have confirmed a strong relationship between beliefs and practice, there are also many studies finding low relationships or even inconsistenc...

Our analysis of data about one learning situation from two theoretical perspectives yields results that on the surface seem to be in conflict. Through networking of two theories we produce a fresh combined analysis tool, which deepens our understand-ing of the data in an integrated way. We elaborate this example to make explicit our two theoretical...

The networking of theories is a promising research practice for connecting theories systematically while valuing their identities. Strategies for the networking of theories and four case studies with different profiles are presented. Theoretical reflections on the cases show that the networking of theories can be regarded as a process that begins w...

In this paper, we consider gestures as part of the resources activated in the mathematics classroom: speech, inscriptions, artifacts, etc. As such, gestures are seen as one of the semiotic tools used by students and teacher in mathematics teaching–learning. To analyze them, we introduce a suitable model, the semiotic bundle. It allows focusing on t...

Problem solving is highly recognised as a basic component of mathematical activity. In addition, recent studies have pointed out the significance of perceptuo-motor activities in mathematics learning, which become more or less active depending on the context. Some years ago in cognitive science the theory of embodied cognition began to stress the f...

Learning and teaching mathematics requires the activation of a variety of resources, which can be grouped around what I consider to be two chief sources of mathematics knowledge: the body and its activity with artifacts, and the activity with signs1. The cognitive significance of the body has become one of the major topics in current psychological...

The paper presents how two different theories—the APC-space and the ATD—can frame in a complementary way the semiotic (or ostensive) dimension of mathematical activity in the way they approach teaching and learning phenomena. The two perspectives coincide in the same subject: the importance given to ostensive objects (gestures, discourses, written...

We propose to operationalize symbol sense by using eye tracking to investigate whether differences in mathematical skills are revealed in eye movement behaviour in tasks where participants are asked to match one mathematical representation to another in a selection of four alternatives.

In this article, we deal with students' algebraic generalizations set in the context of elementary geometric-numeric patterns. Drawing from Vygotsky's psychology, Leont'ev's Activity Theory, and Husserl's phenomenology, we focus on the various semiotic resources mobilized by students in their passage from the particular to the general. Two small gr...

Chapter 1 Introduction 1.1 The Research Problem The overall aim of this research is to analyze the dialectics between the following two chief sources of mathematics knowledge formation: - the body and its activity with artefacts; - the activity with signs. The study focuses in particular on the role of gestures as bodily semiotic resources in te...

In this paper we present a case study referring to an activity at secondary school level, in which students have to perform motions (walking in front of a sensor), in order to obtain a space-time graph (on a calculator), as close as possible to a given graph. The aim is to analyse our empirical data on the students' approach to the two graphs throu...

In this paper we deal with the genesis of students' algebraic generalization of patterns. Our aim is to better understand the way students attend to the perceptually given (e.g. the three first elements of a geometric or numeric sequence) and start moving beyond it in their attempt to grasp a possible general mathematical structure. We provide a mu...

Laboratoire de recherche en sémiotique culturelle et pensée mathématique Université Laurentienne Conséquence, sans doute de l'influence de Piaget, depuis le siècle dernier, les études de psychologie ont reconnu l'importance du rôle du langage et de l'activité kinesthésique dans l'émergence des savoirs, certains concepts mathématiques élémentaires a...

In this paper we deal with the genesis of students' algebraic generalization of patterns. Our aim is to better understand the way students attend to the perceptually given (e.g. the three first elements of a geometric or numeric sequence) and start moving beyond it in their attempt to grasp a possible general mathematical structure. We provide a mu...

This article presents a study based on a teaching experiment realised in the third, fourth and fifth year of Liceo Scientifico. The goal of the teaching experiment is the construction of the concept of integral, in the context of measurements, basing on the theoretical frame of Tall's cognitive roots. Special attention is given to the students' cog...

The paper reports on a case study from a teaching experiment on the construction of meaning of integral at secondary school level, starting from the cognitive root of area. In the considered activity, students are faced with some graphs of functions and are asked to work in group to trace the corresponding integral functions. The analysis considers...

The purpose of this paper is to introduce two perspectives for observing students' activity, taking into account the main elements of their communication and thinking processes: gestures, speech, written words, mathematical signs. These perspectives give origin to two different types of analyses called parallel and serial analysis, in order to dist...

Pursuing our investigation on students' use and understanding of algebraic notations, this paper examines students' cognitive difficulties related to the designation of an indeterminate but fixed object in the context of the generalization of patterns. Stressing the semiotic affinities and differences between unknowns, variables and parameters, we...

In this paper we focus on the role of signs in students' perceptive processes underpinning the generalization of numeric-geometric patterns. Based on a video- taped Grade 9 classroom group activity undertaken by three students and framed by a cultural-semiotic theoretical perspective, we carry out a microgenetic analysis of an elementary form of ma...

The paper tackles the issue of the mediation given by technological devices in learning contexts. In the framework of instrumental analysis outlined by Rabardel, it takes into consideration an activity involving symbolic-graphic calculators to face the notion of integral in the 12 th grade. Basing on the protocol analysis, the problem of transparen...

In this article we present a study relative to a teaching experiment in the third and fourth year of Liceo Scientifico. The focus of the teaching experiment is on the construction of the concept of integral, in the environment of measurements, in the framework of Tall's cognitive roots. We particularly study the students' cognitive processes with t...