Trouche's [Third Computer Algebra in Mathematics Education Symposiums, Reims, France, June 2003] presentation at the Third Computer Algebra in Mathematics Education Symposium focused on the notions of instrumental genesis and of orchestration: the former concerning the mutual transformation of learner and artefact in the course of constructing knowledge with technology; the latter concerning the problem of integrating technology into classroom practice. At the Symposium, there was considerable discussion of the idea of situated abstraction, which the current authors have been developing over the last decade. In this paper, we summarise the theory of instrumental genesis and attempt to link it with situated abstraction. We then seek to broaden Trouche's discussion of orchestration to elaborate the role of artefacts in the process, and describe how the notion of situated abstraction could be used to make sense of the evolving mathematical knowledge of a community as well as an individual. We conclude by elaborating the ways in which technological artefacts can provide shared means of mathematical expression, and discuss the need to recognise the diversity of student's emergent meanings for mathematics, and the legitimacy of mathematical expression that may be initially divergent from institutionalised mathematics.
L ast year, the Ove Arup Foundation commissioned myself and my colleague, Professor Richard Noss, to undertake a small research project to survey the current roles of mathematics in undergraduate engineering education in the UK, with a particular focus on civil engineering, and to identify some visions of future directions for the teaching of mathematics. The research took place from May to December 2002, using a methodology of interviews and visits with universities, professional institutions and civil engineering companies, supported by a literature review and a questionnaire survey of university civil engineering departments. The final report on the research has just been published (see below for download addresses).
We present a detailed bifurcation analysis for the travelling-wave solutions of the Kuramoto-Sivashinsky equation, with an emphasis on periodic solutions. The solutions are described by a I-parameter, reversible third-order ODE. In two previous papers we described new aspects in the observed bifurcations: the 'noose' bifurcation, and a novel kind of 'Shil'nikov' behaviour. This paper brings everything together, and considers the one remaining new aspect, the connected set of period-multiplying k-bifurcations. We offer a possible explanation for this set by considering a 2-parameter, reversible fourth-order ODE that contains the travelling-wave ODE in a particular limit. It is conjectured that the connected set arises from I : n resonances of the eigenvalues of a fixed paint.
Increasingly, companies are taking part in process improvement programmes, which brings about a growing need for employees to interpret and act on data representations. We have carried out case studies in a range of companies to identify the existence and need of what we call Techno-mathematical Literacies (TmL): functional mathematical knowledge mediated by tools and grounded in the context of specific work situations. Based on data gathered from a large biscuit manufacturing and packaging company, we focus our analysis here on semiotic mediation within activity systems and identify two sets of related TmL: the first concerns rendering some invisible aspects visible through the production of mathematical signs; the second concerns developing meanings for action from an interpretation of these signs. We conclude with some more general observations concerning the role that mathematical signs play in the workplace. The need for Techno-mathematical Literacies at work There is a growing movement for industrial companies to modify their production practices according to methodologies collectively known as process improvement. After World War II, Japanese companies such as Toyota developed new manufacturing paradigms (e.g. Lean Manufacturing) under the guidance of American experts, particularly W. E. Deming. Since the 1980s, the Japanese methodologies have been spreading to the West in a major way, in the form of programmes such as Total Quality Management and Total Productive Maintenance (Deming, 1986; Nakajima, 1988). Two American companies, Motorola and General Electric became famous in the 1990s due to their-3-successful development of the Six Sigma process improvement programme (e.g. Pyzdek, 2001). The core of all these programmes is a set of statistical techniques for the collection and interpretation of production data, and the promotion of a workplace culture in which decisions are based on abstractions of work processes in the form of shared, and often computationally represented, data. A key point that emerges in working with process improvement methodologies is that employees at almost all levels are faced with the need to participate in the procedures of data collection and to interpret the charts, tables and graphs that are derived from the data. This faces companies with the question of what knowledge their employees need to participate effectively, and, particularly for our concerns, just how much of the mathematical and technical knowledge which underlies the production of these artefacts it is useful for them to know. In this paper we will look closely at an example of a company attempting to address this issue with its employees 1. Whatever the answer, we take the position that it is considerably more complex than any model based on "skills" or "competences". For example, learning to read graphs is not a straightforward process. Roth and Bowen (2003), for instance, have shown that even professional scientists often misinterpret graphs from their own discipline if they are not sufficiently familiar wit...
The first aim of this paper is to present a characterisation of techno-mathematical literacies needed for effective practice in modern, technology-rich workplaces that are both highly automated and increasingly focused on flexible response to customer needs. The second aim is to introduce an epistemological dimension to activity theory, specifically to the notions of boundary object and boundary crossing. In this paper we draw on ethnographic research in a pensions company and focus on data derived from detailed analysis of the diverse perspectives that exist with respect to one symbolic artefact, the annual pension statement. This statement is designed to facilitate boundary crossing between company and customers. Our study showed that the statement routinely failed in this communicative role, largely due to the invisible factors of the mathematical-financial models underlying the statement that are not made visible to customers, or to the customer enquiry team whose task is to communicate with customers. By focusing on this artefact in boundary-crossing situations, we identify and elaborate the nature of the techno-mathematical knowledge required for effective communication between different communities in one financial services workplace, and suggest the implications of our findings for workplaces more generally.
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