Problem solving has been the theme of mathematics education in Singapore since the 1980s. For the... more Problem solving has been the theme of mathematics education in Singapore since the 1980s. For the past two decades, the Singapore mathematics curriculum has problem solving as its central focus and aims to prepare students to be competent problem solvers. Problem solving, as articulated by the Singapore Mathematics Curriculum Framework is supported by five inter-related components and Metacognition is one of the components. However, there are very few studies to find out how metacognition has worked through the Singapore classrooms and its impact on problem solving. This paper presents findings from a study on metacognitive strategies Singapore Secondary One (Year 7) students (N = 783) employed while solving mathematics problems. Discussion will center on the different methods used to investigate the nature of metacognition during mathematical problem solving, namely survey inventory, retrospective self-report and qualitative interview. Findings from this study suggest that results from different data collection instruments may lead to dissimilarities in the findings but provide a multi-facet perspective of metacognition in mathematical problem solving. As compared, findings based on data from a single instrument may only provide a skew perspective. Findings from this study bear important implications to the interpretation of research findings as well as the research designs for better insights to metacognition employed during mathematical problem solving.
In the first section of this chapter, we review the growing literature on “practices”, focusing o... more In the first section of this chapter, we review the growing literature on “practices”, focusing on the purpose of studying teacher practices in actual classrooms in view of its potential in teacher professional development. Following that, we zoom in to the Singapore situation by reviewing other studies here on mathematics teacher practices. In the second section, we describe an ongoing project and its contribution to research on exemplary practices of Singapore mathematics teachers. In the final section, we discuss the usefulness of this review in relation to the effort of building portraits of Singapore mathematics teacher practices.
This is a workshop on word problems in primary mathematics. The workshop is based on a paper deri... more This is a workshop on word problems in primary mathematics. The workshop is based on a paper derived from an investigation into children' responses to standard and non-standard mathematics word problems before and after an intervention programme. Standard word problems can be solved by identifying the correct operation and performing the necessary computation. The story context does not affect the solution. In solving non-standard word problems the story context is important in obtaining a correct solution. Primary Three children in five Singapore schools participated in a year-long intervention where their teachers used several lessons that included non-standard problems. The children were asked to solve standard and non-standard word problems at the start and at the end of the school year. Among these word problems, there were those that were similar to, those that were similar in the mathematical structure to but different in the superficial features from, and those that were different in mathematical structure from the problems in the intervention programme. The responses from four intact classes were selected for analysis. It was found that the children were able to make sense of their computation results. However, in situations that went beyond computation, many children were not able to make sense. Intervention and use of concrete materials were found to encourage sense-making.
Driven by Universities from both Western and Eastern Economies, the project was grounded within a... more Driven by Universities from both Western and Eastern Economies, the project was grounded within a strong research tradition of rich, case study research. The case study approach provided a depth of context that enabled particular outcomes to be more readily highlighted, and meant that comparisons could take into account the different educational systems. By involving policy makers, practitioners and academics, we have managed to bring additional depth to the
... (Appendix D) The transfer effects were assessed, using the school's English, M... more ... (Appendix D) The transfer effects were assessed, using the school's English, Mathematics and Science mid-year and end-of-year examination results in Primary 3 and Primary 4. ... Page 13 The Control Classes tended to take a retaliatory route in solving the problem and it ...
The Concrete-Pictorial-Abstract (C-P-A) sequence is a key instructional strategy for the developm... more The Concrete-Pictorial-Abstract (C-P-A) sequence is a key instructional strategy for the development of primary mathematics concepts in Singapore. However, the way to go about teaching and learning with the C-PA sequence is unclear. As a result, the benefits of this sequence cannot be fully capitalized. This paper aims to (1) expound from learning theories of Bruner, Dienes and Piaget, and literature on representations, how learning takes place with representations and (2) using the insights gleaned, propose how teaching with representations looks like. It concludes with segments of a series of classroom lesson plans crafted using the proposed model of teaching and learning with representations for the concept of equivalent fractions in Primary Three.
This paper was published in the 2017 proceedings of the 41st Conference of the International Grou... more This paper was published in the 2017 proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education held at Singapore from 17 - 22 July 2017
This chapter first presents the evolving conceptualisation of metacognition since it was first co... more This chapter first presents the evolving conceptualisation of metacognition since it was first coined by Flavell in 1976. In particular, the issue of awareness, monitoring, and regulation of both cognitive and affective resources was examined. The role that metacognition plays in mathematical problem-solving was also examined, leading to a discussion of the role of metacognition in the Singapore School Mathematics Curriculum which has mathematical problem-solving as its central aim. In view of this, the conceptualisation of metacognition as well as the how’s of addressing metacognition in the Singapore mathematics classrooms were discussed from the intended curriculum point of view. Some of the local postgraduate works on metacognition and teaching, and learning of mathematics was also presented to provide an overview of the landscape of the work in this area that has been undertaken thus far. In addition, examples of ongoing works on metacognitive approaches, which have made some inroads in some local schools, were shared to give the reader a glimpse of how research in this area has impacted school practices locally. The chapter concludes with implications for addressing metacognition in the Singapore Mathematics classrooms from the perspective of teachers’ professional development.
This document may be used for private study or research purpose only. This document or any part o... more This document may be used for private study or research purpose only. This document or any part of it may not be duplicated and/or distributed without permission of the copyright owner. The Singapore Copyright Act applies to the use of this document.
The role of assessment in informing the public and parents of the outcome of instruction has, at ... more The role of assessment in informing the public and parents of the outcome of instruction has, at times, become an obstacle to assessment reforms. In Singapore, for example, the public has come to accept the norm-referenced way of reporting scores. On the other hand, assessments tools offered by researchers for the teaching of thinking are often criterionreferenced. There is, therefore, a need to provide a bridge to the two different kinds of assessment tools in order for those that promote thinking to be better received by parents and the public. This is crucial as they are also partners in the education process. The Japanese Open-Ended Approach to teaching mathematics has been developed and researched since 1971. The approach promotes critical and creative thinking within the context of mathematical problem solving. The well-developed assessment tool that has evolved from the approach reflects this emphasis on critical and creative thinking. However, as in many other cases, it is m...
Problem solving has been the theme of mathematics education in Singapore since the 1980s. For the... more Problem solving has been the theme of mathematics education in Singapore since the 1980s. For the past two decades, the Singapore mathematics curriculum has problem solving as its central focus and aims to prepare students to be competent problem solvers. Problem solving, as articulated by the Singapore Mathematics Curriculum Framework is supported by five inter-related components and Metacognition is one of the components. However, there are very few studies to find out how metacognition has worked through the Singapore classrooms and its impact on problem solving. This paper presents findings from a study on metacognitive strategies Singapore Secondary One (Year 7) students (N = 783) employed while solving mathematics problems. Discussion will center on the different methods used to investigate the nature of metacognition during mathematical problem solving, namely survey inventory, retrospective self-report and qualitative interview. Findings from this study suggest that results from different data collection instruments may lead to dissimilarities in the findings but provide a multi-facet perspective of metacognition in mathematical problem solving. As compared, findings based on data from a single instrument may only provide a skew perspective. Findings from this study bear important implications to the interpretation of research findings as well as the research designs for better insights to metacognition employed during mathematical problem solving.
In the first section of this chapter, we review the growing literature on “practices”, focusing o... more In the first section of this chapter, we review the growing literature on “practices”, focusing on the purpose of studying teacher practices in actual classrooms in view of its potential in teacher professional development. Following that, we zoom in to the Singapore situation by reviewing other studies here on mathematics teacher practices. In the second section, we describe an ongoing project and its contribution to research on exemplary practices of Singapore mathematics teachers. In the final section, we discuss the usefulness of this review in relation to the effort of building portraits of Singapore mathematics teacher practices.
This is a workshop on word problems in primary mathematics. The workshop is based on a paper deri... more This is a workshop on word problems in primary mathematics. The workshop is based on a paper derived from an investigation into children' responses to standard and non-standard mathematics word problems before and after an intervention programme. Standard word problems can be solved by identifying the correct operation and performing the necessary computation. The story context does not affect the solution. In solving non-standard word problems the story context is important in obtaining a correct solution. Primary Three children in five Singapore schools participated in a year-long intervention where their teachers used several lessons that included non-standard problems. The children were asked to solve standard and non-standard word problems at the start and at the end of the school year. Among these word problems, there were those that were similar to, those that were similar in the mathematical structure to but different in the superficial features from, and those that were different in mathematical structure from the problems in the intervention programme. The responses from four intact classes were selected for analysis. It was found that the children were able to make sense of their computation results. However, in situations that went beyond computation, many children were not able to make sense. Intervention and use of concrete materials were found to encourage sense-making.
Driven by Universities from both Western and Eastern Economies, the project was grounded within a... more Driven by Universities from both Western and Eastern Economies, the project was grounded within a strong research tradition of rich, case study research. The case study approach provided a depth of context that enabled particular outcomes to be more readily highlighted, and meant that comparisons could take into account the different educational systems. By involving policy makers, practitioners and academics, we have managed to bring additional depth to the
... (Appendix D) The transfer effects were assessed, using the school's English, M... more ... (Appendix D) The transfer effects were assessed, using the school's English, Mathematics and Science mid-year and end-of-year examination results in Primary 3 and Primary 4. ... Page 13 The Control Classes tended to take a retaliatory route in solving the problem and it ...
The Concrete-Pictorial-Abstract (C-P-A) sequence is a key instructional strategy for the developm... more The Concrete-Pictorial-Abstract (C-P-A) sequence is a key instructional strategy for the development of primary mathematics concepts in Singapore. However, the way to go about teaching and learning with the C-PA sequence is unclear. As a result, the benefits of this sequence cannot be fully capitalized. This paper aims to (1) expound from learning theories of Bruner, Dienes and Piaget, and literature on representations, how learning takes place with representations and (2) using the insights gleaned, propose how teaching with representations looks like. It concludes with segments of a series of classroom lesson plans crafted using the proposed model of teaching and learning with representations for the concept of equivalent fractions in Primary Three.
This paper was published in the 2017 proceedings of the 41st Conference of the International Grou... more This paper was published in the 2017 proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education held at Singapore from 17 - 22 July 2017
This chapter first presents the evolving conceptualisation of metacognition since it was first co... more This chapter first presents the evolving conceptualisation of metacognition since it was first coined by Flavell in 1976. In particular, the issue of awareness, monitoring, and regulation of both cognitive and affective resources was examined. The role that metacognition plays in mathematical problem-solving was also examined, leading to a discussion of the role of metacognition in the Singapore School Mathematics Curriculum which has mathematical problem-solving as its central aim. In view of this, the conceptualisation of metacognition as well as the how’s of addressing metacognition in the Singapore mathematics classrooms were discussed from the intended curriculum point of view. Some of the local postgraduate works on metacognition and teaching, and learning of mathematics was also presented to provide an overview of the landscape of the work in this area that has been undertaken thus far. In addition, examples of ongoing works on metacognitive approaches, which have made some inroads in some local schools, were shared to give the reader a glimpse of how research in this area has impacted school practices locally. The chapter concludes with implications for addressing metacognition in the Singapore Mathematics classrooms from the perspective of teachers’ professional development.
This document may be used for private study or research purpose only. This document or any part o... more This document may be used for private study or research purpose only. This document or any part of it may not be duplicated and/or distributed without permission of the copyright owner. The Singapore Copyright Act applies to the use of this document.
The role of assessment in informing the public and parents of the outcome of instruction has, at ... more The role of assessment in informing the public and parents of the outcome of instruction has, at times, become an obstacle to assessment reforms. In Singapore, for example, the public has come to accept the norm-referenced way of reporting scores. On the other hand, assessments tools offered by researchers for the teaching of thinking are often criterionreferenced. There is, therefore, a need to provide a bridge to the two different kinds of assessment tools in order for those that promote thinking to be better received by parents and the public. This is crucial as they are also partners in the education process. The Japanese Open-Ended Approach to teaching mathematics has been developed and researched since 1971. The approach promotes critical and creative thinking within the context of mathematical problem solving. The well-developed assessment tool that has evolved from the approach reflects this emphasis on critical and creative thinking. However, as in many other cases, it is m...
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