Now showing 1 - 10 of 29
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    Promote active learning in mathematics
    The main objective of this project is to explore the various applications of problem posing strategy in promoting active learning in higher mathematics, especially in modern algebra and commutative algebra.
      128  31
  • Publication
    Open Access
    An on-line integration of undergraduate course notes
    (2000-09) ; ;
    Cheang, Gerald
    ;
    Phang, Rosalind Lay Ping
    ;
    Tang, Wee Kee
    Undergraduate mathematics courses in most universities would consist of numerous modules or subjects. The definitions, theorems and concepts taught in one module are very often based on what has been introduced in previously taught modules. This is especially true for the more advanced modules. Although there is usually a prescribed syllabus for each module, it is impossible to list all the details in the syllabus. This may lead to significant differences when a module is taught by different lecturers. One way to address this problem is to provide an easily accessed and linked set of notes for all fundamental modules in reasonable detail. This may serve to prevent omission of certain topics in earlier modules and repetitions in later modules. In this paper, we describe our experience in developing an integrated set of online undergraduate course notes. In addition, we shall also explain how Maple worksheets are merged into the system for students to explore the mathematical concepts dynamically.
      118  127
  • Publication
    Open Access
    When exactly is Scott sober?
    A topological space is sober if every nonempty irreducible closed set is the closure of a unique singleton set. Sobriety is precisely the topological property that allows one to recover completely a topological space from its frame of opens. Because every Hausdor space is sober, sobriety is an overt, and hence unnamed, notion. Even in non-Hausdor settings, sober spaces abound. A well-known instance of a sober space appears in domain theory: the Scott topology of a continuous dcpo is sober. The converse is false as witnessed by two counterexamples constructed in the early 1980's: the first by P.T. Johnstone and the second (a complete lattice) by J. Isbell. Since then, there has been limited progress in the quest for an order-theoretic characterization of those dcpo's for which their Scott topology is sober. This paper provides one answer to this open problem.
      229  193
  • Publication
    Open Access
    First-countability, ω-Rudin spaces and well-filtered determined spaces
    (2021)
    Xu, Xiaoquan
    ;
    Shen, Chong
    ;
    Xi, Xiaoyong
    ;
    In this paper, we investigate some versions of d-space, well-filtered space and Rudin space concerning various countability properties. It is proved that every space with a first-countable sobrification is an ω-Rudin space and every first-countable space is well-filtered determined. Therefore, every ω-well-filtered space with a first-countable sobrification is sober. It is also shown that every irreducible closed subset in a first-countable ω-well-filtered space is countably directed, hence every first-countable ω*-well-filtered d-space is sober.
    WOS© Citations 4Scopus© Citations 4  49  15
  • Publication
    Open Access
    Topologies generated by families of sets and strong poset models
    (2020) ;
    Xi, Xiaoyong
    ;
    Chen, Yixiang
    A poset model of a topological space X is a poset P such that X is homeomorphic to the maximal point space of P (the set Max(P) of all maximal points of P equipped with the relative Scott topology of P). The poset models of topological spaces based on other topologies, such as Lawson topology and lower topology, have also been investigated by other people. These models establish various types of new links between posets and topological spaces. In this paper we introduce the strong Scott topology on a poset and use it to de ne the strong poset model: a strong poset model of a space X is a poset P such that Max(P) (equipped with the relative strong Scott topology) is homeomorphic to X. The main aim is to establish a characterization of T1 spaces with T-generated topologies (such as the Hausdor k-spaces) in terms of maximal point spaces of posets. A poset P is called ME-separated if for any elements x; y of P, x y i " y \ Max(P) "x \ Max(P). We consider the topological spaces that have an ME-separated strong poset model. The main result is that a T1 space has an ME-separated strong poset model i its topology is T-generated. The class of spaces whose topologies are T-generated include all Scott spaces and all Hausdor k-spaces.
      313  151
  • Publication
    Open Access
    Real-life mathematics tasks: A Singapore experience
    (2012)
    Wong, Khoon Yoong
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    Cheang, Wai Kwong
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    Lee, P. Y. (Peng Yee)
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    Yen, Yeen Peng
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    Fan, Lianghuo
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    ;
    Quek, Khiok Seng
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    So, Hyo-Jeong
    ;
    Ng, Yvonne Qiu Ting
    ;
    Cheong, Jim Siew Kuan
    ;
    Oh, Kwang Shin
      842  9420
  • Publication
    Open Access
    Assessing mathematical competencies using disciplinary tasks
    (2012)
    Cheang, Wai Kwong
    ;
    ;
    The Singapore Mathematics Assessment and Pedagogy Project (SMAPP) is a research project conducted by the National Institute of Education and funded by the Ministry of Education. It aims to make assessment practices an integral part of teaching and learning, and broaden student learning outcomes by using authentic disciplinary tasks. As part of the project, some guidelines are provided for designing disciplinary tasks which have the distinctive features of their emphasis on contextual aspects. One of the criteria of a good disciplinary task is its ability to assess multiple mathematical competencies of students. In this paper, we will present some examples to illustrate how these competencies can be assessed. Another aim is to find out to what extent these tasks serve the purpose of assessing these competencies, by analyzing the students’ performance in a sample SMAPP task.
      218  278
  • Publication
    Open Access
    Learning mathematics through exploration and connection
    (2001) ; ;
    Cheang, Gerald
    ;
    Phang, Rosalind Lay Ping
    ;
    Tang, Wee Kee
      127  102
  • Publication
    Open Access
    Problem-posing in teaching university algebra
    (1999-12) ;
    Lee, P. Y. (Peng Yee)
    Posing or raising appropriate problems is necessary and important for active and deep learning in mathematics. However, students rarely make effort to find thinking problems by themselves. Such an attitude and behavior often lead to passive learning and cause various difficulties and problems in mathematics teaching. The main objective of this paper is to explore the ways to develop students’ ability to find and pose good mathematical problems and thus to promote more active learning in mathematics.
      119  107
  • Publication
    Restricted
    Gauges of Baire class one functions
    (2007)
    Tang, Wee Kee
    ;
    Zulijanto Atok
    ;
      346  21