Semicontinuity of Betweenness Functions

A ternary relational structure〈X,[⋅,⋅,⋅]〉, interpreting a notion of betweenness, gives rise to the family of intervals, with interval [a,b] being defined as the set of elements of X between a and b. Under very reasonable circumstances, X is also equipped with some topological structure, in such a way that each interval is a closed nonempty subset of X. The question then arises as to the continuity behavior—within the hyperspace context—of the betweenness function {x,y}↦[x,y]. We investigate two broad scenarios: the first involves metric spaces and Menger\u27s betweenness interpretation; the second deals with continua and the subcontinuum interpretation

Reflection on Practice, in Practice: The Discipline of Noticing

This paper outlines the use of John Mason’s Discipline of Noticing by a group of university level mathematics lecturers. We describe the aims that motivated the study, the challenges we faced in using the Discipline of Noticing to reflect on our teaching, and the progress that we have made

Mathematical Borders? Comparing student teachers’ mathematical identity in Ireland North and South

Mathematical identity is considered as the multi-faceted relationship that an individual has with mathematics, including knowledge and experiences, perceptions of oneself and others. The SCoTENS funded project, named MIST, examined the mathematical identity of student teachers who had chosen to specialise in mathematics in their B.Ed. programme. Students were drawn from two institutions, one in Northern Ireland, one in the Republic and narrative was used as a tool to access mathematical identity. This paper reports on the analysis of similarities and differences between the mathematical identity of these students teachers in each part of the island of Ireland and discusses how this informs our reflections on the mathematical education experienced

Undergraduate topology: a working textbook

This textbook offers an accessible, modern introduction at undergraduate level to an area known variously as general topology, point-set topology or analytic topology with a particular focus on helping students to build theory for themselves. It is the result of several years of the authors' combined university teaching experience stimulated by sustained interest in advanced mathematical thinking and learning, alongside established research careers in analytic topology. Point-set topology is a discipline that needs relatively little background knowledge, but sufficient determination to grasp

Undergraduate topology : a working textbook

xii, 144 p.; 23 c

Bilingual mathematics learners, conceptual mathematical activity and the role of their languages. How best to investigate?

International audienceThe significant role of language in mathematics teaching and learning is not a new phenomenon. Investigating bilingual mathematics learners is complex and research has demonstrated that language switching practices are also complex and involve not only social and cultural aspects, but also cognitive aspects. However, little investigation has been undertaken into the specific role of languages and their influences on conceptual activity at undergraduate level. The framework, and future research directions, presented in this paper aim to investigate further the cognitive aspects of bilingual learners and their use of their languages, when engaged in conceptual mathematical activity

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We prove it to be consistent that there is a poset of cardinality c2 which is not realizable in P(R), ordered by homeomorphic embeddability. This addresses and answers resolutely (and in the negative) the open question of whether there is a ZFC theorem that all posets of cardinality c2 can be represented by subspaces of the real line ordered by homeomorphic embeddability. This question arises from the pioneering work of Banach, Kuratowski and Sierpiński in the area and this result complements the recent work of [A.E. McCluskey, D. Shakhmatov, It is consistent that all posets of cardinality c2 can be realized within P(R) preprint], thereby providing a proof of independence.Copyright 2009 Elsevier B.V. All rights reserved. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://www.elsevier.com/open-access/userlicense/1.0