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

Model Abstraction in Dynamical Systems: Application to Mobile Robot Control

This book investigates abstraction of dynamical systems for the purpose of designing controllers. Abstraction is a means to reduce a system model\u27s complexity while retaining the important behavior of that system. The motivating example throughout this text is the robotic car. Two topics are introduced in this text: control design using abstraction and propagation of uncertainty in abstracted systems. First, this book investigates the conditions for which controllers can be designed in abstracted systems and then transferred to the original dynamical system, taking advantage of design using the simpler model. This book also studies the relationship between the evolution of uncertain initial conditions in abstracted control systems. It is shown that a control system abstraction can capture the time evolution of the uncertainty in the original system by an appropriate choice of control input. This book provides a comprehensive review of the theory behind abstraction and applies the results to general nonlinear dynamical systems. In particular, the following topics are presented: * An overview of the history and current research in mobile robotic control design. * A mathematical review that provides the tools used in this research area. * The development of the robotic car model and both controllers used in the new control design. * A review of abstraction and an extension of these ideas into new system relationship characterizations called traceability and E-traceability. * A framework for designing controllers based on abstraction. * An open-loop control design with simulation results. * An investigation of system abstraction with uncertain initial conditions

Coming Out of the Dungeon: Mathematics and Role-Playing Games

After hiding it for many years, I have a confession to make. Throughout middle school and high school my friends and I would gather almost every weekend, spending hours using numbers, probability, and optimization to build models that we could use to simulate almost anything. That’s right. My big secret is simple. I was a high school mathematical modeler. Of course, our weekend mathematical models didn’t bear any direct relationship to the models we explored in our mathematics and science classes. You would probably not even recognize our regular gatherings as mathematical exercises. If you looked into the room, you’d see a group of us gathered around a table, scribbling on sheets of paper, rolling dice, eating pizza, and talking about dragons, magical spells, and sword fighting. So while I claim we were engaged in mathematical modeling, I suspect that very few math classes built models like ours. After all, how many math teachers have constructed or had their students construct a mathematical representation of a dragon, a magical spell, or a swordfight? And yet, our role-playing games (RPGs) were very much mathematical models of reality — certainly not the reality of our everyday experience, but a reality nonetheless, one intended to simulate a particular kind of world. Most often for us this was the medieval, high-fantasy world of Dungeons & Dragons (D&D), but we also played games with science fiction or modern-day espionage settings. We learned a lot about math, mythology, medieval history, teamwork, storytelling, and imagination in the process. And, when existing games were inadequate vehicles for our imagination, we modified them or created new ones. In doing so, we learned even more about math. Now that I am a mathematics professor, I find myself reflecting on those days as a “fantasy modeler” and considering various questions. What is the relationship between my two interests of fantasy games and mathematics? Does having been a gamer make me a better mathematician or modeler? Does my mathematical experience make me a better gamer? These different aspects of my life may seem mostly unconnected; indeed, the “nerd” stereotype is associated with both activities, but despite public perception, the community of role-players includes many people who are not scientifically-minded. So we cannot say that role-players like math, or math-lovers role-play, because “that is simply what nerds do.” To get at the deeper question of how mathematics and role-playing are related, we first need to look at the processes of gaming, game designing, and modeling

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

Making mathematics phenomenal : Based on an Inaugural Professorial Lecture delivered at the Institute of Education, University of London, on 14 March 2012

Mathematics is often portrayed as an 'abstract' cerebral subject, beyond the reach of many. In response, research with digital technology has led to innovative design in which mathematics can be experienced to some extent like everyday phenomena. I examine how careful design can 'phenomenalise' mathematics - that is to say create mathematical artefacts that can be directly experienced to support not only engagement but also focus on key ideas. I argue that mathematical knowledge gained through interaction with suitably designed tools can prioritise powerful reasons for doing mathematics, imbuing it with a sort of utility and offering learners hooks on which they can gradually develop fluency and connected understanding. Illustrative examples are taken from conventional topics such as number, algebra, geometry and statistics but also from novel situations where mathematical methods are juxtaposed with social values. The suggestion that prioritising utility supports a more natural way of learning mathematics emerges directly from constructionist pedagogy and inferentialist philosophy

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

On Engineering Support for Business Process Modelling and Redesign

Currently, there is an enormous (research) interest in business process redesign (BPR). Several management-oriented approaches have been proposed showing how to make BPR work. However, detailed descriptions of empirical experience are few. Consistent engineering methodologies to aid and guide a BPR-practitioner are currently emerging. Often, these methodologies are claimed to be developed for business process modelling, but stem directly from information system design cultures. We consider an engineering methodology for BPR to consist of modelling concepts, their representation, computerized tools and methods, and pragmatic skills and guidelines for off-line modelling, communicating, analyzing, (re)designing\ud business processes. The modelling concepts form the architectural basis of such an engineering methodology. Therefore, the choice, understanding and precise definition of these concepts determine the productivity and effectiveness of modelling tasks within a BPR project. The\ud current paper contributes to engineering support for BPR. We work out general issues that play a role in the development of engineering support for BPR. Furthermore, we introduce an architectural framework for business process modelling and redesign. This framework consists of a coherent set of modelling concepts and techniques on how to use them. The framework enables the modelling of both the structural and dynamic characteristics of business processes. We illustrate its applicability by modelling a case from service industry. Moreover, the architectural framework supports abstraction and refinement techniques. The use of these techniques for a BPR trajectory are discussed

Kaleidoscope JEIRP on Learning Patterns for the Design and Deployment of Mathematical Games: Final Report

Project deliverable (D40.05.01-F)Over the last few years have witnessed a growing recognition of the educational potential of computer games. However, it is generally agreed that the process of designing and deploying TEL resources generally and games for mathematical learning specifically is a difficult task. The Kaleidoscope project, "Learning patterns for the design and deployment of mathematical games", aims to investigate this problem. We work from the premise that designing and deploying games for mathematical learning requires the assimilation and integration of deep knowledge from diverse domains of expertise including mathematics, games development, software engineering, learning and teaching. We promote the use of a design patterns approach to address this problem. This deliverable reports on the project by presenting both a connected account of the prior deliverables and also a detailed description of the methodology involved in producing those deliverables. In terms of conducting the future work which this report envisages, the setting out of our methodology is seen by us as very significant. The central deliverable includes reference to a large set of learning patterns for use by educators, researchers, practitioners, designers and software developers when designing and deploying TEL-based mathematical games. Our pattern language is suggested as an enabling tool for good practice, by facilitating pattern-specific communication and knowledge sharing between participants. We provide a set of trails as a "way-in" to using the learning pattern language. We report in this methodology how the project has enabled the synergistic collaboration of what started out as two distinct strands: design and deployment, even to the extent that it is now difficult to identify those strands within the processes and deliverables of the project. The tools and outcomes from the project can be found at: http://lp.noe-kaleidoscope.org

Design as conversation with digital materials

This paper explores Donald Schön's concept of design as a conversation with materials, in the context of designing digital systems. It proposes material utterance as a central event in designing. A material utterance is a situated communication act that depends on the particularities of speaker, audience, material and genre. The paper argues that, if digital designing differs from other forms of designing, then accounts for such differences must be sought by understanding the material properties of digital systems and the genres of practice that surround their use. Perspectives from human-computer interaction (HCI) and the psychology of programming are used to examine how such an understanding might be constructed.</p