Now showing 1 - 10 of 64
  • Publication
    Open Access
    Zip product of graphs and crossing numbers
    (2020)
    Ouyang, Zhangdong
    ;
    Huang, Yuanqiu
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    ;
    D. Bokal proved that the crossing number is additive for the zip product under the condition of having two coherent bundles in the zipped graphs. This property is very effective when dealing with the crossing numbers of (capped) Cartesian product of trees with graphs containing a dominating vertex. In this paper, we first prove that the crossing number is still additive for the zip product under a weaker condition. Based on the new condition, we then establish some general expressions for bounding the crossing numbers of (capped) Cartesian product of trees with graphs (possibly without dominating vertex). Exact values of the crossing numbers of Cartesian product of trees with most graphs of order at most five are obtained by applying these expressions, which extend some previous results due to M. Klesc. In fact, our results can also be applied to deal with Cartesian product of trees with graphs of order more than five.
    WOS© Citations 1Scopus© Citations 1  250  81
  • Publication
    Open Access
    Concretisations: A support for teachers to carry out instructional innovations in the mathematics classroom
    (2019) ; ; ;
    Quek, Khiok Seng
    ;
    Yap, Romina Ann Soon
    We recognize that though teachers may participate in various forms of professional development (PD) programmes, learning that they may have gained in the PD may not always lead to corresponding perceivable changes in their classroom teaching. We offer a theoretical re-orientation towards this issue by introducing a construct we term “concretisation”. Concretisations are resources developed in PD settings which can be converted into tangible tools for classroom use. In theorising such resources, we contribute in informing the design process of teacher professional development for better impact into actual classroom practice. We purport principles of design which render concretisations effective. Subsequently, we test these principles by presenting a specific case of teaching mathematical problem solving.
    WOS© Citations 4Scopus© Citations 4  135  164
  • Publication
    Open Access
    Mathematical problem solving for everyone: Infusion and diffusion (MinD)
    (2016) ; ; ;
    Quek, Khiok Seng
    ;
    ;
    Dindyal, Jaguthsing
    ;
    Ho, Foo Him
    ;
    Hang, Kim Hoo
    ;
    Yen, Yeen Peng
      238  185
  • Publication
    Open Access
    Teacher preparation for a problem solving curriculum
    (2009-07) ; ;
    Quek, Khiok Seng
    ;
    Dindyal, Jaguthsing
    ;
    The role of the teacher is central to the success of any curriculum innovation. Thus, teachers’ professional development has become an increasingly important subject of discussion in recent education literature. In the design and implementation of the project reported here, teachers’ preparation for the problem-solving curriculum featured prominently. This paper discusses the challenges of selecting a suitable problem and ways of using it productively within a professional development programme that the authors carried out for the teachers involved in the project.
      296  160
  • Publication
    Metadata only
    Paving the way for mathematical literacy in the 21st century: Pre-service mathematics education, professional development and professional networks
    The quality of any education system is significantly dependent on the quality of teachers in that system (Barber, Mourshed, How the World’s Best-Performing School Systems Come out on Top. McKinsey & Company, New York, 2007). A twenty-first century teacher professionalism requiring specialist knowledge and skills is essential for ensuring the quality of teachers (Darling-Hammond, Journal of Teacher Education 6:35–47, 2010). This chapter presents a multi-faceted and multi-dimensional framework which synergises a teacher education institute, the Ministry of Education, and professional teacher organisations in providing teacher education for a twenty-first century mathematics teacher in Singapore from pre-service through life-long professional development. Singapore’s pragmatic approach in preparing teachers who can adapt to the constantly changing education landscape will be discussed. Directions for future developments towards life-long, life-wide, life-deep, and life-wise Singapore teacher education will be outlined.
      24
  • Publication
    Metadata only
    K-12 school mathematics curriculum: Insights on development, renewal and future orientation
    In Singapore, nationwide educational policies and movements have taken place frequently and within a short space of time from each other. In turn, such educational initiatives get translated into changes in curricula of every school subject – mathematics inclusive. In this chapter, we make an explicit connection between Singapore students’ PISA performance and the aforementioned curricular shifts by highlighting the major changes that have taken place in K-12 Singapore school mathematics curriculum, analysing them in terms of the shifts in curriculum ideologies. Then we map each of the dimensions of the PISA assessment framework with the components of the Singapore Mathematics Curriculum Framework to further substantiate the claim that “the [Singapore] education system and school mathematics curriculum contribute in part towards the success of Singapore’s students in … PISA” (Kaur et al., Mathematics education in Singapore. Springer, Singapore, 2019, p. 134). Additionally, we give some answers to the “Ten Questions for Mathematics Teachers … and how PISA can help answer them” (OECD, PISA, OECD Publishing, Paris, 2016) that are relevant to the Singapore context. Based on the twenty-first-century competencies identified, respectively, by the OECD and Ministry of Education (Singapore), we explore possible new directions the national mathematics curriculum may head towards and hope to peek into the future education landscape for Singapore mathematics.
      16
  • Publication
    Metadata only
    The evolution of mathematics education research in Singapore

    Up until 1990, the Institute of Education in Singapore was primarily a teaching institute involved in training teachers for Singapore schools. Since the inception of the National Institute of Education (NIE) in 1990, as an institute of the Nanyang Technological University, the focus of the institute has been enlarged to include research in education. This chapter examines, through a documentary analysis, how a research culture specifically in mathematics education at the National Institute of Education was nurtured, developed and supported from 1990 onwards. Development of the culture for Mathematics Education Research (MER) has been in tandem with all other areas of research at the NIE. Both top-down and bottom-up approaches have been adopted to support research as part of an academic’s work at the institute. Policies related to recruitment and promotion of academics were developed to ensure that emphasis was on both teaching and research. Development of research, from individually led bite-sized grains to team-based project with coherent themes, was supported. The setting up of the Centre for Research in Pedagogy and Practice in 2004 and dedicated funding from the Ministry of Education Singapore for research of the Singapore education system heralded an era of MER that has made significant contributions both nationally and internationally. This chapter will also illuminate the four main areas of focus and sources on MER through examples of studies carried out in Singapore since 2000. In addition, it briefly outlines the contribution of MER in ASEAN countries.

      12
  • Publication
    Open Access
    Infusing problem solving into mathematics content course for pre-service secondary school mathematics teachers
    (2013) ;
    Quek, Khiok Seng
    ;
    ; ; ;
    Ho, Foo Him
    ;
    Dindyal, Jaguthsing
    This paper presents a re-design of an undergraduate mathematics content course on Introductory Differential Equations for pre-service secondary school mathematics teachers. Based on the science practical paradigm, mathematics practical lessons emphasizing problem-solving processes via the undergraduate content knowledge were embedded within the curriculum delivered through the traditional lecture-tutorial system. The pre-service teachers' performance in six mathematics practical lessons and the mathematics practical test was examined. They were able to respond to the requirements of the mathematics practical to go through the entire process of problem solving and to carry out "Look Back" at their solution: checking the correctness of their solution, offering alternative solutions, and expanding on the given problem. The use of Mathematics Practical has altered the pre-service teachers’ approach in tackling mathematics problems in a positive direction.
      216  485
  • Publication
    Open Access
    Mathematical problem solving for everyone: A new beginning
    (2012)
    Dindyal, Jaguthsing
    ;
    ; ; ;
    Quek, Khiok Seng
    Mathematical problem solving has been at the core of the Singapore mathematics curriculum framework since the 1990s. We report here the features of the Mathematical Problem Solving for Everyone (M-ProSE) project which was carried out in a Singapore school to realise the learning of mathematical problem solving and as described by Pólya and Schoenfeld. A mathematics problem solving package comprising “mathematics practical” lessons and assessment rubric was trialled in the school for Grade 8 in 2009. Responses from three students show mixed perceptions to the module, but an end-of-module assessment shows that the students were able to present their solutions along Pólya’s four stages. We also describe teacher preparation for teaching the module. After the trial period, the school adopted the module as part of the curriculum and it is now a compulsory course for all Grade 8 students in that school.
      548  717
  • Publication
    Open Access
    Diffusion of the mathematics practical paradigm in the teaching of problem solving: Theory and praxis
    (2012)
    Quek, Khiok Seng
    ;
    ; ; ;
    Dindyal, Jaguthsing
    In this paper, we discuss the diffusion (of an innovation) and relate it to our attempt to spread our initial design of a mathematics practical paradigm in the teaching of problem solving.
      248  157