For a learnable mathematics in the digital cultures

I begin with some general remarks concerning the co-evolution of representational forms and mathematical meanings. I then discuss the changed roles of mathematics and novel representations that emerge from the ubiquity of computational models, and briefly consider the implications for learning mathematics. I contend that a central component of knowledge required in modern societies involves the development of a meta-epistemological stance – i.e. developing a sense of mechanism for the models that underpin social and professional discourses. I illustrate this point in relation to recent research in which I am investigating the mathematical epistemology of engineering practice. Finally, I map out one implication for the design of future mathematical learning environments with reference to some data from the `Playground Project. This revised version was published online in July 2006 with corrections to the Cover Date.     

Alternatives to rules for knowledge-based modelling

The technology of knowledge-based systems undoubtedly offers potential for educational modelling, yet its practical impact on today's school classrooms is very limited. To an extent this is because the tools presently used in schools are mostly rule-based expert system shells, which lack usability. We developed three alternative tools, using ideas from knowledge acquisition research, and compared their effectiveness to that of an established rule-based shell. Children working with the new tools produced higher quality models and developed more positive attitudes. We relate these findings to the forms of representation provided by the new tools and present evidence that modelling increases children's representational skill. We conclude that knowledge acquisition systems and alternative forms of representation can contribute to improved forms of knowledge-based modelling.     

Drawing to reason and learn in science

Despite mixed results in research on student learning from drawing in science, there is growing interest in the potential for this visual mode, in tandem with other modes, to enact and enable student reasoning in this subject. Building on current research in this field, and using a micro-ethnographic approach informed by socio-semiotic perspectives, we aimed to identify how and why student drawing can contribute to student reasoning and learning. In our study, secondary school students were challenged to explore and collaboratively create explanatory representations of phenomena including through drawing. Data were generated using multiple wall- and ceiling-mounted cameras capable of continuously tracking groups of students negotiating these representational challenges. Our analysis proceeded through active and iterative viewing of the extensive video record, and the identification of themes to establish possible relationships between drawing and reasoning. Through this process, we (a) identify multiple necessary conditions and varied opportunities for student drawing to enact and enable reasoning, and (b) extend current understandings of how the particular affordances of this mode interact with these conditions to contribute to student learning in science.     

Visual representations of the internet in greek school textbooks and students’ experiences

Our research is concerned with the visual representations found in textbooks used for the teaching of the Internet in Greek secondary schools. Visualization, as both the product and the process of creation, interpretation and reflection upon pictures and images, is considered here to be very important, as it is the only way students gain insight into the nature and function of the Internet, its size, complexity and invisibility. Initially, we attempted to analyze and reflect upon school textbooks’ visual interpretations of the Internet. A scheme of categories of visual representations has been identified and reveals the characteristics of the textbooks’ representations as well as their limitations. Sketch-comics and computer snap-shots are the more popular types of Vrep, although a considerable number of them cannot be characterized as accurate and few of them have an explanatory or complementary function in terms of the content presented in the text. We have also explored the impact on students’ readings of two visual representations in one of the textbooks, used without any caption or textual information. The phenomenological aspects of the VRep seem to attract students’ attention and create obstacles in conceptualizing the main idea conveyed in both representations, but when the field of ICT is implied and not clearly portrayed in the VRep, students face serious problems in recognizing the phenomenon. It seems that it is difficult for an image on its own to paint a thousand words.     

Teaching Image Formation by Extended Light Sources: The Use of a Model Derived from the History of Science

This research, carried out in Greece on pupils aged 12–16, focuses on the transformation of their representations concerning light emission and image formation by extended light sources. The instructive process was carried out in two stages, each one having a different, distinct target set. During the first stage, the appropriate conflict conditions were created by contrasting the subjects’ predictions with the results of experimental situations inspired by the History of Science, with a view to destabilizing the pupils’ alternative representations. During the second stage, the experimental teaching intervention was carried out; it was based on the geometrical optics model and its parameters were derived from Kepler’s relevant historic experiment. For the duration of this process and within the framework of didactical interactions, an effort was made to reorganize initial limited representations and restructure them at the level of the accepted scientific model. The effectiveness of the intervention was evaluated two weeks later, using experimental tasks which had the same cognitive yet different empirical content with respect to the tasks conducted during the intervention. The results of the study showed that the majority of the subjects accepted the model of geometrical optics, that is, the pupils were able to correctly predict and adequately justify the experimental results based on the principle of punctiform light emission. Educational and research implications are discussed.

Convergence of Sequences and Series 2: Interactions Between Nonvisual Reasoning and the Learner’s Beliefs about their own Role

This paper examines the work of students who, when reasoning about real analysis, do so almost exclusively by means of verbal and algebraic reasoning, and tend not to incorporate visual images into their work. It examines the work of students from two parallel courses of introductory real analysis, whose reasoning ranges from those who introduce definitions appropriately and work with them competently, to those who cannot recall definitions and appear to manipulate notation without regard for its reference. It presents a theory that relates the differences to students expectations regarding their role as learners of mathematics. Throughout, the argument is illustrated with interview data from which the theory was inductively generated.     

深入领会“三个代表”的思想努力推进电大改革与发展

本文结合“三个代表”的重要思想和教育“十五”计划的学习,联系电大改革和发展实际,认为：理论联系实际是学习“三个代表”思想的基本原则;要从“三个代表”的思想出发,加深对教育“十五”计划的理解;要在“三个代表”思想的指导下,抓紧制订学校“十五”发展规划;要以“三个代表”思想为强大动力,推进电大新一轮改革和发展。并对新形势下电大发展定位提出了若干意见。     

"三个代表"重要思想指导学校体育发展的研究

认真学习和正确领会江泽民同志“三个代表”重要思想的科学内涵对发展学校体育事业，培养新世纪高素质人才提供支持，明确学校体育事业在国际形势发展变化环境中对人才培养的战略意义和体育学科进一步深化改革的努力方向，加强学校体育学科自身建设与德育，智育，美育等整体的协调发展，使学校体育成为实践“三个代表”重要思想的典范，为建设有中国特色社会主义现代化强国，实现中会民族的伟大复兴，培养全面发展的新人提供支持。     

The uses and abuses of power: teaching school leadership through children's literature

There are relatively few studies of how representations of teachers, schools and educational administrators in popular films and television might be, and are, used in leadership preparation. This paper seeks to add to this small body of work; it reports on an exploratory study of the representation of headteachers in contemporary children's fiction. Thirty-one texts are analysed to ascertain key themes and the major characterisations. The paper draws on children's literature scholars to argue that both the historical school story and its contemporary counterpart focus heavily on the power of the head to control the micro-world of the school. Because these fictional accounts deal with issues of power and justice more openly than many mainstream educational administration texts, this makes them particularly useful in the preparation of potential school leaders.