ion is not inevitable. In fact, shopkeepers have never abstracted the structures of modules which regulate their financial exchanges because they haven’t the motivation to do so. Schematization and formulation The schematization and the formulation of the structure follows the process of identification and of updating. Diénès does not plan any general situations specific to this stage (would something which is well conceived of spell itself out clearly?) but the representation by a graph is often envisaged as a simplified but natural and direct expression of the thought of a child. Representing objects by points and operators by arrows is learned by the use of imitation, like a language. Symbolization Symbolization is the transcription in a new language of properties represented in the preceding stage.

https://link.springer.com/content/pdf/bfm%3A978-0-306-47211-4%2F1.pdf

Theory of didactical situations in mathematics

The pressure towards more school-based teacher education programs, visible in many countries, creates a need to rethink the relationship between theory and practice. The traditional application-of-theory model appears to be rather ineffective and is currently being replaced by other, more reflective approaches. However, until now the variety of different notions and assumptions underlying these new approaches have not provided a sound basis for further development. Two related theoretical bases are presented for a new paradigm in teacher education. The first uses the concepts of episteme and phronesis to introduce a new way of framing relevant knowledge. The second is a more holistic way of describing the relationship between teacher cognition and teacher behavior, leading to a model of three levels in learning about teaching, the Gestalt level, the schema level and the theory level, which are illustrated by interview data. Building on these two theoretical, frameworks, a so-called “realistic approach” to...

https://www.jstor.org/stable/pdfplus/1176444.pdf

Linking Theory and Practice: Changing the Pedagogy of Teacher Education

http://compalg.inf.elte.hu/~tony/Kutatas/BinPacking/Johnson-BinPacking-NP-Column-JAlg-2002.pdf

The NP-completeness column: An ongoing guide

MOST environments are spatially subdivided, or patchy, and there has been much interest in the relationship between the dynamics of populations at the local and regional (metapopulation) scales1 Here we study mathematical models for host-parasitoid interactions, where in each generation specified fractions (µN and µp, respectively) of the host and parasitoid subpopulations in each patch move to adjacent patches; in most previous work, the movement is not localized but is to any other patch2 These simple and biologically sensible models with limited diffusive dispersal exhibit a remarkable range of dynamic behaviour: the density of the host and parasitoid subpopulations in a two-dimensional array of patches may exhibit complex patterns of spiral waves or spatially chaotic variation, they may show static 'crystal lattice' patterns, or they may become extinct This range of behaviour is obtained with the local dynamics being deterministically unstable, with a constant host reproductive rate and no density dependence in the movement patterns The dynamics depend on the host reproductive rate, and on the values of the parameters µN and µp The results are relatively insensitive to the details of the interactions; we get essentially the same results from the mathematically-explicit Nicholon–Bailey model of host-parasitoid interactions, and from a very general 'cellular automaton' model in which only qualitative rules are specified We conclude that local movement in a patchy environment can help otherwise unstable host and parasitoid populations to persist together, but that the deterministically generated spatial patterns in population density can be exceedingly complex (and sometimes indistinguishable from random environmental fluctuations)

Spatial structure and chaos in insect population dynamics

Contents: Preface. Acknowledgements. D. Kirshner, J.A. Whitson, Editors' Introduction. J. Lave, The Culture of Acquisition and the Practice of Understanding. J. Lemke, Cognition, Context, and Learning: A Social Semiotic Perspective. V. Walkerdine, Redefining the Subject in Situated Cognition Theory. P. Agre, Living Math: Lave and Walkerdine on the Meaning of Everday Arithmetic. D. Kirshner, The Development of Logic in Infancy: A Case Study. J.A. Whitson, Cognition as a Semiotic Process: From Situated Mediation to Critical Reflective Transcendence. P. Cobb, K. Gravemeijer, E. Yackel, K. McClain, J. Whitenack, Mathematizing and Symbolizing: The Emergence of Chains of Signification in One First-Grade Classroom. J.P. Gee, Thinking, Learning, and Reading: The Situated Sociocultural Mind. J. St. Julien, Explaining Learning: The Research Trajectory of Situated Cognition and the Implications of Connectionism. C. Bereiter, Situated Cognition and How to Overcome It. Y. Engestrom, M. Cole, Afterword: Situated Cognition in Search of an Agenda.

Situated cognition : social, semiotic, and psychological perspectives

ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.

On numbers and games, by J. H. Conway. Pp ix, 238. £6·50. 1976. SBN 0 12 186350 6 (Academic Press)

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/50B38C6DB166DD611AC72879EF17432C/S0025557200151927a.pdf/div-class-title-span-class-italic-revisiting-mathematics-education-span-by-hans-freudenthal-pp-200-41-1991-isbn-0-7923-1299-6-kluwer-div.pdf

Revisiting mathematics education , by Hans Freudenthal. Pp 200. £41. 1991. ISBN 0-7923-1299-6 (Kluwer)

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/8D567E36F6EE372954942624971D4E29/S0025557200145759a.pdf/div-class-title-span-class-italic-mathematics-teachers-and-children-span-edited-by-d-pimm-pp-314-6-95-1988-isbn-0-340-48756-9-hodder-and-stoughton-div.pdf

Mathematics, teachers and children , edited by D. Pimm. Pp 314. £6·95. 1988. ISBN 0-340-48756-9 (Hodder and Stoughton)

This chapter revisits the notion of an international curriculum, analyzing the various forces that might push countries toward one and reasons why countries should develop their own distinct curricula. We first describe the term curriculum to set the stage for our later discussion. We then discuss, in turn, common influences for curriculum change, common learning goals, common driving forces of public examinations, common emphases and treatments, and common issues for future curriculum development. Although the tendency for countries to include a more-and-more internationally-accepted core selection of topics in their national curricula is to a great extent both to be welcomed and expected, this move has had a potential negative effect on curriculum development. Significant work also remains to be done to explore the way in which new technology (especially digital technology) could affect both the mathematics included in the curriculum and how it could more effectively contribute to the teaching and learning of mathematics in general.

Toward an International Mathematics Curriculum

The paper begins with a reflective, historical overview of the role and development of textbooks and research into their use. Since much of the writing is based on personal experience, there is an obvious UK bias, although experiences there were not dissimilar to what was happening in many other countries at that time. Particular attention is then paid to comparative studies: the difficulties they pose and what can be usefully gained from them. To help in this latter aspect, an attempt is made to distinguish what would appear to be key features of textbooks (or, indeed, other educational materials) which might be assessed in such a study and, in particular, the vital role of the exercises to be found in them.

Geoffrey Howson

Papers

On numbers and games, by J. H. Conway. Pp ix, 238. £6·50. 1976. SBN 0 12 186350 6 (Academic Press)

Revisiting mathematics education , by Hans Freudenthal. Pp 200. £41. 1991. ISBN 0-7923-1299-6 (Kluwer)

Mathematics, teachers and children , edited by D. Pimm. Pp 314. £6·95. 1988. ISBN 0-340-48756-9 (Hodder and Stoughton)

Toward an International Mathematics Curriculum

The development of mathematics textbooks: historical reflections from a personal perspective