Now showing 1 - 10 of 35
  • Publication
    Open Access
    An introduction to topology and its applications: A new approach
    (2012-10)
    The present paper aims to introduce the topic of topology to a reader with little background knowledge of university mathematics. Unlike the traditional approach which begins by abstracting of real analysis to metric spaces and then routing to topological spaces, our present exposition of opology exploits some basic experience with computations, scientific measurements and observations, such as those encountered in the use of a scientific calculator, a computer program or a measuring device. It is hoped that, in the present age of computers and information technology, topology, which is known to be an abstract mathematical subject, can be made palatable and relevant to a wider audience. Conjugate to our main discourse, we introduce some necessary knowledge of functional programming which is needed by the reader to understand how applications of topology can be realized in a concrete manner.
      136  212
  • Publication
    Open Access
    The groundhog problem
    (2020) ;
    Liljedahl, Peter
    The Groundhog Problem is stated as follows: “A groundhog has made an infinite number of holes roughly a metre apart in a straight line in both directions on an infinite plane. Every day it travels a fixed number of holes in one direction. A farmer would like to catch the groundhog by shining a torch, but only once a night, into one of the holes at midnight when it is asleep. What strategy can the farmer use to ensure that he catches the groundhog eventually?” It turns out that the solution of this problem relies on a set-theoretic concept usually taught at the tertiary level. One key purpose of this paper is to explicitly articulate the problem solving trajectory of a professional mathematician who is cognizant of his/her own problem solving disposition and thinking.
      272  290
  • Publication
    Metadata only
    The WF-rank of topological spaces
    (2022)
    Li, Qingguo
    ;
    Liu, Bei
    ;
    The class of well-filtered spaces is one of the important class of topological spaces in domain theory. Recently, the existence of well-filterification of T0-spaces has been established. In this paper, we introduce the definition of wf-rank and prove that for any ordinal α, there exists a T0-space whose wf-rank equals α.
      34
  • Publication
    Metadata only
    Applications of Scott-closed sets in convex structures
    (2022)
    Yue, Yueli
    ;
    Yao, Wei
    ;
    The aim of this paper is to show that Scott-closed sets have a good application in convex structures. Firstly, based on remotehood system, we give a characterization of convex structure by the Kleisli monoid with respect to the Scott-closed set monad. Then, by an op-canonical lax extension of the Scott-closed set monad, we introduce convex convergence spaces and prove that convex structures are precisely the reflexive and transitive lax algebras. Finally, we study the relationship between ordered structures and separated convex structures.
    WOS© Citations 6Scopus© Citations 5  42
  • Publication
    Open Access
    The impact of online video suite on the Singapore pre-service teachers’ buying-in to innovative teaching of factorisation via Algecards
    (2015) ; ;
    Ho, Foo Him
    A group of pre-service teachers at one institute of education was assigned to record their reflections as they viewed an online video suite which recorded (1) a secondary mathematics teacher employing a Concrete-Pictorial-Abstract (CPA)-based approach to teaching quadratic expansion and factorisation to a group of low-ability secondary 2 (14+) mathematics students; (2) students using concrete manipulative, AlgeCards, and pictorial representations, Rectangle Diagram, to perform quadratic expansion and factorisation; and (3) teachers, including a Head of Department (HOD), sharing their teaching experience in the use of the teaching package based on the abovementioned CPA approach. This chapter studied the impact of the online video suite on the pre-service teachers’ willingness to buy-in to innovative teaching of factorisation via AlgeCards through inspecting the written responses of these teachers to preset Milestone Tasks placed at different points along the progression of the video watching. The term ‘buy-in’ refers to the evidential shift of teacher’s belief in the applicability of the lesson innovation. We identified the salient features of the video suite which were responsible for bringing about pre-service teachers’ buying-in to the use of AlgeCards in teaching factorisation. Based on these findings, this chapter makes some recommendations on the design of teacher preparation method course using video technology.
    WOS© Citations 1Scopus© Citations 2  100  56
  • Publication
    Open Access
    Flipping the topology classroom
    (National Institute of Education (Singapore), 2017)
      45  33
  • Publication
    Open Access
    Pre-service teachers’ use of symmetry of quadratic graphs in problem solving
    (2012) ;
    Ho, Foo Him
    ;
    Dindyal, Jaguthsing
    This paper reports on the performance of pre-service mathematics teachers with regards to the use of symmetry in problem solving. The current study reveals that pre-service teachers do not make use of symmetry as their main problem-solving tool, even in situations where symmetry is the obvious notion to consider. In addition, a quantitative comparison of the effectiveness of problem solving among approaches (those relying on symmetry versus those relying on other conventional methods) is reported herein. This formal comparison validates the opinion that active usage of symmetry in problem solving significantly enhances the chance of solving non-routine problems.
      154  143
  • Publication
    Open Access
      209  132
  • Publication
    Open Access
    Putnam grasshopper problem: Collaborative problem solving, generalisations, and computational thinking
    (Association of Mathematics Educators, 2023) ; ; ;
    Tong, Cherng Luen
    This paper considers several generalisations of an interesting problem posed in the 2021 William Lowell Putnam Mathematical Competition. We describe how the authors embarked on a collaborative problem-solving journey which resulted in solving two of these generalisations and how computational thinking guided our approaches.
      8  68
  • Publication
    Open Access
    Passing a proof message: Student-teacher communication through a commognitive lens
    This study employs Sfard’s (2008) socio-cultural theory of Commognition to analyse student teachers’ thinking and communicating practices. Specifically, we investigate the effectiveness of the student teachers’ communication of a particular mathematical proof with reference of the four features of the commognitive framework, i.e., word use, visual mediators, narrative and routines. In this paper, we can report on the routine of the discourse to analyse the quality of mathematical discourse in two situations of “Expert-to-Novice” and “Novice-to-Novice”.
      218  194