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"

The mathematical components of engineering expertise: the relationship between doing and understanding mathematics

this paper are extracts from our interviews with engineers.) Where, then, is the complex mathematics that certainly exists in modern engineering? Throughout all aspects of engineering design, computer software has an overwhelming presence. Also, in the particular firm that we visited, there a small number of analytical specialists (a few per cent of the professional engineers employed) who act as consultants for the mathematical/analytical problems which the general design engineers cannot readily solve. (In general in structural engineering, such specialist work is often carried out by external consultants, eg. academic researchers

Towards a narrative-oriented framework for designing mathematical learning

This paper proposes a narrative-oriented approach to the design of educational activities, as well as a CSCL system to support them, in the context of learning mathematics. Both Mathematics and interface design seem unrelated to narrative. Mathematical language, as we know it, is devoid of time and person. Computer interfaces are static and non-linear. Yet, as Bruner (1986; 1990) and others show, narrative is a powerful cognitive and epistemological tool. The questions we wish to explore are - - If, and how, can mathematical meaning be expressed in narrative forms - without compromising rigour? - What are the narrative aspects of user interface? How can interface design be guided by notions of narrative? - How can we harness the power of narrative in teaching mathematics, in a CSCL environment? We begin by giving a brief account of the use of narrative in educational theory. We will describe the environment and tools used by the WebLabs project, and report on one of our experiments. We will then describe our narrative-oriented framework, by using it to analyze both the environment and the experiment described

Designing for mathematical abstraction

Our focus is on the design of systems (pedagogical, technical, social) that encourage mathematical abstraction, a process we refer to as designing for abstraction. In this paper, we draw on detailed design experiments from our research on children's understanding about chance and distribution to re-present this work as a case study in designing for abstraction. Through the case study, we elaborate a number of design heuristics that we claim are also identifiable in the broader literature on designing for mathematical abstraction. Our previous work on the micro-evolution of mathematical knowledge indicated that new mathematical abstractions are routinely forged in activity with available tools and representations, coordinated with relatively naïve unstructured knowledge. In this paper, we identify the role of design in steering the micro-evolution of knowledge towards the focus of the designer's aspirations. A significant finding from the current analysis is the identification of a heuristic in designing for abstraction that requires the intentional blurring of the key mathematical concepts with the tools whose use might foster the construction of that abstraction. It is commonly recognized that meaningful design constructs emerge from careful analysis of children's activity in relation to the designer's own framework for mathematical abstraction. The case study in this paper emphasizes the insufficiency of such a model for the relationship between epistemology and design. In fact, the case study characterises the dialectic relationship between epistemological analysis and design, in which the theoretical foundations of designing for abstraction and for the micro-evolution of mathematical knowledge can co-emerge. © 2010 Springer Science+Business Media B.V

Changing patterns of transition from school to university mathematics

There has been widespread concern over the lack of preparedness of students making the transition from school to university mathematics and the changing profile of entrants to mathematical subjects in higher education has been well documented. In this paper, using documentary analysis and data from an informal case study, we argue the antecedents of this changed profile in the general shift across all subjects to a more utilitarian higher education, alongside the more specific changes in A-level mathematics provision which have been largely market driven. Our conclusions suggest that, ironically, changes put in place to make mathematics more widely useful may result in it losing just those features that make it marketable

Situating graphs as workplace knowledge

We investigate the use and knowledge of graphs in the context of a large industrial factory. We are particularly interested in the question of "transparency", a question that has been extensively considered in the general literature on tool use, and more recently, by Michael Roth and his colleagues in the context of scientific work. Roth uses the notion of transparency to characterise instances of graph use by highly educated scientists in cases where the context was familiar: the scientists were able to read the situation "through" the graph. This paper explores the limits of the validity of the transparency metaphor. We present two vignettes of actual graph use by a factory worker, and contrast his actions and knowledge with that of a highly-qualified process engineer working on the same production line. We note that in neither case were the graphs transparent. We argue that a fuller account that describes a spectrum of transparency is needed, and we seek to achieve this by adopting some elements of a semiotic approach that enhance a strictly activity-theoretical view

The visibility of models: using technology as a bridge between mathematics and engineering

Engineering mathematics is traditionally conceived as a set of unambiguous mathematical tools applied to solving engineering problems, and it would seem that modern mathematical software is making the toolbox metaphor ever more appropriate. We question the validity of this metaphor, and make the case that engineers do in fact use mathematics as more than a set of passive tools—that mathematical models for phenomena depend critically on the settings in which they are used, and the tools with which they are expressed. The perennial debate over whether mathematics should be taught by mathematicians or by engineers looks increasingly anachronistic in the light of technological change, and we think it is more instructive to examine the potential of technology for changing the relationships between mathematicians and engineers, and for connecting their respective knowledge domains in new ways

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

Improving work processes by making the invisible visible

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 nee