Big Ideas in Mathematics

Yearbook 2019, Association of Mathematics Educators

Tin Lam Toh, Joseph B W Yeo

• Publisher: WSPC
• Year: 2019
• ISBN:
  9789811205392
• Topic: Mathematik
• Subtopic: Mathematik im Bildungswesen

The new emphasis in the Singapore mathematics education is on Big Ideas (Charles, 2005). This book contains more than 15 chapters from various experts on mathematics education that describe various aspects of Big Ideas from theory to practice. It contains chapters that discuss the historical development of mathematical concepts, specific mathematical concepts in relation to Big Ideas in mathematics, the spirit of Big Ideas in mathematics and its enactment in the mathematics classroom.

This book presents a wide spectrum of issues related to Big Ideas in mathematics education. On the one end, we have topics that are mathematics content related, those that discuss the underlying principles of Big Ideas, and others that deepen the readers' knowledge in this area, and on the other hand there are practice oriented papers in preparing practitioners to have a clearer picture of classroom enactment related to an emphasis on Big Ideas.

Contents:

• Big Ideas in Mathematics (Joseph B W YEO & TOH Tin Lam)
• General Perspectives of Big Ideas in Mathematics and Mathematics Education:
  • Some Great Breakthrough Ideas in Mathematics (KOH Khee Meng & TAY Eng Guan)
  • On Characteristics of Statistical Thinking (ZHU Ying)
  • Situated Mathematics: Positioning Mathematics Ideas as Human Ideas (David WAGNER)
  • Big Ideas of Primary Mathematics: It's All about Connections (Chris HURST)
  • Teaching towards Big Ideas: Challenges and Opportunities (CHOY Ban Heng)
• Content Knowledge and Teaching Ideas of Selected Big Ideas in Mathematics:
  • Big Ideas about Equivalence in the Primary Mathematics Classroom (YEO Kai Kow Joseph)
  • Teaching Mathematics: A Modest Call to Consider Big Ideas (TAY Eng Guan)
  • Invariance as a Big Idea (TOH Pee Choon)
  • Empirical Motivation for Teaching Functions and Modelling (YAP Von Bing)
  • Mathematical Modeling in Problem Solving: A Big Idea across the Curriculum (Padmanabhan SESHAIYER & Jennifer SUH)
  • Unpacking the Big Idea of Proportionality: Connecting Ratio, Rate, Proportion and Variation (Joseph B W YEO)
  • Teaching towards Big Ideas: Deepening Students' Understanding of Mathematics (CHUA Boon Liang)
  • Teaching Pre-University Calculus with Big Ideas in Mind (TOH Tin Lam)
• Pedagogical Practices in Teaching Towards Big Ideas in Mathematics:
  • Making Big Ideas Explicit Using the Teaching through Problem Solving Approach (Pearlyn LIM Li Gek)
  • Mathematical Vocabulary: A Bridge that Connects Ideas (Berinderjeet KAUR, WONG Lai Fong, TOH Wei Yeng Karen & TONG Cherng Luen)
  • Engaging Students in Big Ideas and Mathematical Processes in the Primary Mathematics Classrooms (CHENG Lu Pien & Vincent KOH Hoon Hwee)
  • Enhancing Students' Reasoning in Mathematics: An Approach using Typical Problems (Jaguthsing DINDYAL)
  • Big Ideas from Small Ideas (LEONG Yew Hoong)
  • Enhancing Mathematics Teaching with Big Ideas (Cynthia SETO, CHOON Ming Kwong & PANG Yen Ping)
  • Making Vertical Connections when Teaching Towards Big Ideas (LOW Leng & WONG Lai Fong)
• Contributing Authors

Readership: Graduate students, researchers, practitioners and teachers in mathematics.Big Ideas;Mathematics Instruction;Connections0 Key Features:

• New emphasis on Big Ideas
• The book is both theory based and practice oriented
• To the best of the editors' knowledge, so far no competing titles for this book

Chapter 1

Big Ideas in Mathematics

Joseph B. W. YEOTOH Tin Lam

This chapter provides an overview of all the other 20 chapters presented in this book. The remaining chapters in this book are classified into three main categories. The first section contains five chapters which present to readers general perspectives of Big Ideas in mathematics and mathematics education. The second section consists of eight chapters that discuss the content knowledge and teaching ideas of selected Big Ideas in mathematics. The last section contains seven chapters on general pedagogical practices in teaching towards Big Ideas in mathematics. This chapter concludes with some observations on Big Ideas and implications for teachers.

1Introduction

Research on the practices of effective teachers has received much attention of mathematics education researchers (e.g. Ma, 1999; Weiss, Heck, & Shimkus, 2004). For example, effective teachers ask effective questions and are able to engage students in co-constructing the mathematics discourse in the classrooms. In terms of instructional materials, these teachers make teaching materials explicit to their students (Leong, Cheng, Toh, Kaur, & Toh, 2018).

Other studies emphasize the importance of teachers having solid content knowledge for teaching. Turner-Bisset (1999) identified content knowledge as one important attribute of expert teachers. There are many other studies which emphasize the importance of content knowledge for teachers (e.g. Chapman, 2005; Kilpatrick, 2001; Schmidt, 2002; Toh, 2017).

While the above are two important aspects of effective or expert teachers, it has been contended that pedagogical practice and mathematics content knowledge of a teacher must be grounded on a set of Big Mathematics Ideas (Charles, 2005).

2Big Ideas in Mathematics

As early as the beginning of this century, the National Council of Teachers of Mathematics (NCTM) have stressed the importance of Big Ideas of mathematics. According to the Principles and Standards for School Mathematics (NCTM, 2000), “[t]eachers need to understand the big ideas of mathematics and be able to represent mathematics as a coherent and connected enterprise (p. 17). Charles (2005) gave a formal definition of a Big idea:

A Big idea is a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole (p. 10).

In brief, one can appreciate that Big Ideas are about connections of various ideas across mathematics. Charles (2005) further stressed the importance of Big Ideas as a grounding for mathematics teachers.

In the most recent curriculum revision of the Singapore mathematics curriculum, the Singapore Ministry of Education (MOE, 2018) highlights that one of the three key emphases of this review is to “develop a greater awareness of the nature of mathematics and the big ideas that are central to the discipline” (p. 9) in order to achieve coherence across different topics. In other words, this is similar to Charles’ (2005) notion of a Big Idea as one linking various “understandings into a coherent whole”.

By using a word count on the 2020 secondary mathematics syllabuses (MOE, 2018), the word “connection” occurs 37 times. It is thus not an exaggeration to state that Big Ideas are about connections. From the perspective of classroom practice, the emphasis of Big Ideas in mathematics can be translated to conscious effort of getting students to view mathematics as a highly connected system of thinking and concepts across various topics, rather than seeing these as isolated unrelated concepts with little or no relation with others.

Readers should note that this emphasis on connection is not entirely new in the Singapore Mathematics curriculum review. In fact, as early as 2006, the syllabus documents (MOE, 2006a; 2006b) have already highlighted that it is crucial that students are able to “[r]ecognise and use connections among mathematical ideas, and between mathematics and other disciplines” (p. 11). Rather, the new emphasis in the latest curriculum revision to emphasize that the recognition and use of connections are based on a sound understanding of the nature (or disciplinarity) of mathematics.

This book, which is a derivative of the lectures and workshops from the annual Mathematics Teacher Conference 2018, builds on the synergy of mathematicians and mathematics educators to provide researchers and mathematics educators a diverse perspective of Big Ideas in mathematics and, more importantly, how these Big Ideas can be translated to effective practices in the mathematics classroom. We believe that the notion of Big Ideas in Mathematics is an opportunity in which mathematicians and mathematics educators are able to offer differing (and often complementary) perspectives as an effort to offer readers a whole full spectrum of understanding of this topic.

This book consists of 21 chapters, of which the first chapter is the introduction of the book while the remaining 20 chapters are classified into three main categories. Section One provides the general perspectives of Big Ideas in mathematics (and statistics) and mathematics education. Consisting of five chapters, this section sets the tone for the next two sections by discussing two broad views of mathematics (and statistics) and the development of Big Ideas in mathematics education from three different perspectives.

Section Two consists of eight chapters that deal directly with the content knowledge and teaching ideas of selected Big Ideas (e.g. equivalence, invariance, functions, models, proportionality) in the 2020 Singapore school mathematics syllabuses, and other Big Ideas such as those that are specific within a branch of mathematics (e.g. calculus).

On the other hand, Section Three, consisting of seven chapters, explores various pedagogical practices that might help teachers teach towards Big Ideas in mathematics. Although some of the authors may illustrate their pedagogy with one or two Big Ideas in mathematics (e.g. measures), the teaching method is generic enough to be applied to other Big Ideas.

3Section One on General Perspectives of Big Ideas in Mathematics and Mathematics Education

The focus of the five chapters in Section One is on various general viewpoints of Big Ideas in mathematics and mathematics education. The first two chapters present two broad views of mathematics (including statistics) while the next three chapters look at the development of Big Ideas in mathematics education from different perspectives.

This section begins with Chapter 2, where Koh and Tay describe three breakthrough ideas in the history of mathematics, framed in the contexts of three “great crises” in mathematics – the discovery of irrational numbers, the foundations of calculus, and the foundations of mathematics. Through the lens of the development of mathematics in history, Koh and Tay hope to enable teachers to appreciate the significance and influence of these breakthrough ideas, so as to facilitate teachers to build connections across these ideas.

On the other hand, Zhu proposes statistical thinking as an alternative Big Idea to understand the world around us: one that is different from the kind of deterministic mathematics described in Chapter 2. In Chapter 3, readers will learn to appreciate the differences between statistical and mathematical thinking, some key characteristics of statistical thinking and its importance in guiding our thinking, learning and daily life in this era of big data in the 21^st century.

In Chapter 4, Wagner looks at the development of Big Ideas in mathematics education from a different perspective: he believes that these ideas develop as responses to human concerns, and so the teaching of mathematics should be positioned as being responsive to community and environmental concerns in order to support responsible citizenship oriented around sustainable development and peace.

Hurst, in Chapter 5, hopes to make use of Big Idea thinking to re-conceptualize mathematics education. He highlights the importance of connections within and between Big Ideas as the essence of Big Idea thinking, and the expression of these connections in terms of learning sequences and trajectories. Hurst further classifies Big Ideas into three broad categories of sequential Big Ideas, umbrella Big Ideas and process Big Ideas using an extensive model.

In Singapore, Big Ideas are categorized differently in the 2020 Singapore school mathematics syllabus (MOE, 2018): they are framed around four recurring themes in mathematics (namely, properties and relationships; operations and algorithms; representations and communications; and abstractions and applications). The syllabus document specifies eight clusters of Big Ideas (namely, equivalence, invariance, functions, models, diagrams, notations, measures and proportionality), but it emphasizes that these eight Big Ideas are not meant to be authoritative and comprehensive. In Chapter 6, Choy examines the four recurring themes and the eight clusters of Big Ideas in the syllabus document and presents a notion of teaching towards Big Ideas. He also discusses some challenges and suggests what teachers can do to mitigate some of the difficulties surrounding the worthwhile task of teaching towards Big Ideas.

4Section Two on Content Knowledge and Teaching Ideas of Selected Big Ideas in Mathematics

In Section Two, we zoom in on some Big Ideas in mathematics in the 2020 Singapore school mathematics syllabuses in order to understand more about these Big Ideas and to suggest some teaching ideas. As the eight clusters of Big Ideas specified in the syllabus document are not meant to be exhaustive, this section will also examine some other Big Ideas.

We begin this section with the Big Idea of equivalence in Chapter 7. First, Yeo K. K. J. reviews literature on Big Ideas in mathematics, Big Ideas for teaching and learning, and the Big Idea of equivalence. Then he proposes three activities on equivalence that can be introduced in the primary mathematics classroom.

In Chapter 8, Tay also discusses the Big Idea of equivalence but he uses examples from the Singapore pre-university mathematics curriculum. According to Tay, one must have a deep consideration of the nature of mathematics in order to engage in discussion on Big Ideas in mathematics. This chapter also looks at the Big Idea behind solving equations (namely, the technique of ‘isolating the unknown’) and the Big Ideas of definition and notation.

Closely related to the Big Idea of equivalence is the Big Idea of invariance. In Chapter 9, Toh P. C. demonstrates how invariance threads across several topics horizontally across content strands and vertically across levels by giving examples of invariance in geometry at the primary and secondary levels, and number theory at the pre-university level.

The Big Ideas of functions and mathematical modelling in secondary school mathematics are discussed in Chapter 10. Yap believes that the introduction of empirical flavour in the teaching of related topics will be able to illustrate to students the essence of these topics. This chapter also provides two possible lesson plans incorporating the ideas that he has proposed.

In Chapter 11, Seshaiyer and Suh also advocate the use of mathematical modelling as a Big Idea in teaching mathematics to solve real world tasks that involve not only mathematics but other disciplines as well. They give an example of a modelling task given to a class in an elementary school during a lesson study and discusses the solutions from some students. They believe that this kind of tasks can provide students with the opportunity to engage in the four pillars of 21^st century skills: communication, collaboration, critical thinking and creative problem solving.

Next, Yeo J. B. W. unpacks the Big Idea of proportionality in Chapter 12 by examining how the concepts of proportion, ratio, rate and variation are connected to one another, so that primary and secondary school teachers are better equipped not only to explain to their students the similarities and differences among these concepts but also to connect these ideas into a coherent whole and to appreciate how proportionality is used in real life.

Chapter 13 also deals with Big Idea of proportionality but from another perspective. In this chapter, Chua suggests presenting secondary school students with different real-world situations in order for them to identify those that depict proportionality. He then discusses the underlying idea behind the concept of proportionality and its connections to other topics such as gradient, trigonometry and statistics.

All the above chapters in Section 2 deal with Big Ideas across topics. But there are also Big Ideas within a specific branch of mathematics, e.g. calculus. In Chapter 14, Toh T. L. paints a portrait of how pre-university calculus lessons would appear for a teacher who has a deep understanding of “Big Idea of calculus education”. The examples on building up students’ reper...