Designing to see and share structure in number sequences

This paper reports on a design experiment in the domain of number sequences conducted in the course of the WebLabs project. We iteratively designed and tested a set of activities and tools in which 10-14 year old students used the ToonTalk programming environment to construct models of sequences and series, and then shared their models and their observations about them utilising a webbased collaboration system. We report on the evolution of a design pattern (programming method) called ‘Streams’ which enables students to engage in the process of summing and ‘hold the series in their hand’, and consequently make sophisticated arguments regarding the mathematical structures of the sequences without requiring the use of algebra. While the focus of this paper is mainly on the design of activities, and in particular their epistemological foundations, some illustrative examples of one group of students’ work indicate the potential of the activities and tools for expressing and reflecting on deep mathematical ideas

Jorge A. Swieca's contributions to quantum field theory in the 60s and 70s and their relevance in present research

After revisiting some high points of particle physics and QFT of the two decades from 1960 to 1980, I comment on the work by Jorge Andre Swieca. I explain how it fits into the quantum field theory during these two decades and draw attention to its relevance to the ongoing particle physics research. A particular aim of this article is to direct thr readers mindfulness to the relevance of what at the time of Swieca was called "the Schwinger Higgs screening mechanism". which, together with recent ideas which generalize the concept of gauge theories, has all the ingredients to revolutionize the issue of gauge theories and the standard model.Comment: 49 pages, expansion and actualization of text, improvement of formulations and addition of many references to be published in EPJH - Historical Perspectives on Contemporary Physic

Windows on the infinite : constructing meanings in a Logo-based microworld.

This thesis focuses on how people think about the infinite. A review of both the\ud historical and psychological/educational literature, reveals a complexity which\ud sharpens the research questions and informs the methodology. Furthermore, the areas\ud of mathematics where infinity occurs are those that have traditionally been presented\ud to students mainly from an algebraic/symbolic perspective, which has tended to make\ud it difficult to link formal and intuitive knowledge. The challenge is to create situations\ud in which infinity can become more accessible. My theoretical approach follows the\ud constructionist paradigm, adopting the position that the construction of meanings\ud involves the use of representations; that representations are tools for understanding;\ud and that the learning of a concept is facilitated when there are more opportunities of\ud constructing and interacting with external representations of a concept, which are as\ud diverse as possible.\ud Based on this premise, I built a computational set of open tools — a\ud microworld — which could simultaneously provide its users with insights into a range\ud of infinity-related ideas, and offer the researcher a window into the users' thinking\ud about the infinite. The microworld provided a means for students to construct and\ud explore different types of representations — symbolic, graphical and numerical — of\ud infinite processes via programming activities. The processes studied were infinite\ud sequences and the construction of fractals. The corpus of data is based on case studies\ud of 8 individuals, whose ages ranged from 14 to mid-thirties, interacting as pairs with\ud the microworld. These case studies served as the basis for an analysis of the ways in\ud which the tools of the microworld structured, and were structured by, the activities.\ud The findings indicate that the environment and its tools shaped students'\ud understandings of the infinite in rich ways, allowing them to discriminate subtle\ud process-oriented features of infinite processes, and permitted the students to deal with\ud the complexity of the infinite by assisting them in coordinating the different\ud epistemological elements present. On a theoretical level, the thesis elaborates and\ud refines the notion of situated abstraction and introduces the idea of "situated proof"

Knowledge used for teaching counting: A case study of the treatment of counting by two Grade 3 teachers situated in schools serving working class communities in the Western Cape Province of South Africa

Knowing how to correctly count, is fundamental to the future mathematics success of young children. Earlier studies show that many South African primary school students underperform in mathematics even when evaluated with task below grade level. Reports suggest that this is a problem stemming from the poor pedagogic, and or content knowledge of classroom mathematics teachers. Shulman (1986; 1987) refers to this area of knowledge as Pedagogic Content Knowledge (PCK). In the field of mathematics teaching and learning, Ball, Thames and Phelps (2008) refer to it as Mathematics Knowledge for Teaching (MKfT). Teachers' mathematics PCK, comprises of three core knowledge domain: (i) Teacher's Knowledge of Content and Teaching (KCT); (ii) Teacher's Knowledge of Content and Student (KCS); and (iii) teacher's Knowledge of Content and Curriculum (KCC). Teachers' KCS was considered in this study as it concerns what teachers know about what learners know and how they learn. The general interest of this project was to study the construction of experience of mathematics (non-core domain knowledge) by genetic endowment on the basis of contextual data. More specifically, the particular interest of the study is on the construction of the experience of counting in the pedagogic situations of Grade 3 schooling. For that purpose, video records of mathematics teaching in two schools situated in working-class communities were analysed. The study adopted an Integrated Causal Model approach which drew on resources from different disciplines such as mathematics education, cognitive science, evolutionary psychology and mathematics. The study was partly framed by Bernstein's pedagogic device, particularly with respect to his notion of evaluation, as well as the inter-related constructs of PCK, MKfT and KCS. The theoretical resources used to describe computations were drawn largely from Davis (2001, 2010b, 2011a, 2012, 2013a, 2015, 2018) and related work on the use of morphisms as elaborated in Baker et al. (1971), Gallistel & King, (2010), Krause (1969) and Open University (1970). These resources were used to produce the analytic framework for the production of and analysis of data. The analysis describes the computational activities of teachers and learners during the recorded lessons, specifically the computational domains made available pedagogically. In so doing, I was able to provide more illumination on what is described as teacher's KCS for teaching counting at the Grade 3 level. From the generated data, the study finds that counting proper was restricted to the constitution and identification of very small ordered discrete aggregates which can be handled by human core domain object tracking system and approximate number system, and that an implicit reliance on numerical order derived from computations on aggregates was central to the teaching and learning of counting

Investigating the foundations of Turkish elementary mathematics education through an analysis of a late ottoman textbook

Ankara : The Program of Curriculum and Instruction İhsan Doğramacı of Bilkent University, 2015.Thesis (Master's) -- Bilkent University, 2015.Includes bibliographical references leaves 70-83.Developing an understanding of the foundations of the educational tradition of the Turkish Republic is connected to an exploration of the specifics of Ottoman education. This qualitative study explored an Ottoman mathematics textbook published in the early twentieth century. Under the influence of naturalistic inquiry, the textbook was analyzed in terms of content, organization, and principles of elementary mathematics education. It was concluded that the textbook is successfully presented multiple representations and real-life examples while the development of content did not provide opportunities to develop reasoning skills.Yaprak, EsraM.S

Developing understanding of triangle

As children develop concepts of shape they move from a visual understanding to a property based approach to classification. In this study two cohorts, one a longitudinal study from grade 1 to 4 and the other a sample across a school from pre-school to grade 8, were asked to identify triangles. The resulting data shows errors of inclusion are greater than errors of exclusion and suggests an order in which particular properties are attended to as children learn