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

Mitigating the water footprint of export cut flowers from the Lake Naivasha Basin, Kenya

Kenya’s cut-flower industry has been praised as an economic success as it contributed an annual average of US$141 million foreign exchange (7$ 352 million in 2005 alone. The industry also provides employment, income and infrastructure such as schools and hospitals for a large population around Lake Naivasha. On the other hand, the commercial farms have been blamed for causing a drop in the lake level and for putting the lake’s biodiversity at risk. The objective of this study is to quantify the water footprint within the Lake Naivasha Basin related to cut flowers and assess the potential for mitigating this footprint by involving cut-flower traders, retailers and consumers overseas. The water footprint of one rose flower is estimated to be 7-13 litres. The total virtual water export related to export of cut flowers from the Lake Naivasha Basin was 16 Mm3/yr during the period 1996-2005 (22% green water; 45% blue water; 33% grey water). Our findings show that, although the commercial farms around the lake have contributed to the decline in the lake level through water abstractions, both the commercial farms and the smallholder farms in the upper catchment are responsible for the lake pollution due to nutrient load. The\ud observed decline in the lake level and deterioration of the lake’s biodiversity calls for sustainable management of the basin through pricing water at its full cost and other regulatory measures

Between social physics and phenomenology

In recent years, both phenomenology and social physics, which might be thought of asoccupying the humanistic and scientific poles of urban discourse, have taken an interestin space syntax as a means of furthering their academic aims. Here we suggest that thiscould presage a far deeper theoretical integration of the multi-disciplinary study of citiesthan either currently envisage

Mitigating the water footprint of export cut flowers from the Lake Naivasha Basin, Kenya

Kenya’s cut-flower industry has been praised as an economic success as it\ud contributed an annual average of US$141 million foreign exchange (7$ 352 million in 2005 alone. The industry also provides employment, income and infrastructure such as schools and hospitals for a large population around Lake Naivasha. On the other hand, the commercial farms have been blamed for causing a drop in the lake level, polluting the lake and for possibly affecting the lake’s biodiversity. The objective of this study is to quantify the water footprint within the Lake Naivasha Basin related to cut flowers and analyse the possibility to mitigate this footprint by involving cut-flower traders, retailers and consumers overseas. The water footprint of one rose flower is estimated to be 7–13 litres. The total virtual water export related to export of cut flowers from the Lake Naivasha Basin was 16 Mm3/yr during the period 1996–2005 (22 % green water; 45 % blue water; 33 % grey water). Our findings show that, although the decline in the lake level can be attributed mainly to the commercial farms around the lake, both the commercial farms and the smallholder farms in the upper catchment are responsible for the lake pollution due to nutrient load. The observed decline in the lake level and deterioration of the lake’s biodiversity calls for sustainable management of the basin through pricing water at its full cost and other regulatory measures. Pricing water at full marginal cost is important, but the conditions in Kenya are unlikely to result in serious steps to full-cost pricing, since many farmers resist even modest water price increases and government is lacking means of enforcement. We propose an alternative in this study that can be implemented with a focus on sustainable water use in flower farming around Lake Naivasha alone. The proposal involves a water-sustainability agreement between major agents along the cut-flower supply chain and includes a premium to the final product at the retailer end of the supply chain. Such a ‘water sustainability premium’ will raise awareness among flower consumers and—when channelled back to the farmers—facilitate the flower farms to install the necessary equipment and implement the right measures to use water in a sustainable manner. The collected premiums will generate a fund that can be used for financing measures to reduce the water footprint and to improve watershed managemen

MetaSpace II: Object and full-body tracking for interaction and navigation in social VR

MetaSpace II (MS2) is a social Virtual Reality (VR) system where multiple users can not only see and hear but also interact with each other, grasp and manipulate objects, walk around in space, and get tactile feedback. MS2 allows walking in physical space by tracking each user's skeleton in real-time and allows users to feel by employing passive haptics i.e., when users touch or manipulate an object in the virtual world, they simultaneously also touch or manipulate a corresponding object in the physical world. To enable these elements in VR, MS2 creates a correspondence in spatial layout and object placement by building the virtual world on top of a 3D scan of the real world. Through the association between the real and virtual world, users are able to walk freely while wearing a head-mounted device, avoid obstacles like walls and furniture, and interact with people and objects. Most current virtual reality (VR) environments are designed for a single user experience where interactions with virtual objects are mediated by hand-held input devices or hand gestures. Additionally, users are only shown a representation of their hands in VR floating in front of the camera as seen from a first person perspective. We believe, representing each user as a full-body avatar that is controlled by natural movements of the person in the real world (see Figure 1d), can greatly enhance believability and a user's sense immersion in VR.Comment: 10 pages, 9 figures. Video: http://living.media.mit.edu/projects/metaspace-ii

Leveling transparency via situated intermediary learning objectives (SILOs)

When designers set out to create a mathematics learning activity, they have a fair sense of its objectives: students will understand a concept and master relevant procedural skills. In reform-oriented activities, students first engage in concrete situations, wherein they achieve situated, intermediary learning objectives (SILOs), and only then they rearticulate their solutions formally. We define SILOs as heuristics learners devise to accommodate contingencies in an evolving problem space, e.g., monitoring and repairing manipulable structures so that they model with fidelity a source situation. Students achieve SILOs through problem-solving with media, instructors orient toward SILOs via discursive solicitation, and designers articulate SILOs via analyzing implementation data. We describe the emergence of three SILOs in developing the activity Giant Steps for Algebra. Whereas the notion of SILOs emerged spontaneously as a framework to organize a system of practice, i.e. our collaborative design, it aligns with phenomenological theory of knowledge as instrumented action

Diversity in virtual reality landscape modelling

This paper examines the range of decisions that landscape architects need to consider in communicating the future of the landscape. The objective of this research paper was to investigate the issues affecting the landscape modelling process. It also aimed to provide a better understanding of the creators of landscape visualisations and the role they play in determining what audiences, often from many disciplines, see and debate. Interviews were conducted to gather ideas from practice and academia. The paper acknowledges that this was a limited exploration with the intention of establishing a base for a further study. Issues relating to the diversity of audiences, diversity of interpretation, appropriate levels of representation and representing the passage of time in VR models and technologies currently available for VR modelling were the focus of the interviews

Developing a Generic Debugger for Advanced-Dispatching Languages

Programming-language research has introduced a considerable number of advanced-dispatching mechanisms in order to improve modularity. Advanced-dispatching mechanisms allow changing the behavior of a function without modifying their call sites and thus make the local behavior of code less comprehensible. Debuggers are tools, thus needed, which can help a developer to comprehend program behavior but current debuggers do not provide inspection of advanced-\ud dispatching-related language constructs. In this paper, we present a debugger which extends a traditional Java debugger with the ability of debugging an advanced-dispatching language constructs and a user interface for inspecting this