Nurturing Reflective Learners In Mathematics: Yearbook 2013, Association Of Mathematics Educators

Yearbook 2013, Association of Mathematics Educators

Berinderjeet Kaur

• Publisher: WSPC
• Year: 2013
• ISBN:
  9789814472777
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This fifth volume in the series of yearbooks by the Association of Mathematics Educators in Singapore entitled Nurturing Reflective Learners in Mathematics is unique in that it focuses on a single theme in mathematics education. The objective is to encourage teachers and researchers to advance reflection among students and teachers in mathematics classrooms.

Several renowned international and Singapore researchers in the field have published their work in this volume. The fifteen chapters of the book illustrate evidence-based practices that school teachers and researchers can experiment with in their own classrooms to bring about meaningful learning outcomes. Three broad themes, namely fundamentals, instructional tools, and approaches to teaching for nurturing reflective learners in mathematics classrooms, shape the ideas in these chapters. The book makes a significant contribution towards the learning of mathematics. It is a good resource for mathematics teachers, educators and research students.

Contents:

• Nurturing Reflective Learners in Mathematics: An Introduction (Berinderjeet Kaur)
• The Neurocognition of Reflection: The Mystery in Learning, the Essence of Teaching, From Mystery to Mastery (Frank Chee Tet Voon)
• Working with the Whole Psyche: Nurturing Reflective Learners (John Mason)
• Knowledge and Beliefs for Nurturing Reflective Learners of Rational Number Concepts (Kim Beswick)
• Metacognitive Reflection at Secondary Level (Wong Khoon Yoong)
• Reflecting on an Excellent Teacher's Video Recorded Mathematics Lesson: What Can We Learn? (Lim Chap Sam and Chew Cheng Meng)
• Learning from Student Reflections (Barry Kissane)
• Reflecting on Calculation: When Drilling Becomes Fulfilling (Anne Watson)
• Developing Reflective Learners Through Solving Non-Routine Problems (Marian Kemp)
• Mathematics Competition Questions and Mathematical Tasks for Instructional Use (Toh Tin Lam)
• Mathematical Tasks in the Advanced Mathematics Class for Nurturing Reflection (Oh Nam Kwon and Ji Eun Lee)
• Using Codes to Facilitate Metacognition in Mathematics (Wong Oon Hua)
• Mathematics Lessons Stimulating Reflective Learning: Japanese Perspective (Keiko Hino)
• Insights from Students' Process of Understanding Mathematics for Nurturing Reflective Learners (Masataka Koyama)
• Mathematics Learning Episode that Promotes Reflective Thinking Among Elementary Pupils (Auxencia Alarcon Limjap)

Readership: Mathematics educators, teachers and research students.

Chapter 1

Nurturing Reflective Learners in Mathematics:
An Introduction

Berinderjeet KAUR

This introductory chapter provides an overview of the chapters in the book. The chapters are organised according to three broad themes: fundamentals, instructional tools, and approaches to teaching for nurturing reflective learners in mathematics classrooms. It ends with some concluding thoughts that readers may want to be cognizant of while reading the book and also using it for reference and further work.

1 Introduction

This yearbook of the Association of Mathematics Educators (AME) in Singapore focuses on Nurturing Reflective Learners in Mathematics. Like three of our past yearbooks, Mathematical Problem Solving (Kaur, Yeap, & Kapur, 2009), Mathematical Applications and Modelling (Kaur & Dindyal, 2010), and Reasoning, Communication and Connections in Mathematics (Kaur & Toh, 2012) the theme of this book is also shaped by the framework, shown in Figure 1, of the school mathematics curriculum in Singapore. The primary goal of school mathematics in Singapore is mathematical problem solving and amongst the five inter-related components that the framework places emphasis on is metacognition. In elaborating the framework, for both the primary and secondary students, the Ministry of Education (MOE) (2012a, 2012b), syllabus documents clarify that:

Metacognition, or thinking about thinking, refers to the awareness of, and the ability to control one's thinking processes, in particular the selection and use of problem-solving strategies. It includes monitoring of one's own thinking, and self-regulation of learning. To develop metacognitive awareness and strategies, and know when and how to use the strategies, students should have opportunities to solve non-routine and open-ended problems, to discuss their solutions, to think aloud and reflect on what they are doing, and to keep track of how things are going and make changes when necessary (Ministry of Education, 2012a, p. 17; 2012b, p. 16).

Figure 1. Framework of the Singapore school mathematics curriculum

In addition the syllabus documents outline three principles of mathematics teaching and three phases of mathematics learning in the classrooms. The three principles of teaching are as follows:

Principle 1 — Teaching is for learning; learning is for understanding; understanding is for reasoning and applying and, ultimately problem solving.

Principle 2 — Teaching should build on students’ knowledge; take cognizance of students’ interests and experiences; and engage them in active and reflective learning.

Principle 3 — Teaching should connect learning to the real world, harness ICT tools and emphasise 21st century competencies (Ministry of Education, 2012a, p. 21; 2012b, p. 23).

The three phases of mathematics learning in the classrooms are as follows:

Phase I — Readiness

Student readiness to learn is vital to learning success. In the readiness phase of learning, teachers prepare students so that they are ready to learn. This requires considerations of prior knowledge, motivating contexts, and learning environment.

Phase II — Engagement

This is the main phase of learning where teachers use a repertoire of pedagogies to engage students in learning new concepts and skills. Three pedagogical approaches, activity-based learning, teacher-directed inquiry, and direct instruction, form the spine that supports most of the mathematics instruction in the classroom. They are not mutually exclusive and could be used in different parts of a lesson or unit. For example, the lesson or unit could start with an activity, followed by teacher-led inquiry and end with direct instruction.

Phase III — Mastery

This is the final phase of learning where teachers help students consolidate and extend their learning. The mastery approaches include: motivated practice, reflective review and extended learning. (Ministry of Education, 2012a, pp. 22-25; 2012b, pp. 24-27).

The document further expands reflective review and elaborates that “It is important that students consolidate and deepen their learning through tasks that allow them to reflect on their learning. This is a good habit that needs to be cultivated from an early age and it supports the development of metacognition ” (Ministry of Education, 2012a, p. 25; 2012b, p. 27).

The framework, learning principles and phases of learning in the Singapore school mathematics syllabus documents signal very clearly the need for teachers to engage their students in reflecting about their actions before, during or after solving mathematical tasks. Such practice will allow students to create habits of the mind that lead to monitoring their own thinking and also regulating their own learning. According to Kriewaldt (2001), ‘reflection is an integral element of metacognition as it is the means by which one monitors thinking processes’ (p. 3). In addition, reflection often contains elements of self-regulation that are stimulated through questions such as “how would I do this differently the next time? ”

In reviewing Dewey's work, Rodgers (2002) states that, “reflection is a particular way of thinking and cannot be equated with mere haphazard ‘mulling’ over something ” (p. 849). The importance of reflecting on ‘something’ is also emphasised by Wheatley (1992), who argues that “reflection plays a critically important role in mathematics learning and that just completing tasks is insufficient ” (p. 529). In attempting to address the challenge of how teachers, in Singapore schools, may nurture reflective learners in their mathematics classrooms the theme of the 2011 conference for teachers, jointly organised by the Association of Mathematics Educators (AME) and the Singapore Mathematical Society (SMS), was appropriately Nurturing Reflective Learners.

The following 14 peer-reviewed chapters resulted from the keynote and invited lectures delivered during the conference. The authors of the chapters were asked to focus on evidence-based practices that school teachers can experiment in their lessons to bring about meaningful learning outcomes. The chapters are categorised into three main sections, namely the fundamentals, instructional tools, and approaches to teaching for nurturing reflective learners in mathematics classrooms. It must be noted that the 14 chapters do give a reader some ideas about the why, what and how of nurturing reflective learners in mathematics lessons. However, in no way are the 14 chapters a collection of all the know-how of the subject.

2 Fundamentals for Nurturing Reflective Learners

Learning results from students’ active participation in activities. In the classroom, teachers guide students in their learning through instructional activities. Often teacher's knowledge and beliefs guide them in shaping the activities. So, in attempting to nurture reflective learners in mathematics a few fundamentals necessary are explored in this book. They are the cognitive aspect of reflection, engagement of the whole psyche in aspects of reflection, knowledge and beliefs of teachers necessary for the development of reflective learners, metacognitive reflection and education of teachers for advancing reflection in their classrooms. It is befitting that in chapter 2, Voon explains in simple language the cognitive aspect of reflection that is the basis of learning and mastery from the perspective of a professor who teaches medical and dental students, doctors, dentists, obstetricians, psychiatrists and surgeons at the National University of Singapore. He advocates the habitual practice of reflection, an intellectual exploration of the various pathways of thought using our conscious mind, to form and strengthen new neural pathways. Mason in the next chapter, draws the attention of the readers for the need of working with the whole psyche when nurturing reflective learners. He notes that the whole person is an intricate interweaving of intellect-cognition, emotion-affect and behaviour-enaction with attention and will. Therefore by working on some mathematics together and reflecting on actions, one can learn from the experience of doing the mathematics. This is often done by withdrawing from the activity at some point and adopting a reflexive stance, asking oneself questions such as ‘what was ineffective and what was effective? ……what did I learn about myself (dispositions, propensities, habits, etc…)?’

Beswick in chapter 4, highlights that both teachers’ knowledge and beliefs are crucial to the development of reflective learners. Using rational number concepts as a basis she draws on learning episodes to illustrate behaviours of reflective learners and teacher actions that appear to support reflective learning and the teacher knowledge and beliefs that appear to underpin these. In chapter 5, Wong coins the word metacognitive reflection to include elements of metacognition (such as awareness and regulation) and reflection (as in looking back in the Polya's model of problem solving) to examine roles for mathematics instruction at secondary level. He examines two aspects of metacognitive reflection: metacognition during problem solving and regulation of learning. He reviews local studies and also draws on findings from literature thus outlining approaches for teaching metacognition and regulation of learning.

To nurture reflective learners, teachers must be reflective learners themselves. Hence, as part of teacher education teachers must have opportunities to engage in reflection that may take any of the forms: refection-in-action (minute by minute decisions a teacher may make on his or her actions when executing a lesson), reflection-on-action (looking back at a lesson that has passed) and reflection-for-action (desired outcomes guiding the plan for a lesson) (Schön, 1983; Killion&Todnem, 1991 cited in Harford, MacRuairc,&McCartan, 2010). Lim and Chew, in chapter 6, demonstrate how they used a video recorded lesson of a primary mathematics excellent teacher to promote reflective thinking among in-service teachers in Malaysia and Singapore. Lastly in the next chapter, Kissane claims that good teachers are often described as ‘reflective practitioners’ and that nurturing reflective learning in the school depends critically on teachers being appropriately reflective themselves. He describes in the chapter some experiences that were designed to encourage pre-service teachers to reflect on aspects of their learning and consider what their teacher educator might learn from it also.

3 Instructional Tools for Nurturing Reflective Learners

In the context of this book, instructional tools are limited to basically mathematical tasks and a teacher designed strategy that may be used to engage students in reflecting about their work in the mathematics classroom.

It is apparent that central to all mathematics lessons are mathematical tasks. A mathematical task is defined as a set of problems or a single complex problem that focuses students’ attention on a particular mathematical idea (Stein, Grover,&Henningsen, 1996). From the TIMSS Video Study (NCES, 2003), in which Australia, Czech Republic, Hong Kong, Japan, Netherlands, Switzerland, and the United States participated, it was found that students spent over 80% of their time in mathematics class working on mathematical tasks. According to Doyle (1988), “the work students do, defined in large measure by the tasks teachers assign, determines how they think about a curricular domain and come to understand its meaning ” (p. 167). Hence different kinds of tasks lead to different types of instruction, which subsequently lead to different opportunities for student learning (Doyle, 1988).

Watson, in chapter 8, shows that reflection on a sequence of work can enable learners to recognise similarities at a higher, more abstract, level than they had experienced while doing the work. She illustrates with examples of routine calculation practice how teachers can engage students to reflect on the effects of actions and uncover underlying relations. In chapter 9, Kemp constructs a framework comprising six criteria that teachers may use to craft suitable mathematical tasks for engaging students in reflective thinking. The six criteria for tasks are: be unfamiliar or non-routine to the students, have different possible approaches or methods of solution, require some thinking about constraints or assumptions, be appropriate to the level of mathematical knowledge and experience of the students, be potentially interesting and motivating to the students; and require justification or explanation of the reasons for the chosen approach and solution. She illustrates through three examples the use of the framework and interpretations of the six criteria. In the following chapter, Toh draws on mathematics competition questions and illustrate how teachers may adapt them for students to work and reflect on, thereby engaging in higher order thinking and also challenging them to re-examine their originally acquired mathematical notions and beliefs. He draws on several algebra competition questions to illustrate how appropriate tasks for classroom use may be crafted. In chapter 11, Kwon and Lee focus on an aspect of reflective thinking, i.e. looking back on one's problem solving and thinking to analyse mathematical tasks on the topic: Polar coordinates in advanced mathematics. These tasks are suitable for nurturing reflective thinking amongst high school students of mathematics.

The last chapter in this category shows an attempt by a primary school teacher to devise a strategy for engaging students to reflect on their errors in mathematics written work. Wong, in chapter 12, shows how he adopted an idea used mainly by English Language teachers in Singapore to grade written compositions and give their students feedback for use in his mathematics lessons. He devised codes and used them to give his students feedback about their errors when grading their written work.

4 Approaches to Teaching for Nurturing Reflective Learners

A lesson that has as its focus dissemination of knowledge by the teacher would certainly have little or no scope for teacher questions or knowledge construction by students. In contrast, a lesson that has as its focus the development of reflective attributes amongst students would involve significant student participation in the lesson. The three chapters in this category each illustrate a variation of the types of teacher actions that promote reflective learning. Hino, in chapter 13, shows the dominant presence of “reflective learning ” in mathematics lessons in Japan. This is verified by the vocabulary that exists to describe teacher actions. In the chapter she describes the Japanese problem solving approach and several perspectives for enhancing reflective thinking and learning in classroom practice of mathematics lessons. She also draws on ...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Contents
• chapter 1 Nurturing Reflective Learners in Mathematics: An Introduction
• chapter 2 The Neurocognition of Reflection: The Mystery in Learning, the Essence of Teaching, From Mystery to Mastery
• chapter 3 Working with the Whole Psyche: Nurturing Reflective Learners
• chapter 4 Knowledge and Beliefs for Nurturing Reflective Learners of Rational Number Concepts
• chapter 5 Metacognitive Reflection at Secondary Level
• chapter 6 Reflecting on an Excellent Teacher's Video Recorded Mathematics Lesson: What Can We Learn?
• chapter 7 Learning from Student Reflections
• Chapter 8 Reflecting on Calculation: When Drilling Becomes Fulfilling
• Chapter 9 Developing Reflective Learners Through Solving Non-Routine Problems
• Chapter 10 Mathematics Competition Questions and Mathematical Tasks for Instructional Use
• Chapter 11 Mathematical Tasks in the Advanced Mathematics Class for Nurturing Reflection
• Chapter 12 Using Codes to Facilitate Metacognition in Mathematics
• Chapter 13 Mathematics Lessons Stimulating Reflective Learning: Japanese Perspective
• Chapter 14 Insights from Students’ Process of Understanding Mathematics for Nurturing Reflective Learners
• chapter 15 Mathematical Tasks in the Advanced Mathematics Class for Nurturing Reflection
• Contributing Authors