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

Knowledge Construction of 3D Geometry Concepts and Processes Within a Virtual Reality Learning Environment

A consensus has emerged within the mathematics education community about the limitations of traditional approaches for teaching and learning 3D geometry. Therefore, it has been suggested that new approaches based on the use of computers need to be adopted. One such new approach that has been proposed utilises Virtual Reality Learning Environment (VRLE). This paper reports on the initial phases of a research study whose major aim is to design and evaluate a VRLE to facilitate the construction of knowledge about 3D geometry concepts and processes. This research study investigates two primary school students’ construction of 3D geometry knowledge whilst engaged within a VRLE developed by the researcher. A design experiments research methodology was employed in this study. This is research that iterates through cycles of design and research with the objective of arriving at theoretical and design principles that will have application both within and beyond the immediate research study. Therefore, the results being reported in this paper will be used to inform the modification not only of the VRLE but also of theoretical frameworks underlying the design and implementation of VRLEs

Knowledge Building of 3D Geometry Concepts and Processes within a Virtual Reality Learning Environment

This paper reports on a pilot study for a prototype VRLE (Virtual Reality Learning Environment) named VRMath. The two primary school students who were involved in this study engaged in two VRMath learning activities designed by the researchers. The results indicated that 3D navigation within the VR 3D space was difficult. However, it could be aided with the navigation aids designed within VRMath. The 3D navigation within the 3D virtual space also caused the participants confusion in terms of their spatial visualisation and orientation abilities. The construction of 3D geometrical objects within VRMath was also difficult especially when the participants were operating the 3D rotation mentally and physically with respect to their body (i.e., the egocentric frame of reference). It was found that the simultaneously use of different frames of reference could help the construction of 3D geometrical objects. During the learning activities, issues about the usability of VRMath were also explored

The Team Balancing Act - Enhancing Knowledge - Building Activity in On-Line Learning Communities

Online learning in the university sector is a given. Constructivist views of learning (often team based) and the notion of knowledge-building, mediated through the use of ICTs seemingly address many of the imperatives to equip individuals for emergent knowledge-age work practice. While teamwork has many perceived advantages, teams also inexplicably fail despite the apparent quality of the participants. Teams are successful when members address what is a relatively narrow range of actions. However, even within this limited range of actions individuals demonstrate definite preferences towards certain activities and roles. This paper reports on the findings from a study that investigated if knowledge-building activity can be enhanced in tertiary education CSCL environments through the use of groups balanced by Team Role Preference (Margerison & McCann, 1995, 1998). The study found that higher quality knowledge-building activity was more likely to occur in balanced groups than in random groups. The analysis of data revealed that a diversity of ideas was more likely to emerge from within balanced groups than from within random groups particularly when the random groups were heavily skewed towards one team role preference. This provided a compelling reason for explaining why balanced groups may lead to better knowledge-building activity

Production system model of children\u27s development of number concepts

The purpose of the present research study was to produce a global, cumulative model of number concept development for children between the ages of two and eight years old. The theoretical and methodological orientation of this study was greatly influenced by Richard Young\u27s production system analysis of seriation by young children (Young, 1971, 1976) and by Newell\u27s (1973) seminal paper, ‘You can\u27t play twenty questions with nature and win’. The methodology used in this investigation thus was as follows. A series of complex number tasks encompassing many aspects of the concept of number were developed. Five children aged between three and seven years then were videotaped while performing some of these complex number tasks. From a detailed protocol analysis of the video-recordings, computer simulation models written in the production system language PSS3 (Ohlsson, 1979) were produced. Specific production system models were produced for each of following aspects of the children\u27s number knowledge: (i) sharing of discrete quantities; (ii) comparison of shares; and (iii) conservation/addition/subtraction of number. These domain-specific models were based on the converging experimental evidence obtained from each of the children’s responses to variants of the complex number tasks. Each child thus received a different set of problems which were chosen systematically in order to clarify particular features of the child\u27s abilities. After a production system model for each child had been produced within a domain, these models were compared and contrasted. From this analysis, developmental trends within the domain were identified and discussed. The research and educational implications of these developmental trends then were discussed. In the concluding parts of this study, the children\u27s domain-specific production system models were cumulated into global, comprehensive models which accurately represented their behaviour in a variety of number tasks. These comprehensive models were compared and contrasted and general developmental trends in young children\u27s number knowledge were identified and discussed

VRMath: A 3D Microworld for Learning 3D Geometry

This paper reports on the design of a prototype VRLE (Virtual Reality Learning Environment) named VRMath. VRMath is an online application that utilises VR (Virtual Reality)technology combined with the power of a Logo-like programming language and hypermedia and the Internet to facilitate the learning of 3-Dimensional (3D) geometry concepts and processes. VRMath is being designed within the framework of a design-experiment (The Design-Based Research Collective, 2003) during which VRMath will evolve through a series of iteration cycles of design à enactment à reflection à redesign into an educational tool that will provide mathematics educators with new and more powerful ways of facilitating the construction of 3D geometry knowledge. During these iteration cycles, in addition to many new ways about thinking and doing mathematics being identified, implications to inform the future design of this and other VRLEs also will emerge

Growth of Teacher Knowledge: The Promise of CSCL

To ensure that universities meet the needs of their learners more completely, teaching and learning strategies should be adopted to make educational provision more flexible. This study investigated how a lesson-planning task within the context of a computer-supported collaborative learning (CSCL) environment facilitated the growth of teacher knowledge, specifically the subject-matter knowledge and pedagogical content knowledge about the teaching of ratios and fractions. This study used a CSCL environment called Knowledge Forum(r) with a cohort of preservice teachers collaborating in a lesson planning task. The social interaction within the computer-mediated community in this study contributed to the growth of teacher knowledge by providing a new social context for learning that prompted students to articulate their ideas and make ideas visible for peer inspection. Through peer-to-peer interactions like asking questions, requesting clarification, revising interpretations, or elaborating ideas, the students learnt both the limits and utility of different models to explain mathematical notions. These on-line social interactions supported knowledge integration by helping to broaden students' initial repertoire of instructional representations and mathematical constructs, demonstrating personal utility for particular ideas, and encouraging students to refine their understanding of mathematics

Modelling Within Mathematical Online Knowledge Building Communities

In this paper we present an alternative perspective on interactive and digital media learning environments (IDM) involving mathematical modeling. We argue that we need to move beyond using "add-on" and "up-grading" strategies to enrich learning, towards a focus on the generation of conceptual artifacts and the building of knowledge. To facilitate knowledge building, we advocate extending students beyond basic discourse to include a range of thinking and reasoning processes that can assist them in constructing, analyzing, refining, and conveying mathematical ideas. We also identify a number of knowledge-building enabling characteristics of an IDM-based learning environment and propose two sets of interrelated principles to inform the design of IDM environments and the accompanying ends-in-view activities. We illustrate our ideas with examples of online communities for mathematics education

Developing Maths-Confidence in Sixteen Maths-Anxious Preservice Student Teachers

Large numbers of primary preservice student teachers' in Australia lack confidence in their own mathematical abilities and skills when entering teacher education courses. This study investigated the development of maths-confidence in sixteen self-identified maths-anxious preservice student teachers. These students were engaged in the development of their mathematical repertoires within the context of a supportive computer-supported collaborative learning (CSCL) environment. The design of the Intervention Program used in the study was informed by a theoretical framework derived from the literature in the fields of learning environments, novel open-ended mathematical activities, computer supported collaborative learning, community of learners and negative beliefs about learning. The findings from this study indicate that the continuous support from their group members via the computer-mediated Knowledge Forum community, and the support they received from the researcher and facilitator within the non-intimidating workshop environments was crucial in the development of maths-confidence in these preservice student teachers