Lee, Ngan Hoe

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-P-A 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 reports on the work to review the key pedagogical approach advocated in Singapore Primary Mathematics Curriculum – the Concrete- Pictorial-Abstract Approach or C-P-A Approach in view of the changes brought about by technological advances. In particular, the Concrete- Virtual-Pictorial-Abstract Approach or C-V-P-A Approach is proposed to take into account the role that virtual manipulatives play in enriching the representations of mathematical concepts in the mathematics classrooms. Through a case study, the study sought to determine the possible impact of this proposed revised approach on teachers’ delivery of lessons.

Mathematics education in Singapore is a shared responsibility of the Ministry of Education (MOE) and the National Institute of Education (NIE) . The MOE overseas the intended, implemented and attained curriculum in all schools while the NIE is involved in teacher preparation and development and also research in mathematics education. Therefore this report has two sections respectively , the first describes the education system and school mathematics curricula while the second briefly provides relevant information on teacher preparation and development and mathematics education research in Singapore.

Characterised by increased automation and digitalisation of work processes, the Fourth Industrial Revolution (4IR) has displaced and redesigned many existing jobs, and will create new occupations that are currently non-existent. To prepare a future workforce that is adaptive amid a volatile employment landscape, schools should provide the necessary learning experiences to help students today develop transferrable competencies, which encompass deep conceptual understanding of domain-specific knowledge and 21^st century competencies in the cognitive, intrapersonal, and interpersonal domains. In this paper, we study this possibility in the context of mathematics learning and propose a constructivist learning design (CLD) that affords students to engage in deeper learning processes. In the proposed CLD, students first work collaboratively to solve a complex problem targeting a math concept that they have yet to learn, before being engaged in instruction that builds upon their solutions in the teaching of the concept, and practices that reinforce these ideas. Testing CLD in mathematics learning at secondary level via a quasi-experimental design, we found out that (1) CLD facilitates deeper learning as it encouraged students to apply their cognitive, intrapersonal, and interpersonal competencies, and (2) CLD students (n=23) outperformed their Direct Instruction counterparts (n=18) on mathematical conceptual understanding and transfer. Overall, this study suggests that the CLD has the potential to cultivate competencies that allow students to transfer in novel situations, rendering it as a possible learning environment to better prepare students for the 4IR.

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.

In the light of global trends of change and the need to adapt, there is a need to ensure that our young can think for themselves so that they can find their own solutions to new problems. This importance of teaching thinking has been reflected in the Singapore Mathematics Curriculum. In particular, metacognition – thinking about thinking – has been an important aspect of the Singapore Mathematics Curriculum since 1992.

Research in addressing the issue of metacognition in the mathematics classroom is not lacking. However, such studies have generalisability limitations for the Singapore classrooms, while local researchers tend to concentrate on the academically stronger students. Furthermore, in most cases, studies were carried out using only one or two of the metacognitive instructional strategies in a rather clinical context, rendering implementation of such strategies in the real classroom difficult.

The purpose of the study is to plan an intervention programme, based on metacognitive instructional strategies that have been found to be successful in addressing metacognition in the mathematics classrooms, to address the needs of students identified to be mathematically weak in an actual classroom context.

The study aimed to investigate the effect of the Metacognitive Instructional Strategies (MIS) on the Secondary One Normal (Academic) students’

● general self-concept (overall and intellectual)
● mathematics efficacy
● self-regulated learning strategies for mathematics
● problem-solving performance – strategies in reading mathematics problems, perseverance in problem solving, success in problem solving
● mathematics achievement

The study also examines the impact of MIS on Malays, the ethnic group that has often been singled out as mathematically underachieving, compared with that on the non-Malays.

This quasi-experimental study employed an adaptation of the 4 × 2 Factorial Pretest-Posttest Control Group Design. The four levels of treatment included three comparison classes and an experimental class. The two levels of ethnicity are referred to as non-Malays and Malays.

Basically, the MIS Curriculum used an integration of the following approaches with the Problem Wheel (p.29) as a background:

● Mathematics log writing
● Effective questioning
● Identification of structural properties of problems
● Pair / group problem-solving

Though the study was too brief to produce any significantly powerful results, evidences do point towards several positive outcomes of MIS, as summarised below:

● MIS contributes to the positive change in the intellectual self-concept and mathematics self-efficacy among the students. The impact is greater for the intellectual self-concept than mathematics self- efficacy among the Malay students.

● MIS contributes to the positive change in the acquisition of the selfregulated learning strategies for mathematics. The change appeared to be more pronounced among the Malay students.

● MIS contributes to an increased level of perseverance in problem solving. This is accompanied with an improvement in problem solving performance and better reading strategies used in reading mathematics. However, the level of improvement in mathematics solving performance is more pronounced among the non-Malay students.

● MIS contributes to the positive change in mathematics achievement.

As teachers begin to construct an understanding of ‘inquiry’, and incorporate IBL into their classroom practice, they are challenged to be sensitive to the ‘constructivist’ nature of the CLD. This paper presents a reflection model structured to trigger thinking about beliefs in teaching and learning in order for teachers to re-examine their practice and adopt new pedagogies. The reflections by two secondary school mathematics teachers are presented as they experiment with inquiry-based learning in the CLD. The teachers showed awareness, monitoring and regulation of their teaching practices including new and existing beliefs.