Mathematical Modelling: From Theory To Practice

From Theory to Practice

Ngan Hoe Lee, Dawn Kit Ee Ng

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814546935
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

This book documents the journey undertaken by educators from the Mathematics and Mathematics Education (MME) Academic Group in the National Institute of Education (NIE) and Singapore schools during a Mathematical Modelling Outreach (MMO) event in June 2010 under the guidance of renowned experts in the field of mathematical modelling. The main goal of MMO was to reach out to Singapore primary and secondary schools and introduce the potentials of mathematical modelling as a platform for eliciting mathematical thinking, communication, and reasoning among students. This book contributes to the expanding literature on mathematical modelling by offering voices from the Singaporean context. It suggests how theoretical perspectives on mathematical modelling can be transformed into actual practice in schools, all within the existing infrastructure of the current Singapore mathematics curriculum. More importantly, the book provides documentary evidence on how plans put in place through MMO in 2010 have since been realised.The publication of this book is hence timely at this juncture. Not only does the book record how MMO was among the first pebbles launched into the pond, it also serves as a bridge over which educators can stand upon to view how the ripple effect had developed from the initial MMO pebble and the directions it may continue to extend. Perhaps in the process, other ripples in the teaching, learning, and research of mathematical modelling can be created. Contents:

• Introduction: Mathematical Modelling Outreach in Singapore (NG Kit Ee Dawn and LEE Ngan Hoe)
• Setting the Stage for Mathematical Modelling in Schools:
  • Promotion of Mathematical Modelling Competencies in the Context of Modelling Projects (Gabriele KAISER and Susanne GR?NEWALD)
  • Problem Finding and Problem Posing for Mathematical Modelling (Gloria STILLMAN)
  • Mathematical Modelling in Singapore Schools: A Framework for Instruction (ANG Keng Cheng)
  • Mathematical Modelling in Australia (Vincent GEIGER)
  • Mathematical Modelling in Japan (Toshikazu IKEDA)
• Fostering Mathematical Modelling in Schools:
  • Learning through Modelling in the Primary Years (Lyn D ENGLISH)
  • Fostering Mathematical Modelling in Secondary Schools (Gloria STILLMAN)
  • Mathematical Modelling — An Example from an Inter-School Modelling Challenge (Raymond BROWN, Trevor REDMOND, Joanne SHEEHY and Dawn LANG)
• Mathematical Learning through Modelling Tasks:
  • Learning through "Designing a Café" (YEO Kai Kow Joseph)
  • Learning through "Plane Punctuality" (HO Weng Kin)
  • Learning through "Mobile Phone Plan" (CHENG Lu Pien and CHUA Kwee Gek)
  • Learning through "The Best Paper Plane" (SOON Wan Mei Amanda, CHIOK Hwee Fen and KONG May Hua Maybelline)
  • Learning through "Designing a Tent" (HO Siew Yin)
  • Learning through "Dream Home" (CHAN Chun Ming Eric)
  • Learning through "The Unsinkable Titanic" (JAGUTHSING Dindyal and FOO Him Ho)

Readership: Graduate students and researchers in mathematics education; mathematics educators. Key Features:

• This book discusses the use of mathematical modelling activities for teaching and learning in Singapore classrooms, drawing upon experiences from other countries
• It provides focused discussions on the practicalities of conducting modelling activities in the classroom based on actual implementation in the Singapore context, drawing on the Singapore Mathematics Curriculum and other curricula
• It also proposes the way forward in addressing the issue of mathematical modelling for teacher education and curricula planning in the Singapore context

CHAPTER 1

INTRODUCTION: MATHEMATICAL MODELLING OUTREACH IN SINGAPORE

NG Kit Ee Dawn

LEE Ngan Hoe

This book documents the journey undertaken by educators from the Mathematics and Mathematics Education (MME) Academic Group in the National Institute of Education (NIE) and Singapore schools during a Mathematical Modelling Outreach (MMO) event in June 2010 under the guidance of renowned experts in the field of mathematical modelling. In this introductory chapter, we provide the context in which MMO was organised, the theoretical framework of mathematical modelling adopted for MMO, and outline the focuses of mathematical modelling during MMO in relation to the Singapore mathematics curriculum framework. We will also present an overview of the chapters in this book, summarise the progress of modelling efforts in schools to date, and draw implications for future directions on mathematical modelling in schools.

Keywords: Facilitation, Mathematical Modelling Outreach, Modelling Cycle, Singapore Schools, Teacher Education.

1.1. Mathematical Modelling Outreach (MMO)

According to the Organisation for Economic Co-Operation and Development (OECD), mathematical literacy refers to “an individual’s capacity to formulate, employ, and interpret mathematics in a variety of contexts” (OCED, 2013, p. 5). Mathematical literacy also includes the ability to reason mathematically and to make appropriate choices in use of mathematical concepts, skills, and tools to describe, explain and predict real-world situations. In other words, mathematical literacy is perceived as a part of and within the real-world as an individual harnesses the mathematics he or she possesses for decision making.

Although for many years, Singapore students have been top scorers in mathematics for international studies such as Trends in International Mathematics and Science Study (TIMSS) and Programme for International Student Assessment (PISA) (Ministry of Education [MOE], 2011; 2014a), educators in Singapore recognise the importance of preparing students towards mathematical literacy along with equipping students with 21st century competencies (MOE, 2014b). Mathematical modelling is one of the ways towards this goal.

Proponents of mathematical modelling have argued that the mathematical modelling promote connections between school-based mathematics and real-world problem solving (Blum, 2002; English, 2009; Kaiser & Maaß, 2007; Lesh & Doerr, 2003; Niss, 2010; Stillman, Brown, & Galbraith, 2008). In particular, modelling eliciting activities involving complex data embedded within real-world interdisciplinary situations have been reported as rich platforms to inculcate mathematical reasoning and sense making, making learning more meaningful to younger children (English, 2009).

1.1.1. Context for MMO in Singapore

Mathematical modelling has become the foundation for the PISA 2015 mathematics framework (OECD, 2013) and it has been incorporated into the curricula of education systems all over the world. This is seen from the growing participation of the biennial International Community of Teachers of Mathematical Modelling and Applications (ICTMA) conferences which showcase representative research, teaching, and learning on modelling from different continents. For example, in Germany, Professor Gabriele Kaiser and her colleagues from The University of Hamburg have been organising modelling weeks for upper and lower secondary students for more than a decade. During these modelling weeks, students work in small groups guided by their mentors (e.g., student-teachers and educators from the university) to solve real-world problems. Chapter 2 of this book provides an excellent description of the modelling week. In Australia, Associate Professor Raymond Brown from the Griffith University at the Gold Coast has been collaborating with Associate Professors Gloria Stillman and Vincent Geiger from the Australian Catholic University along with other mathematics educators such as Mr Trevor Redmond previously from A. B. Paterson College to facilitate Mathematical Modelling Challenges for many years. Modelling Challenges involves primary, secondary, and pre-university students working in groups on a range of modelling activities in week-long infusion events under the tutelage of experts in the field of mathematical modelling. Representatives from Singapore secondary schools have begun to participate in Modelling Challenges regularly since 2009. Australian schools in Queensland have long infused mathematical modelling in their curriculum. Chapters 5 and 9 in this book provide readers with insights into teaching and learning with mathematical modelling in the Australian context.

In Singapore, mathematical modelling was first incorporated in the mathematics curriculum framework in 2003. However, the introduction to mathematical modelling was subtle, with limited teacher education on the value of modelling activities. Schools were free to interpret the learning outcomes from modelling activities which were often viewed as optional enrichment tasks. Singapore teachers and students face constraints in time management due to traditional high stakes assessment systems. This in turn reduces the options to various mathematical learning activities. If modelling tasks were not integrated into the mainstream curriculum focuses in teaching, learning, and assessment, it would be challenging for Singapore teachers to harness the potentials of such tasks (Ang, 2010a). Furthermore, Singapore teachers were mostly unfamiliar with what mathematical modelling is, with many associating it with drawings of bar models commonly used for problem solving in schools (Chan, 2013). Many teachers were also unable to distinguish between applications and modelling as both refer to problem solving in real-world contexts (Ng, 2013a). Stillman, Brown, and Galbraith (2008) provided an effective way differentiating between applications and modelling. Application tasks are adopted when the teacher already has in mind specific taught mathematical knowledge and skills to be used and searches for appropriate real-world situations to showcase the application. On the other hand, mathematical modelling starts with the real-world context which draws upon different mathematical knowledge and skills depending on the interpretations of the modellers during model development to represent the problem solving context. Whilst application tasks have been used frequently in Singapore schools for many years, modelling was still a predominantly new idea which requires a mind set change in teachers’ conceptions of the problem solving process as well as pedagogical focuses (Ng, 2010). The open-ended non-linear problem solving process during mathematical modelling poses varying challenges to Singapore teachers and their students. Teachers generally lack awareness of the facilitation focuses during the cyclical modelling stages, specifically how students can be guided in their mathematisation processes between these stages. With reports on the impact of modelling activities on students’ engagement with mathematics few and far between, mathematical modelling has remained quite a mystery to Singapore schools.

The lack of teacher readiness in mathematical modelling both in mind set and facilitation repertoire needed to be addressed (Ng, 2010; 2013b). In a concerted push towards the infusion of mathematical modelling in schools, a range of efforts from the Mathematics and Mathematics Education Academic Group (MME) in the National Institute of Education (NIE) and Singapore Ministry of Education started in 2009 (Ang, 2013; Balakrishnan, Yen, & Goh, 2010, Balakrishnan, 2011; Curriculum Development and Planning Division [CPDD], 2009; Lee, 2013; Ng & Lee, 2010; Tan & Ang, 2013). First, opportunities needed to be provided for primary and secondary school teachers to experience mathematical modelling lessons and witness students’ mathematisation processes during model development. This was the initial impetus for Singapore’s first Mathematical Modelling Outreach (MMO) event organised by MME at NIE Singapore in June 2010. Associate Professor Ang Keng Cheng, a pioneer researcher on mathematical modelling in Singapore, as well as modelling experts in Australia and Germany became pillars of support for MMO.

1.1.2. Objectives and structure of MMO

MMO was set to achieve three main objectives. Its main goal was to reach out to Singapore primary and secondary schools and introduce the potentials of mathematical modelling as a platform for eliciting mathematical thinking, communication, and reasoning among students. Another purpose was to make connections between school-based mathematics and real-world problem solving more explicit for students through model development. It is hoped that students would build upon their existing mathematics repertoire whilst creating mathematical arguments for specific real-world contexts in the modelling process. A third objective of MMO was to provide teachers with preliminary teacher education on (a) what mathematical modelling is, (b) how to facilitate the modelling process, and (c) task design guidelines for crafting modelling problems.

MMO was designed to be a three-day immersion programme for school representatives from 1st to 3rd June 2010. A total of 13 and 15 Singapore primary and secondary schools respectively participated in MMO along with one primary school from Bandung, Indonesia, and one secondary school from the Gold Coast, Australia. Each school sent a team of one teacher and four students for MMO. A two-pronged approach was adopted during MMO. Firstly, students formed inter-school working groups of four at primary and secondary levels to solve modelling problems facilitated by NIE pre-service teachers from the third year of Bachelor of Arts and Science Degree programme. The pre-service teachers were selected based on their mathematics results and teaching performance. They worked in pairs and each pair is placed under the tutelage of a mathematics educator from MME to prepare for the facilitation of an assigned modelling problem. The pre-service teachers and their mentors underwent several sessions of mathematical modelling problem solving and facilitation process conducted by the first author before they prepared for their assigned modelling task in MMO. Student-groups from schools were provided with scaffolding worksheets and resources to complete their modelling tasks over the first two days of MMO. They spent the third day preparing for their sharing and giving verbal presentations of their journey in developing the mathematical model for the given real-world problem. Feedback on the mathematical models created by student-groups was provided by the pre-service teachers and their MME mentors. Secondly, the accompanying teachers for MMO were invited to participate in teacher workshops on mathematical modelling with respect to (a) to (c) above as well as seminars presented by invited speakers. Details about the teacher workshops can be found in Chan (2013), Lee (2013), and Ng (2013a). The workshops and seminars were planned with a view of sharing ideas on how mathematical modelling can be infused in Singapore mathematics classrooms. Representatives from the Singapore Ministry of Education who were crafting guidelines to schools on mathematical modelling then also contributed to the question-and-answer sessions in each workshop. In addition, the teachers were also invited to sit in during some of the facilitation sessions conducted by the pre-service teachers so that they could experience first-hand how a modelling activity can be conducted. Student work from MMO were presented on 4th June 2010 during the second Lee Peng Yee Symposium where renowned modelling experts were also invited as keynote speakers to provide more holistic views about what modelling can be interpreted as at the primary and secondary levels in different education systems around the world and offer practical suggestions for teachers on how to infuse modelling tasks in their day-to-day teaching activities.

1.2. Theoretical Framework for Mathematical Modelling in MMO

Mathematical modelling is referred to as a process of representing real-world problems in mathematical terms in an attempt to understand and find solutions to the problems (Ang, 2010b). Blum and Niss (1991) had considered this as the “applied problem solving process” (p. 38). There have been many diagrammatic representations of this applied problem process or mathematical modelling process discussed in modelling research throughout the years. Chapter 2 in this book by Kaiser and Grünewald provides a comprehensive but critical history of the conceptual frameworks underlying the myriad diagrams. In this section, we cite de Lange (2006) as the starting platform for our theoretical framework in MMO. There are parallel concepts between this framework and those used in PISA’s mathematics framework (OECD, 2013) and the current Singapore mathematics syllabus (CPDD, 2012a).

Figure 1.1 is a simplistic adaptation of de Lange’s conceptual framework of the mathematical modelling process which he referred to as the “mathematisation cycle” (de Lange, 2006, p. 17). de Lange emphasised that mathematisation is key to mathematical literacy which involves the appropriate and flexible use of mathematics within real-world contexts. Hence, mathematisation process always begins from an applied problem or the real-world problem. This has been the case for most diagrammatic representations of the mathematical modelling process. The real-world problem is then analysed using mathematical lenses. Here, the modeller selects appropriate mathematical concepts, skills, and tools based on his or her interpretation of the context the real-world problem is situated in. This results in a mathematical problem which is essentially a simplified version of the actual real-world problem with assumptions made, conditions identified, and key variables quantified mathematically. de Lange (2006) highlighted the gradual “trimming away” (p. 18) of reality when formulating the mathematical problem as a necessary procedure but cautioned that the mathematical problem should represent the real-world problem faithfully. A mathematical solution follows where the modeller attempts to bring fruition to the preliminary mathematical model. The modeller later tries to make sense of the solution in terms of the real-world context and examines the solution for its actual applicability in reality. A real-world solution develops from the preliminary model. In essence, the process of mathematisation is cyclical because the modeller has to move between the four components in Figure 1.1 when reviewing the limitations of the initial mathematical model and refining it.

Figure 1.1. An adaptation of the mathematisation cycle from de Lange (2006, p. 17).

1.2.1. Theoretical framework of modelling MMO in relation to the Singapore mathematics curriculum

In the design and selection of tasks for MMO, we adopted two theoretical perspectives of mathematical modelling in educational contexts outlined by Kaiser and Sriraman (2006): realistic modelling and contextual modelling. Realistic modelling serves pragmatic-utilitarian goals which promote the solving of real-world problems to support the relevance of school mathematics and to nurture modelling competencies (e.g., Kaiser, Maaß, 2007). Contextual modelling views mathematical models as “purposeful conceptual systems” (Lesh, 2003, p. 44) which describes, explains, or predicts real-world phenomena. Proponents of this perspective (e.g., English, 2009; Lesh & Doerr, 2003) emphasise the mathematisation of a situation in which modellers devise or select appropriate symbolic mathematical representations (i.e., model development) for the given real-world situation in meaningful ways (e.g., explaining, justifying, predicting, conjecturing, representing, and quantifying). An important goal of the contextual modelling perspective is for modellers to develop, critique, and validate their own mathematical models in view of the assumptions and conditions se...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Authors’ Biography
• Acknowledgements
• Chapter 1 Introduction: Mathematical Modelling Outreach in Singapore: NG Kit Ee Dawn and Lee Ngan Hoe
• Section 1: Setting the Stage for Mathematical Modelling in Schools
• Section 2: Fostering Mathematical Modelling in Schools
• Section 3: Mathematical Learning through Modelling Tasks