Toward a Semiotic Framework for Using Technology in Mathematics Education: The Case of Learning 3D Geometry

This paper proposes and examines a semiotic framework to inform the use of technology in mathematics education. Semiotics asserts that all cognition is irreducibly triadic, of the nature of a sign, fallible, and thoroughly immersed in a continuing process of interpretation (Halton, 1992). Mathematical meaning-making or meaningful knowledge construction is a continuing process of interpretation within multiple semiotic resources including typological, topological, and social-actional resources. Based on this semiotic framework, an application named VRMath has been developed to facilitate the learning of 3D geometry. VRMath utilises innovative virtual reality (VR) technology and integrates many semiotic resources to form a virtual reality learning environment (VRLE) as well as a mathematical microworld (Edwards, 1995) for learning 3D geometry. The semiotic framework and VRMath are both now being evaluated and will be re-examined continuously

Next steps in implementing Kaput's research programme

We explore some key constructs and research themes initiated by Jim Kaput, and attempt to illuminate them further with reference to our own research. These 'design principles' focus on the evolution of digital representations since the early nineties, and we attempt to take forward our collective understanding of the cognitive and cultural affordances they offer. There are two main organising ideas for the paper. The first centres around Kaput's notion of outsourcing of processing power, and explores the implications of this for mathematical learning. We argue that a key component for design is to create visible, transparent views of outsourcing, a transparency without which there may be as many pitfalls as opportunities for mathematical learning. The second organising idea is that of communication, a key notion for Kaput, and the importance of designing for communication in ways that recognise the mutual influence of tools for communication and for mathematical expression

The technological mediation of mathematics and its learning

This paper examines the extent to which mathematical knowledge, and its related pedagogy, is inextricably linked to the tools – physical, virtual, cultural – in which it is expressed. Our goal is to focus on a few exemplars of computational tools, and to describe with some illustrative examples, how mathematical meanings are shaped by their use. We begin with an appraisal of the role of digital technologies, and our rationale for focusing on them. We present four categories of digital tool-use that distinguish their differing potential to shape mathematical cognition. The four categories are: i. dynamic and graphical tools, ii. tools that outsource processing power, iii. new representational infrastructures, and iv. the implications of highbandwidth connectivity on the nature of mathematics activity. In conclusion, we draw out the implications of this analysis for mathematical epistemology and the mathematical meanings students develop. We also underline the central importance of design, both of the tools themselves and the activities in which they are embedded

Designing as Construction of Representations: A Dynamic Viewpoint in Cognitive Design Research

This article presents a cognitively oriented viewpoint on design. It focuses on cognitive, dynamic aspects of real design, i.e., the actual cognitive activity implemented by designers during their work on professional design projects. Rather than conceiving de-signing as problem solving - Simon's symbolic information processing (SIP) approach - or as a reflective practice or some other form of situated activity - the situativity (SIT) approach - we consider that, from a cognitive viewpoint, designing is most appropriately characterised as a construction of representations. After a critical discussion of the SIP and SIT approaches to design, we present our view-point. This presentation concerns the evolving nature of representations regarding levels of abstraction and degrees of precision, the function of external representations, and specific qualities of representation in collective design. Designing is described at three levels: the organisation of the activity, its strategies, and its design-representation construction activities (different ways to generate, trans-form, and evaluate representations). Even if we adopt a "generic design" stance, we claim that design can take different forms depending on the nature of the artefact, and we propose some candidates for dimensions that allow a distinction to be made between these forms of design. We discuss the potential specificity of HCI design, and the lack of cognitive design research occupied with the quality of design. We close our discussion of representational structures and activities by an outline of some directions regarding their functional linkages

The Mechanics of Embodiment: A Dialogue on Embodiment and Computational Modeling

Embodied theories are increasingly challenging traditional views of cognition by arguing that conceptual representations that constitute our knowledge are grounded in sensory and motor experiences, and processed at this sensorimotor level, rather than being represented and processed abstractly in an amodal conceptual system. Given the established empirical foundation, and the relatively underspecified theories to date, many researchers are extremely interested in embodied cognition but are clamouring for more mechanistic implementations. What is needed at this stage is a push toward explicit computational models that implement sensory-motor grounding as intrinsic to cognitive processes. In this article, six authors from varying backgrounds and approaches address issues concerning the construction of embodied computational models, and illustrate what they view as the critical current and next steps toward mechanistic theories of embodiment. The first part has the form of a dialogue between two fictional characters: Ernest, the �experimenter�, and Mary, the �computational modeller�. The dialogue consists of an interactive sequence of questions, requests for clarification, challenges, and (tentative) answers, and touches the most important aspects of grounded theories that should inform computational modeling and, conversely, the impact that computational modeling could have on embodied theories. The second part of the article discusses the most important open challenges for embodied computational modelling

Syntax-Aware Multi-Sense Word Embeddings for Deep Compositional Models of Meaning

Deep compositional models of meaning acting on distributional representations of words in order to produce vectors of larger text constituents are evolving to a popular area of NLP research. We detail a compositional distributional framework based on a rich form of word embeddings that aims at facilitating the interactions between words in the context of a sentence. Embeddings and composition layers are jointly learned against a generic objective that enhances the vectors with syntactic information from the surrounding context. Furthermore, each word is associated with a number of senses, the most plausible of which is selected dynamically during the composition process. We evaluate the produced vectors qualitatively and quantitatively with positive results. At the sentence level, the effectiveness of the framework is demonstrated on the MSRPar task, for which we report results within the state-of-the-art range.Comment: Accepted for presentation at EMNLP 201

Experimentation and Representation in Architecture: analyzing one’s own design activity

Architects materialize ideas on physical supports to register their thoughts and to discover new possibilities from hints and suggestions in their own drawings. Uncertainty is inherent to creative processes encouraging the production of different ideas through testing. This research brings to light that the re-examination of artefacts from new points of view allows for the review and generation of design ideas and decisions, capacitating students to make yet new discoveries from what they have done so far. Tacit knowledge aids specific decisions. Student reports become analytical records of their material registers (sketches, physical and virtual models) making it explicit that which is implicit in those artefacts. This apparently confirms previous studies that suggest that knowledge per se not always triggers or controls decisions in design. Many physical as well as perceptive actions actually lead the initial steps and play a crucial role in the whole course of production. Besides serving as external representations, sketches and models provide visual hints that will be checked later, favouring the upcoming of the unexpected, stimulating creativity. The intent here is to point out how these different means of representation and expression contribute in a peculiar manner to the whole process of discovery and solution to problems in architecture. The authors propose here a reflection on the process of design and its uncertainties in its initial phase, concentrating on sketches and real models as experimentations. They consider these means not from a graphic and physical register stand point, but in terms of conception and concepts they embody, as records of students thinking and knowledge. Keywords: Experimentation; Uncertainty; Representation; Design Process; Cognition; Education</p

Tangible user interfaces : past, present and future directions

In the last two decades, Tangible User Interfaces (TUIs) have emerged as a new interface type that interlinks the digital and physical worlds. Drawing upon users' knowledge and skills of interaction with the real non-digital world, TUIs show a potential to enhance the way in which people interact with and leverage digital information. However, TUI research is still in its infancy and extensive research is required in or- der to fully understand the implications of tangible user interfaces, to develop technologies that further bridge the digital and the physical, and to guide TUI design with empirical knowledge. This paper examines the existing body of work on Tangible User In- terfaces. We start by sketching the history of tangible user interfaces, examining the intellectual origins of this field. We then present TUIs in a broader context, survey application domains, and review frame- works and taxonomies. We also discuss conceptual foundations of TUIs including perspectives from cognitive sciences, phycology, and philoso- phy. Methods and technologies for designing, building, and evaluating TUIs are also addressed. Finally, we discuss the strengths and limita- tions of TUIs and chart directions for future research

DISCOURSE-DRIVEN MEANING CONSTRUCTION IN NEOSEMANTIC NOUN-TO-VERB CONVERSIONS [MEANING CONSTRUCTION IN NOUN-TO-VERB CONVERSIONS]

Neosemantic noun-to-verb conversions such as beer → to beer, door → to door, pink → to pink, etc., constitute a particularly interesting field of study for Cognitive Linguistics in that they call for a discourse-guided and context-based analysis of meaning construction. The present article takes a closer look at the cognitive motivation for the conversion process involved in the noun-verb alterations with a view to explaining the semantics of some conversion formations in relation to the user-centred discourse context. The analysis developed in this article draws from the combined insights of Fauconnier and Turner’s (2002) Conceptual Integration Theory and Langacker’s (2005, 2008) Current Discourse Space