Embracing the Crisis in the Foundations of Mathematics

The crisis in the foundations of mathematics is a conceptual crisis. I suggest that we embrace the crisis and adopt a pluralist position towards foundations. There are many foundations in mathematics. However, ‘many foundations’ (for one building) is an oxymoron. Therefore, we shift vocabulary to say that mathematics, as one discipline, is composed of many different theories. This entails that there are no absolute mathematical truths, only truths within a theory. There is no unified, consistent ontology, only ontology within a theory.

Models to illustrate the foundations of mathematics /

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Investigating functional thinking in the elementary classroom : foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.

Politics, Power, PISA: a genealogy of mathematics education policy at second level in Ireland at the beginning of the 21st century

This thesis traces a genealogy of the discourse of mathematics education reform in Ireland at the beginning of the twenty first century at a time when the hegemonic political discourse is that of neoliberalism. It draws on the work of Michel Foucault to identify the network of power relations involved in the development of a single case of curriculum reform – in this case Project Maths. It identifies the construction of an apparatus within the fields of politics, economics and education, the elements of which include institutions like the OECD and the Government, the bureaucracy, expert groups and special interest groups, the media, the school, the State, state assessment and international assessment. Five major themes in educational reform emerge from the analysis: the arrival of neoliberal governance in Ireland; the triumph of human capital theory as the hegemonic educational philosophy here; the dominant role of OECD/PISA and its values in the mathematics education discourse in Ireland; the fetishisation of western scientific knowledge and knowledge as commodity; and the formation of a new kind of subjectivity, namely the subjectivity of the young person as a form of human-capital-to-be. In particular, it provides a critical analysis of the influence of OECD/PISA on the development of mathematics education policy here – especially on Project Maths curriculum, assessment and pedagogy. It unpacks the arguments in favour of curriculum change and lays bare their ideological foundations. This discourse contextualises educational change as occurring within a rapidly changing economic environment where the concept of the State’s economic aspirations and developments in science, technology and communications are reshaping both the focus of business and the demands being put on education. Within this discourse, education is to be repurposed and its consequences measured against the paradigm of the Knowledge Economy – usually characterised as the inevitable or necessary future of a carefully defined present.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Theoretical Foundations of Developing Modeling Instruction Curriculum for College Biology Courses

Modeling Instruction (MI) has been successfully implemented in high school science classes. Moreover, MI curriculum for introductory physics has also been developed at a university level. Noticing the gap, the author will provide theoretical foundations to support the statement that MI curriculum should be developed for college biology courses.

Getting skilled for education, training and employment : Indigenous Cultural Knowledges and mainstream knowledges of mathematics

This abstract is a preliminary discussion of the importance of blending of Indigenous cultural knowledges with mainstream knowledges of mathematics for supporting Indigenous young people. This import is emphasised in the documents Preparing the Ground for Partnership (Priest, 2005), The Indigenous Education Strategic Directions 2008–2011 (Department of Education, Training and the Arts, 2007) and the National Goals for Indigenous Education (Department of Education, Employment and Work Relations, 2008). These documents highlight the contextualising of literacy and numeracy to students’ community and culture (see Priest, 2005). Here, Community describes “a culture that is oriented primarily towards the needs of the group. Martin Nakata (2007) describes contextualising to culture as about that which already exists, that is, Torres Strait Islander community, cultural context and home languages (Nakata, 2007, p. 2). Continuing, Ezeife (2002) cites Hollins (1996) in stating that Indigenous people belong to “high-context culture groups” (p. 185). That is, “high-context cultures are characterized by a holistic (top-down) approach to information processing in which meaning is “extracted” from the environment and the situation. Low-context cultures use a linear, sequential building block (bottom-up) approach to information processing in which meaning is constructed” (p.185). In this regard, students who use holistic thought processing are more likely to be disadvantaged in mainstream mathematics classrooms. This is because Westernised mathematics is presented as broken into parts with limited connections made between concepts and with the students’ culture. It potentially conflicts with how they learn. If this is to change the curriculum needs to be made more culture-sensitive and community orientated so that students know and understand what they are learning and for what purposes.

Theories of Mathematics Education : Seeking New Frontiers

This inaugural book in the new series Advances in Mathematics Education is the most up to date, comprehensive and avant garde treatment of Theories of Mathematics Education which use two highly acclaimed ZDM special issues on theories of mathematics education (issue 6/2005 and issue 1/2006), as a point of departure. Historically grounded in the Theories of Mathematics Education (TME group) revived by the book editors at the 29th Annual PME meeting in Melbourne and using the unique style of preface-chapter-commentary, this volume consist of contributions from leading thinkers in mathematics education who have worked on theory building. This book is as much summative and synthetic as well as forward-looking by highlighting theories from psychology, philosophy and social sciences that continue to influence theory building. In addition a significant portion of the book includes newer developments in areas within mathematics education such as complexity theory, neurosciences, modeling, critical theory, feminist theory, social justice theory and networking theories. The 19 parts, 17 prefaces and 23 commentaries synergize the efforts of over 50 contributing authors scattered across the globe that are active in the ongoing work on theory development in mathematics education.

Surveying theories and philosophies of mathematics education

Any theory of thinking or teaching or learning rests on an underlying philosophy of knowledge. Mathematics education is situated at the nexus of two fields of inquiry, namely mathematics and education. However, numerous other disciplines interact with these two fields which compound the complexity of developing theories that define mathematics education. We first address the issue of clarifying a philosophy of mathematics education before attempting to answer whether theories of mathematics education are constructible? In doing so we draw on the foundational writings of Lincoln and Guba (1994), in which they clearly posit that any discipline within education, in our case mathematics education, needs to clarify for itself the following questions: (1) What is reality? Or what is the nature of the world around us? (2) How do we go about knowing the world around us? [the methodological question, which presents possibilities to various disciplines to develop methodological paradigms] and, (3) How can we be certain in the “truth” of what we know? [the epistemological question]