Practice-Oriented Research in Tertiary Mathematics Education

Contents
Chapter 1: Practice-Oriented Research in Tertiary Mathematics Education - An Introduction
	1.1 Context of This Book
	1.2 Overall Structure of the Book
		1.2.1 Section 1: Research on the Secondary-Tertiary Transition
		1.2.2 Section 2: Research on University Students´ Mathematical Practices
		1.2.3 Section 3: Research on Teaching and Curriculum Design
		1.2.4 Section 4: Research on University Students´ Mathematical Inquiry
		1.2.5 Section 5: Research on Mathematics for Non-specialists
	References
Part I: Research on the Secondary-Tertiary Transition
Chapter 2: Emotions in Self-Regulated Learning of First-Year Mathematics Students
	2.1 Introduction: The Transition from School to University in Mathematics
	2.2 Theory
		2.2.1 Self-Regulated Learning in Undergraduate Mathematics
		2.2.2 Achievement Emotions and Control-Value Theory
		2.2.3 An Integrated Model of Achievement Emotions and Self-Regulated Learning
	2.3 Research Interest and Research Questions
	2.4 Methods and Research Design
		2.4.1 Institutional Context of the Study
		2.4.2 Data Collection
		2.4.3 Data Analysis
	2.5 Results
		2.5.1 Joy, Relief, Anxiety, and Hopelessness in the First Year of Study
		2.5.2 The Roles of Perceived Control and Subjective Values in the Emergence of Joy, Relief, Anxiety, and Hopelessness
		2.5.3 Joy, Relief, Anxiety, Hopelessness, and Self-Regulated Learning
	2.6 Discussion
		2.6.1 Discussion of Results
		2.6.2 Implications, Limitations, and Outlook
	References
Chapter 3: The Unease About the Mathematics-Society Relation as Learning Potential
	3.1 Prelude
	3.2 Subject-Scientific Approach
		3.2.1 Fundamental Assumptions and Subject-Scientific Categories
		3.2.2 Subject-Scientific Understanding of Learning
	3.3 Unease to Be Identified as a Mathematician (Only)
		3.3.1 Against Being Identified as Becoming a Constricted One-Track-Specialist
		3.3.2 Against Being Identified as Just Being Mathematically Able
		3.3.3 Interlude
	3.4 Belief Research
		3.4.1 Change of Teacher Beliefs
		3.4.2 In-/Consistencies Between Teacher Beliefs and Teaching Practices
	3.5 Current Trends in Belief Research from a Subject-Scientific Perspective
	3.6 Discussion
	References
Chapter 4: Collaboration Between Secondary and Post-secondary Teachers About Their Ways of Doing Mathematics Using Contexts
	4.1 Introduction
	4.2 The Secondary to Postsecondary Transition in Mathematics
		4.2.1 A Need for Dialogue Between Secondary and Postsecondary Teachers
		4.2.2 The Use of Contexts in Mathematics and the Secondary to Postsecondary Transition
		4.2.3 Research Questions
	4.3 The Theoretical Perspective
	4.4 Methodology
		4.4.1 An Investigation in Two Phases
		4.4.2 The Dialogue Organized Around a Reflexive Activity
		4.4.3 The Overall Analytical Process
	4.5 Results
		4.5.1 Phase 1: Two Territories Established Around the Use of Contexts
			4.5.1.1 Secondary Level Territory: Contextual Mathematics
				Three Specificities to Characterize the Territory at the Secondary Level
			4.5.1.2 The Territory of Postsecondary Mathematics: Illustrated Mathematics
				Four Specificities to Characterize the Territory at the Postsecondary Level
		4.5.2 Phase 2: A Process of Rapprochement Between Levels
			4.5.2.1 Moment 1: Explaining Their Respective Ways of Doing
			4.5.2.2 Moment 2: The Establishment of Common Elements
			4.5.2.3 Moment 3: Revisiting One´s Territory in Light of the Other
			4.5.2.4 Moment 4: Joint Planning
	4.6 Conclusion
	References
Chapter 5: Framing Goals of Mathematics Support Measures
	5.1 Supporting Students in the Secondary-Tertiary Transition and the WiGeMath Project
	5.2 Development of the Goal Categories in the WiGeMath Framework
		5.2.1 The Underlying Concept of Theory-Driven Evaluation
		5.2.2 The Purpose of a Framework Model for Goal Categories
		5.2.3 Main Steps in Developing the Model
		5.2.4 Presentation of the Goal Categories
			5.2.4.1 Educational Goals
			5.2.4.2 System-Related Goals
	5.3 Using the Goal Categories of the Framework to Compare Measures
		5.3.1 Background and Methods
		5.3.2 Pre-University Bridging Courses
		5.3.3 Redesigned Lectures
		5.3.4 Mathematics Learning Support Centres
		5.3.5 Comparing Different Types of Measures
	5.4 Discussion
		5.4.1 The Framework Model
		5.4.2 Using the Framework Model to Evaluate Measures
		5.4.3 Further Use of the Framework Model
	Appendix
	References
Part II: Research on University Students´ Mathematical Practices
Chapter 6: ``It Is Easy to See´´: Tacit Expectations in Teaching the Implicit Function Theorem
	6.1 Introduction
	6.2 Theoretical Framework and Research Questions
	6.3 Context of the Study
	6.4 Mathematical Analysis of Student Tasks in the Exercise Class
	6.5 Methodology
	6.6 Results - Students´ Solutions and Reflections
	6.7 Results - Interviews with Teachers
	6.8 Conclusions and Further Perspectives
	References
Chapter 7: University Students´ Development of (Non-)Mathematical Practices: The Case of a First Analysis Course
	7.1 Introduction
	7.2 Theoretical Framework
		7.2.1 Mathematical and Non-Mathematical Practices
		7.2.2 The Progressive Development of Practices
	7.3 Methodology
	7.4 Results
		7.4.1 Suggested Practice Associated with T2
		7.4.2 Practices Enacted by Participants for Solving T2
			7.4.2.1 The Identification of T2 with a Type of Task and Technique
			7.4.2.2 The Implementation of a Technique to Accomplish T2
			7.4.2.3 The Explanation of a Technique for Accomplishing T2
	7.5 Discussion
		7.5.1 Answer to the Research Questions and Contribution of the Study to Research in University Mathematics Education
		7.5.2 Limitations and Directions for Future Research
	References
Chapter 8: The Mathematical Practice of Learning from Lectures: Preliminary Hypotheses on How Students Learn to Understand Def...
	8.1 Introduction
	8.2 Literature Review
		8.2.1 What Do We Mean by Learning from Lectures?
		8.2.2 Research on Lecturing in Advanced Mathematics
		8.2.3 The Inadequacy of a Transmission Model of Learning
		8.2.4 The Importance of Modeling During Lectures
		8.2.5 Goals of This Chapter
	8.3 Data and Analysis
		8.3.1 A Data Corpus of Lectures in Advanced Mathematics
		8.3.2 Analysis
	8.4 Results
		8.4.1 When Learning a Definition, One Should Justify Why the Definition Has Desirable Attributes
		8.4.2 When a New Definition Is Proposed, One Should Actively Explore the Definition
		8.4.3 When a New Definition Is Provided, One Should Exemplify this Definition in Many Ways
		8.4.4 How Should Students Study New Definitions That Are Presented in Lectures?
	8.5 Discussion
	References
Chapter 9: Supporting Students in Developing Adequate Concept Images and Definitions at University: The Case of the Convergenc...
	9.1 Introduction and Overview
	9.2 Theoretical Background and Literature Review
	9.3 Research Questions
	9.4 Context of the Study
	9.5 The Design of the Initial Learning Environment
		9.5.1 The Set of Examples and Non-examples and Its Anticipated Use
		9.5.2 The Initial Task Formulation
		9.5.3 Anticipated Obstacles and Prepared Support
	9.6 Design of the Study, Sample, Collected Data, Methods of Data Analysis
		9.6.1 Instructional Design of the Workshop
		9.6.2 Iterative Analysis from the Perspective of Design Research
	9.7 Results
		9.7.1 Changes in the Set of Examples/Non-examples and the Anticipated Use
		9.7.2 Changes in the Prepared Support for the Second Cycle Based on Retrospective Analysis of Cycle 1
		9.7.3 Changes in the Prepared Support for the Third Cycle Based on Retrospective Analysis of Cycle 2
		9.7.4 Changes in the Task Formulation
	9.8 Discussion
	References
Chapter 10: Investigating High School Graduates´ Basis for Argumentation: Considering Local Organisation, Epistemic Value, and...
	10.1 Introduction
	10.2 Theoretical Background
		10.2.1 Set of Accepted Statements, Local Organisation, and the Basis for Argumentation
		10.2.2 The Epistemic Value of Statements
		10.2.3 Toulmin´s Model for Structuring Argumentation
		10.2.4 Basis for Argumentation, Local Organisation, and Epistemic Value
		10.2.5 Findings from the Literature
	10.3 Research Questions
	10.4 Methodology
		10.4.1 Research Instruments
			10.4.1.1 Task Analysis and Expected Solution
			10.4.1.2 Construction of the Interview Guide
		10.4.2 Procedure
		10.4.3 Piloting the Research Instrument
		10.4.4 Data Collection
		10.4.5 Data Analysis
	10.5 Results
		10.5.1 Results concerning the Elements of the Basis for Argumentation used in the Proof Constructions
		10.5.2 Results concerning the Embeddedness of the Statements used in a Local Organisation
		10.5.3 Results concerning the Epistemic Value Assigned to the Statements, Rules, and Definitions used
		10.5.4 Results on the Effects of Epistemic Values on the Conclusion´s Modal Qualifier
	10.6 Discussion
		10.6.1 Elements of the Basis for Argumentation used in the Proof Constructions
		10.6.2 Statements Embedded in a Local Organisation
		10.6.3 The Epistemic Value Assigned to the Statements and Definitions used
		10.6.4 Effects of Epistemic Values on the Conclusions´ Modal Qualifier
		10.6.5 Limitations
		10.6.6 Conclusions
	References
Chapter 11: Proving and Defining in Mathematics Two Intertwined Mathematical Practices
	11.1 Introduction
	11.2 Defining to be Able to Prove - The Case of Irrational Numbers
		11.2.1 Defining Irrational Numbers by Cuts (Dedekind, 1872)
		11.2.2 Defining Rational Numbers as Fundamental Sequences (Cantor, 1872)
		11.2.3 Impact of the Way of Defining Real Numbers on Proving
		11.2.4 A Didactic Situation to Address Issues Related to -Completeness Versus -Incompleteness
	11.3 Enumeration, Infinite Sets, and Diagonal Proofs
		11.3.1 How to Define Infinite Sets?
	11.4 Infinite Sets as Non-finite Sets
	11.5 Infinite Sets as Violating the Principle the ``Whole is Greater Than the Part.´´
		11.5.1 Impact of the Ways of Defining on Proving That a Set Is Infinite
	11.6 How Big Is Infinity?
		11.6.1 The Diagonal Proof That  Is Denumerable
		11.6.2 The Diagonal Proof That  Is Not Denumerable
	11.7 Didactic Implications
	11.8 Conclusion
	Appendix
	References
Part III: Research on Teaching and Curriculum Design
Chapter 12: Developing Mathematics Teaching in University Tutorials: An Activity Perspective
	12.1 Introduction
	12.2 Practice-Oriented/Close-to-Practice Research
	12.3 Our Use of Activity Theory
	12.4 Meaning Making
	12.5 Methodology
		12.5.1 Analysis of Data
	12.6 Analysis of Dialogue in Key Episodes
		12.6.1 Tutoring for Students´ Meaning-Making - Actions and Goals
		12.6.2 The Practice of Tutoring - Summary of Tutorial - Key Points
		12.6.3 The Episodes and the Grounded Analysis
		12.6.4 Synthesizing/Exposing: Building on Students´ Solutions to Present the General Solution Method - (Episode 5)
		12.6.5 The Tensions Manifested in the Three Episodes
	12.7 Analyzing the Episodes from an Activity Theory Perspective
	12.8 In Conclusion
	References
Chapter 13: Lecture Notes Design by Post-secondary Instructors: Resources and Priorities
	13.1 Theoretical Tools
	13.2 Methods
		13.2.1 The Textbooks
		13.2.2 Participants
		13.2.3 Data Collected
		13.2.4 Analysis
	13.3 Results
		13.3.1 Maps
		13.3.2 Resources
		13.3.3 Lecture Notes
		13.3.4 Instrumentation and Instrumentalization of the Resources
	13.4 Discussion and Conclusion
	References
Chapter 14: Creating a Shared Basis of Agreement by Using a Cognitive Conflict
	14.1 The `Flow of a Proof´ and Its Rhetorical Features
	14.2 Theoretical Framework - The New Rhetoric
	14.3 Cognitive Conflict and Mathematics Education
	14.4 The Study
		14.4.1 Objectives
		14.4.2 Setting
		14.4.3 Analysis
			14.4.3.1 Interviews Analysis
			14.4.3.2 PNR Analysis
	14.5 Findings
		14.5.1 Findings from the Lecturer Interviews
		14.5.2 Scope and Organization of the Lesson
		14.5.3 Analysis of an Episode from Module V - Cognitive Conflict, and Dissociation
	14.6 Discussion and Implications
	References
Chapter 15: Teaching Mathematics Education to Mathematics and Education Undergraduates
	15.1 Mathematics Education Courses in the University Curriculum
	15.2 Challenges in the Transition from Studies in Mathematics or Education to Mathematics Education
	15.3 Theoretical Underpinnings of Undergraduate RME Course Design
	15.4 Design, Delivery and Assessment of Two RME Courses
		15.4.1 The BMath Course
		15.4.2 The BEd Course
	15.5 The Interplay of Research and Practice in Welcoming Two Different Communities of Learners - from Mathematics and from Edu...
	References
Chapter 16: Inquiry-Oriented Linear Algebra: Connecting Design-Based Research and Instructional Change Research in Curriculum ...
	16.1 Background Theory and Literature
		16.1.1 Realistic Mathematics Education
		16.1.2 Inquiry-Oriented Instruction
		16.1.3 Instructional Change at the University Level
	16.2 The Design Research Spiral
		16.2.1 Design Phase
		16.2.2 PTE Phase
		16.2.3 CTE Phase
		16.2.4 OWG Phase
		16.2.5 Web Phase
	16.3 Discussion
	References
Chapter 17: Profession-Specific Curriculum Design in Mathematics Teacher Education: Connecting Disciplinary Practice to the Le...
	17.1 Profession-Specific Teaching Designs: Introducing Theory Elements for Reflecting on Design and Content Decisions
		17.1.1 Facets of Teacher Knowledge as Categorial and Normative Theory Elements
		17.1.2 Learning Abstract Algebra: What We Learn from Previous Research for Answering How-Questions
		17.1.3 Learning Abstract Algebra: What We Learn from Previous Research for Answering What-Questions
	17.2 Design Principles and Design Elements for Enhancing Profession-Specificity in an Abstract Algebra Class for Prospective T...
		17.2.1 First Design Experiment Cycle
		17.2.2 Second Design Experiment Cycle: Scaffolding Guided Reinvention and Noticing Connections
	17.3 Outlook on the Third Cycle and Discussion
	References
Chapter 18: Drivers and Strategies That Lead to Sustainable Change in the Teaching and Learning of Calculus Within a Networked...
	18.1 Introduction
	18.2 Theoretical Background
	18.3 Methods
	18.4 Findings and Results
		18.4.1 California State University East Bay (CSUEB)
			18.4.1.1 Shared Tools and Resources
			18.4.1.2 Professional Development
			18.4.1.3 Policies and Structures
			18.4.1.4 Networking
		18.4.2 Kennesaw State University (KSU)
			18.4.2.1 Shared Tools and Resources
			18.4.2.2 Professional Development
			18.4.2.3 Policies and Structures
			18.4.2.4 Networking
		18.4.3 The Ohio State University (OSU)
			18.4.3.1 Shared Tools and Resources
			18.4.3.2 Professional Development
			18.4.3.3 Policies and Structures
			18.4.3.4 Networking
	18.5 Reflections and Synthesis
		18.5.1 Shared Tools and Resources
		18.5.2 Professional Development
		18.5.3 Policies and Structures
		18.5.4 Networking
	18.6 Implications and Limitations
	References
Part IV: Research on University Students´ Mathematical Inquiry
Chapter 19: Real or Fake Inquiries? Study and Research Paths in Statistics and Engineering Education
	19.1 Introduction
	19.2 Theoretical Framework, Research Questions and Empirical Methodology
	19.3 An SRP in Elasticity
	19.4 An SRP in Statistics
	19.5 Conclusions and New Open Questions
		19.5.1 The Choice of the Generating Question and the Curriculum Constraint
		19.5.2 Taking Q Seriously and Creating Adidacticity During the Inquiry Process
		19.5.3 Changing the Generating Question or Changing the Situation in Which It Arises?
		19.5.4 The Inclusion of TDS Notions into ATD Analyses
	References
Chapter 20: Fostering Inquiry and Creativity in Abstract Algebra: The Theory of Banquets and Its Reflexive Stance on the Struc...
	20.1 Introduction
	20.2 Inquiry and Creativity in Abstract Algebra Teaching and Learning
		20.2.1 Inquiry
		20.2.2 Creativity
		20.2.3 The Objects-Structures Dialectic
	20.3 The Theory of Banquets: A Didactic Engineering
		20.3.1 Mathematical Presentation of the Theory of Banquets
		20.3.2 A Priori Analysis of the Classification Tasks
	20.4 Learning Affordances of the Theory of Banquets
		20.4.1 What Is a Banquet? Students´ Creative Processes in Making Sense of a Formal System of Axioms
		20.4.2 What Does It Mean to Classify Banquets? Students´ Creative Processes in Developing a Structuralist Point of View
	20.5 Conclusion and Perspectives
	References
Chapter 21: Following in Cauchy´s Footsteps: Student Inquiry in Real Analysis
	21.1 Introduction
	21.2 Context and Brief Description of the Instructional Sequence
		21.2.1 Intermediate Value Theorem as Starting Point
		21.2.2 Context: The Course and the Participating Students
		21.2.3 Starting Point and Cauchy´s Proof of IVT
		21.2.4 Data Analysis
	21.3 Classroom Inquiry: From the Bisection Method to Least Upper Bounds
		21.3.1 Developing a Shared Understanding of the Two Approximation Methods
		21.3.2 Connecting the Approximation Method to Formal Mathematical Language and Notation
		21.3.3 Eliciting Student Reasoning: Conjectures About Sequences Generated by the Bisection Method
		21.3.4 Building on Students´ Ways of Reasoning: General Conjectures About Sequence Convergence
		21.3.5 Building on Students´ Ways of Reasoning: Investigating the False General Conjectures
		21.3.6 Building on Students´ Ways of Reasoning: Investigating the True General Conjecture
		21.3.7 Generating Student Ways of Reasoning: Brainstorming Why the Least Upper Bound Will Be the Limit
	21.4 Conclusion
	References
Chapter 22: Examining the Role of Generic Skills in Inquiry-Based Mathematics Education - The Case of Extreme Apprenticeship
	22.1 Introduction
	22.2 Generic Skills and Their Role in Mathematics Curricula
	22.3 Extreme Apprenticeship, a Form of Inquiry-Based Mathematics Education
	22.4 Research Problem and Hypotheses
	22.5 Method
	22.6 Context and Sources
	22.7 Results
		22.7.1 Generic Skills as Learning Objectives
		22.7.2 Communicating the Generic Skills
		22.7.3 Interplay of Objectives, Methods and Assessment
		22.7.4 Programme-Level Development
	22.8 Concluding Remarks
	References
Chapter 23: On the Levels and Types of Students´ Inquiry: The Case of Calculus
	23.1 Introduction
	23.2 Theoretical Background and Framework
	23.3 The Levels of Inquiry in Calculus Textbooks
		23.3.1 The Structured and Guided Inquiries
		23.3.2 The Confirmation and Open Inquiries: Two Extremes
	23.4 Additional Types of Activities That Promote Students´ Inquiry
		23.4.1 Classifying Mathematical Objects
		23.4.2 Interpreting Multiple Representations
		23.4.3 Evaluating Mathematical Statements
		23.4.4 Creating Problems
		23.4.5 Analysing Reasoning and Solutions
		23.4.6 Different Types of Activities and Milieu Construction
	23.5 Concluding Discussion
	References
Chapter 24: From ``Presenting Inquiry Results´´ to ``Mathematizing at the Board as Part of Inquiry´´: A Commognitive Look at F...
	24.1 Introduction
	24.2 Boards, Inquiry, and Mathematics
	24.3 Mathematizing at the Board from the Commognitive Standpoint
		24.3.1 Commognition in a Nutshell
		24.3.2 Mathematizing at the Board
	24.4 From a Broad Practice to More Focused Routines
		24.4.1 Chalk Talk
		24.4.2 Audience
		24.4.3 What Is Said and What Is Written
	24.5 Illustrations
		24.5.1 Jonah´s Proof
			24.5.1.1 Coordinating Between Written and Oral Narratives
			24.5.1.2 Accounting for the Audience
			24.5.1.3 Meta-Mathematizing
		24.5.2 Virginia´s Proof
			24.5.2.1 Coordinating Between Written and Oral Narratives
			24.5.2.2 Accounting for the Audience
			24.5.2.3 Meta-Mathematizing
	24.6 Summary
	References
Chapter 25: Preservice Secondary School Teachers Revisiting Real Numbers: A Striking Instance of Klein´s Second Discontinuity
	25.1 Introduction
	25.2 Formulating Klein´s Double Discontinuity Within the ATD
	25.3 Real Numbers in Capstone Mathematics for Future High School Teachers
	25.4 Context of the Capstone Course UvMat and Methodology for the Case Study
	25.5 Student Work on the Task T
	25.6 Students´ Perceptions
	25.7 Discussion and Conclusions
	References
Part V: Research on Mathematics for Non-specialists
Chapter 26: Challenges for Research on Tertiary Mathematics Education for Non-specialists: Where Are We and Where Are We to Go?
	26.1 Introduction
	26.2 A Historical Perspective
		26.2.1 A First Historical Lens: The École Polytechnique
		26.2.2 A Second Historical Lens: CIEM/ICMI Studies
	26.3 Mathematics Education for Non-specialists Through the Lens of the Encyclopaedia of Mathematics Education
	26.4 Mathematical Training for Non-specialists from an Institutional Perspective
		26.4.1 Selected Theoretical Elements of the ATD
		26.4.2 Mathematical Praxeologies in Workplaces
		26.4.3 Mathematics and Major Discipline Courses
		26.4.4 Didactic Proposals for Mathematical Training for Non-specialists
	26.5 Conclusion
	References
Chapter 27: Mathematics in the Training of Engineers: Contributions of the Anthropological Theory of the Didactic
	27.1 Introduction
	27.2 Problems with Mathematics Courses for Engineers
	27.3 Some Key Notions from ATD
	27.4 Practices in Engineering Courses
	27.5 SRPs in Engineering Programs
		27.5.1 Epistemological Tools for Designing and Managing an SRP: An Example in Statistics
		27.5.2 Teaching Formats of SRPs and Their Ecology: Two Examples of Engineering Courses
	27.6 Conclusions
	References
Chapter 28: Modifying Exercises in Mathematics Service Courses for Engineers Based on Subject-Specific Analyses of Engineering...
	28.1 Introduction
	28.2 Theoretical Perspective and Previous Research
		28.2.1 Concepts of the Anthropological Theory of the Didactic
		28.2.2 Mathematical Practices in Signal Theory
			28.2.2.1 Amplitude Modulation and the Role of Complex Numbers in Electrical Engineering and in Mathematics Service Courses
			28.2.2.2 ATD Analyses of the Lecturer´s Sample Solution and Student Solutions
	28.3 From Analyses of Engineering Mathematical Practices to Modifying Exercises in Mathematics Service Courses
	28.4 Discussion
	Appendix: Exercise with Lecturer Sample Solution
	References
Chapter 29: Learning Mathematics in a Context of Electrical Engineering
	29.1 Introduction
	29.2 Background and Context of the Study
	29.3 Theory and Methodology
	29.4 Previous Relevant Research
	29.5 Analysis of Data
		29.5.1 Example: An Electric Circuit
		29.5.2 Opportunities for Connections
	29.6 Discussion
	References
Chapter 30: Towards an Institutional Epistemology
	30.1 Introduction
	30.2 Theoretical Framework
	30.3 From Mathematics to Land Surveying, an Example of Transpositive Effects
	30.4 Aspects of an Industrial Epistemology
		30.4.1 General Conditions and Constraints
		30.4.2 Measurement System Analysis
		30.4.3 Process Capability Analysis
	30.5 Making the PageRank Algorithm Intelligible
	30.6 Conclusion
	References
Chapter 31: Concept Images of Signals and Systems: Bringing Together Mathematics and Engineering
	31.1 Background
	31.2 Literature Review
		31.2.1 Engineering and Mathematics
		31.2.2 Use of Representations as Contexts
	31.3 Signals and Systems Courses in Engineering
	31.4 Interviews and Analysis
	31.5 Signals and Systems Concept Image and Conceptual Problems
		31.5.1 Concept Image for Signals and Systems
	31.6 Analysis of Concept Inventory Questions
		31.6.1 Students´ Descriptions of Frequency
		31.6.2 Connections Between Graphical and Symbolic Contexts
		31.6.3 Applications and the Concept Image
	31.7 Discussion
	31.8 Limitations
	31.9 Conclusion
	References
Chapter 32: Analyzing the Interface Between Mathematics and Engineering in Basic Engineering Courses
	32.1 Introduction
	32.2 Theoretical Backgrounds
		32.2.1 Different Mathematical Practices and Disparities in Mathematics and Engineering Courses
		32.2.2 Conceptions of Mathematical Modeling
		32.2.3 Conceptions of Problem-Solving
		32.2.4 Conceptualizations About the Use of Mathematics in Physics
	32.3 Synthesis of Frameworks with a View Towards the Electrotechnical Tasks
	32.4 Research Questions
	32.5 Methodology and Data Collection
		32.5.1 Overview
		32.5.2 Goals and Methods for Interviewing EE Experts About the Tasks
		32.5.3 Methods for Developing the Student-Expert-Solution
	32.6 The Exercise on Oscillating Current as an Example and Its Solution Outline for the First Two Subtasks
		32.6.1 The Subtasks B1 and B2 and the Official Solution Outline
		32.6.2 Summary of the SES of Subtasks B1 and B2
	32.7 Development of the Student Expert Solution for Exercise B3: Setting Up the Differential Equation)
		32.7.1 Official Solution Outline of Subtask B3
		32.7.2 SES1 Object-Level: Extended Structured Solution Outline, Knowledge from EE-Theory Relevant for the Solving Process
		32.7.3 SES1 Meta-Level: Viewing the Solution According to the Theoretical Approaches and Identifying Cognitive Resources
		32.7.4 Developing SES2 of B3 Based on the Expert Interviews
	32.8 Development of the Student Expert Solution for Exercise B4 (the Solving of the Differential Equation)
		32.8.1 Official Solution Outline
		32.8.2 SES1 Object-Level: Extended Structured Solution Outline, Knowledge from EE-Theory Relevant for the Solving Process
		32.8.3 SES1 Meta-Level: Structuring the Solution According to the Theoretical Approaches and Identifying Cognitive Resources
		32.8.4 Developing SES2 to B4, Based on the Expert Interviews
	32.9 Summary and Outlook
	References
Chapter 33: Tertiary Mathematics Through the Eyes of Non-specialists: Engineering Students´ Experiences and Perceptions
	33.1 Introduction - Students´ Perceptions and the Curriculum
	33.2 Theoretical Horizon - Mathematics as a Service Subject, Disintegrated Practice and the Position of Engineering Students
	33.3 Research Questions, Context and Data
	33.4 Findings
		33.4.1 Recognising a ``Mathematics Text´´
		33.4.2 The Usefulness and Role of Mathematics
			33.4.2.1 General Mathematico-Logical Thinking
			33.4.2.2 Schema for Learning
			33.4.2.3 Ways of Thinking for Systematic Problem-Solving
			33.4.2.4 Understanding of the Mathematical Underpinnings of Activities at Work Place
			33.4.2.5 Understanding of Mathematical Underpinnings of Other Academic Subjects
			33.4.2.6 Applications of Mathematics at Workplace
			33.4.2.7 Applications of Mathematics in Other Academic Subjects
		33.4.3 Ways of Studying Mathematics Compared to the Other Subjects
			33.4.3.1 Knowledge Structures, Criteria for Accomplishment and Intellectual Demands
			33.4.3.2 Forms and Habits of Working and Thinking, and Their Appreciation
			33.4.3.3 Investment of Time and Effort, and the Worth of Credit Points
	33.5 Discussion
	References
Chapter 34: Early Developments in Doctoral Research in Norwegian Undergraduate Mathematics Education
	34.1 Examples of Doctoral Research in RUME Supported by MatRIC
		34.1.1 Researching Flipped Classroom Approaches by Helge Fredriksen
		34.1.2 Researching Online and Blended Learning Approaches in Mathematics for Engineering Students by Shaista Kanwal
		34.1.3 Researching Learning with a Visualisation and Simulation Program by Ninni Marie Hogstad
		34.1.4 Researching Economics Students´ Performance in Mathematics by Ida Landgärds
		34.1.5 Researching the Development of Mathematical Competency of Biology Students by Yannis Liakos
		34.1.6 Researching Biology Students´ Use of Mathematics by Floridona Tetaj
		34.1.7 Researching Relationships Between Prior Knowledge, Self-Efficacy and Approaches to Learning Mathematics of Engineering ...
	34.2 Conclusion
	References