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

The construction of meanings for trend in active graphing

The development of increased and accessible computing power has been a major agent in the current emphasis placed upon the presentation of data in graphical form as a means of informing or persuading. However research in Science and Mathematics Education has shown that skills in the interpretation and production of graphs are relatively difficult for Secondary school pupils. Exploratory studies have suggested that the use of spreadsheets might have the potential to change fundamentally how children learn graphing skills. We describe research using a pedagogic strategy developed during this exploratory work, which we call Active Graphing, in which access to spreadsheets allows graphs to be used as analytic tools within practical experiments. Through a study of pairs of 8 and 9 year old pupils working on such tasks, we have been able to identify aspects of their interaction with the experiment itself, the data collected and the graphs, and so trace the emergence of meanings for trend. © 2000 Kluwer Academic Publishers

The Micro-Evolution of Mathematical Knowledge: The Case of Randomness

In this paper we explore the growth of mathematical knowledge and in particular, seek to clarify the relationship between abstraction and context. Our method is to gain a deeper appreciation of the process by which mathematical abstraction is achieved and the nature of abstraction itself, by connecting our analysis at the level of observation with a corresponding theoretical analysis at an appropriate grain size. In this paper we build on previous work to take a further step towards constructing a viable model of the micro-evolution of mathematical knowledge in context. The theoretical model elaborated here is grounded in data drawn from a study of 10-11 year olds’ construction of meanings for randomness in the context of a carefully designed computational microworld, whose central feature was the visibility of its mechanisms-how the random behavior of objects actually worked. In this paper, we illustrate the theory by reference to a single case study chosen to illuminate the relationship between the situation (including, crucially, its tools and tasks) and the emergence of new knowledge. Our explanation will employ the notion of situated abstraction as an explanatory device that attempts to synthesize existing micro- and macro-level descriptions of knowledge construction. One implication will be that the apparent dichotomy between mathematical knowledge as de-contextualized or highly situated can be usefully resolved as affording different perspectives on a broadening of contextual neighborhood over which a network of knowledge elements applies

The distributed developmental network - d2n: a social configuration to support design pattern generation

DiSessa et al. (2004) conducted a comparative study of how research teams design, develop and evaluate TEL software, in the context of component-based educational programming. They identified the issue of the social configuration of the production team as a critical family of issues that are easily marginalized (p.117). These social configurations are loosely equivalent to what Activity Theorists refer to as the rules and division of labour (Engeström, 1987) in the activity system of TEL production. DiSessa et al. (2004) studied four such configurations in detail and noted their relationship with the evolution of the technology and its use. These models suggest different ways of bringing the various participants involved in TEL development together. Based on the definition of interdisciplinarity (van den Besselaar and Heimeriks, 2001; Gibbons, 1994), in this chapter we detail how to support participants from different disciplines to work together in small, product-oriented groups, using design patterns. Our patterns were developed in the context of the Learning patterns for the design and deployment of mathematical games project, funded under the Kaleidoscope Network of Excellence of the European Union. Our primary aim was to develop patterns that worked at the interface between disciplines. They were focused on pragmatic ways to have teachers and technologists productively engage with each other. Furthermore, many patterns were developed from the use of particular tools in educational contexts, where the tools were developed from scratch as outputs of research projects. There was a reflection in the patterns of the need for participants to understand each others practices in order to achieve integrated development. DiSessa et al. (ibid) reflect on the fact that teachers can find it difficult and sometimes intimidating to participate as equal contributors in a technology-based development process and suggest that effective management of collaboration can address this problem. As distinct from DiSessas four models, we identified a somewhat more complex emerging structure, that of a development network, where distributed groups with local expertise use a pattern language to share their expertise, sometimes in collaborative long-term projects, sometimes in ad-hoc exchanges. A detailed analysis of this model is presented in this chapter. What is clear at this stage is that a successful model needs to empower all partners in the design process, avoiding producer-consumer and sage-laymen relationships

Connecting the equals sign

Children tend to view the equals sign as an operator symbol bereft of the rich relational properties of equality statements. It has been argued by some that this restricted view of the equals sign is due to cultural or cognitive factors. We suggest a significant factor is that rich relational meanings lack relevance within the context of paper-based arithmetic. One possible way to allow learners access to relational meanings is through interaction with technologically supported utilities for the equals sign. We report upon a trial in which two students draw on existing and emerging notions of mathematical equivalence in order to connect an onscreen = object with other arithmetical objects

Three utilities for the equals sign

We compare the activity of young children using a microworld and a JavaScript relational calculator with the literature on children using traditional calculators. We describe how the children constructed different meanings for the equal sign in each setting. It appears that the nature of the meaning constructed is highly dependent on specificities of the task design and the tools available. In particular, the microworld offers the potential for children to adopt a meaning of equivalence for the equal sign

Putting the learning back into e-learning

The design of web-based learning environments is primarily focused on the production and delivery of content to a learner. The principles of constructionism are intended to guide the development of learning environments where the learner has more control. In this paper, we describe characteristics of constructionist and learning environments that can foster the learning of mathematics. Our experiences are drawn from the development of microworlds for an e-museum. Reflecting on this process turns out to provide some fresh insights into how e-learning environments might be re-conceptualised in the future

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

A Design Guide for Open Online Courses

This guide is a comprehensive summary of how we went about creating Citizen Maths, an open online maths course and service. The guide shares our design principles and the techniques we used to put them into practice. Our aim is to provide – with the appropriate ‘translation’ – a resource that will be useful to to other teams who are developing online education initiatives