Gabriele Kaiser, Werner Blum, Rita Borromeo Ferri, Gloria Stillman
Springer | 2011 | 747 páginas | PDF | 15,5 MB
Applications and modelling and their learning and teaching in school and university have become a prominent topic in the last decades in view of the world-wide importance of the usage of mathematics in science, technology and everyday life. "Trends in Teaching and Learning of Mathematical Modelling "provides the reader ship with an overview on the newest international trends and developments on the teaching and learning of modelling from various theoretical and practical perspectives. The comprehensive overview on the most recent empirical research reflecting the development and promotion of modelling competencies at various age levels allows insight into possible affective and cognitive blockages and barriers for students modelling processes and its teaching. The papers on the usage of technology describe new possibilities, how the usage of technology can inspire the teaching and learning of modelling. International modelling projects offer chances and possibilities to enrich the teaching and learning of mathematical modelling at secondary and tertiary level and describe challenging modelling examples and their possible usage in school and university. The necessary change of teacher education towards an inclusion of mathematical modelling is reflected from different perspectives and challenging examples are given. The contributing authors are influential members of the group International Community of Teachers of Modelling and Applications and important researchers in mathematics and mathematics education. The book will be of interest to mathematics educators, teacher educators, researchers, education administrators, curriculum developers, teachers and student teachers.
Common terms and phrases:
mathematical modelling mathematics education metacognitive praxeologies Springer Science+Business Media
Common terms and phrases:
mathematical modelling mathematics education metacognitive praxeologies Springer Science+Business Media
Cover
International Perspectives on the Teaching and Learning of Mathematical Modelling 1
Trends in Teaching and Learning of Mathematical Modelling
ISBN 9789400709096
Series Preface
Contents
Chapter 1: Trends in Teaching and Learning of Mathematical Modelling – Preface
References
Part I: Modelling from Primary to Upper Secondary School: Findings of Empirical Research
Chapter 2: Modelling from Primary to Upper Secondary School: Findings of Empirical Research – Overview
Chapter 3: Can Modelling Be Taught and Learnt? Some Answers from Empirical Research
1 A Cognitive View on Mathematical Modelling
2 How Do Students Deal with Modelling Tasks?
3 How Do Teachers Treat Modelling in the Classroom?
4 Some Ideas for Teaching Modelling
References
Chapter 4: “Can Modelling Be Taught and Learnt?” – A Commentary
1 Introduction
2 View of Modelling
3 Technology and Modelling
4 Critical Perspective and Cognition
5 Teachers and Modelling
6 Final Considerations
References
Chapter 5: Upper Secondary Students’ Handling of Real-World Contexts.*
1 Introduction
2 Some Aspects of the Current Discussion and Research Question
3 Methodology and Methods
3.1 Methodological Remarks
3.2 Methods
3.3 Tasks
4 Results
4.1 General Results
4.2 Ideal Types
4.3 Case Synopsis
5 Final Remarks
References
Chapter 6: Word Problem Classification: A Promising Modelling Task at the Elementary Level
1 Theoretical and Empirical Background
2 Method
2.1 Subjects, Tasks and Procedure
2.2 Analysis
3 Results
3.1 Solution Task
3.2 Classification Task
4 Conclusions and Discussion
References
Chapter 7: Understanding and Promoting Mathematical Modelling Competencies: An Applied Perspective
1 Introduction
2 Methodology
3 Results
3.1 Finding Similar Examples or Phenomena
3.2 Connecting Physical Phenomena with Abstract Concepts
3.3 Building Modelling from the Ground up
3.4 Communicating Broader Context of a Modelling Solution
4 Discussion and Summary
References
Chapter 8: Secondary Teachers’ Beliefs About Teaching Applications – Design and Selected Results of a Qualitative Case Study
1 The Call for Applications in Mathematics Curriculum is Quite Old!
2 A First Root Cause Analysis: Focusing on Teachers
3 Why a Qualitative Case Study?: Methodology and Methods
3.1 Theoretical Constructs
3.2 Study Design
4 Discussion and Some Selected Results
5 Conclusion
References
Chapter 9: Secondary Teachers’ Beliefs on Modelling in Geometry and Stochastics
1 Teachers’ Beliefs and Individual Curricula
2 Theoretical Background, Data, and Evaluation
3 Applications and Model Building
4 Geometry
4.1 Elementary Geometry
4.2 Analytical Geometry
5 Stochastics
6 Conclusions
References
Chapter 10: Examining Mathematising Activities in Modelling Tasks with a Hidden Mathematical Character
1 Introduction
2 Framework
2.1 ATD and Modelling
2.2 Structural Analysis
3 Empirical Design
4 A Priori Analysis
5 Findings
6 Summary and Conclusions
References
Chapter 11: The Sun Hour Project
1 Introduction
2 Classroom Circumstances
2.1 Classroom Experiences in Sweden
2.2 Classroom Experiences in Germany
3 Researchers’ View of the Project
3.1 Theoretical Frameworks
3.2 Analyzing the Work in the Swedish Classroom
3.3 Analyzing the Work in the German Classroom
4 Conclusions
References
Chapter 12: Mathematical Knowledge Application and Student Difficulties in a Design-Based Interdisciplinary Project
1 Background of Research
2 Rationale for Research
3 Research Design
3.1 Research Task
3.2 Setting and Sample
3.3 Data Collection Methods
3.4 Analysis Procedures
4 Findings
4.1 Coverage of Mathematical Knowledge and Skills
4.2 Mathematical Difficulties
5 Discussion and Conclusion
References
Chapter 13: Evaluation of Teaching Activities with Multi-Variable Functions in Context
1 The Background of the Research
2 The Educational Course Content
2.1 Two-Variable Functions – Analytic Geometry Approach
2.2 Mathematical Modelling
2.2.1 The Vehicle Stopping Distance Example
2.2.2 The Sound Propagation Example
3 Results from an Evaluation Questionnaire
3.1 Content of Investigation
3.2 Results from the Evaluation Questionnaire and Discussion
4 Conclusion
Reference
Chapter 14: Mathematical Modelling in Secondary Education: A Case Study
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions and Recommendations
References
Chapter 15: Students Overcoming Blockages While Building a Mathematical Model: Exploring a Framework
1 Introduction
2 Theoretical Framework
2.1 Modelling
2.2 Mathematising
2.3 Opportunities and Blockages
3 Method
4 Results
4.1 The Swimming Pool Task
4.2 The Horizon Task
4.3 Additional Results
5 Conclusion and Discussion
References
Chapter 16: What Did Taiwan Mathematics Teachers Think of Model-Eliciting Activities and Modelling Teaching?
1 Background
2 Theoretical Frame
3 Methodological Approach
3.1 Samples
3.2 Research Process
3.3 Data Collection
3.4 Data Analysis
3.4.1 Positive Thinking about MEAs and Modelling Teaching
Close-in Real Life Situation
Enhancement of Mathematical Competencies
Advantages of Modelling Teaching
MEAs as Supplementary Materials
3.4.2 Negative Thinking about Modelling Teaching
Out of School Curriculum
Out of Entrance Examinations
Other Obstacles of Modelling Teaching
3.4.3 Weaknesses of Designing MEAs
4 Conclusions and Implications
References
Part II: Looking Deeper into Modelling Processes: Studies with a Cognitive Perspective
Chapter 17: Looking Deeper into Modelling Processes: Studies with a Cognitive Perspective – Overview
References
Chapter 18: Applying Metacognitive Knowledge and Strategies in Applications and Modelling Tasks at Secondary School
1 Introduction
2 Modelling and Metacognition
2.1 The Modelling Process and Reflection
2.2 Meta-Metacognition and Modelling
2.3 Productive Metacognitive Acts and Modelling
2.3.1 Routine Metacognitive Activity
2.3.2 Responses to Red Flag Situations
3 Development of Modellers’ Reflective Metacognitive Activity Through Meta-Metacognition
4 Conclusions
References
Chapter 19: Effective Mathematical Modelling without Blockages – A Commentary
1 Meta-Cognition – A Fuzzy Word?
2 Effective Modelling Without Blockages: But How?
3 Connecting and Acting
References
Chapter 20: Modelling Tasks: Insight into Mathematical Understanding
1 Prior Knowledge of Task Context
2 Ways of Dealing with the Context
3 Higher Order Thinking
4 Methods
4.1 The Tasks
5 Findings
5.1 Evidence of Prior Knowledge of Task Context
5.2 Busse’s Ideal Types
5.3 Modelling and Higher Order Thinking
6 Discussion and Conclusion
References
Chapter 21: Mathematical Modelling of Daily Life in Adult Education: Focusing on the Notion of Knowledge
1 Introduction
2 Theoretical Framework
2.1 A Discussion on the Real World
2.2 Knowledge, Practice and Context
2.3 A Connection with Modelling and Competence
3 Methodology and Data Analysis
4 Results
4.1 Students’ First Chains of Meanings
4.2 Responses to Question 1 – Number Magnitude
4.3 A “Different” Answer in the Group
4.4 Responses to Question 2 – The Best Flavour
4.5 Explaining to the Group – The Medium Flavour
4.6 Responses to Question 3 – The Size of the Glass
4.7 Responses to Question 4 – Part/Whole Relation Versus a Total of 120
5 Discussion and Final Comments
References
Chapter 22: Students’ Modelling Routes in the Context of Object Manipulation and Experimentation in Mathematics
1 Introduction
2 Connecting Modelling to Experimentation in Mathematics Classroom
2.1 From the Point of View of Mathematics Education
2.2 From the Point of View of Learning by Doing
3 Theoretical Perspectives on Applications and Modelling
4 The Research Empirical Work
5 Description and Data Analysis
6 Synthesis of Findings
References
Chapter 23: Engineering Model Eliciting Activities for Elementary School Students
1 Introduction
2 Theoretical Framework
3 The Present Study
3.1 Participants and Procedures
3.2 Data Sources and Analysis
4 Results
4.1 Model A
4.2 Model B
4.3 Model C
4.4 Model D
5 Conclusions
References
Chapter 24: Project Modelling Routes in 12–16-Year-Old Pupils*
1 Context and Aims
2 Theoretical Background
3 Study Methodology
4 From Analysis to Results
5 Results
6 Conclusions
References
Part III: Mathematical Modelling in Teacher Education
Chapter 25: Mathematical Modelling in Teacher Education – Overview
1 Introduction
2 Research Using Different Perspectives of Modelling
3 Reflections
References
Chapter 26: Models and Modelling Perspectives on Teaching and Learning Mathematics in the Twenty-First Century
1 Introduction
2 MMP Research Investigates What It Means to “Understand” Important Concepts and Abilities
3 Do Model-Eliciting Activities (MEAs) Work?
4 In What Ways Do MEAs (and MMP in General) Provide Alternatives to Traditional Research?
4.1 MEAs Provide Alternatives to Naturalistic Observations
4.2 MEAs Provide Alternatives to Expert-Novice Studies
5 Teaching from a Models and Modelling Perspective
5.1 The Nature of Teachers’ Knowledge
5.1.1 Engaging Students with Modelling Tasks
5.1.2 Recognizing and Responding to Students’ Thinking
5.2 The Development of Teachers’ Knowledge
5.2.1 The Collaborative Design of Sequences of Modelling Tasks
5.3 Principles for the Design of Model Development Experiences for Teachers
6 Concluding Remarks
References
Chapter 27: Mathematical Modelling in a Distance Course for Teachers
1 Introduction
2 MM Course: Main and Common Events
2.1 Modelling in the Classroom: Possibilities and Challenges
3 Conclusion
References
Chapter 28: In-Service and Prospective Teachers’ Views About Modelling Tasks in the Mathematics Classroom – Results of a Quantitative Empirical Study
1 Introduction
2 Theoretical Background
3 Research Questions
4 Design and Sample
5 Results
6 Discussion and Conclusions
References
Chapter 29: Pre-service Secondary Mathematics Teachers’ Affinity with Using Modelling Tasks in Teaching Years 8–10
1 Introduction
2 Background
3 Theoretical Framework
4 The Study
5 Findings
5.1 Diagnostic Competencies with Respect to Modelling
5.2 Competencies in Didactical Reflections About Modelling – Appropriateness of Task
5.3 Affinity with Modelling in Teaching in Years 8–10
5.4 Affinity with Modelling Related to Beliefs About the Nature of Mathematics
6 Discussion and Conclusion
References
Part IV: Using Technologies: New Possibilities of Teaching and Learning Modelling
Chapter 30: Using Technologies: New Possibilities of Teaching and Learning Modelling – Overview
1 Modelling Using Digital Tools
2 Empirical Studies and Experiences on Modelling Using Digital Tools
References
Chapter 31: Factors Affecting Teachers’ Adoption of Innovative Practices with Technology and Mathematical Modelling
1 Introduction
2 The Use of CAS in Mathematical Modelling
3 Models of the Use of Technology in Mathematical Modelling
4 Context of the Study
4.1 Curriculum Contexts
4.2 Teachers’ Backgrounds
5 Method and Approach to Data Collection
6 A Tale of Two Cities and Two Teachers
6.1 Teacher 1
6.2 Teacher 2
7 Discussion and Conclusions
References
Chapter 32: Modelling Considering the Influence of Technology
1 Introduction
2 Alcohol in Blood – A Prospective Example for Modelling with Technology
2.1 Modelling a Theoretical Concentration
2.2 Absorption and Reduction Shown as a Mathematical Process
2.2.1 First Model – Linear Approach
2.2.2 Second Model – Semi-linear Approach
2.3 The Role of Technology in the Modelling Cycle
2.4 An Example Where Technology Is Helpful to Get an Idea
2.5 The Fuel Tank – An Example for Using Technology to Validate
3 Examination Tasks – With Modelling Problems and Use of Technology?
4 Conclusion
References
Chapter 33: Improving Learning in Science and Mathematics with Exploratory and Interactive Computational Modelling
1 Introduction
2 Course Organisation, Methodology and Student Evaluation Procedures
3 Computational Modelling Activities with Modellus
4 Conclusions
References
Part V: Modelling Competency: Teaching, Learning and Assessing Competencies
Chapter 34: Modelling Competency: Teaching, Learning and Assessing Competencies – Overview
1 Presentation of the Papers
References
Chapter 35: Drivers for Mathematical Modelling: Pragmatism in Practice
1 Being in Touch with the Real World
2 How Well Do Students Link Mathematical Knowledge to the Task at Hand?
3 How Far Away Is the Real World?
3.1 Kidney Dialysis
3.2 Rocket Satellite Systems
3.3 Aggregation of Slime Mould Amoebae
3.4 Road Traffic Flows
3.5 Local Models
4 Is Mathematical Modelling a Driver for Mathematical Modelling?
References
Chapter 36: Identifying Drivers for Mathematical Modelling – A Commentary
1 How Well Do Students Link Mathematical Knowledge to the Task at Hand?
2 How Far Away Is the Real World?
3 Is Mathematical Modelling a Driver for Mathematical Modelling?
4 What the Title Made Me Think of…
References
Chapter 37: Documenting the Development of Modelling Competencies of Grade 7 Mathematics Students
1 Introduction
2 A Perspective for Modelling
3 What Is Competence and What Are Modelling Competencies?
4 Methodology
5 Results
References
Chapter 38: Students’ Reflections in Mathematical Modelling Projects
1 Introduction
2 Internal and External Reflections in Mathematical Modelling Competency
3 Students’ Reflections in Modelling Projects
3.1 The Institutional Context of the Project Work at Roskilde University
3.2 The Use of a Traffic Model in the City of Roskilde – The Case of ‘Ny Østergade’
3.3 Modelling in Scientific Investigations: The Project of the HPA-Axis
4 Concluding Remarks
References
Chapter 39: From Data to Functions: Connecting Modelling Competencies and Statistical Literacy
1 Introduction
2 Modelling Competencies and Statistical Literacy
3 Empirical Evidence on Modelling Competencies and Statistical Thinking
4 Results
4.1 Discussion and Outlook on Further Results
5 Conclusions
References
Chapter 40: First Results from a Study Investigating Swedish Upper Secondary Students’ Mathematical Modelling Competencies
1 Introduction and Purpose
2 Methodology, Theoretical Considerations and Method
2.1 Mathematical Modelling and Modelling Competencies
2.2 Developing an Instrument
2.3 Statistical Analysis
3 Results
4 Discussion
5 Conclusions
References
Chapter 41: Why Cats Happen to Fall from the Sky or on Good and Bad Models
1 The Operation ‘Cat Airdrop’
2 Modelling in School: Chances and Obstacles
3 The Problem Field ‘Central Examinations
4 The Use of Computers
5 The Professional Development and Motivation of Teachers
6 Conclusion
References
Chapter 42: Assessing Modelling Competencies Using a Multidimensional IRT Approach
1 Introduction
2 Modelling Competency
3 Test Instrument
4 Data Scaling
5 Results of the Evaluation
6 Discussion
References
Part VI: Modelling in Tertiary Education
Chapter 43: Modelling in Tertiary Education – Overview
References
Chapter 44: The Mathematical Expertise of Mechanical Engineers: Taking and Processing Measurements
1 Introduction
2 Method of Investigation and Task
3 Approach of Students
4 Findings and Discussion
4.1 Benefits and Problems of the Method
4.2 Modelling Qualifications
4.3 Data Interpretation and Model Validation Qualifications
4.4 Comparison with Other Research
5 Conclusions for Education
References
Chapter 45: Mathematical Modelling Skills and Creative Thinking Levels: An Experimental Study
1 Introduction
2 Mathematical Modelling Skills
2.1 Test Questions
2.2 Implementation
2.3 Results
3 Creative Thinking Levels
3.1 Test Questions
3.2 Implementation
3.3 Results
3.4 Relationship with Mathematical Modelling Skills
4 Knowledge in Basic Mathematics
4.1 Score in Basic Mathematical Courses
4.2 Relationship with Mathematical Modelling Skills
5 Summary
References
Chapter 46: Modelling the Evolution of the Belgian Population Using Matrices, Eigenvalues and Eigenvectors
1 Introduction
2 The Teaching Sequence
2.1 Calculations with Authentic Data
2.2 The Matrix Model
2.3 Two Observations Concerning the Long Term Evolution of the Population
2.4 Mathematical Treatment of the Observations
2.5 Eigenvalues and Eigenvectors
3 Experiences
3.1 During the ‘Science Week’
3.2 In Mathematics Teacher Education
3.3 In an Introductory Mathematics Course for Bachelor Students in Applied Economics
4 Conclusion
References
Chapter 47: Modelling and the Educational Challenge in Industrial Mathematics
1 Computational Technology
2 Educational Challenge
3 Sphere of Applications
4 Modelling as a Course Subject
5 Modelling Problems to Challenge Undergraduates
6 Modelling Education: How Much and When?
References
Chapter 48: Modelling of Infectious Disease with Biomathematics: Implications for Teaching and Research
1 Introduction and Framework
2 The Study
2.1 The Models
2.2 The Questionnaire
3 The Responses
3.1 Students
3.2 Lecturers
3.3 Analysis of the Responses
4 Conclusions
References
Chapter 49: Using Response Analysis Mapping to Display Modellers’ Mathematical Modelling Progress
1 Introduction
2 Components Based on Experiences
3 Applied Response Analysis Mapping as an Analysis Method
4 Research Setting
5 Modelling Progress Using Applied Response Analysis Mapping
5.1 The Case of a Graduate School Student
5.2 The Case of an Electronics Expert
6 Discussion
6.1 Focus on CRE
6.2 Focus on CME
7 Conclusion
Appendix 1: Mathematical Modelling Progress of NT
Appendix 2: Mathematical Modelling Progress of KN
References
Part VII: Modelling Examples and Modelling Projects: Concrete Cases
Chapter 50: Modelling Examples and Modelling Projects – Overview
1 The Challenge
1.1 Analysis of the Challenge
1.2 Helping Teachers
1.3 Situations for Modelling
Chapter 51: Modelling Chemical Equilibrium in School Mathematics with Technology
1 Introduction
2 The Case
3 Models of Chemical Equilibria
3.1 Chemical Equilibria in Our Case
4 Aims of the Students’ Work
5 Methods for Examination of Data
6 Results
6.1 Students’ Perception of Modelling in Mathematics and in Chemistry
6.2 The Students’ Understanding of the System of Chemical Equilibrium
6.3 Students’ Understanding of Connections Between Theory and Practice
6.4 Technology as a Means to Make the Modelling Process More Explicit
7 Conclusion
7.1 Technical Obstacles
7.2 Few Students’ Reflections
7.3 Little Focus on Modelling
8 Perspectives
References
Chapter 52: Real-World Modelling in Regular Lessons: A Long-Term Experiment
1 Idea of the Experiment
2 Design of the Experiment
3 Choice of Five Real-World Modelling Tasks
3.1 First Task: A Guided Modelling as an Introduction
3.2 Second Task: Is the Olympic Medals Table Fair?
3.3 Third Task: How to Type on a Container?
3.4 Fourth Task: Building of an ICE-Track 9
3.5 Fifth Task: How to Do an Optimal Free Throw in Basketball 11 ?
3.6 Difficulty and Attractiveness of the Projects
4 Detailed Discussion of Task Number 4: ICE-Track
5 Evaluation
5.1 Concept of Questionnaires
5.2 Development from Second to Fourth Task and Final Judgements
6 Comparison Project: Setting and Results
7 Summary and Conclusion
References
Chapter 53: Modelling Tasks at the Internet Portal “Program for Gifted”
1 Fostering of Gifted Students
2 The Interests of Gifted Students: A Bottom-up Approach
3 Examples Containing Modelling Components
3.1 From Cones to Higher Algebraic Curves and Back
3.2 From the Lottery to the Pascal Triangle
4 Summary
References
Chapter 54: Modelling at Primary School Through a French–German Comparison of Curricula and Textbooks
1 Origin, Method and Theoretical Framework of the Study
2 Comparison of Curricula: Is Modelling a Knowledge to Be Taught?
3 Articulation Between Real and Mathematical World in Textbooks
3.1 Real World Knowledge
3.2 Mathematical World Knowledge
3.3 Representation Involved in the Tasks
3.4 Whole Competencies and Partial Competencies: Didactical Functions of Tasks
4 Conclusion: Challenges for Modelling Resources and Teacher Training
References
Chapter 55: Modifying Teachers’ Practices: The Case of a European Training Course on Modelling and Applications
1 Introduction
2 Teacher Education on Modelling and Applications: From a Teachers’ Problem to a Professional Problem
3 A Theoretical Framework to Describe Teaching Actions
3.1 Modelling the Teaching Activity
4 LEMA Professional Development: Changing Teachers’ Practices
5 Conclusions and Implications
References
Chapter 56: Google’s PageRank: A Present-Day Application of Mathematics in the Classroom *
1 Introduction
2 The a Directed Graph and the Description by Transition Matrices
2.1 How Can We Measure the Relevance of a Site?
2.2 Now to an Example Slightly More Complicated
2.3 The Crucial Attribute of U
2.4 Explicit Solution (Formula)
3 Summary and Reflections
References
Chapter 57: Authentic Modelling Problems in Mathematics Education
1 Theoretical Framework for Modelling in Mathematics Education
2 Framework and Structure of the Modelling Week
3 Students Modelling the Spread of Disease in a Population of Ladybirds
4 Evaluation of the Modelling Week
References
Chapter 58: Using Modelling Experiences to Develop Japanese Senior High School Students’ Awareness of the Interrelations between Mathematics and Science
1 Introduction
2 An Example Using Modelling: Teaching Materials for Kepler’s Law for High School Students Becoming Scientists
2.1 Premodel
2.2 Mathematical Development Model
3 Students’ Evaluation and Impressions
3.1 The Class Treated the Premodel
3.2 The Class Treated the Mathematical Development Model
4 Conclusion and Future Subjects
References
Chapter 59: Stochastic Case Problems for the Secondary Classroom with Reliability Theory
1 Introduction
2 Reliability Theory
2.1 Brief History
2.2 A Summary of the Reliabilities of Simple Fundamental Systems
2.2.1 Series Systems
2.2.2 Parallel Systems
2.2.3 Combined Systems
3 Reliability Theory Problems for the Secondary Classroom
3.1 Prerequisites for Reliability Theory
3.2 Examples of Problems for Secondary School
3.2.1 Level I Problems
3.2.2 Level II Problems
3.2.3 Sample Project
4 Conclusion: The Mathematical Residue of Reliability Tasks (RT)
References
Chapter 60: LEMA – Professional Development of Teachers in Relation to Mathematical Modelling*
1 Theoretical Background
2 Design of the Course of Professional Development
3 Design of the Evaluation
4 Results and Discussion
References
Chapter 61: Modelling in the Classroom: Obstacles from the Teacher’s Perspective
1 Basic Theory
1.1 Mathematical Modelling
1.2 Obstacles to the Integration of Modelling
1.3 Research Questions
2 Methodology
2.1 Instruments for the Study
2.2 Study Design
2.3 Sample
3 Questionnaire Development
3.1 Questionnaire Development
3.2 Format of Questionnaire
4 Results
4.1 I Have Too Little Material
4.2 Performance Assessment is Too Complex
4.3 I Don’t Have Enough Time for Modelling
5 Discussion
References
Chapter 62: Teachers’ Professional Learning: Modelling at the Boundaries
1 Introduction
2 Classroom Practice and Its Transformation
3 Knowledge for Teaching and Learning in Modelling Classrooms
4 Further Theoretical Reflections
References
Part VIII: Theoretical and Curricular Ref lections on Mathematical Modelling
Chapter 63: Theoretical and Curricular Reflections on Mathematical Modelling – Overview
1 Paper Summaries
References
Chapter 64: Making Connections Between Modelling and Constructing Mathematics Knowledge: An Historical Perspective
1 Aims in This Study
2 Mathematics Textbooks Before World War II in Japan
3 Roles of Real World Situations
4 Repeated Instances of the Same Contexts
5 Making Connections Between Real World Problems
6 Implications for Teaching of Mathematics Today Including Modelling
7 Conclusion
References
Chapter 65: Practical Knowledge of Research Mathematicians, Scientists, and Engineers About the Teaching of Modelling
1 Introduction
2 The Case Study
2.1 Theoretical Framework
2.2 Design of the Case Study
3 Summary of Interview Findings
3.1 Teaching Goals of Modelling in Tertiary Education
3.2 Modelling Courses in Tertiary Education
3.3 Students’ Modelling Competencies
3.4 Opinions about Modelling in Secondary Education
4 Discussion and Conclusions
4.1 Similarities Between Interviewees and Education Researchers
4.2 Differences Between Interviewees and Education Researchers
4.3 Differences Among Interviewees
4.4 Possible Implications for Secondary Education
References
Chapter 66: Evolution of Applications and Modelling in a Senior Secondary Curriculum
1 Introduction
2 Mathematical Modelling and Applications in Queensland
2.1 Syllabus Objectives
2.2 Implementation in Schools
3 Research Methods
4 Findings
4.1 Reasons Applications and Modelling Valuable Initiative at Senior Secondary
4.2 Distinction Between Applications and Modelling
4.3 Embedding Applications and Modelling in Current Practice
4.4 Designing Tasks
5 Discussion and Conclusion
References
Chapter 67: Sense of Reality Through Mathematical Modelling.*
1 Introduction
2 Brief Description of Modelling in Colombian Educational Regulations
3 The Project
3.1 The Context
3.2 Methodological Approach
4 Results
4.1 What is Sense of Reality ?
4.2 What is Reality for Alberto and Alexander?
4.3 Sense of Reality in School Mathematics
4.4 A First Approach
5 Discussion
6 Conclusions
References
Chapter 68: What Is ‘Authentic’ in the Teaching and Learning of Mathematical Modelling?
1 Introduction
2 Variations of Tasks in Mathematics Education
3 Definitions of ‘Authenticity’ in Mathematics Education
4 Problems with Defining ‘Authenticity’
5 Pragmatically Constructing Authenticity in Mathematical Modelling
References
International Perspectives on the Teaching and Learning of Mathematical Modelling 1
Trends in Teaching and Learning of Mathematical Modelling
ISBN 9789400709096
Series Preface
Contents
Chapter 1: Trends in Teaching and Learning of Mathematical Modelling – Preface
References
Part I: Modelling from Primary to Upper Secondary School: Findings of Empirical Research
Chapter 2: Modelling from Primary to Upper Secondary School: Findings of Empirical Research – Overview
Chapter 3: Can Modelling Be Taught and Learnt? Some Answers from Empirical Research
1 A Cognitive View on Mathematical Modelling
2 How Do Students Deal with Modelling Tasks?
3 How Do Teachers Treat Modelling in the Classroom?
4 Some Ideas for Teaching Modelling
References
Chapter 4: “Can Modelling Be Taught and Learnt?” – A Commentary
1 Introduction
2 View of Modelling
3 Technology and Modelling
4 Critical Perspective and Cognition
5 Teachers and Modelling
6 Final Considerations
References
Chapter 5: Upper Secondary Students’ Handling of Real-World Contexts.*
1 Introduction
2 Some Aspects of the Current Discussion and Research Question
3 Methodology and Methods
3.1 Methodological Remarks
3.2 Methods
3.3 Tasks
4 Results
4.1 General Results
4.2 Ideal Types
4.3 Case Synopsis
5 Final Remarks
References
Chapter 6: Word Problem Classification: A Promising Modelling Task at the Elementary Level
1 Theoretical and Empirical Background
2 Method
2.1 Subjects, Tasks and Procedure
2.2 Analysis
3 Results
3.1 Solution Task
3.2 Classification Task
4 Conclusions and Discussion
References
Chapter 7: Understanding and Promoting Mathematical Modelling Competencies: An Applied Perspective
1 Introduction
2 Methodology
3 Results
3.1 Finding Similar Examples or Phenomena
3.2 Connecting Physical Phenomena with Abstract Concepts
3.3 Building Modelling from the Ground up
3.4 Communicating Broader Context of a Modelling Solution
4 Discussion and Summary
References
Chapter 8: Secondary Teachers’ Beliefs About Teaching Applications – Design and Selected Results of a Qualitative Case Study
1 The Call for Applications in Mathematics Curriculum is Quite Old!
2 A First Root Cause Analysis: Focusing on Teachers
3 Why a Qualitative Case Study?: Methodology and Methods
3.1 Theoretical Constructs
3.2 Study Design
4 Discussion and Some Selected Results
5 Conclusion
References
Chapter 9: Secondary Teachers’ Beliefs on Modelling in Geometry and Stochastics
1 Teachers’ Beliefs and Individual Curricula
2 Theoretical Background, Data, and Evaluation
3 Applications and Model Building
4 Geometry
4.1 Elementary Geometry
4.2 Analytical Geometry
5 Stochastics
6 Conclusions
References
Chapter 10: Examining Mathematising Activities in Modelling Tasks with a Hidden Mathematical Character
1 Introduction
2 Framework
2.1 ATD and Modelling
2.2 Structural Analysis
3 Empirical Design
4 A Priori Analysis
5 Findings
6 Summary and Conclusions
References
Chapter 11: The Sun Hour Project
1 Introduction
2 Classroom Circumstances
2.1 Classroom Experiences in Sweden
2.2 Classroom Experiences in Germany
3 Researchers’ View of the Project
3.1 Theoretical Frameworks
3.2 Analyzing the Work in the Swedish Classroom
3.3 Analyzing the Work in the German Classroom
4 Conclusions
References
Chapter 12: Mathematical Knowledge Application and Student Difficulties in a Design-Based Interdisciplinary Project
1 Background of Research
2 Rationale for Research
3 Research Design
3.1 Research Task
3.2 Setting and Sample
3.3 Data Collection Methods
3.4 Analysis Procedures
4 Findings
4.1 Coverage of Mathematical Knowledge and Skills
4.2 Mathematical Difficulties
5 Discussion and Conclusion
References
Chapter 13: Evaluation of Teaching Activities with Multi-Variable Functions in Context
1 The Background of the Research
2 The Educational Course Content
2.1 Two-Variable Functions – Analytic Geometry Approach
2.2 Mathematical Modelling
2.2.1 The Vehicle Stopping Distance Example
2.2.2 The Sound Propagation Example
3 Results from an Evaluation Questionnaire
3.1 Content of Investigation
3.2 Results from the Evaluation Questionnaire and Discussion
4 Conclusion
Reference
Chapter 14: Mathematical Modelling in Secondary Education: A Case Study
1 Introduction
2 Methodology
3 Results and Discussion
4 Conclusions and Recommendations
References
Chapter 15: Students Overcoming Blockages While Building a Mathematical Model: Exploring a Framework
1 Introduction
2 Theoretical Framework
2.1 Modelling
2.2 Mathematising
2.3 Opportunities and Blockages
3 Method
4 Results
4.1 The Swimming Pool Task
4.2 The Horizon Task
4.3 Additional Results
5 Conclusion and Discussion
References
Chapter 16: What Did Taiwan Mathematics Teachers Think of Model-Eliciting Activities and Modelling Teaching?
1 Background
2 Theoretical Frame
3 Methodological Approach
3.1 Samples
3.2 Research Process
3.3 Data Collection
3.4 Data Analysis
3.4.1 Positive Thinking about MEAs and Modelling Teaching
Close-in Real Life Situation
Enhancement of Mathematical Competencies
Advantages of Modelling Teaching
MEAs as Supplementary Materials
3.4.2 Negative Thinking about Modelling Teaching
Out of School Curriculum
Out of Entrance Examinations
Other Obstacles of Modelling Teaching
3.4.3 Weaknesses of Designing MEAs
4 Conclusions and Implications
References
Part II: Looking Deeper into Modelling Processes: Studies with a Cognitive Perspective
Chapter 17: Looking Deeper into Modelling Processes: Studies with a Cognitive Perspective – Overview
References
Chapter 18: Applying Metacognitive Knowledge and Strategies in Applications and Modelling Tasks at Secondary School
1 Introduction
2 Modelling and Metacognition
2.1 The Modelling Process and Reflection
2.2 Meta-Metacognition and Modelling
2.3 Productive Metacognitive Acts and Modelling
2.3.1 Routine Metacognitive Activity
2.3.2 Responses to Red Flag Situations
3 Development of Modellers’ Reflective Metacognitive Activity Through Meta-Metacognition
4 Conclusions
References
Chapter 19: Effective Mathematical Modelling without Blockages – A Commentary
1 Meta-Cognition – A Fuzzy Word?
2 Effective Modelling Without Blockages: But How?
3 Connecting and Acting
References
Chapter 20: Modelling Tasks: Insight into Mathematical Understanding
1 Prior Knowledge of Task Context
2 Ways of Dealing with the Context
3 Higher Order Thinking
4 Methods
4.1 The Tasks
5 Findings
5.1 Evidence of Prior Knowledge of Task Context
5.2 Busse’s Ideal Types
5.3 Modelling and Higher Order Thinking
6 Discussion and Conclusion
References
Chapter 21: Mathematical Modelling of Daily Life in Adult Education: Focusing on the Notion of Knowledge
1 Introduction
2 Theoretical Framework
2.1 A Discussion on the Real World
2.2 Knowledge, Practice and Context
2.3 A Connection with Modelling and Competence
3 Methodology and Data Analysis
4 Results
4.1 Students’ First Chains of Meanings
4.2 Responses to Question 1 – Number Magnitude
4.3 A “Different” Answer in the Group
4.4 Responses to Question 2 – The Best Flavour
4.5 Explaining to the Group – The Medium Flavour
4.6 Responses to Question 3 – The Size of the Glass
4.7 Responses to Question 4 – Part/Whole Relation Versus a Total of 120
5 Discussion and Final Comments
References
Chapter 22: Students’ Modelling Routes in the Context of Object Manipulation and Experimentation in Mathematics
1 Introduction
2 Connecting Modelling to Experimentation in Mathematics Classroom
2.1 From the Point of View of Mathematics Education
2.2 From the Point of View of Learning by Doing
3 Theoretical Perspectives on Applications and Modelling
4 The Research Empirical Work
5 Description and Data Analysis
6 Synthesis of Findings
References
Chapter 23: Engineering Model Eliciting Activities for Elementary School Students
1 Introduction
2 Theoretical Framework
3 The Present Study
3.1 Participants and Procedures
3.2 Data Sources and Analysis
4 Results
4.1 Model A
4.2 Model B
4.3 Model C
4.4 Model D
5 Conclusions
References
Chapter 24: Project Modelling Routes in 12–16-Year-Old Pupils*
1 Context and Aims
2 Theoretical Background
3 Study Methodology
4 From Analysis to Results
5 Results
6 Conclusions
References
Part III: Mathematical Modelling in Teacher Education
Chapter 25: Mathematical Modelling in Teacher Education – Overview
1 Introduction
2 Research Using Different Perspectives of Modelling
3 Reflections
References
Chapter 26: Models and Modelling Perspectives on Teaching and Learning Mathematics in the Twenty-First Century
1 Introduction
2 MMP Research Investigates What It Means to “Understand” Important Concepts and Abilities
3 Do Model-Eliciting Activities (MEAs) Work?
4 In What Ways Do MEAs (and MMP in General) Provide Alternatives to Traditional Research?
4.1 MEAs Provide Alternatives to Naturalistic Observations
4.2 MEAs Provide Alternatives to Expert-Novice Studies
5 Teaching from a Models and Modelling Perspective
5.1 The Nature of Teachers’ Knowledge
5.1.1 Engaging Students with Modelling Tasks
5.1.2 Recognizing and Responding to Students’ Thinking
5.2 The Development of Teachers’ Knowledge
5.2.1 The Collaborative Design of Sequences of Modelling Tasks
5.3 Principles for the Design of Model Development Experiences for Teachers
6 Concluding Remarks
References
Chapter 27: Mathematical Modelling in a Distance Course for Teachers
1 Introduction
2 MM Course: Main and Common Events
2.1 Modelling in the Classroom: Possibilities and Challenges
3 Conclusion
References
Chapter 28: In-Service and Prospective Teachers’ Views About Modelling Tasks in the Mathematics Classroom – Results of a Quantitative Empirical Study
1 Introduction
2 Theoretical Background
3 Research Questions
4 Design and Sample
5 Results
6 Discussion and Conclusions
References
Chapter 29: Pre-service Secondary Mathematics Teachers’ Affinity with Using Modelling Tasks in Teaching Years 8–10
1 Introduction
2 Background
3 Theoretical Framework
4 The Study
5 Findings
5.1 Diagnostic Competencies with Respect to Modelling
5.2 Competencies in Didactical Reflections About Modelling – Appropriateness of Task
5.3 Affinity with Modelling in Teaching in Years 8–10
5.4 Affinity with Modelling Related to Beliefs About the Nature of Mathematics
6 Discussion and Conclusion
References
Part IV: Using Technologies: New Possibilities of Teaching and Learning Modelling
Chapter 30: Using Technologies: New Possibilities of Teaching and Learning Modelling – Overview
1 Modelling Using Digital Tools
2 Empirical Studies and Experiences on Modelling Using Digital Tools
References
Chapter 31: Factors Affecting Teachers’ Adoption of Innovative Practices with Technology and Mathematical Modelling
1 Introduction
2 The Use of CAS in Mathematical Modelling
3 Models of the Use of Technology in Mathematical Modelling
4 Context of the Study
4.1 Curriculum Contexts
4.2 Teachers’ Backgrounds
5 Method and Approach to Data Collection
6 A Tale of Two Cities and Two Teachers
6.1 Teacher 1
6.2 Teacher 2
7 Discussion and Conclusions
References
Chapter 32: Modelling Considering the Influence of Technology
1 Introduction
2 Alcohol in Blood – A Prospective Example for Modelling with Technology
2.1 Modelling a Theoretical Concentration
2.2 Absorption and Reduction Shown as a Mathematical Process
2.2.1 First Model – Linear Approach
2.2.2 Second Model – Semi-linear Approach
2.3 The Role of Technology in the Modelling Cycle
2.4 An Example Where Technology Is Helpful to Get an Idea
2.5 The Fuel Tank – An Example for Using Technology to Validate
3 Examination Tasks – With Modelling Problems and Use of Technology?
4 Conclusion
References
Chapter 33: Improving Learning in Science and Mathematics with Exploratory and Interactive Computational Modelling
1 Introduction
2 Course Organisation, Methodology and Student Evaluation Procedures
3 Computational Modelling Activities with Modellus
4 Conclusions
References
Part V: Modelling Competency: Teaching, Learning and Assessing Competencies
Chapter 34: Modelling Competency: Teaching, Learning and Assessing Competencies – Overview
1 Presentation of the Papers
References
Chapter 35: Drivers for Mathematical Modelling: Pragmatism in Practice
1 Being in Touch with the Real World
2 How Well Do Students Link Mathematical Knowledge to the Task at Hand?
3 How Far Away Is the Real World?
3.1 Kidney Dialysis
3.2 Rocket Satellite Systems
3.3 Aggregation of Slime Mould Amoebae
3.4 Road Traffic Flows
3.5 Local Models
4 Is Mathematical Modelling a Driver for Mathematical Modelling?
References
Chapter 36: Identifying Drivers for Mathematical Modelling – A Commentary
1 How Well Do Students Link Mathematical Knowledge to the Task at Hand?
2 How Far Away Is the Real World?
3 Is Mathematical Modelling a Driver for Mathematical Modelling?
4 What the Title Made Me Think of…
References
Chapter 37: Documenting the Development of Modelling Competencies of Grade 7 Mathematics Students
1 Introduction
2 A Perspective for Modelling
3 What Is Competence and What Are Modelling Competencies?
4 Methodology
5 Results
References
Chapter 38: Students’ Reflections in Mathematical Modelling Projects
1 Introduction
2 Internal and External Reflections in Mathematical Modelling Competency
3 Students’ Reflections in Modelling Projects
3.1 The Institutional Context of the Project Work at Roskilde University
3.2 The Use of a Traffic Model in the City of Roskilde – The Case of ‘Ny Østergade’
3.3 Modelling in Scientific Investigations: The Project of the HPA-Axis
4 Concluding Remarks
References
Chapter 39: From Data to Functions: Connecting Modelling Competencies and Statistical Literacy
1 Introduction
2 Modelling Competencies and Statistical Literacy
3 Empirical Evidence on Modelling Competencies and Statistical Thinking
4 Results
4.1 Discussion and Outlook on Further Results
5 Conclusions
References
Chapter 40: First Results from a Study Investigating Swedish Upper Secondary Students’ Mathematical Modelling Competencies
1 Introduction and Purpose
2 Methodology, Theoretical Considerations and Method
2.1 Mathematical Modelling and Modelling Competencies
2.2 Developing an Instrument
2.3 Statistical Analysis
3 Results
4 Discussion
5 Conclusions
References
Chapter 41: Why Cats Happen to Fall from the Sky or on Good and Bad Models
1 The Operation ‘Cat Airdrop’
2 Modelling in School: Chances and Obstacles
3 The Problem Field ‘Central Examinations
4 The Use of Computers
5 The Professional Development and Motivation of Teachers
6 Conclusion
References
Chapter 42: Assessing Modelling Competencies Using a Multidimensional IRT Approach
1 Introduction
2 Modelling Competency
3 Test Instrument
4 Data Scaling
5 Results of the Evaluation
6 Discussion
References
Part VI: Modelling in Tertiary Education
Chapter 43: Modelling in Tertiary Education – Overview
References
Chapter 44: The Mathematical Expertise of Mechanical Engineers: Taking and Processing Measurements
1 Introduction
2 Method of Investigation and Task
3 Approach of Students
4 Findings and Discussion
4.1 Benefits and Problems of the Method
4.2 Modelling Qualifications
4.3 Data Interpretation and Model Validation Qualifications
4.4 Comparison with Other Research
5 Conclusions for Education
References
Chapter 45: Mathematical Modelling Skills and Creative Thinking Levels: An Experimental Study
1 Introduction
2 Mathematical Modelling Skills
2.1 Test Questions
2.2 Implementation
2.3 Results
3 Creative Thinking Levels
3.1 Test Questions
3.2 Implementation
3.3 Results
3.4 Relationship with Mathematical Modelling Skills
4 Knowledge in Basic Mathematics
4.1 Score in Basic Mathematical Courses
4.2 Relationship with Mathematical Modelling Skills
5 Summary
References
Chapter 46: Modelling the Evolution of the Belgian Population Using Matrices, Eigenvalues and Eigenvectors
1 Introduction
2 The Teaching Sequence
2.1 Calculations with Authentic Data
2.2 The Matrix Model
2.3 Two Observations Concerning the Long Term Evolution of the Population
2.4 Mathematical Treatment of the Observations
2.5 Eigenvalues and Eigenvectors
3 Experiences
3.1 During the ‘Science Week’
3.2 In Mathematics Teacher Education
3.3 In an Introductory Mathematics Course for Bachelor Students in Applied Economics
4 Conclusion
References
Chapter 47: Modelling and the Educational Challenge in Industrial Mathematics
1 Computational Technology
2 Educational Challenge
3 Sphere of Applications
4 Modelling as a Course Subject
5 Modelling Problems to Challenge Undergraduates
6 Modelling Education: How Much and When?
References
Chapter 48: Modelling of Infectious Disease with Biomathematics: Implications for Teaching and Research
1 Introduction and Framework
2 The Study
2.1 The Models
2.2 The Questionnaire
3 The Responses
3.1 Students
3.2 Lecturers
3.3 Analysis of the Responses
4 Conclusions
References
Chapter 49: Using Response Analysis Mapping to Display Modellers’ Mathematical Modelling Progress
1 Introduction
2 Components Based on Experiences
3 Applied Response Analysis Mapping as an Analysis Method
4 Research Setting
5 Modelling Progress Using Applied Response Analysis Mapping
5.1 The Case of a Graduate School Student
5.2 The Case of an Electronics Expert
6 Discussion
6.1 Focus on CRE
6.2 Focus on CME
7 Conclusion
Appendix 1: Mathematical Modelling Progress of NT
Appendix 2: Mathematical Modelling Progress of KN
References
Part VII: Modelling Examples and Modelling Projects: Concrete Cases
Chapter 50: Modelling Examples and Modelling Projects – Overview
1 The Challenge
1.1 Analysis of the Challenge
1.2 Helping Teachers
1.3 Situations for Modelling
Chapter 51: Modelling Chemical Equilibrium in School Mathematics with Technology
1 Introduction
2 The Case
3 Models of Chemical Equilibria
3.1 Chemical Equilibria in Our Case
4 Aims of the Students’ Work
5 Methods for Examination of Data
6 Results
6.1 Students’ Perception of Modelling in Mathematics and in Chemistry
6.2 The Students’ Understanding of the System of Chemical Equilibrium
6.3 Students’ Understanding of Connections Between Theory and Practice
6.4 Technology as a Means to Make the Modelling Process More Explicit
7 Conclusion
7.1 Technical Obstacles
7.2 Few Students’ Reflections
7.3 Little Focus on Modelling
8 Perspectives
References
Chapter 52: Real-World Modelling in Regular Lessons: A Long-Term Experiment
1 Idea of the Experiment
2 Design of the Experiment
3 Choice of Five Real-World Modelling Tasks
3.1 First Task: A Guided Modelling as an Introduction
3.2 Second Task: Is the Olympic Medals Table Fair?
3.3 Third Task: How to Type on a Container?
3.4 Fourth Task: Building of an ICE-Track 9
3.5 Fifth Task: How to Do an Optimal Free Throw in Basketball 11 ?
3.6 Difficulty and Attractiveness of the Projects
4 Detailed Discussion of Task Number 4: ICE-Track
5 Evaluation
5.1 Concept of Questionnaires
5.2 Development from Second to Fourth Task and Final Judgements
6 Comparison Project: Setting and Results
7 Summary and Conclusion
References
Chapter 53: Modelling Tasks at the Internet Portal “Program for Gifted”
1 Fostering of Gifted Students
2 The Interests of Gifted Students: A Bottom-up Approach
3 Examples Containing Modelling Components
3.1 From Cones to Higher Algebraic Curves and Back
3.2 From the Lottery to the Pascal Triangle
4 Summary
References
Chapter 54: Modelling at Primary School Through a French–German Comparison of Curricula and Textbooks
1 Origin, Method and Theoretical Framework of the Study
2 Comparison of Curricula: Is Modelling a Knowledge to Be Taught?
3 Articulation Between Real and Mathematical World in Textbooks
3.1 Real World Knowledge
3.2 Mathematical World Knowledge
3.3 Representation Involved in the Tasks
3.4 Whole Competencies and Partial Competencies: Didactical Functions of Tasks
4 Conclusion: Challenges for Modelling Resources and Teacher Training
References
Chapter 55: Modifying Teachers’ Practices: The Case of a European Training Course on Modelling and Applications
1 Introduction
2 Teacher Education on Modelling and Applications: From a Teachers’ Problem to a Professional Problem
3 A Theoretical Framework to Describe Teaching Actions
3.1 Modelling the Teaching Activity
4 LEMA Professional Development: Changing Teachers’ Practices
5 Conclusions and Implications
References
Chapter 56: Google’s PageRank: A Present-Day Application of Mathematics in the Classroom *
1 Introduction
2 The a Directed Graph and the Description by Transition Matrices
2.1 How Can We Measure the Relevance of a Site?
2.2 Now to an Example Slightly More Complicated
2.3 The Crucial Attribute of U
2.4 Explicit Solution (Formula)
3 Summary and Reflections
References
Chapter 57: Authentic Modelling Problems in Mathematics Education
1 Theoretical Framework for Modelling in Mathematics Education
2 Framework and Structure of the Modelling Week
3 Students Modelling the Spread of Disease in a Population of Ladybirds
4 Evaluation of the Modelling Week
References
Chapter 58: Using Modelling Experiences to Develop Japanese Senior High School Students’ Awareness of the Interrelations between Mathematics and Science
1 Introduction
2 An Example Using Modelling: Teaching Materials for Kepler’s Law for High School Students Becoming Scientists
2.1 Premodel
2.2 Mathematical Development Model
3 Students’ Evaluation and Impressions
3.1 The Class Treated the Premodel
3.2 The Class Treated the Mathematical Development Model
4 Conclusion and Future Subjects
References
Chapter 59: Stochastic Case Problems for the Secondary Classroom with Reliability Theory
1 Introduction
2 Reliability Theory
2.1 Brief History
2.2 A Summary of the Reliabilities of Simple Fundamental Systems
2.2.1 Series Systems
2.2.2 Parallel Systems
2.2.3 Combined Systems
3 Reliability Theory Problems for the Secondary Classroom
3.1 Prerequisites for Reliability Theory
3.2 Examples of Problems for Secondary School
3.2.1 Level I Problems
3.2.2 Level II Problems
3.2.3 Sample Project
4 Conclusion: The Mathematical Residue of Reliability Tasks (RT)
References
Chapter 60: LEMA – Professional Development of Teachers in Relation to Mathematical Modelling*
1 Theoretical Background
2 Design of the Course of Professional Development
3 Design of the Evaluation
4 Results and Discussion
References
Chapter 61: Modelling in the Classroom: Obstacles from the Teacher’s Perspective
1 Basic Theory
1.1 Mathematical Modelling
1.2 Obstacles to the Integration of Modelling
1.3 Research Questions
2 Methodology
2.1 Instruments for the Study
2.2 Study Design
2.3 Sample
3 Questionnaire Development
3.1 Questionnaire Development
3.2 Format of Questionnaire
4 Results
4.1 I Have Too Little Material
4.2 Performance Assessment is Too Complex
4.3 I Don’t Have Enough Time for Modelling
5 Discussion
References
Chapter 62: Teachers’ Professional Learning: Modelling at the Boundaries
1 Introduction
2 Classroom Practice and Its Transformation
3 Knowledge for Teaching and Learning in Modelling Classrooms
4 Further Theoretical Reflections
References
Part VIII: Theoretical and Curricular Ref lections on Mathematical Modelling
Chapter 63: Theoretical and Curricular Reflections on Mathematical Modelling – Overview
1 Paper Summaries
References
Chapter 64: Making Connections Between Modelling and Constructing Mathematics Knowledge: An Historical Perspective
1 Aims in This Study
2 Mathematics Textbooks Before World War II in Japan
3 Roles of Real World Situations
4 Repeated Instances of the Same Contexts
5 Making Connections Between Real World Problems
6 Implications for Teaching of Mathematics Today Including Modelling
7 Conclusion
References
Chapter 65: Practical Knowledge of Research Mathematicians, Scientists, and Engineers About the Teaching of Modelling
1 Introduction
2 The Case Study
2.1 Theoretical Framework
2.2 Design of the Case Study
3 Summary of Interview Findings
3.1 Teaching Goals of Modelling in Tertiary Education
3.2 Modelling Courses in Tertiary Education
3.3 Students’ Modelling Competencies
3.4 Opinions about Modelling in Secondary Education
4 Discussion and Conclusions
4.1 Similarities Between Interviewees and Education Researchers
4.2 Differences Between Interviewees and Education Researchers
4.3 Differences Among Interviewees
4.4 Possible Implications for Secondary Education
References
Chapter 66: Evolution of Applications and Modelling in a Senior Secondary Curriculum
1 Introduction
2 Mathematical Modelling and Applications in Queensland
2.1 Syllabus Objectives
2.2 Implementation in Schools
3 Research Methods
4 Findings
4.1 Reasons Applications and Modelling Valuable Initiative at Senior Secondary
4.2 Distinction Between Applications and Modelling
4.3 Embedding Applications and Modelling in Current Practice
4.4 Designing Tasks
5 Discussion and Conclusion
References
Chapter 67: Sense of Reality Through Mathematical Modelling.*
1 Introduction
2 Brief Description of Modelling in Colombian Educational Regulations
3 The Project
3.1 The Context
3.2 Methodological Approach
4 Results
4.1 What is Sense of Reality ?
4.2 What is Reality for Alberto and Alexander?
4.3 Sense of Reality in School Mathematics
4.4 A First Approach
5 Discussion
6 Conclusions
References
Chapter 68: What Is ‘Authentic’ in the Teaching and Learning of Mathematical Modelling?
1 Introduction
2 Variations of Tasks in Mathematics Education
3 Definitions of ‘Authenticity’ in Mathematics Education
4 Problems with Defining ‘Authenticity’
5 Pragmatically Constructing Authenticity in Mathematical Modelling
References
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