Mathematical Competencies in the Digital Era

Foreword
Acknowledgements
Contents
Mathematical Competencies in the Digital Era: An Introduction
1 Rationale Behind the Book
2 The Structure of the Book
3 Introduction to Setting the Scene
4 Introduction to The Eight Mathematical Competencies
5 Introduction to The Three Types of Overview and Judgement
6 Introduction to Broadening the Scene
7 A Platform for Further Discussion and Research
References
Setting the Scene
On the Mathematical Competencies Framework and Its Potentials for Connecting with Other Theoretical Perspectives
1 Introduction
2 What Are Theoretical Frameworks in Mathematics Education?
3 Placing the KOM Framework in the Landscape of Theoretical Frameworks
4 An Example Related to Mathematical Modelling
5 An Example Related to Overview and Judgement Regarding the Historical Development of Mathematics
6 An Example Related to the Didactico-Pedagogical Competency of Uncovering Learning
7 Potentials for Connecting and Networking with Other Theoretical Constructs
8 Towards Mutual Fertilisation
References
The Mathematical Competencies Framework and Digital Technologies
1 Introduction
2 Mathematical Competencies
2.1 First Set: Posing and Answering Questions in and by Means of Mathematics
2.2 Second Set: Handling the Language, Constructs and Tools of Mathematics
2.3 Exemplifying Technology Influence and Impact on Mathematical Competencies: Representation and Reasoning Competencies
2.4 Exploration and Argumentation Using GeoGebra
3 Meta-Competencies
3.1 Exemplifying Technology Influence and Impact on Mathematical Meta-Competencies: Proofs Using Computers
4 Teacher Competencies
4.1 Curriculum Competency—Being Able to Evaluate and Draw up Curricula
4.2 Teaching Competency—Being Able to Think Out, Plan and Carry Out Teaching
4.3 Competency of Revealing Learning—Being Able to Reveal and Interpret Students’ Learning
4.4 Assessment Competency—Being Able to Reveal, Evaluate and Characterize the Students’ Mathematical Yield and Competencies
4.5 Cooperation Competency—Being Able to Cooperate with Colleagues and Others Regarding Teaching and Its Boundary Conditions
4.6 Professional Development Competency—Being Able to Develop one’s Competency as a Mathematics Teacher
4.7 Exemplifying Technological Influence on Teacher Mathematical Competencies—Introducing Computational Thinking into Mathematics; Curriculum Competency and Professional Development Competency
5 Mathematical Competencies Under the Influence of Digital Affordances
References
The Eight Mathematical Competencies
Processes of Mathematical Thinking Competency in Interactions with a Digital Tool
1 Introduction
2 The Mathematical Thinking Competency of the KOM Framework
3 Instrumental Genesis and Conceptual Fields
4 Semiotic Mediation
5 Networking of Theories and the Roles of the Selected Theoretical Perspectives
6 Method and Selection of the Case
7 Data: Exploring Differentiability Using Secant Lines
8 Analysis 1: Exercised Processes of the Mathematical Thinking Competency
9 Analysis 2: Beginning Instrumental Genesis
10 Analysis 3: Signs of Semiotic Mediation
11 Discussion and Conclusion
References
Mathematical Competencies Framework Meets Problem-Solving Research in Mathematics Education
1 Introduction
2 Mathematical Competencies: An Overarching Framework
2.1 Problem-Handling Competency
2.2 Mathematical Aids and Tools Competency
2.3 Facets and Dimensions of a Mathematical Competency
3 Networking of Theories in Mathematics Education
4 Some Theoretical Developments from Problem-Solving Research in Mathematics Education
4.1 A Framework for the Systematic Use of Technology in Mathematical Problem Solving
4.2 Coordination of Notions from Problem-Solving Research with the Mathematical Competencies Framework
5 A Virtual Course on Euclidean Geometry for Prospective Mathematics Teachers
5.1 A Geometric Problem and Some Prerequisites to Tackle It
5.2 Anna’s Solving Process of the Geometric Problem
5.3 Anna Formulates and Proves Conjecture 1
5.4 Anna Formulates Conjectures 2 and 3 and Proposes Three Problems
6 A Networked Analysis of Anna’s Solving Process
6.1 Mathematical Competencies Framework as an Overarching Framework for Analyzing Anna’s Problem-Solving Process
6.2 Producing a More Fine-Grained Analysis Through the Coordination of Theoretical Notions
7 Conclusion
References
Mathematical Modelling and Digital Tools—And How a Merger Can Support Students’ Learning
1 Introduction
2 First Case Study—Pirates of the Caribbean
3 The Second Case Study—Anaesthesia
4 Media-Milieu Dialectic Through the Herbartian Schema
4.1 The Dialectics and Piracy
4.2 Dialectics and Anaesthetics
5 Mathematical Digital Competency and Modelling
5.1 Competencies and Piracy
5.2 Competencies and Anaesthesia
6 Discussion and Concluding Remarks
References
Lower Secondary Students’ Reasoning Competency in a Digital Environment: The Case of Instrumented Justification
1 How Does the Use of Digital Technology Influence Students’ Mathematical Reasoning Competency?
1.1 Reasoning Competency in the KOM Framework
1.2 Designing an Analytical Tool from a Complex Theoretical Panorama
2 Method
2.1 Task Design
2.2 Presentation of the Case
3 Analysis of the Students’ Justification Process
4 Results and Discussion
4.1 Claims and Qualifier Change
4.2 Instrumented Techniques, Data, and Warrants
5 Discussion
5.1 How Does Isa and Em’s Use of Digital Technology Influence Their Reasoning Competency?
5.2 Gaining Insights into Argumentation Processes in a Digital Environment
5.3 Theoretical Implications of Adapting Toulmin’s Model Through the Scheme-Technique Duality
6 Concluding Remarks
References
Mathematical Representation Competency in the Era of Digital Representations of Mathematical Objects
1 Introduction
2 Conceptual Frameworks and Constructs
2.1 Representation Competency in the KOM Framework
2.2 Representations of Mathematical Objects—from Paper and Pencil to DGE
2.3 The Fundamental Role of Geometrical Knowledge in Dealing with Representations
3 Method and Context
4 Presentation of Data and Ensuing Analysis
5 Discussion
6 Networking Representation Competency and Other Theoretical Constructs
7 Conclusion
References
New Demands on the Symbols and Formalism Competency in the Digital Era
1 Introduction
2 The Nature of Symbol Systems
3 The Symbol Grounding Problem
4 Networking KOM and the Symbol Grounding Theory into a Coordinated Conceptual Framework
5 Symbolism and Formalism as a Mathematical Competency
6 Challenges in the Digital Era
7 Conclusion
Appendix
References
Activating Mathematical Communication Competency When Using DGE—Is It Possible?
1 Introduction
2 Theoretical Framework
2.1 Mathematical Communication
2.2 Instrumental Genesis and Instrumentation Profiles
2.3 Previous Results
3 Methodology
3.1 The Task
3.2 Data Collection
3.3 Presenting and Analysing Data
4 Case A: Andrea and Bea
4.1 Analysing Mathematical Communication
4.2 Analysing Instrumentation Profiles
4.3 Summarising Case A
5 Case B: Caroline and Diane
5.1 Analysing Mathematical Communication
5.2 Analysing Instrumentation Profiles
5.3 Summarising Case B
6 Case C: Emma and Frida
6.1 Analysing Mathematical Communication
6.2 Analysing Instrumentation Profiles
6.3 Summarising Case C
7 Summarising the Results
8 Reflections on the Theory Networking Conducted in This Study
8.1 Four Types of Tool-Based Mathematical Communication
8.2 Tool-Based Mathematical Communication: A Case of Local Integration
9 Conclusion
References
An Embodied Cognition View on the KOM-Framework’s Aids and Tools Competency in Relation to Digital Technologies
1 Introduction
2 The Mathematical Competency of Aids and Tools
3 The (Embodied) Instrumental Approach
3.1 Embodied Cognition
3.2 Body-Artefact System
4 First Example: Slope Fields
4.1 Analysis of the First Example
5 Second Example: Virtual Manipulatives
5.1 Analysis of the Second Example
6 Discussion of Connecting Theoretical Perspectives
References
The Three Types of Overview and Judgement
Mathematics in Action: On the Who, Where and How of the Constructions and Use of Mathematical Models in Society
1 Introduction
2 OJ1, Internal and External Reflections
2.1 A Competency and an Overview
3 Networking of Theories and Digital Tools
4 Statistical Models and the Use of Hydroxychloroquine
5 Epidemiology as a Domain for Students’ Modelling Projects
5.1 A Bachelor Project on the Modelling of Influenza Epidemics
6 Discussion
References
Perspectives on Embedding the Historical Development of Mathematics in Mathematical Tasks
1 The Second Type of Overview and Judgement
2 The Anthropological Theory of the Didactic
3 Hypothesis and Research Goal
4 Combined and Coordinated Theories
5 The Empirical Cases
5.1 Analysis of the Empirical Cases
5.2 Part 1—Tasks from the Perspective of Praxeologies
5.3 Part 2—Case 1 Analysis
5.4 Part 2—Case 2 Analysis
6 Discussion and Conclusion
References
Facilitating Teachers’ Reflections on the Nature of Mathematics Through an Online Community
1 Introduction
2 Mathematical Overview and Judgment
3 Beliefs About Mathematics
4 The Relationship Between OJ, Beliefs, and Knowledge
5 Method
5.1 Context and Setting
5.2 Data Sources and Analysis
6 The Case of Teacher Li
7 Discussion
8 Final Remarks on Networking of Theories
References
Broadening the Scene
Teachers’ Facilitation of Students’ Mathematical Reasoning in a Dynamic Geometry Environment: An Analysis Through Three Lenses
1 Introduction
2 KOM’s Mathematics Teacher Competencies
3 Instrumental Orchestration
4 Justificational Mediations
5 Methods
6 The Teaching Sequence
7 A Series of Episodes in Light of Justificational Mediations
8 Analysis from a TIO and JM Perspective
9 Analysis from a KOM Teacher Competencies and JM Perspective
10 Potentials for Networking KOM and TIO Bridged by JM
11 Conclusion
References
Mathematical Competencies and Programming: The Swedish Case
1 Introduction
2 Programming in the Swedish School Curricula
3 Relating the KOM Competencies to the Swedish National Curriculum
4 Research Results
4.1 Substudy 1: Programming in Teaching Materials for Mathematics
4.2 Substudy 2: Teachers’ Views on Programming in Mathematics
4.3 Substudy 3: A Mathematical Pattern in Code, a Teacher’s Example
5 Concluding Remarks
References
Coordinating Mathematical Competencies and Computational Thinking Practices from a Networking of Theories Point of View
1 Introduction
2 Related Work
3 CT and KOM
4 Method
5 Analysis—Coordinating CT Practices and KOM
5.1 Data Practices
5.2 Modelling and Simulation Practices
5.3 Computational Problem-Solving Practices
5.4 Systems Thinking Practices
6 Discussion
7 Conclusion
References
A Rich View of Mathematics Education and Assessment: Mathematical Competencies
1 Introduction
2 PISA and the KOM Competencies
3 The Appearance and Application of Competencies in the PISA Context
3.1 Use of Competencies to Describe Mathematical Accomplishments in PISA
3.2 Research, Exploring the Difficulty of PISA Assessment Questions Against the Different Mathematical Competencies
4 Use of the Competencies in PISA 2012 Test Development
4.1 Targeting the Difficulty Level of Assessment Tasks
4.2 Enhancing the Spread of Cognitive Demands Across Assessment Tasks
4.3 Applying the Competencies in PISA 2012
5 Using the Competencies to Highlight Some Differences Between Digital and Paper-Based Assessment Tasks
5.1 The Representation Competency
5.2 A Comparison of Two Items
5.3 Using Mathematical Tools
6 Rating a Task Using the Competencies
7 Benefits of Using the Competencies as a Lens in PISA 2012
8 Conclusions
Appendix: Item Rating Scheme Using PISA Fundamental Mathematical Capabilities (27 Feb, 2012)
References
Index