Search

Teaching and learning Calculus concepts and procedures, particularly the definite integral concept, is a challenge for teachers and students in their academic careers. In this research, we supplement the analysis made by different authors, applying the theoretical and methodological tools of the Onto-Semiotic Approach to mathematical knowledge and instruction. The goal is to understand the diverse meanings of the concept of the definite integral and potentials semiotic conflicts based on the given data. We focus attention on a first intuitive meaning, which involves mainly arithmetic knowledge, and the definite integral formal meaning as Riemann's sums limit predominantly in the curricular guidelines. The recognition of the onto-semiotic complexity of mathematics objects is considered as a key factor in explaining the learning difficulties of concepts, procedures and its application for problem-solving, as well as to make grounded decisions on teaching. The methodology analysis of a mathematical text, which we exemplify in this work applying the tools of Onto-Semiotic Approach, provides a microscopic level of analysis that allows us to identify some semiotic-cognitive facts of didactic interest. This also allows for the identification of some epistemic strata, that is, institutional knowledge that should have been previously studied, which usually goes unnoticed in the teaching process.

In this article we present the results of a qualitative investigation into the teaching and learning of Taylor series and local approximations. In order to perform a comparative analysis, two investigations are conducted: the first is historical and epistemological, concerned with the pedagogical evolution of semantics, syntax and semiotics; the second is a contemporary institutional investigation, devoted to the results of a review of curricula, textbooks and course handouts in Tunisia and the United States. Our results show that contemporary curricula emphasize the algebraic and analytic approaches to the theory but omit the graphical and numerical registers, which both played a large role in the historical development. We contend that by presenting a more historically accurate didactic approach, student learning can be enriched to give a deeper and more conceptually accurate understanding of the topic.

Reporting on a Grade 4 teaching and learning sequence, we highlight foundational constructs of measurement and data modelling which are fundamental to competence development in both science and mathematics. The sequence involved students generating and representing measures of their teacher's arm-span, with a focus on the invention and refinement of data representations characteristic of STEM epistemic practices. We undertook a micro-ethnographic practical epistemology analysis of video data of classroom and group discussions and student work, to explore the teaching and learning processes driving students' understandings of data variation and measures of central tendency. Through this analysis we appreciated aesthetic experiences as fundamental for students' growing epistemic and conceptual understandings and the teacher's pedagogical practices. We draw on the pragmatist aesthetic perspectives of Dewey and Peirce to show how students' sign-making shifted from an everyday interest in the arm span measuring process to an interest in the potentialities of the data set as an exploratory field. We consider this a shift from an everyday 'art' aesthetic to a 'mathematics' disciplinary aesthetic. We argue that such an aesthetic perspective, in which feeling and meaning are intertwined, provides a powerful way to understand the role of affect in learning.

There has been longstanding interest in students' aesthetic feelings in engaging with school science, but new multimodal accounts of meaning-making, and changing curricular goals where science is integrated with other subjects, pose new methodological challenges for researching these feelings. In this paper, we aim to synthesise relevant literature as the bases for proposing methodological practices to address these new research challenges. We draw on past and current pragmatist and semiotic perspectives to justify our methodological approach. We claim that research on aesthetic feelings needs to identify: (a) the complex nature of aesthetic meaning/feeling processes and multiple influences on these processes and (b) short-term and longer-term consequences on participants' taste for, and learning in, science. Building on a practical epistemology, we propose that these questions can be addressed through multimodal analyses of aesthetic intentions and responses when embedded in teacher and student inquiry purposes, discursive interactions, artefact-making, and evaluative feelings/judgements. To illustrate the scope and challenges of these methodological strategies, we draw on examples of analyses in companion papers in this special issue, both when science is taught alone and when integrated with other subjects.

The notions of meaning and sense, which are closely related to understanding, play an essential role in educational processes in general and, therefore, in the teaching and learning of mathematics. However, the terms 'meaning' and 'sense' are unevenly used in the theory of language, semiotics, philosophy and psychology. After describing the main characteristics of the referential and pragmatic theories of meaning, in this article, we detail various approaches to the notion of meaning, and present the perspective suggested by the onto-semiotic approach (OSA) of mathematical knowledge and instruction. This proposal partially incorporates the semiotic, ontological and epistemological assumptions of three general semiotic theories (by Hjelmslev, Peirce and Wittgenstein), with a holistic approach to the notion of object, sign and meaning, and takes into account some referential and operational theories, as well as other cognitive and sociocultural theories, about meaning. We also analyse how the meaning of mathematical objects is addressed in three models that have an impact on mathematics education: Frege's logical-realist position, Vergnaud's conceptual triplet and Steinbring's epistemological triangle. We finally suggest some implications for instructional design and teacher education of the OSA holistic approach to the meaning.

Over a three-year collaborative partnership involving university researchers, government curriculum specialists, and school division teams and educators, this post-qualitative study has engaged a diffractive methodology to research pedagogical change in relationship to a renewed provincial curriculum framework in English Language Arts (ELA). In the data from focus groups, online surveys, and interviews, assessment has emerged as an ethical space of pedagogical, epistemological, and ontological tension and concern. Drawing on research in applied linguistics and digital literacies, and theoretical work in new materialism and Indigenous philosophies in education, a conceptual framework is proposed for emplacing assessment that assumes a relational ontology. Understanding language and literacy practices as produced in and through dynamic social-material-semiotic environments, this article explores how emplacing assessment opens up a dialogue for creating more ethical and responsive relationships in assessment, teaching, and learning.

Disciplines are by their very "origin and nature" unstable--always already differed and deferred from themselves--due to their genetic relation to what Jacques Derrida calls "différance". In a sense, "différance" is what makes disciplines possible in the first place, and it can be thought of as the economy or mode of operation of disciplines as discursive knowledge formations. In light of this view, interdisciplinary endeavors are also always already subject to the conditions of "différance". Interdisciplinary integration--necessitated by the differing and deferring effects of "différance"--should then be thought of as a matter of creatively producing relations-in-"différance".

We propose a new relational direction in higher education that acknowledges external and internal images as integrated in thinking and learning. We expand educational theory and practice that commonly rely on discrete conceptual developments that exclude images. Our argument epistemologically relies on certain semiotic views that consider the role of iconic signs and iconicity (meaning making by the virtue of similarity) as significant in relation to knowledge and learning. The analogical and imaginative work required to discover similarity between external pictures and any educational concept is a form of iconic mind work that opens a space for transformational thinking and creative solutions. The novelty of our argument lies in the schematic philosophy of mind by C.S. Peirce, and specifically, an adaptation of Peirce's triadic sign diagram for learning, exemplified via an 'inquiry graphic' sign and reflective method. This outlook seeks connections in diversity, arguing for an education that acknowledges and teaches intrinsic human relatedness to other humans, beings and things in the environment, thus challenging any notions of anthropocentrism and superiority. It calls for a reaction in international higher education to acknowledge the iconicity of concepts and thinking and embrace integrated image-concept signs for deep and critical teaching-learning, towards a modest educational contribution to counteracting social and environmental crises.

At an overarching level this paper attempts to draw attention to emerging trends in the humanities where alternative ways of doing science reconfigure epistemological traditions and research methodologies, the role of intellectuals and their engagement with current conditions of the world, including ways in which scholars gazes are constituted. Drawing on what we call a Second Wave of Southern Perspectives (SWaSP), that sees the entanglements of two clusters -- the first of which comprises contemporary ways of reading anticolonial, postcolonial and decolonial thinkers with offerings of Southern perspectives, and a second where contemporary theories about language and communication that considers their cultural and social dimensions, this paper calls for a mobile global-centric gazing. More specifically this paper actualises ontoepistemological trajectories that feed into the scholarship about multilingualism, looking at its different possible beings and becomings that enable a variety of ways of conceptualising multilingual practices. We do this by first presenting a brief review about recent discussions related to the concept of repertoires in the field of multilingualism and pathways that can move these debates in different directions. After this, we present possible ways to go beyond the sociolinguistics of multilingualism, by considering contemporary challenges in the knowledge production enterprise.

A subject of growing interest in mathematics education is the affective domain and its effects on the teaching and learning processes, giving rise to different models of its components and conditioning factors. In this paper, we apply the ontological and semiotic categories from the Onto-Semiotic Approach (OSA) to research in mathematics education, to build an inclusive and systematic model to consider affective situations, practices, objects and processes, as well as the corresponding dualities: personal -- institutional, ostensive -- non-ostensive, extensive -- intensive, unitary -- systemic, expression -- content. The dynamic character of affects (emotions, attitudes, beliefs and values) and their relations with the epistemic, cognitive, interactional and resources is modelled by the didactical configuration and didactical trajectory notions, theoretical tools which include the affective sub-configuration and sub-trajectory as key components. Another result obtained from this work is the revision of the indicators of affective suitability proposed in previous works.

In the 21st century, meaning making is a multimodal act; we communicate what we know and how we know it using much more than printed text on a blank page. As a result, qualitative researchers need new methodologies, methods, and tools for working with the complex artifacts that our research subjects produce. In this article we describe the co-development of an analytic methodology and a tool for working with youth produced films as multimodal artifacts of youth engagement with identity. Specifically, we describe how to employ this multimodal framework in data analysis, with an emphasis on how different modes interact with one another, and how new meanings are made possible through multimodal interactions. (Contains 1 figure and 1 table.)

This case study focuses on a Native American family's experience on a walk in an urban forest preserve. Drawing on interaction analysis traditions, we analyze video data and transcript data to characterize how learning unfolds in place, in this case an urban forest. We build on this analysis, as well as the work of Indigenous scholars, to theoretically develop "walking," "reading," and "storying land" as a methodology for making sense of physical and biological worlds.

The inclusion of visual images in current educational literacy discussions tends to contextualise them within more semiotic, socio-critical and textually focussed theoretical traditions. These particular traditions privilege and emphasise the structures and "language-like" aspects of visual images, and include the broader social and cultural structural frames, such as gender and class, as well as the specific codes and "grammars" of individual images. While there are strong benefits in employing these approaches, the nature of visual images themselves may require a broader, interdisciplinary approach. This paper will include discussion of the field of visual culture in general, the unique nature of images, the role of philosophy in regard to image, the inclusion of the individual's hermeneutic role in meaning-making, and the attendant educational implications when applying such work to contemporary educational literacy practice. (Contains 1 footnote and 4 figures.)

This article proposes a methodology to create a multimodal corpus that can be shared with a group of researchers in order to analyze synchronous online pedagogical interactions. Epistemological aspects involved in studying online interactions from a multimodal and semiotic perspective are addressed. Then, issues and challenges raised by corpus creation and sharing are examined with a particular focus on ethics. Basing my discussion and analysis around a particular research project, the steps involved in the creation of a multimodal and shareable corpus are described and the scientific benefits of a collective exploration of data and their subsequent multimodal representations are discussed and illustrated. It is finally claimed that corpus sharing can contribute vastly to the field of computer-assisted language learning by enhancing its scientific robustness as it favors a multidisciplinary, systematic, and in-depth analysis of multimodal data.

This article explores the underlying processes involved as two experienced Portuguese as a foreign language instructors, who are novices to 3D technologies, became immersed in the epistemology of teaching in a 3D context. The two instructors undertook a challenging initiative to develop and deliver two sections of oral Portuguese in a 3D environment. Through the contingent support of a European community of practitioners, the two instructors explored how the game-based elements and other semiotic resources of SL could be used to enact affordances for the development of goal-oriented tasks, collaborative activities, and interactions in order to guide students in building their oral proficiency in Portuguese. The skill and knowledge development were examined with the trajectories of Compton's framework and epistemological theory where teaching in virtual contexts is contingent upon the construction of technological, pedagogical, and evaluation skills.

Despite boasting its self-characterization as the "land of the free," US American "freedom" is, at times, tainted with historical amnesia, hypocrisy, and inhumanity. This article examines today's socio-political climate by drawing from de Tocqueville's (2003) prediction that American democracy is a tyranny of the majority. Because tyranny relies on gaslighting and dismissing facts, no definitive portraiture of freedom is made; therefore, the tyrannized wonder whether they are truly living within freedom or, instead, in collective submission to its illusion. This article examines this phenomenon in order to investigate how whiteness (re)produces conditions of disillusionment and tyranny. By employing hermeneutics of whiteness as a methodology, the authors investigate how whiteness infiltrates thoughts/epistemology, speech/rhetoric, emotions/emotionality, and nationalistic symbols/semiotics. The authors analyze implications for US education and offer a plea for all to consider.

This paper aims to explain how semiotics and constructivism can collaborate in an educational epistemology by developing a joint approach to prescientific conceptions. Empirical data and findings of constructivist research are interpreted in the light of Peirce's semiotics. Peirce's semiotics is an anti-psychologistic logic (CP 2.252; CP 4.551; W 8:15; Pietarinen in "Signs of logic," Springer, Dordrecht, 2006; Stjernfelt in "Diagrammatology. An investigation on the borderlines of phenomenology, ontology and semiotics," Springer, Dordrecht, 2007) and relational logic. Constructivism was traditionally developed within psychology and sociology and, therefore, some incompatibilities can be expected between these two schools. While acknowledging the differences, we explain that constructivism and semiotics share the assumption of realism that knowledge can only be developed upon knowledge and, therefore, an epistemological collaboration is possible. The semiotic analysis performed confirms the constructivist results and provides a further insight into the teacher-student relation. Like the constructivist approach, Peirce's doctrine of agapism infers that the personal dimension of teaching must not be ignored. Thus, we argue for the importance of genuine sympathy in teaching attitudes. More broadly, the article also contributes to the development of postmodern humanities. At the end of the modern age, the humanities are passing through a critical period of transformation. There is a growing interest in semiotics and semiotic philosophy in many areas of the humanities. Such a case, on which we draw, is the development of a theoretical semiotic approach to education, namely edusemiotics (Stables and Semetsky, "Pedagogy and edusemiotics: theoretical challenge/practical opportunities," Sense Publishers, Rotterdam, 2015).

Theorizing flows of community practices implicates three interrelated themes: (1) theorizing practices rather than practicing theories, (2) theorizing our stories, and (3) research as responsibility to the communities in which we work. One way to claim literacy research as a principled epistemological stance is to confront the real-life effects of literacy theorizing and the practicing of our theories. We must challenge analyses that disadvantage youth from nondominant populations by deconstructing the categories that devalue communities and educational possibilities. To keep our insights, only within the domain of theory and not the practice of theory can allow the reproduction of symbolic violence. Theories like the "culture of poverty" and the language gap promise a facile explanatory power for complex multilayered issues. By engaging with multilingualism, transnational literacies, and community communicative flows, we can theorize communicative practices as part of an array of the multiple and multimodal semiotic systems and resources that flow in and through communities. It is argued that we all translanguage across the ecologies of our everyday practices through the full and ample use of our linguistic repertoires. In that sense, we are all translingual. If we think of moving across language and literacy borders as a space for pedagogical potential, we can underscore that learning is about engagement with diverse voices and conceptualizations. Students can develop a translingual awareness that perceives languages as plural and fluid, and this awareness can help students succeed outside of monolingual norms.

According to our approach to mathematics education, the optimal aim of the teaching of mathematics is to assist students in achieving efficient mathematical work. But, what does efficient exactly mean in that case? And how can teachers reach this objective? The model of Mathematical Working Spaces with its three dimensions--semiotic, instrumental, discursive--allows us to address these questions in an original way based on a multidimensional approach to the use of tools and instruments and on the notion of complete mathematical work. The Mathematical work is considered complete when a genuine relationship exists between epistemological and cognitive aspects, and when the three dimensions of the model are appropriately articulated. Two teaching situations in probability for Grades 9 and 10 (age 14 and 15) are used to illustrate how the model can help identify either misunderstandings that are not acknowledged by the teacher, or complete mathematical work despite some differences between intended and actual work.

The aim of this study is to analyze and document the extent to which high school teachers rely on a set of technology affordances to articulate epistemological and cognitive actions in problem solving approaches. Participants were encouraged to construct dynamic representations of tasks and always to look for different ways to identify and support mathematical relations. To this end, three interrelated geneses (instrumental, semiotic, discursive) proposed in the Mathematical Working Space frame are used to document ways in which the participants constructed dynamic configuration of mathematical tasks as a means initially to explore invariants and eventually formulate conjectures based on empirical arguments. Technology affordances that involve dragging objects, quantifying parameters, graphing loci and using sliders were important for the participants to articulate semiotic and discursive geneses. Thus, the participants' approaches that appeared individual, small group and plenary discussions contributed to identifying initial conjectures that later were analyzed via empirical, geometric or algebraic arguments. In addition, the participants recognized that a working space to understand mathematical ideas and to solve problems should foster mathematical discussions beyond formal settings. The use of digital technologies plays an important role in both representing and exploring mathematical tasks and in following the discussion outside the formal class. As a result, the participants outlined a possible route to implement a route for learners to integrate activities to account for instrumental, semiotic, and discursive genesis.

The theoretical and methodological model of Mathematical Working Space (MWS) is introduced in this paper. For over 10 years, the model has been the object of collaborative research among various researchers, generally coming from French and Spanish speaking countries. Articulating epistemological and cognitive aspects, the MWS model is aimed at providing a tool for the specific study of mathematical work in which students and teachers are effectively engaged during mathematics sessions. The abstract space thus conceived refers to a structure organized in a way that allows the analysis of the mathematical activity of individuals dealing with mathematical problems. Thus, analyzing mathematical work through the lens of MWS enables tracking down how meaning is progressively constructed, as a process of bridging the epistemological and the cognitive perspectives--these being modelled as two planes at different levels in the diagrammatic structure--in accordance with different specific yet intertwined genetic developments. Each is identified as a genesis related to a specific dimension in the model: semiotic, instrumental and discursive geneses. A general overview of the different papers included in this issue is given, and shows how the model can be used to study different tasks, teaching situations and activities set in specific mathematical fields or domains. Some perspectives are finally drawn, while reflecting on the possibility of "networking" different theoretical frames with the MWS framework. Indeed, the latter is not proposed as a holistic theory, but rather should function as a tool interacting strongly with other approaches.

The past decade has seen videogames become an important facet in the economic and cultural tapestry of the 21st century. However, while the Australian Curriculum: English (ACE) advocates the teaching of multimodal texts (ACARA, 2016), videogames have been neglected within the curriculum. Nevertheless, such a significant aspect of popular 21st century entertainment culture warrants attention by educationalists and consideration as a highly motivating curriculum resource. This paper aims to explore the linguistic and visual semiotic depictions of value positions in the videogame "Watch Dogs" (Ubisoft, 2014). Despite the ubiquity and popularity of videogames, and the growing use of Systemic Functional Linguistic (SFL) and related broader semiotic theory in educational research, limited research has been conducted on games from an SFL perspective. This paper will identify content descriptions from the ACE and how suitable videogames similar to Watch Dogs could be used for the teaching of these descriptions. Semiotic systems such as the appraisal system (Martin & White, 2005) and various approaches to image analyses (Kress & van Leeuwen, 2006; Painter, Martin, & Unsworth, 2013) will be utilised in the identification and explication of game characters' value positions. The logogenetic synergy between the meaning making systems of language and image will then be canvassed in relation to their implications for Systemic Functional Semiotic accounts of inter-modal meaning-making, as well as implications for curriculum and pedagogy in the 21st century.

In line with contemporary constructivist pedagogies, students are frequently expected to learn through interaction in physical education (PE). There is a relatively sophisticated body of literature focusing on learning in groups, peer teaching, and cooperative learning. Current research has not, however, focused on how the body is implicated in interactional learning. This is surprising given that much learning in PE is expected to take place in the physical domain. The aim of this paper is to contribute to current theorizing by examining social interactions in PE practice. By drawing on symbolic interactionist theory, we put forward a framework for considering how inter-student interactions occur in a multimodal sense. Key ideas relate to (1) the sequential organization of interactions; (2) the ways in which semiotic resources in different fields are used to elaborate each other; (3) the importance of interpretation as a driver of interaction; (4) the creation of local environments in which participants attend to and work together within a shared world of perception; and (5) the influence of material environments on social interaction. The specific concepts employed are "epistemic ecology," "epistemic position," and "learning trajectory." The paper includes observational data from an investigation of learning in Swedish PE to demonstrate the explanatory power and limitations of the theoretical tenets presented. The paper is concluded with practical implications of understanding group work in a multimodal manner.

Information design has practical and theoretical components. As an academic discipline we may view information design as a combined discipline, a practical theory, or as a theoretical practice. So far information design has incorporated facts, influences, methods, practices, principles, processes, strategies, and tools from a large number of disciplines. Information design also needs to incorporate theories, or parts of theories, from other disciplines. This essay presents seven information design theories with more than hundred facts, hypotheses, and postulates.

This paper proposes an ecological methodology in order to re-think the concept of situatedness in ways that can take into account that we live in relation to, and are of, a more-and-other-than-human world. In doing so, the paper proposes that situatedness should be understood in terms of processes of co-invention that, fractally and recursively, open onto other co-inventions that include the non-human. The paper illustrates this through the concept of patterning. It advances a number of terms--cutting, knotting, contrasting, figuring--as potential practices that can be drawn on to provide analyses of dynamic and multiple relations that cross the boundaries between human and non-human forces.

Compared with research on the role of student engagement with expert representations in learning science, investigation of the use and theoretical justification of student-generated representations to learn science is less common. In this paper, we present a framework that aims to integrate three perspectives to explain how and why representational construction supports learning in science. The first or semiotic perspective focuses on student use of particular features of symbolic and material tools to make meanings in science. The second or epistemic perspective focuses on how this representational construction relates to the broader picture of knowledge-building practices of inquiry in this disciplinary field, and the third or epistemological perspective focuses on how and what students can know through engaging in the challenge of representing causal accounts through these semiotic tools. We argue that each perspective entails productive constraints on students' meaning-making as they construct and interpret their own representations. Our framework seeks to take into account the interplay of diverse cultural and cognitive resources students use in these meaning-making processes. We outline the basis for this framework before illustrating its explanatory value through a sequence of lessons on the topic of evaporation. (Contains 6 figures.)

In this article, I attempt to describe how certain theoretical constructions of semiotics could be applied in educational theoretical work. First I introduce meaning as a basic concept of semiotics, thus also touching on concepts such as action, competence and causality. I am then able to define learning as a change of competences, and also refer to the pedagogical concept of learning i.e. Bildung, which can be roughly defined as valuable human learning. I then take up the problem of education as pedagogical direction and communication. Finally, I conclude with some considerations on the famous Greimassian semiotic square.

We present epistemological ruptures that have occurred in mathematical history and in the transformation of using technology in mathematics education in the twenty-first century. We describe how such changes establish a new form of digital semiotics that challenges learning paradigms and mathematical inquiry for learners today. We focus on drawing analogies between the emergence of non-Euclidean geometry with recent advances in technological environments that are dynamic and interactive both visually and haptically. This analysis yields a new digital semiotic theory.

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. Based on empirical qualitative analysis we identify three different kinds of semiotic actions featuring the organization of the argumentations, and related to different epistemological status of the arguments. In such semiotic actions, the students' argumentation and proof activities are managed and guided according to two intertwined modalities of control, which we call semiotic and theoretic control. The former refers to decisions concerning the selection and implementation of semiotic resources; the latter refers to decisions concerning the selection and implementation of a more or less explicit theory or parts of it. The structure of the model allows us to pinpoint a dialectical dynamics between the two.

There is an over-arching consensus that the use of the history of mathematics should decidedly improve the quality of mathematics teaching. Mathematicians and mathematics educators show here a rare unanimity. One deplores, however, and in a likewise general manner, the scarcity of positive examples of such a use. This paper analyses whether there are shortcomings in the--implicit or explicit--conceptual bases, which might cause the expectations not to be fulfilled. A largely common denominator of various approaches is some connection with the term "genetic." The author discusses such conceptions from the point of view of a historian of mathematics who is keen to contribute to progress in mathematics education. For this aim, he explores methodological aspects of research into the history of mathematics, based on--as one of the reviewers appreciated--his "life long research."

Fictionalisation seems to render ethnographic inquiry suspect as research appropriate to the social sciences because it subverts a claim of realism or objectivity. This leads some ethnographers to propose alternative criteria for validation, such as participation and evocation, which offer alignment with realism; others suggest that scientific values of veridical observation should be replaced by the literary values of stylistic representation. This article argues that fictionalisation plays an essential and constructive role in both scientific and literary thinking and discourse to which philosophers have paid attention for centuries. In prioritising mediation between writer and reader rather than sociological description, ethnography may be considered from a cultural-historical perspective. How we make worlds out of our words becomes an important methodological question.

The main objective of this paper is to apply the onto-semiotic approach to analyze the mathematical concept of different coordinate systems, as well as some situations and university students' actions related to these coordinate systems. The identification of objects that emerge from the mathematical activity and a first intent to describe an epistemic network that relates to this activity were carried out. Multivariate calculus students' responses to questions involving single and multivariate functions in polar, cylindrical, and spherical coordinates were used to classify semiotic functions that relate the different mathematical objects.

This article reports on a case study from a design-based research project that investigated how students make sense of the disciplinary tools they are taught to use, and specifically, what personal, interpersonal, and material resources support this process. The probability topic of binomial distribution was selected due to robust documentation of widespread student error in comparing likelihoods of possible events generated in random compound-event experiments, such as flipping a coin four times, for example, students erroneously evaluate HHHT as more likely than HHHH, whereas in fact these are 2 of 16 equiprobable elemental events in the sample space of this experiment. The study's conjecture was that students' intuitive reasoning underlying these canonical errors is nevertheless in accordance with mathematical theory: student intuition is couched in terms of an unexpanded sample space--that is, five heteroprobable aggregate events (no-H, 1H, 2H, 3H, 4H), and therefore students' judgments should be understood accordingly as correct, for example, the combination "3H, 1T" is indeed more likely than "4H," because "3H, 1T" can occur in four different orders (HHHT, HHTH, HTHH, THHH) but "4H" has only a single permutation (HHHH). The design problem was how to help students reconcile their mathematically correct 5 aggregate-event intuition with the expanded 16 elemental-event sample space. A sequence of activities was designed involving estimation of the outcome distribution in an urn-type quasi-binomial sampling experiment, followed by the construction and interpretation of its expanded sample space. Li, whose experiences were typical of a total of twenty-eight Grade 4-6 participants in individual semi-structured clinical interviews, successfully built on his population-to-sample expectation of likelihood in developing the notion of the expanded sample space. Drawing on cognitive-science, sociocultural, and cultural-semiotics theories of mathematical learning, I develop the construct "semiotic leap" to account for how Li appropriated as a warrant for his intuitive inference an artifact that had initially made no sense to him. More broadly, I conclude that students can ground mathematical procedures they are taught to operate even when they initially do not understand the rationale or objective of these cultural artifacts (i.e., students who are taught a procedure can still be guided to re-invent the procedure-"as-instrument"). (Contains 20 footnotes, 1 table, and 6 figures.)

This article analyses the role of educational opportunity in a time of globalisation, a new economy and life in a multicultural society, and gives an epistemological and semantic account of the concept "intercultural education", distinguishing it from multicultural and transcultural education. Starting with a historic overview of various conceptualisations of meetings/clashes among people with different linguistic, religious, cultural or ethnic features, distinctive theoretical elaborations are reviewed, above all in a European context and in the educational field. After outlining the development of intercultural education (main contents, methods and objectives, as well as limits), this article supports the thesis that education, in an intercultural sense, is currently the most appropriate answer to globalisation and interdependence. (Contains 4 notes.)

The aim of this article is to introduce some theoretical frameworks which may develop the research going on within phenomenography and variation theory. Central concepts from the epistemological and cognitive theories of Charles S. Peirce, Niklas Luhmann and Margaret Boden are presented and their implications for phenomenography and variation theory are discussed. Peirce's concept of the semiotic triad clarifies the interrelations of conceptions, their linguistic expressions and their meanings or referents. Luhmann's concepts of primary and secondary distinctions point out an implicit hierarchy of dimensions of variation. Boden's notion of conceptual spaces as grounds for explorative and transformative creativity points to the relevance and value of becoming familiar with the conceptional spaces of the outcomes of phenomenographic studies. Finally, I point to the possibility of including the notion of a "hidden curriculum" of the discipline as an implicit dimension of variation in classroom studies based on variation theory. (Contains 1 table and 7 notes.)

Starting with Lyotard's characterisation of postmodernity as incredulity, this is related to another of his key concepts--that of "performativity". Lyotard appears to deploy performativity to characterise those technologies that bring about the optimisation of efficient performance. However, there is another sense of performativity where it is linked to "performance". Performance conditions the possibility of any and all performatives, or to put it another way, as performance is itself enabled by performativity, so too performativity is realised through its performance. Both senses of performativity and the linkage between them are clearly manifested in the space of knowledge production that is the contemporary university. This linkage is itself a feature of the semiotic process within which contemporary knowledge and knowledge production is located and which Lyotard himself recognised, albeit implicitly. The implication of this therefore is that Lyotard himself performs his text and in so doing both manifests and contributes to realising both senses of performativity.

This comment attempts to identify different "semiotic perspectives" proposed by the authors of this special issue according to the problems they discuss. These problems can be distinguished as problems concerning the representation of mathematical knowledge, the definition and objectivity of meaning, epistemological questions of learning and activity in mathematics, and the social dimension of sign processes. The contributions are discussed so as to make visible further research perspectives with regard to "semiotics in mathematics education".

Mathematical signs and symbols have a decisive role for coding, constructing and communicating mathematical knowledge. Nevertheless these mathematical signs do not already contain mathematical meaning and conceptual ideas themselves. The contribution will present basic elements of an epistemology of mathematical knowledge and then apply these theoretical ideas for analyzing case studies of two teaching episodes from elementary mathematics teaching. In this way different roles of mathematical signs as means of communication (oral function), of indicating (deictic function) and of writing (symbolic function) will be elaborated.

Learning is better than knowing, generalization is more illuminating than abstract generality or universality because we perceive and thus become conscious of change or development only. Signs and representations establish the dialectic of fixation on the one hand and transformation on the other, which is so essential to learning and cognition. Mathematical epistemology from a semiotic point of view therefore is above all a genetical epistemology. All real mathematical activity is concerned with representations of mathematical entities rather than with things in themselves and with the processes of continuous transformation of a given representation into others. This paper tries to give an overview of the essential relationships between activity theory, epistemology and mathematical education, using the semiotics of Charles S. Peirce as a unifying reference. It is certainly beyond the scope of such a paper to spell out all the questions involved in every detail. Much of what is said in the four short sections to follow is calling for further concretization and research.

Meaning is one of the recent terms which have gained great currency in mathematics education. It is generally used as a correlate of individuals' intentions and considered a central element in contemporary accounts of knowledge formation. One important question that arises in this context is the following: if, in one way or another, knowledge rests on the intrinsically subjective intentions and deeds of the individual, how can the objectivity of conceptual mathematical entities be guaranteed? In the first part of this paper, both Peirce's and Husserl's theories of meaning are discussed in light of the aforementioned question. I examine their attempts to reconcile the subjective dimension of knowing with the alleged transcendental nature of mathematical objects. I argue that transcendentalism, either in Peirce's or Husserl's theory of meaning, leads to an irresolvable tension between subject and object. In the final part of the article, I sketch a notion of meaning and conceptual objects based on a semiotic-cultural approach to cognition and knowledge which gives up transcendentalism and instead conveys the notion of contextual objectivity.