Now showing 1 - 10 of 29
  • Publication
    Open Access
    Constructing an exemplary mathematical justification: Why is it so hard for mathematics teachers?
    (National Institute of Education (Singapore), 2019)
    Chua, Boon Liang
    ;
    ;
    Yap, Von Bing
      106  87
  • Publication
    Open Access
    Twelve erroneous proofs
    Assessing how well students understand proofs is difficult. One way to achieve this is to present an erroneous proof and require students to identify the error. Twelve erroneous proofs on various topics in mathematics at the Secondary school level are presented as examples.
      100  77
  • Publication
    Open Access
    Scaffolding cards: A strategy for facilitating groups in problem solving
    (2013) ;
    Dindyal, Jaguthsing
    ;
    Ho, Foo Him
    Problem solving task design is not only the design of a non-routine problem to be solved by the students. Our task design also requires a supporting document, the practical worksheet, which would act as a cognitive scaffold for the students in the initial stages of the problem solving process before they can internalize the metacognitive strategies and automate the use of these strategies when faced with a new problem. A further enhancement of the scaffolding that can be provided by the teacher as she facilitates forty or more students working on the practical worksheet is a set of scaffolding cards. In this paper, we describe the cards and the preliminary use of these cards to facilitate problem solving for teachers in a professional development workshop.
      121  110
  • Publication
    Open Access
    Fine-tuning in a design experiment
    (2013)
    Ho, Foo Him
    ;
    ;
    Quek, Tay, Toh, Leong, and Dindyal (2011) proposed that a design-theory-practice troika should always be considered for a designed package to be acceptable to the research users who, in this case, are teachers and schools. This paper describes the fine-tuning to the MProSE problem-solving design made by the teachers in the school after first round of teaching. This process involved teacher input from their experience, and detailed time-consuming discussions and learning between the researcher-designers and the teacher-implementers.
      112  148
  • Publication
    Open Access
    Use of practical worksheet in teacher education at the undergraduate and postgraduate levels
    (2012) ; ;
    Ho, Foo Him
    ;
    Quek, Khiok Seng
    We have applied the ‘practical paradigm’ in teaching problem solving to secondary school students. The key feature of the practical paradigm is the use of a practical worksheet to guide the students’ processes in problem solving. In this paper, we report the diffusion of the practical paradigm to university level courses for prospective and practising teachers. The higher level of mathematics content would demand higher order thinking skills. Learners without a model of problem solving would often revert to solving by referring to many examples of the same ‘type’ of problem. Polya-type problem solving skills framed by the practical worksheet was used as an attempt to elicit more effective problem solving behaviour from them. Preliminary findings show that they were able to use the practical worksheet to model their solution of problems in the courses.
      196  141
  • Publication
    Open Access
    Calculus for teaching and learning (CASTLE): An exploratory study
    (National Institute of Education (Singapore), 2022) ; ; ; ;
    Tan, Victor
    ;
    Tang, Wee Kee
      271  117
  • Publication
    Open Access
    Mathematical problem solving for everyone: Infusion and diffusion (MInD)
    (Office of Education Research, National Institute of Education, Singapore, 2020) ; ; ;
    Quek, Khiok Seng
    ;
    ;
    Dindyal, Jaguthsing
    ;
    Ho, Foo Him
    This research project is an attempt to realise the ideals of mathematical problem solving, which is at the heart of the Singapore mathematics curriculum in the daily practices of mainstream mathematics classrooms. This work builds on the foundation of M-ProSE (OER 32/08 TTL) to diffuse the findings to the mainstream school curriculum. Our work involves three steps: (1) initialisation of problem solving as an essential part of the mathematics curriculum in a school at the foundational year; (2) infusion of problem solving as an embedded regular curricular and pedagogical practice across all year levels in the school, and (3) diffusion of this innovation from this school to the full range of schools in Singapore. In each of the above steps, we take a complex systems approach and include curriculum, instructional practices, assessment and teacher professional development in our overall design research process. Our current project builds upon the initial foundation of MProSE to scale out (infuse) and scale up (diffuse) the innovation to mainstream schools in Singapore, hence the project is named MInD. With the experience and data collected from MProSE research school, the design needs to be re-adjusted in order for problem solving to be diffused throughout the mainstream schools. The importance and relevance of this research project to schools is readily observed by the schools' responses: To the researchers' pleasant surprise, four mainstream schools readily expressed their commitment to participate in this research as the school leaders see the relevance of this project to their school curriculum. Further, the Principal of MProSE research school expressed his interest to get his school involved for the infusion phase(step (2)) of the research. The research team of MInD consists of the original researchers from MProSE and two more new team members. The entire team consists of expertise from different fields: mathematicians, mathematics educator, educational psychologist, curriculum specialist, senior teacher, a school principal (who is also a mathematician), an expert of change management and leadership studies, a senior MOE curriculum specialist.
      120  20
  • 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.
      219  495
  • Publication
    Open Access
    How formal should calculus in the school mathematics curriculum be: Reflections arising from an error in a calculus examination question
    This paper examines the calculus curriculum in the current Singapore secondary and pre-university levels. Two concepts, (1) increasing and decreasing functions and their derivatives, and (2) the second derivative test for the nature of stationary points, are elaborated. An example of an incorrect calculus item in a national examination is brought up in relation to conditional reasoning involving calculus concepts. We reckon that the current emphasis on procedural knowledge in calculus is useful. However, we argue that formal conditional reasoning should not be introduced prematurely for school students.
      274  174
  • Publication
    Open Access
    On pre-service teachers' content knowledge of school calculus: An exploratory study
    This paper reports an exploratory study on the pre-service teachers’ content knowledge on school calculus. A calculus instrument assessing the pre-service teachers’ iconic thinking, algorithmic thinking and formal thinking related to various concepts in school calculus was administered to a group of pre-service mathematics teachers. Their performance on five of the items is reported in this paper. Other than their good performance in the iconic recognition of stationary points, their recognition on points of inflexion, differentiability and notion of minimum points was relatively poor. In addition, they appeared to lack the algorithmic flexibility in testing the nature of stationary points and the formal thinking about definition of an extremum point. The implications of the findings are discussed.
      98  115