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.     

Coordinating visualizations of polysemous action: values added for grounding proportion

We contribute to research on visualization as an epistemic learning tool by inquiring into the didactical potential of having students visualize one phenomenon in accord with two different partial meanings of the same concept. 22 Grade 4–6 students participated in a design study that investigated the emergence of proportional-equivalence notions from mediated perceptuomotor schemas. Working as individuals or pairs in tutorial clinical interviews, students solved non-symbolic interaction problems that utilized remote-sensing technology. Next, they used symbolic artifacts interpolated into the problem space as semiotic means to objectify in mathematical register a variety of both additive and multiplicative solution strategies. Finally, they reflected on tensions between these competing visualizations of the space. Micro-ethnographic analyses of episodes from three paradigmatic case studies suggest that students reconciled semiotic conflicts by generating heuristic logico-mathematical inferences that integrated competing meanings into cohesive conceptual networks. These inferences hinged on revisualizing additive elements multiplicatively. Implications are drawn for rethinking didactical design for proportions.     

Learning about multiplication by comparing algorithms: “One times one,but actually they are ten times ten”

Multiplication algorithms in primary school are still frequently introduced with little attention to meaning. We present a case study focusing on a third grade class that engaged in comparing two algorithms and discussing “why they both work”. The objectives of the didactical intervention were to foster students' development of mathematical meanings concerning multiplication algorithms, and their development of an attitude to judge and compare the value and efficiency of different algorithms. Underlying hypotheses were that it is possible to promote the simultaneous unfolding of the semiotic potential of two algorithms, considered as cultural artifacts, with respect to the objectives of the didactical intervention, and to establish a fruitful synergy between the two algorithms. As results, this study sheds light onto the new theoretical construct of “bridging sign”, illuminating students’ meaning-making processes involving more than one artifact; and it provides important insight into the actual unfolding of the hypothesized potential of the algorithms.     

From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation

Research on the use of artifacts such as textbooks and digital technologies has shown that their use is not a straight forward process but an activity characterized by mutual participation between artifact and user. Taking a socio-cultural perspective, we analyze the role of artifacts in the teaching and learning of mathematics and argue that artifacts influence the didactical situation in a fundamental way. Therefore, we believe that understanding the role of artifacts within the didactical situation is crucial in order to become aware of and work on the relationships between the teacher, their students and the mathematics and, therefore, are worthwhile to be considered as an additional fundamental aspect in the didactical situation. Thus, by expanding the didactical triangle, first to a didactical tetrahedron, and finally to a ??socio-didactical tetrahedron??, a more comprehensive model is provided in order to understand the teaching and learning of mathematics.     

Embodied Semiotic Activities and Their Role in the Construction of Mathematical Meaning of Motion Graphs

This paper examines the relation between bodily actions, artifact-mediated activities, and semiotic processes that students experience while producing and interpreting graphs of two-dimensional motion in the plane. We designed a technology-based setting that enabled students to engage in embodied semiotic activities and experience two modes of interaction: 2D freehand motion and 2D synthesized motion, designed by the composition of single variable function graphs. Our theoretical framework combines two perspectives: the embodied approach to the nature of mathematical thinking and the Vygotskian notion of semiotic mediation. The article describes in detail the actions, gestures, graph drawings, and verbal discourse of one pair of high school students and analyzes the social semiotic processes they experienced. Our analysis shows how the computerized artifacts and the students’ gestures served as means of semiotic mediation. Specifically, they supported the interpretation and the production of motion graphs; they mediated the transition between an individual’s meaning of mathematical signs and culturally accepted mathematical meaning; and they enable linking bodily actions with formal signs.     

Definitions and examples in elementary calculus: the case of monotonicity of functions

This paper is devoted to the investigation of students’ understanding and handling of examples in the framework of an example-based introductory mathematics undergraduate course. The plan of the course included a wide use of graphs in standard lectures, tutoring sessions as well as in examinations. This study deals with the notion of increasing function, which has been introduced by means of both the standard definition and a range of examples and non-examples, most often conveyed through graphs. We have analysed students’ interpretations of the notion of increasing function as they applied them in a set of written examination tests. The data gathered have been completed by a number of interviews of students whose answers were difficult to interpret. The outcomes underline the importance of linguistic and semiotic competence and suggest that the design of innovative teaching paths should take care of the linguistic and semiotic skills needed to handle the representations involved.     

Debugging an Artifact, Instrumenting a Bug: Dialectics of Instrumentation and Design in Technology-Rich Learning Environments

This article explores ways of conceptualizing the design of innovative learning tools as emergent from dialectics between designers and learner-users of those tools. More specifically, I focus on the reciprocities between a designer’s objectives for student learning and a user’s situated activity in a learning environment, as these interact and co-develop in cycles of design-based research. Recent investigations of technology-supported mathematics learning conducted from an ‘instrumental’ perspective provide a powerful framework for analyzing the process through which classroom artifacts become conceptual tools, simultaneously characterizing the ways students come to both implement and understand a device in the context of a task. Similarly, design-based approaches to investigating instructional activity offer epistemological grounds for treating the process of designing artifacts to support learning as unfolding in concert with rather than concluding prior to situated student use. Drawing on each of these perspectives, I describe the design and initial implementation of a set of software artifacts intended to support students’ collaborative problem solving through locally networked handheld computers. Through detailed analyses of three classroom episodes, I report on the ways one student group’s innovative and unexpected use of these tools served as an opportunity to both examine student learning in the context of that novelty and to refine the software design. This account provides an empirical example through which to consider the potential for instrumental genesis to inform design, and for design research epistemology to broaden the scope of instrumental theory.

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.     

Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus

University and older school students following scientific courses now use complex calculators with graphical, numerical and symbolic capabilities. In this context, the design of lessons for 11th grade pre-calculus students was a stimulating challenge.In the design of lessons, emphasising the role of mediation of calculators and the development of schemes of use in an 'instrumental genesis' was productive. Techniques, often discarded in teaching with technology, were viewed as a means to connect task to theories. Teaching techniques of use of a complex calculator in relation with 'traditional' techniques was considered to help students to develop instrumental and paper/pencil schemes, rich in mathematical meanings and to give sense to symbolic calculations as well as graphical and numerical approaches.The paper looks at tasks and techniques to help students to develop an appropriate instrumental genesis for algebra and functions, and to prepare for calculus. It then focuses on the potential of the calculator for connecting enactive representations and theoretical calculus. Finally, it looks at strategies to help students to experiment with symbolic concepts in calculus.This revised version was published online in September 2005 with corrections to the Cover Date.     

Exploiting the potential of primary historical sources in primary school: a focus on teacher’s actions

Integrating history of mathematics in classes could be a hard task with young pupils. Indeed, original historical sources have a language that is far from the modern one. Such texts represent cultural artefacts that can give access to mathematical knowledge. The teacher can exploit such potential acting as a mediator between the mathematical signs of the source and those signs that are accessible to students. Through a case study, we investigate the role of the teacher in the process of semiotic mediation during a collective discussion. The analysed intervention is made of two phases: firstly, students work collaboratively and secondly, the teacher mediates a discussion aimed at institutionalizing the knowledge. During the discussion, working on a text from Tartaglia’s translation of Euclid’s Elements, a group of fifth graders constructs a definition of prime numbers. Referring to the Theory of Semiotic Mediation, we analyse the role of the teacher in building up semiotic chains linking students’ productions to an institutionalized knowledge emerging from the collective discussion. We highlight how teacher’s focalization on students’ words allows the progress of the discussion: the potential of the historical text is exploited fostering a definition that is close to culturally shared mathematics.     

“No! He starts walking backwards!”: interpreting motion graphs and the question of space, place and distance

This article deals with the interpretation of motion Cartesian graphs by Grade 8 students. Drawing on a sociocultural theoretical framework, it pays attention to the discursive and semiotic process through which the students attempt to make sense of graphs. The students’ interpretative processes are investigated through the theoretical construct of knowledge objectification and the configuration of mathematical signs, gestures, and words they resort to in order to achieve higher levels of conceptualization. Fine-grained video and discourse analyses offer an overview of the manner in which the students’ interpretations evolve into more condensed versions through the effect of what is called in the article “semiotic contractions” and “iconic orchestrations.”     

Influences on Science Fair Participant Research Design Selection and Success

The purpose of this investigation is to gain a better understanding of the characteristics of pre-college researchers by examining the influences on research design selection and success in competition. The 22 participants were finalists at the 44th International Science and Engineering Fair, where they completed a questionnaire describing the influences on their research activities and methodologies. Additional data consisted of artifacts collected from the participants and the Science Service. Data were analyzed using an interpretive methodology. Assertions generated from the analysis indicated that students are largely accurate in self-reports of design, interacted with mentors on several levels, outside influences reflect shifts in motivational orientation, and success attributions reflect internalization of influences. Recommendations for future research with science fair participants are included.     

Digital Tools for Algebra Education: Criteria and Evaluation

In the Netherlands, as in many other countries, the algebraic expertise of students graduating from secondary education is an issue. The use of digital tools for algebra education is expected to change epistemologies, activity structures and student achievement. Therefore, a study was set up to investigate in what way the use of ICT in upper secondary education might enhance the algebraic expertise of students. One of the first decisions to be made concerned the choice of appropriate digital tools. This paper describes the process of designing and using an instrument for evaluating digital tools. The conceptual framework guiding this process includes notions on symbol sense, instrumental genesis and formative assessment. The evaluation instrument is designed through a Delphi method and provides a blueprint of tool features that are relevant for the purpose of this study. The results show that such an evaluation instrument is valuable both for choosing appropriate digital tools and for making concrete the aims and expectations that researchers have on the issue of integrating technology in algebra education. The final instrument is presented and illustrated through examples implemented in different digital algebra tools.     

Ideological Complex Systems: Mathematical Theory

The goal of this article is to build an abstract mathematical theory rather than a computational one of the process of transmission of ideology. The basis of much of the argument is Patten's Environment Theory that characterizes a system with its double environment (input or stimulus and output or response) and the existing interactions among them. Ideological processes are semiotic processes, and if in Patten's theory, the two environments are physical, in this theory ideological processes are physical and semiotic, as are stimulus and response. © 2014 Wiley Periodicals, Inc. Complexity 21: 47–65, 2015     

Try to See It My Way: The Discursive Function of Idiosyncratic Mathematical Metaphor

What are the nature, forms, and roles of metaphors in mathematics instruction? We present and closely analyze three examples of idiosyncratic metaphors produced during one-to-one tutorial clinical interviews with 11-year-old participants as they attempted to use unfamiliar artifacts and procedures to reason about realistic probability problems. Our interpretations of these episodes suggest that metaphor is both spurred by and transformative of joint engagement in situated activities: metaphor serves individuals as semiotic means of objectifying and communicating their own evolving understanding of disciplinary representations and procedures, and its multimodal instantiation immediately modifies interlocutors' attention to and interaction with the artifacts. Instructors steer this process toward normative mathematical views by initiating, modifying, or elaborating metaphorical constructions. We speculate on situation parameters affecting students' utilization of idiosyncratic resources as well as how socio-mathematical license for metaphor may contribute to effective instructional discourse.     

Young children’s multimodal mathematical explanations

This paper investigates how three children provided mathematical explanations whilst playing with a set of glass jars in a Swedish preschool. Using the idea of semiotic bundles combined with the work on multimodal interactions, the different semiotic resources used individually and in combinations by the children are described. Given that the children were developing their verbal fluency, it was not surprising to find that they also included physical arrangements of the jars and actions to support their explanations. Hence, to produce their explanations of different attributes such as thin and sameness, the children drew on each other’s gestures and actions with the jars. This research has implications for how the relationship between verbal language and gestures can be viewed in regard to young children’s explanations.     

The activity structure of technology-based mathematics lessons: a case study of three teachers in English secondary schools

ABSTRACT

This article examines patterns of classroom organisation and interaction associated with the use of a particular type of digital technology – the dynamic software GeoGebra – in the lessons of an opportunity sample of three English secondary-school mathematics teachers. The concept of activity structure is used to organise this study, further informed by the concept of instrumental orchestration. While the case study analysis identifies structures already reported in those earlier papers, it also draws attention to the prevalence of a Predict-and-test format in tasks carried out by students at the computer. This study also shows how synthesising the activity structure and instrumental orchestration frameworks may be productive.     

Living in Enclaves

What are the consequences of living in small isolated communities vs living in larger environments as members of bigger communities? If the resources are scarce and highly spatially and temporally variable, living in isolated communities can lead to extinction because of temporary food scarcity and less efficient genetic selection. Artifacts that increase the energy extracted from natural resources can avoid extinction of local populations, but still choosing the artifacts to be reproduced from the restricted artifact pool of the local community rather than from a larger pool leads to less good artifacts and smaller global population size. © 2002 Wiley Periodicals, Inc.     

Seeing chance: perceptual reasoning as an epistemic resource for grounding compound event spaces

The mathematics subject matter of probability is notoriously challenging, and in particular the content of random compound events. When students analyze experiments, they often omit to discern variations as distinct events, e.g., HT and TH in the case of flipping a pair of coins, and thus infer erroneous predictions. Educators have addressed this conceptual difficulty by engaging students in actual experiments whose outcomes contradict the erroneous predictions. Yet whereas empirical activities per se are crucial for any probability design, because they introduce the pivotal contents of randomness, variance, sample size, and relations among them, empirical activities may not be the unique or best means for students to accept the logic of combinatorial analysis. Instead, learners may avail of their own pre-analytic perceptual judgments of the random generator itself so as to arrive at predictions that agree rather than conflict with mathematical analysis. I support this view first by detailing its philosophical, theoretical, and didactical foundations and then by presenting empirical findings from a design-based research project. Twenty-eight students aged 9?C11 participated in tutorial, task-based clinical interviews that utilized an innovative random generator. Their predictions were mathematically correct even though initially they did not discern variations. Students were then led to recognize the formal event space as a semiotic means of objectifying these presymbolic notions. I elaborate on the thesis via micro-ethnographic analysis of key episodes from a paradigmatic case study.