Lee, Ngan Hoe

This chapter discusses how a lesson closure, which should be present in every Mathematics lesson, can be carried out effectively to promote offline metacognition. According to the theory of constructivism, learning is achieved through making connection between the new concept and a learner’s prior knowledge, or schema. Lesson closure is crucial to provide a time and space for students to consolidate their learning and acquire the mathematical concepts. We will introduce a strategy to close a lesson in which the teacher uses students’ reflection of their learning to create a visual representation, known as a closure diagram. This activity provides a structure that allows students to either integrate the knowledge into their schema or identify gaps in their learning. Both situations involve students regulating their thinking, which is metacognitive in nature. Variations of the closure strategy for different purposes will also be discussed and illustrated with artefacts from classroom lessons.

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.

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.

A popular topic for local mathematical research is investigating the factors underlying difficulties encountered by weak pupils in word problem solving. With the emphasis on infusing thinking skills into English, Mathematics, Science and Social Studies in primary classes, there is now an urgent need to look for alternative ways of helping weak pupils to learn thinking skills through mathematics word problem solving. Thinking strategies such as the use of graphic organizers that build thinking skills have been successfully used to teach English, Science and Social Studies to slow learners. The organizers have helped pupils to decompose problems into smaller parts for easier understanding, to organize information into schemata and to establish links between the schemata. This paper attempts to show that weak pupils in primary school could be helped to learn and think in mathematics classes through the use of graphic organizers while solving word problems. The thinking processes illustrated are part-whole, sequencing, comparing and contrasting, decision making and predicting.

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.