Artifacts and signs after a Vygotskian perspective: the role of the teacher

The notion of mediation, widely used in the current mathematics education literature, has been elaborated into a pedagogical model describing the contribution of integrating tools to the human activity, and to teaching and learning mathematics in particular. Following the seminal idea of Vygotsky, and elaborating on it, we postulate that an artifact can be exploited by the teacher as a tool of semiotic mediation to develop genuine mathematical signs, that are detached from the use of the artifact, but that nevertheless maintain with it a deep semiotic link. The teaching organization proposed in this paper is modeled by what we have called the didactical cycle. Starting from assuming the centrality of semiotic activities, collective mathematical discussion plays a crucial role: during a mathematical discussion the intentional action of the teacher is focused on guiding the process of semiotic mediation leading to the expected evolution of signs. The focus of the paper is on the role of the teacher in the teaching–learning process centered on the use of artifacts and in particular a dynamic geometry environment. Some examples will be discussed, drawn from a long-term teaching experiment, carried out over the past years as part of a National project. The analysis is accomplished through a Vygotskian perspective, and it mainly focuses on the process of semiotic mediation centered on the use of artifacts and on the role of the teacher in this process.     

Aristotle’s Non-Logical Works and the Square of Oppositions in Semiotics

This paper aims to highlight some peculiarities of the semiotic square, whose creation is due in particular to Greimas’ works. The starting point is the semiotic notion of complex term, which I regard as one of the main differences between Greimas’ square and Blanché’s hexagon. The remarks on the complex terms make room for a historical survey in Aristotle’s texts, where one can find the philosophical roots of the idea of middle term between two contraries and its relation to notions such as difference, position and motion. In the Stagirite’s non-logical works, the theory of the intermediate, or middle term, represents an important link between opposition issues and ethics: this becomes a privileged perspective from which to reconsider the semiotic use of the square, i.e., its inclusion in the semio-narrative structures articulating the sense of texts.      

Synergy between manipulative and digital artefacts: a teaching experiment on axial symmetry at primary school

This paper presents a study aimed at investigating the didactic potentiality of the combined use of two different kinds of artefacts for the purpose of constructing and conceptualizing mathematical meanings related to the notion of axial symmetry. In our view, the process of meanings construction can be fostered by the use of adequate artefacts, but it requires a teaching/learning model, which explicitly takes care of the evolution of meanings, from those personal, emerging through the activities, to the mathematical ones, aims of the teaching intervention. The main hypothesis of this study is that a potential synergy may occur between the use of different artefacts, synergy that can foster the integration of different and complementary meanings providing a rich support to the development of the expected mathematical meaning. The Theory of Semiotic Mediation offers the theoretical framework suitable to design the teaching sequence and to analyze the collected data. Specifically, the construct of semiotic potential provides the tool for describing the potentialities of the two artefacts, while that of didactic cycle offers a model for the organization of the different activities. The paper reports on a teaching sequence and its implementation in a teaching experiment, involving pupils at fourth grade level. We describe them, within the chosen theoretical framework, and provide the analysis of key episodes of the teaching sequence. We show evidence supporting our main hypothesis about the combined use of an artefact that can be manipulated (paper and pin), and a digital artefact (Dynamic Geometry Environment) in the development of the notion of axial symmetry and its properties: the combined, intentional and controlled use of the two artefacts may develop a synergy, so that each activity enhances the potential of the other.     

Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.     

Introducing students to geometric theorems: how the teacher can exploit the semiotic potential of a DGS

Since their appearance new technologies have raised many expectations about their potential for innovating teaching and learning practices; in particular any didactical software, such as a Dynamic Geometry System (DGS) or a Computer Algebra System (CAS), has been considered an innovative element suited to enhance mathematical learning and support teachers’ classroom practice. This paper shows how the teacher can exploit the potential of a DGS to overcome crucial difficulties in moving from an intuitive to a deductive approach to geometry. A specific intervention will be presented and discussed through examples drawn from a long-term teaching experiment carried out in the 9th and 10th grades of a scientific high school. Focusing on an episode through the lens of a semiotic analysis we will see how the teacher’s intervention develops, exploiting the semiotic potential offered by the DGS Cabri-Géomètre. The semiotic lens highlights specific patterns in the teacher’s action that make students’ personal meanings evolve towards the mathematical meanings that are the objective of the intervention.     

RESEARCH IN MATHEMATICS EDUCATION: SOME ISSUES AND SOME EMERGING INFLUENCES

In this article, I introduce a typology of forms of algebraic thinking. In the first part, I argue that the form and generality of algebraic thinking are characterised by the mathematical problem at hand and the embodied and other semiotic resources that are mobilised to tackle the problem in analytic ways. My claim is based not only on semiotic considerations but also on new theories of cognition that stress the fundamental role of the context, the body and the senses in the way in which we come to know. In the second part, I present some concrete examples from a longitudinal classroom research study through which the typology of forms of algebraic thinking is illustrated.     

Promoting fundamental change in mathematics teaching: a theoretical, methodological, and empirical approach to the problem

Mathematics education reform, as conceptualized in the United States and a number of other countries, represents a fundamental change in the teaching of mathematics and the results it would produce for students. Whereas there are data that suggest some progress is being made in the direction of reform, teacher education and professional development during the last two decades have been largely unsuccessful in preparing teachers to enact the reform vision. In this article, I present a theoretical construct, major assimilatory structures, that can contribute to explaining the difficulty of promoting change in mathematics teaching. I describe a methodology—accounts of practice—for identifying major assimilatory structures of teachers and present an example of a major assimilatory structure, perception-based perspective, that emerged from our empirical work.     

Book Reviews

The making of pictures and the use of mathematics are often considered as activities carried out by two different classes of people.

It may be true that the artist can get on without mathematics, but the converse is far less true.

The operation which an artist terms ‘drawing’, might be described by a mathematician as ‘the mapping of a three‐dimensional network into a two‐dimensional one’.

This article attempts to show how the mathematically minded student can use his mathematics to manipulate pictures. In doing so it introduces him to the tasks which a computer must perform in picture manipulation.

The article is in two parts:

Part A, discusses the use of three‐dimensional sketching and the role it plays in the preparation of ‘orthographic’ working drawings.

It describes how a designer transfers his thoughts about spacial objects to paper, thus assisting himself to refine them and enabling others to perceive them.

A case is made for encouraging perspective sketching in the teaching of engineering drawing.

Part B describes a technique for plotting perspective sketches by numerical methods, which may be useful in motivating numerically inclined students towards involvement with perspective sketching.     

Designing a duo of material and digital artifacts: the pascaline and Cabri Elem e-books in primary school mathematics

This paper focuses on a duo of artifacts, constituted by a physical artifact and its digital counterpart. It deals with the theoretically and empirically underpinned design process of the digital artifact, the e-pascaline developed with Cabri Elem technology, in reference to a physical artifact, the pascaline. The theoretical frameworks of the instrumental approach and the theory of semiotic mediation together with the analysis of teaching experiments with the pascaline support the design in terms of continuity and discontinuity between the two artifacts. The components of the digital artifact were chosen from among the components of the physical artifact that are the object of instrumental genesis by the students and that are analyzed as having a semiotic potential that contributes to didactical aims. Components instrumented by students which had inadequate semiotic potential were eliminated. With the resulting duo, each artifact adds value to the use of the other artifact for mathematical learning.     

Mathematics teaching expertise development approaches and practices: similarities and differences between Western and Eastern countries

This paper comments on the other papers in this issue related to how “mathematics teaching expertise” is conceptualized and the approaches employed to facilitate its development in Western and Eastern countries. Similarities and differences are found to exist in the conceptualization of mathematics teaching expertise and the development approaches employed. The papers in this issue share the similarity of exploring mathematics teaching expertise from the perspective of knowledge. Under the influence of this perspective, the approaches mentioned in the papers mainly focus on the development of teachers’ knowledge. A feature in common among teacher development approaches employed in Western countries is to let teachers attend some courses or training programs designed or organized by mathematics teacher educators at universities. In contrast, teacher development approaches employed in Eastern countries, particularly those employed in Mainland China, are relatively more practical in nature and directly related to teachers’ needs, like learning from observing exemplary teaching. This shows that the conception of mathematics teaching expertise and development approaches are culturally and contextually dependent. It is argued that a broader perspective of mathematics teaching expertise should be taken to explore mathematics teaching expertise and its development, and teacher expertise development should be conceptualized as a complex system rather than as some separated knowledge, skills and techniques.     

The ostensive dimension through the lenses of two didactic approaches

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 symbols, etc.) not only as signs but also as essential tools of mathematical practices. On the one hand, APC-space starts from a general semiotic analysis in terms of “semiotic bundles” that is to be integrated into a more specific epistemological analysis of mathematical activity. On the other hand, ATD proposes a general model of mathematical knowledge and practice in terms of “praxeologies” that has to include a more specific analysis of the role of ostensive objects in the development of mathematical activities in the classroom. The articulation of both theoretical perspectives is proposed as a contribution to the development of suitable frames for Networking Theories in mathematics education.     

Using the K5Connected Cognition Diagram to analyze teachers’ communication and understanding of regions in three-dimensional space

This paper reports on a study that introduces and applies the K₅Connected Cognition Diagram as a lens to explore video data showing teachers’ interactions related to the partitioning of regions by axes in a three-dimensional geometric space. The study considers “semiotic bundles” ( Arzarello, 2006), introduces “semiotic connections,” and discusses the fundamental role each plays in developing individual understanding and communication with peers. While all teachers solved the problem posed, many failed to make or verbalize connections between the types of semiotic resources introduced during their discussions.     

Seventh-grade students’ representations for pictorial growth and change problems

The purpose of this teaching experiment was to investigate eight seventh-grade pre-algebra students’ development of algebraic thinking in problems related in growth and change pattern structure. The teaching experiment was designed to help students (1) identify and generalise patterns in relationships between quantities in the pictorial growth and change problems, (2) represent these generalisations in verbal and symbolic representations, and (3) build effective connections between their external and internal representations for pattern finding and generalising. Findings from the study demonstrated that the students recognized patterns in related problems that enabled them to describe generalised quantitative relationships in the problems. Students modeled their thinking using different external representations—drawing diagrams, creating tables, writing verbal generalisations, and constructing generalised symbolic expressions. Seven of the eight students primarily created and interpreted diagrams as a way to generalise verbally and then symbolically.     

Mathematics education research embracing arts and sciences

As a young field in its own right (unlike the ancient discipline of mathematics), mathematics education research has been eclectic in drawing upon the established knowledge bases and methodologies of other fields. Psychology served as an early model for a paradigm that valorized psychometric research, largely based in the theoretical frameworks of cognitive science. More recently, with the recognition of the need for sociocultural theories, because mathematics is generally learned in social groups, sociology and anthropology have contributed to methodologies that gradually moved away from psychometrics towards qualitative methods that sought a deeper understanding of issues involved. The emergent perspective struck a balance between research on individual learning (including learners’ beliefs and affect) and the dynamics of classroom mathematical practices. Now, as the field matures, the value of both quantitative and qualitative methods is acknowledged, and these are frequently combined in research that uses mixed methods, sometimes taking the form of design experiments or multi-tiered teaching experiments. Creativity and rigor are required in all mathematics education research, thus it is argued in this paper, using examples, that characteristics of both the arts and the sciences are implicated in this work.     

Rethinking the role of algorithms in school mathematics: a conceptual model with focus on cognitive development

Starting from the context of mathematics learning in the East and West, this paper discusses the position and role of algorithms within school mathematics and argues that learning of algorithms has suffered from an alleged dichotomy between procedures and understanding, in that algorithms have been associated with low-level cognition. The paper first introduces a broad perspective about algorithms in school mathematics, and then, partially drawing on Bloom’s taxonomy and Säljö’s categorization of learning, proposes a model for the learning of algorithms with focus on students’ cognitive development. The model consists of three cognitive levels: (1) Knowledge and Skills, (2) Understanding and Comprehension, and (3) Evaluation and Construction. The model suggests that the learning of algorithms does not simply imply a low level of cognition, and provides a new perspective and framework to analyse the learning of algorithms. Following the model, we present examples to demonstrate the three levels and discuss related teaching strategies. We propose that the model can be used as an analysis tool to reconceptualize the role of algorithms in school mathematics and pose some questions for further research and scholarly discourse in this direction.     

Designing an intelligent teaching simulator for learning to teach by practicing

Learning to teach is difficult for prospective teachers because of the complex nature of the work of teaching. Practicing (Lampert in J Teach Educ 61(1–2):21–34, 2010), interacting with the practice of teaching from a first-person perspective, may give them a unique experience in learning to teach. Computer-based simulators in which the apprentice teacher can interact with virtual students may be used to create that kind of experience. In this paper, we show how to apply techniques in artificial intelligence to design an intelligent learning environment. We show how to model the apprentice’s decision making and resources that can help him or her improve the practice of teaching.     

Elementary mathematics specialists in “departmentalized” teaching assignments: Affordances and constraints

In this article, we describe the experiences of three Elementary Mathematics Specialists (EMS) who were part of a larger project investigating the impact of EMS certification and assignment (self-contained or “departmentalized”) on teaching practices and student achievement outcomes. All three of the teachers were “departmentalized,” in the sense that each was responsible for teaching mathematics to at least two groups of students, and accordingly, did not teach all subjects as would a typical self-contained elementary teacher. Each teacher had recently earned an Elementary Mathematics Specialist certificate through completion of a 24-credit, graduate-level program designed to build pedagogical content knowledge and leadership capacity in mathematics. Through a series of observations and interviews over the course of one school year, we examined how the teachers described and navigated specific affordances and constraints they encountered in their particular contexts. Common affordances included opportunities to revise and learn from instruction, and constraints included reduced flexibility introduced by the need to schedule multiple classes of mathematics. Despite these common features, we found important differences between the three models of departmentalization, which we describe as team approach, class swap, and grade-level mathematics teacher. For example, some of the models provided more opportunities for collaboration while others made it difficult for teachers to address potential inequities in learning opportunities across sections. Despite the constraints of their respective models, we found evidence of the EMS-certified teachers drawing on professional expertise in mathematics to meet student needs.     

Theories in practice: mathematics teaching and mathematics teacher education

Whilst research on the teaching of mathematics and the preparation of teachers of mathematics has been of major concern in our field for some decades, one can see a proliferation of such studies and of theories in relation to that work in recent years. This article is a reaction to the other papers in this special issue but I attempt, at the same time, to offer a different perspective. I examine first the theories of learning that are either explicitly or implicitly presented, noting the need for such theories in relation to teacher learning, separating them into: socio-cultural theories; Piagetian theory; and learning from practice. I go on to discuss the role of social and individual perspectives in authors’ approach. In the final section I consider the nature of the knowledge labelled as mathematical knowledge for teaching (MKT). I suggest that there is an implied telos about ‘good teaching’ in much of our research and that perhaps the challenge is to study what happens in practice and offer multiple stories of that practice in the spirit of “wild profusion” (Lather in Getting lost: Feminist efforts towards a double(d) science. SUNY Press, New York, 2007).     

Where There's a Model,There's a Metaphor: Metaphorical Thinking in Students' Understanding of a Mathematical Model

The central question addressed in this article concerns the ways in which applied problem situations provide distinctive conditions to support the production of meaning and the understanding of mathematical topics. In theoretical terms, a first approach is rooted in C. S. Peirce's perspective on semiotic mediation, and it represents a standpoint from which the notion of interpretation is taken as essential to learning. A second route explores metaphorical thinking and undertakes the position according to which human understanding is metaphorical in its own nature. The connection between the two perspectives becomes a fundamental issue, and it entails the conception of some hybrid constructs. Finally, the analysis of empirical data suggests that the activity on applied situations, as it fosters metaphorical thinking, offer students' reasoning a double anchoring (or a duplication of references) for mathematical concepts.