Objectifying early-number : a visual nomenclature to express mathematical domain knowledge

To address issues of divisive ideologies in the Mathematics Education community and to subsequently advance educational practice, an alternative theoretical framework and operational model is proposed which represents a consilience of constructivist learning theories whilst acknowledging the objective but improvable nature of domain knowledge. Based upon Popper’s three-world model of knowledge, the proposed theory supports the differentiation and explicit modelling of both shared domain knowledge and idiosyncratic personal understanding using a visual nomenclature. The visual nomenclature embodies Piaget’s notion of reflective abstraction and so may support an individual’s experience-based transformation of personal understanding with regards to shared domain knowledge. Using the operational model and visual nomenclature, seminal literature regarding early-number counting and addition was analysed and described. Exemplars of the resultant visual artefacts demonstrate the proposed theory’s viability as a tool with which to characterise the reflective abstraction-based organisation of a domain’s shared knowledge. Utilising such a description of knowledge, future research needs to consider the refinement of the operational model and visual nomenclature to include the analysis, description and scaffolded transformation of personal understanding. A detailed model of knowledge and understanding may then underpin the future development of educational software tools such as computer-mediated teaching and learning environments.

A visual description of early-number counting, addition and subtraction knowledge

Early-number is a rich fabric of interconnected ideas that is often misunderstood and thus taught in ways that do not lead to rich understanding. In this presentation, a visual language is used to describe the organisation of this domain of knowledge. This visual language is based upon Piaget’s notion of reflective abstraction (Dubinsky, 1991; Piaget, 1977/2001), and thus captures the epistemological associations that link the problems, concepts and representations of the domain. The constructs of this visual language are introduced and then applied to the early-number domain. The introduction to this visual language may prompt reflection upon its suitability and significance to the description of other domains of knowledge. Through such a process of analysis and description, the visual language may serve as a scaffold for enhancing pedagogical content knowledge and thus ultimately improve learning outcomes.

A Popperian consilience : modelling mathematical knowledge and understanding

Goldin (2003) and McDonald, Yanchar, and Osguthorpe (2005) have called for mathematics learning theory that reconciles the chasm between ideologies, and which may advance mathematics teaching and learning practice. This paper discusses the theoretical underpinnings of a recently completed PhD study that draws upon Popper’s (1978) three-world model of knowledge as a lens through which to reconsider a variety of learning theories, including Piaget’s reflective abstraction. Based upon this consideration of theories, an alternative theoretical framework and complementary operational model was synthesised, the viability of which was demonstrated by its use to analyse the domain of early-number counting, addition and subtraction.

Visualising mathematical knowledge and understanding

Contemporary mathematics education attempts to instil within learners the conceptualization of mathematics as a highly organized and inter-connected set of ideas. To support this, a means to graphically represent this organization of ideas is presented which reflects the cognitive mechanisms that shape a learner’s understanding. This organisation of information may then be analysed, with the view to informing the design of mathematics instruction in face-to-face and/or computer-mediated learning environments. However, this analysis requires significant work to develop both theory and practice.

Rock, paper, scissors : reflective practice for design process in the novice landscape architect

We describe a pedagogical approach that addresses challenges in design education for novices. These include an inability to frame new problems and limited-to-no design capability or domain knowledge. Such challenges can reduce student engagement with design practice, cause derivative design solutions as well as the inappropriate simplification of design assignments and assessment criteria by educators. We argue that a curriculum that develops the student’s design process will enable them to deal with the uncertain and dynamic situations that characterise design. We describe how this may be achieved and explain our pedagogical approach in terms of methods from Reflective Practice and theories of abstraction and creativity. We present a landscape architecture unit, recently taught, as an example. It constitutes design exercises that require little domain or design expertise to support the development of conceptual thinking and a design rationale. We show how this approach (a) leveraged the novice’s existing spatial and thinking skills while (b) retaining contextually-rich design situations. Examples of the design exercises taught are described along with samples of student work. The assessment rationale is also presented and explained. Finally, we conclude by reflecting on how this approach relates to innovation, sustainability and other disciplines.

Building a supportive environment for teaching and learning : a reflective commentary

Universities are increasingly caught in the transition between college institutions of independant academics to become managed businesses of research and teaching and learning. This is introducing substantial issues with the work, approach and personal development needs of individual academics. It is causing even greater concerns for managers within universities. The developments in University Management have increasingly become driven towards issues of finance, quality and marketing. Organizational development in teaching and learning practices has been less commonly a point of focus. This paper outlines developments of this nature at the University of Salford. Through its combination of authors it does so at whole University, Faculty and School levels. It outlines the variety of ways in which teaching and learning developments can be supported within an organisation.

Formative and reflective measurement and validation mismatch in survey research : an archival analysis of information systems success constructs 1985-2007

The generic IS-success constructs first identified by DeLone and McLean (1992) continue to be widely employed in research. Yet, recent work by Petter et al (2007) has cast doubt on the validity of many mainstream constructs employed in IS research over the past 3 decades; critiquing the almost universal conceptualization and validation of these constructs as reflective when in many studies the measures appear to have been implicitly operationalized as formative. Cited examples of proper specification of the Delone and McLean constructs are few, particularly in light of their extensive employment in IS research. This paper introduces a four-stage formative construct development framework: Conceive > Operationalize > Respond > Validate (CORV). Employing the CORV framework in an archival analysis of research published in top outlets 1985-2007, the paper explores the extent of possible problems with past IS research due to potential misspecification of the four application-related success dimensions: Individual-Impact, Organizational-Impact, System-Quality and Information-Quality. Results suggest major concerns where there is a mismatch of the Respond and Validate stages. A general dearth of attention to the Operationalize and Respond stages in methodological writings is also observed.

A reflective model for colour studies learning

In a university context how should colour be taught in order to engage students? Entwistle states, ‘What we learn depends on how we learn, and why we have to learn it.’ Therefore, there is a need to address the accumulating evidence that highlights the effects of learning environments on the quality of student learning when considering colour education. It is necessary to embrace the contextual demands while ensuring that the student knowledge of colour and the joy of discovering its characteristics in practice are enhanced. Institutional policy is forcing educators to re-evaluate traditional studio’s effectiveness and the intensive 'hands-on' interactive approach that is embedded in such an approach. As curriculum development involves not only theory and project work, the classroom culture and physical environment also need to be addressed. The increase in student numbers impacting the number of academic staff/student ratio, availability of teaching support as well as increasing variety of student age, work commitments, learning styles and attitudes have called for positive changes to how we teach. The Queensland University of Technology’s restructure in 2005 was a great opportunity to re-evaluate and redesign the approach to teaching within the design units of Interior Design undergraduate program –including colour. The resultant approach “encapsulates a mode of delivery, studio structure, as well as the learning context in which students and staff interact to facilitate learning”1 with a potential “to be integrated into a range of Interior Design units as it provides an adaptive educational framework rather than a prescriptive set of rules”.

Developing mathematics understanding and abstraction : the case of equivalence in the elementary years

Generalising arithmetic structures is seen as a key to developing algebraic understanding. Many adolescent students begin secondary school with a poor understanding of the structure of arithmetic. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding and abstraction in the elementary school context. The particular focus is on the use of models and representations to construct an understanding of equivalence. The results of a longitudinal intervention study with five elementary schools, following 220 students as they progressed from Year 2 to Year 6, informed the development of this theory. Data were gathered from multiple sources including interviews, videos of classroom teaching, and pre-and post-tests. Data reduction resulted in the development of nine conjectures representing a growth in integration of models and representations. These conjectures formed the basis of the theory.

Reflective learning through social network sites in design education

Reflective learning is vital for successful practice-led education such as animation, multimedia design and graphic design, and social network sites can accommodate various learning styles for effective reflective learning. In this paper, the researcher studies reflective learning through social network sites with two animation units. These units aim to provide students with an understanding of the tasks and workflows involved in the production of style sheets, character sheets and motion graphics for use in 3D productions for film and television and game design. In particular, an assessment in these units requires students to complete their online reflective journals throughout the semester. The reflective learning has been integrated within the unit design and students are encouraged to reflect weekly learning processes and outcomes. A survey evaluating for students’ learning experience was conducted, and its outcomes indicate that social network site based reflective learning will not be effective without considering students’ learning circumstances and designing peer-to-peer interactions.

Blog and design education : facilitating reflective learning

This paper discusses about effectiveness of blogs for reflective learning in design education. Students in two animation units were asked to complete their online journal via blog in terms of reflective learning. Students were encouraged to respond their weekly outcomes and project development process to their blog and share it with other students. A survey was undertaken to evaluate their learning experience and one of the key outcomes indicates that interaction design for social network is significantly important to blog based learning design.