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 uses of ethnography in the science of cultural evolution’. Commentary on Mesoudi, A., Whiten, A. and K. Laland ‘Toward a unified science of cultural evolution’

There is considerable scope for developing a more explicit role for ethnography within the research program proposed in the article. Ethnographic studies of cultural micro-evolution would complement experimental approaches by providing insights into the “natural” settings in which cultural behaviours occur. Ethnography can also contribute to the study of cultural macro-evolution by shedding light on the conditions that generate and maintain cultural lineages

Use of DNA-based genetic markers in plant breeding

Genetic markers have been used since the beginnings of plant breeding, but the concept of linkage and recently the availability of molecular markers have offered new and powerful tools that can help to perform the traditional tasks of selection or that can change the traditional breeding process. Markers can either be used in a descriptive manner to identify varieties, to study the ‘micro-evolution’ of composite crosses or variety mixtures or to analyse the breeding progress retrospectively in order to learn from the past. The operative use of markers in plant breeding is connected to the selection of parental lines and progeny lines. The possible implementation of these processes stretches from the introgression of specific chromosome fragments to ‘marker-based idiotype breeding’

Computer simulations of history of life: speciation, emergence of complex species from simpler organisms, and extinctions

We propose a generic model of eco-systems, with a {\it hierarchical} food web structure. In our computer simulations we let the eco-system evolve continuously for so long that that we can monitor extinctions as well as speciations over geological time scales. {\it Speciation} leads not only to horizontal diversification of species at any given trophic level but also to vertical bio-diversity that accounts for the emergence of complex species from simpler forms of life. We find that five or six trophic levels appear as the eco-system evolves for sufficiently long time, starting initially from just one single level. Moreover, the time intervals between the successive collections of ecological data is so short that we could also study ``micro''-evolution of the eco-system, i.e., the birth, ageing and death of individual organisms.Comment: 7 pages, including 4 EPS figures, REVTE

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 Inclusion of False, Falsified, and Falsifiable Data that Favor an Evolutionary Worldview in the High School Science Curriculum of Public and Private Schools in the Philippines

This paper examines a curricular issue in general within public and private high schools in the Philippines. This paper presents six known and documented errant points found in textbooks that promote an evolutionary worldview and are commonly found in local high school biology textbooks. The proposed solutions present several questions designed to facilitate formation of the readers’ own views regarding this phenomenon

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

Unified "micro"- and "macro-" evolution of eco-systems: Self-organization of a dynamic network

Very recently we have developed a dynamic network model for eco-systems that achieved ``unification'' of ``micro'' and ``macro''-evolution. We now propose an extension of our model so as to stabilize the eco-system and describe {\it speciation} in a more realistic manner.Comment: 7 pages with 3 figures; for Max Born Symposium, Poland, Sept. 200