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Teaching Mathematical Modelling: Connecting to Research and Practice

Gloria Ann Stillman, Gabriele Kaiser, Werner Blum and Jill P. Brown, editors
Publication Date: 
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International Perspectives in the Teaching and Learning of Mathematical Modelling
[Reviewed by
Peter Olszewski
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Teaching Mathematical Modelling: Connecting to Research and Practice is a vast collection of international research on teaching mathematical modeling of applied problems. The book explores both pedagogical and teaching practices for the interested reader. The students’ ages range from elementary to secondary schooling years, with a heavy focus on how teachers can use real-world examples in their mathematics classes.

The book is divided into seven parts:

I. Innovative Practices in Modelling Education Research and Teaching

II. Research into, or Evaluation of, Teaching Practice

III. Pedagogical Issues for Teaching and Learning

IV. Influences of Technologies

V. Assessment in Schools

VI. Applicability at Different Levels of Schooling, Vocational Education, and in Tertiary Education

VII. Modelling and Applications in Business and the Lived Environment.

The book’s foundation is laid out by the discussion of the editors on the International Community of Teachers of Mathematical Modelling and Application. Here, the authors describe the need and importance of modeling in mathematics education in order for students to be globally competitive. As pointed out over the course of the book, exposing students to mathematical models and real-life situations should start as early as possible for students to develop critical thinking skills for the future. Each contributor to the text outlines his/her model with a clear procedure, model, and outcome.

The book has a global perspective, with authors ranging from Australia and Japan to Brazil. Some of the most interesting models are outlined below along with some of my critics.

  1. The Parachuting MO model described on page 211 is a great example of a problem that requires developing two outcomes. This is a great problem for the traditional mathematics major to see, along with physics and engineering students. The research questions studied for this model are very relevant for today’s classroom lessons, as we want to see how students handle a problem with multiple outcomes. Here, the students had a difficult time in “understanding, making assumptions, idealizing and structuring a situation model as well as in constructing a mathematical model.” The model did have a few positive outcomes as students were motivated to use Pythagoras’s Theorem and they were able to see the problem did have two possible outcomes.
  2. The Bluefin Tuna model in chapter 20 is a great example of how data collection can be used and modeled on a graph. It was very interesting to see how the data was given to the first period students but only part of the data was given to the second group. Here, students had to think about how to recover the number of bluefin tuna. The modification for second period students was a great idea but fell short of the goal, as class time was insufficient. The model was extended to year 8 and 9 students. Year 9 students used the CASIO fx-9700GE graphing calculator and made the connection to a sequence. Using a recurrence relation, students were able to find the rate of increase of Bluefin Tuna. This was a great model for several reasons. Firstly, students are able to make connections between linear models and then to sequences, secondly, students are able to use technology to graph their results and be able to make further assumptions, and thirdly, as pointed out on page 237, “many Japanese students are interested in mathematics.” With a projects like this, students can be exposed to mathematics through real-world problems and be able to see the power and beauty of mathematics.
  3. For the Pension Tax Issue model that Noboru Yoshimura and Akira Yanagimoto explain in chapter 21, we see once again how effective modeling can be. After completing the model, students were more inclined to find other possible solutions and were more likely to study the topic than before. In short, it was an eye-opening model and a win-win for both teachers and students.
  4. In chapter 38, Gilbert Greefrath and Michael Rieß examine some of the results of the CASI-Project in German schools. Many problems were posed to the students about the copier task and the Roundabout task. It would have been nice to see some more of the procedures and results students had and obtained. According to the comparison charts on page 454, students gained a much deeper appreciation of mathematics and see how important mathematics is in the real world. This point of view is only obtained through the presentation of applied models.

As we can see, these are only a brief selections of the models presented in the book. There are 52 chapters. Some of the ideas for models are excellent, while others fall short of expected outcomes, either due to time constraints on class periods or not enough of the students’ results outlined by their authors. In any case, this book gives the reader a more diverse outlook on what others are doing with models at an international level. The true test is trying out these models in our own classroom and seeing what works and what doesn’t. This, of course, is how we should continually evolve in our teaching, getting students motivated to think on their own. Independent discovery is an important skill that students of the 21st century need in order to be competitive with others and with other countries.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He can be reached at Webpage: Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.

Series Preface:  Gabriele Kaiser and Gloria Stillman

Chapter 1 Mathematical Modelling: Connecting to Teaching and Research Practices – the impact of globalisation:  Gloria Stillman, Gabriele Kaiser, Werner Blum, and Jill Brown

Part I Innovative Practices in Modelling Education Research and Teaching
Chapter 2 From Conference to Community: An ICTMA Journey—The Ken Houston Inaugural Lecture: Peter Galbraith
Chapter 3 Modelling from the Perspective of Commognition – An Emerging Framework: Jonas Bergman Ärlebäck and Peter Frejd
Chapter 4 Should Interpretation Systems be Considered to be Models if They only Function Implicitly?:  Rita Borromeo Ferri and Richard Lesh
Chapter 5 Mathematical Modelling, Mathematical Content and Tensions in Discourses: Andréia Maria Pereira de Oliveira and Jonei Cerqueira Barbosa
Chapter 6 Ethnomodelling as a Methodology for Ethnomathematics: Milton Rosa and Daniel Clark Orey
Chapter 7 Dual Modelling Cycle Framework for Responding to the Diversities of Modellers: Akihiko Saeki and Akio Matsuzaki
Chapter 8 The Eyes to See: Theoretical Lenses for Mathematical Modelling Research: Nils Buchholtz
Chapter 9 Strässer’s Didactic Tetrahedron as a Basis for Theorising Mathematical Modelling Activity within Social Contexts: Vince Geiger
Chapter 10 Ethnomodelling as a Research Lens on Ethnomathematics and Modelling: Milton Rosa and Daniel Clark Orey

Part II Research into, or Evaluation of, Teaching Practice
Chapter 11 Real-life Modelling within a Traditional Curriculum: Lessons from a Singapore Experience: Ang Keng Cheng
Chapter 12 Students’ Mathematical Learning in Modelling Activities: Morten Blomhøj and Tinne Hoff Kjeldsen
Chapter 13 Students' Designing an Ideal Tourism Route as Mathematical Modelling: Chan Chun Ming Eric
Chapter 14 Comparison of Mathematical Modelling Skills of Secondary and Tertiary Students: Juntao Fu and Jinxing Xie
Chapter 15 Taking Advantage of Incidental School Events to Engage with the Applications of Mathematics: The Case of Surviving the Reconstruction: Vince Geiger, Merrilyn Goos, and Shelley Dole
Chapter 16 The Development of Modelling Competencies by Year 9 Students: Effects of a Modelling Project: Susanne Grünewald
Chapter 17 Evidence of a Dual Modelling Cycle: Through a Teaching Practice Example for Pre-service Teachers: Akio Matsuzaki and Akihiko Saeki
Chapter 18 Considering Multiple Solutions for Modelling Problems – Design and First Results from the MultiMa-Project: Stanislaw Schukajlow and André Krug
Chapter 19 Challenges in Modelling Challenges: Intents and Purposes: Gloria Stillman, Jill Brown, and Peter Galbraith
Chapter 20 Mathematical Modelling of a Real-world Problem: The Decreasing Number of Bluefin Tuna: Akira Yanagimoto and Noboru Yoshimura
Chapter 21 Mathematical Modelling of a Social Problem: Pension Tax Issues: Noboru Yoshimura and Akira Yanagimoto

Part III Pedagogical Issues for Teaching and Learning
Chapter 22 Pedagogical Reflections on the Role of Modelling in Mathematics Instruction: Toshikazu Ikeda
Chapter 23 Complex Modelling Problems in Co-Operative, Self-directed Learning Environments: Gabriele Kaiser and Peter Stender
Chapter 24 Inducting Year 6 Students into “a Culture of Mathematising as a   Practice”: Jill Brown
Chapter 25 A Whole Week of Modelling – Examples and Experiences of Modelling for Students in Mathematics Education: Nils Buchholtz and Sarah Mesrogli
Chapter 26 Teachers’ Self-Perceptions of their Pedagogical Content Knowledge Related to Modelling – An Empirical Study with Austrian Teachers: Sebastian Kuntze, Hans-Stefan Siller, and Christiane Vogl
Chapter 27 A Cross-Sectional Study about Modelling Competency in Secondary School: Matthias Ludwig and Xenia-Rosemarie Reit
Chapter 28 Teacher Readiness in Mathematical Modelling: Are there Differences between Pre-service and In-service Teachers? Kit Ee Dawn Ng
Chapter 29 Exploring the Relationship between Mathematical Modelling and Classroom Discourse: Trevor Redmond, Raymond Brown, and Joanne Sheehy
Chapter 30 The Role of Textbooks in Developing a Socio-critical Perspective on Mathematical Modelling in Secondary Classrooms: Gloria Stillman, Jill P Brown, Rhonda Faragher, Vince Geiger and Peter Galbraith
Chapter 31 Pre-service Secondary School Teachers’ Knowledge in Mathematical Modelling – A Case Study: Tan Liang Soon and Ang Keng Cheng
Chapter 32 How Students Connect Descriptions of Real-world Situations to Mathematical Models in Different Representational Modes: Wim Van Dooren, Dirk De Bock, and Lieven Verschaffel
Chapter 33 Pre-service Teacher Learning for Mathematical Modelling: Mark Winter
Chapter 34 Initial Perspectives of Teacher Professional Development on Mathematical Modelling in Singapore: Problem Posing and Task Design: Lee Ngan Hoe
Chapter 35 Initial Perspectives of Teacher Professional Development on Mathematical Modelling in Singapore: Conceptions of Mathematical Modelling: Chan Chun Ming Eric
Chapter 36 Initial Perspectives of Teacher Professional Development on Mathematical Modelling in Singapore: A Framework for Facilitation: Kit Ee Dawn Ng
Chapter 37 Teacher Professional Development on Mathematical Modelling: Initial Perspectives from Singapore: Vince Geiger

Part IV       Influences of Technologies
Chapter 38 Reality Based Test Tasks with Digital Tools at Lower Secondary: Gilbert Greefrath and Michael Rieß
Chapter 39 On Comparing Mathematical Models and Pedagogical Learning: Janeen Lamb and Jana Visnovska

Part V Assessment in Schools
Chapter 40 Formative Assessment in Everyday Teaching of Mathematical Modelling: Implementation of Written and Oral Feedback to Competency-Oriented Tasks: Michael Besser, Werner Blum, and Malte Klimczak
Chapter 41 Assessment of Modelling in Mathematics Examination Papers: Ready-made Models and Reproductive Mathematising: Pauline Vos

Part VI Applicability at Different Levels of Schooling, Vocational Education, and in Tertiary Education
Chapter 42 Complex Modelling in the Primary and Middle School Years: An Interdisciplinary Approach: Lyn D. English
Chapter 43 Modelling in Brazilian Mathematics Teacher Education Courses: Maria Salett Biembengut
Chapter 44 The Development of Mathematical Concept Knowledge and of the Ability to use this Concept to Create a Model: César Cristóbal Escalante and Verónica Vargas Alejo
Chapter 45 Problem Posing: A Possible Pathway to Mathematical Modelling: Ann Downton
Chapter 46 A Study of the Effectiveness of Mathematical Modelling of Home:  Delivery Packaging on Year 12 Students' Function Education: Tetsushi Kawasaki and Yoshiki Nisawa
Chapter 47 How to Introduce Mathematical Modelling in Industrial Design Education? Geert Langereis, Jun Hu, and Loe Feijs
Chapter 48 Rationality of Practice and Mathematical Modelling – On Connections, Conflicts, and Codifications: Lars Mouwitz
Chapter 49 Extending Model Eliciting Activities (MEAs) beyond Mathematics Curricula in Universities: Mark Schofield
Chapter 50 Building Awareness of Mathematical Modelling in Teacher Education: A Case Study in Indonesia: Wanty Widjaja

Part VII Modelling and Applications in Business and the Lived Environment
Chapter 51: Mathematics and the Pharmacokinetics of Alcohol: Michael Jennings and Peter Adams. - Chapter 52 Beyond the Modelling Process: An Example to Study the Logistic Model of Customer Lifetime Value in Business Marketing: Issic K. C. Leung

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