Encyclopedia of mathematics education

Foreword
Preface to Second Edition
Preface to First Edition
List of Topics
About the Section Editors
Contributors
A
	Ability Grouping in Mathematics Classrooms
		Definitions
		Introduction
		Ability Grouping Practices in Different Countries
		Ability Grouping, Achievement, and Equity
		Conclusion
		References
	Abstract Algebra Teaching and Learning
		Definition
		Historical and Epistemological Landmarks
		Structures in Mathematics Education
			Idealization, Thematization, and Reflective Thinking
			Structures
			FUGS Concepts and Didactical Consequences
		Instructional Approaches
			A Cognitive Approach with the Support of Technology
			Guided Reinvention of Abstract Algebra Concepts
			Structuralist Thinking and Praxeologies
		Conclusion
		Cross-References
		References
	Abstraction in Context
		Definition
		The Emergence of the Theoretical Framework
		Model and Methodology
		Research Directions
		Cross-References
		References
	Abstraction in Mathematics Education
		Definition
		Elaboration
		Perspectives
		Cross-References
		References
	Actions, Processes, Objects, Schemas (APOS) in Mathematics Education
		The Mental Structures and Mechanisms of APOS Theory
		Studies That Use APOS Theory
		APOS-Based Instruction
		Results
		Ongoing and Future Work
		Cross-References
		References
	Activity Theory in Mathematics Education
		Definition
		History of Activity Theory
		Minimum Unit
		Structure of Activity
		Subjectification and Personality
		A Holistic Psychology
		Cross-References
		References
	Adults Learning Mathematics
		Definitions: Contested Meanings in an Emerging Field
		Characteristics of an Emerging Field
		Adults as Workers
		Adults as Critical Citizens
		Adults as Thinking Beings with Emotions and Experiences
		Teaching, Learning, and Professional Development
		Concluding Remarks
		Cross-References
		References
	Affect in Mathematics Education
		Definition
		From Anxiety and Problem-Solving to Affective Systems
		Defining Emotions
		The Role of Emotional States in Self-Regulation
		The Spectrum of Emotional Traits
		Regulation of Emotions
		Creating an Emotionally Supportive Learning Environment
		Cross-References
		References
	Algebra Teaching and Learning
		Definition
		Evolving Perspectives on School Algebra and Its Research Over the Years
		What Does Research Tell Us About the Learning of School Algebra?
		What Research Says About the Teaching of Algebra
		For Further Study and Reflection
		Cross-References
		References
	Algorithmics
		Definition
		Algorithms in Mathematics Education Research
		Students´ Understanding of Algorithmic Structures and Languages
		Algorithmics and Programming Competencies
		Perspectives
		Cross-References
		References
	Algorithms
		Definition
		Cross-References
		References
	Analysis Teaching and Learning
		Introduction
		Foundations: Real Numbers
		Foundations: Limits and Derivatives
		The Calculus-Analysis Transition at University
		Developments and Perspectives
		Cross-References
		References
	Anthropological Theory of the Didactic (ATD)
		Introduction
		The Dialectic of Persons and Institutions
		Relations to Objects
		The Theory of Praxeologies
		Conditions and Constraints
		The (Possibly) Didactic
		Study Paradigms
		Cross-References
		References
	Argumentation in Mathematics
		Definition
		Characteristics of Argumentation
		Cross-References
		References
	Argumentation in Mathematics Education
		Definition
		Mathematics Classrooms and Argumentation
		Approaches to Argumentation in Mathematics Education
		Cross-References
		References
	Assessment of Mathematics Teacher Knowledge
		Definition
		Certification of New Teachers
		Promotion and Recognition of Practicing Teachers
		Research on Teacher Knowledge
		Cross-References
		References
	Authority and Mathematics Education
		Definition
		Characteristics
		Cross-References
		References
	Autism, Special Needs, and Mathematics Learning
		Definition
		Characteristics
		Cross-References
		References
B
	Bilingual/Multilingual Issues in Learning Mathematics
		Definition
		Characteristics
			Theoretical Perspectives
		Bilingualism
		Research Findings
		History
		Issues in Designing Research
		Future Issues and Questions
		Cross-References
		References
	Blind Students, Special Needs, and Mathematics Learning
		Characteristics
		Cross-References
		References
	Bloom´s Taxonomy in Mathematics Education
		Definition
		Overview
		Historical Development
		Influence on Mathematics Education
		Contemporary Classroom Applications
		Cross-References
		References
C
	Calculus Teaching and Learning
		Definition: What Teaching and Learning Calculus Is About
		Calculus Curriculum
		Early Research in Learning Calculus: The Cognitive Difficulties
		Research and Alternative Approaches to Teaching and Learning Calculus
		The Role of Technology
		The Role of Historical Perspective and Other Approaches
		The Role of the Teacher
		The Transition Between Secondary and Tertiary Education
		New Directions of Research
		Cross-References
		References
	Collaborative Learning in Mathematics Education
		References
	Commognition
		Definition
		Origins
		Foundations
		Method
		Commognitive Theory of the Development of Mathematical Discourses
		Commognitive Theory of Factors that Shape the Learning of Mathematics
		Contributions of Commognitive Research: Past and Future
		Cross-References
		References
	Communities of Inquiry in Mathematics Teacher Education
		Definition
		Characteristics
		Cross-References
		References
	Communities of Practice in Mathematics Education
		Definition and Originators
		How CoP Connects to Developments in Theories of Learning Mathematics
		History of Use
		Perspectives on Issues in Different Cultures/Places
		Gaps That Need to Be Filled
		Cross-References
		References
	Communities of Practice in Mathematics Teacher Education
		Definition
		Background
		Communities of Practice as a Framework for Understanding Mathematics Teacher Learning and Development
			Learning as Participation in Practices
			Learning as Developing an Identity
		Issues for Future Research
		Cross-References
		References
	Competency Frameworks in Mathematics Education
		Definition
		Characteristics
		Cross-References
		References
	Complexity in Mathematics Education
		Definition/Introduction
		Complexity as an Epistemological Discourse
		Complexity as a Historical Discourse
		Complexity as a Disciplinary Discourse
		Complexity as a Pragmatic Discourse
		Cross-References
		References
	Computational/Algorithmic Thinking
		Introduction
		Defining CT and AT and Relating Them
		State of Research on CT/AT
		Curricular Aspects and Emerging Implications for Mathematics Education
		Conclusion
		Cross-References
	Acknowledgment
		References
	Concept Development in Mathematics Education
		Characteristics
		Cross-References
		References
	Constructivism in Mathematics Education
		Background
		Radical and Social Constructivism in Mathematics Education
		Truth and Objectivity
		A Short List: Impact of Constructivism in Mathematics Education
		Cross-References
		References
	Constructivist Teaching Experiment
		Introduction
		Characteristics: The Elements of Constructivist Teaching Experiments
			Teaching Episodes
			Exploratory Teaching
			Meanings of ``Experiment´´
		Living, Experiential Models of Students´ Mathematics
		Meanings of Teaching in a Teaching Experiment
			Responsive and Intuitive Interaction
			Analytical Interaction
			The Role of a Witness of the Teaching Episodes
		Retrospective Conceptual Analysis
		Cross-References
	Acknowledgment
		Appendix: Example Studies Using Teaching Experiment Methodology
		References
	Cooperative Didactic Engineering
		Definition
		Origin
		CE´s Background Assumptions
		Principles
		CE as a Form of Both Anthropological and Engineering Research
		CE: An Epistemology of Paradigmatic Analogy, Toward the Ascent from the Abstract to the Concrete
		A Method of Documenting Practice and Research on Practice: The PTHAS
		Cross-References
		References
	Creativity in Mathematics Education
		Definition
		Creativity
		Giftedness and Ability
		Giftedness and Creativity
		Empirical Research
		Conceptual Relationships
		Giftedness and Creativity in Psychology and Neuroscience
		Teaching for Creativity
		Cross-References
		References
	Critical Mathematics Education
		Definition
		Characteristics
			Critical Education
			Steps into Critical Mathematics Education
			Some Issues in Critical Mathematics Education
			Some Notions in Critical Mathematics Education
			Critical Mathematics Education for the Future
		Cross-References
		References
	Critical Thinking in Mathematics Education
		Definition
		Introduction
			Critical Thinking and Mathematical Reasoning
			Critical Thinking and Applications of Mathematics
			Limitations of Developing CT Through Mathematics Education
			Further Unresolved Issues
		Cross-References
		References
	Cultural Diversity in Mathematics Education
		Introduction
		Diversity and Uses of Cultural Mathematical Tools
		Diversity and Cultural and Mathematical Identities
		Diversity and Teachers´ Social Representations of Cultural Differences
		Conclusion
		Cross-References
		References
	Cultural Influences in Mathematics Education
		Definition
		Introduction
		Features of Mathematics
		Perceptions of Mathematical Success
		Summary
		Cross-References
		References
	Curriculum Resources and Textbooks in Mathematics Education
		Definition of Curriculum Resources
		Design and ``Use´´ of Mathematics Curriculum Resources
			Design and Quality of Curriculum Resources
			The ``Use´´ of Curriculum Resources
		Future Research Perspectives
		Cross-References
		References
D
	Data Handling and Statistics Teaching and Learning
		Definition
		Changing Views on Teaching Statistics Over the Years
		A Focus on Statistical Literacy and Reasoning
		What Does Research Tell Us About Teaching and Learning Statistics?
		Teaching and Learning
		For Further Research
		Cross-References
		References
	Deaf Children, Special Needs, and Mathematics Learning
		Characteristics
		Learning Mathematics in Primary School
		Counting
		Early Mathematical Reasoning and Arithmetic Operations
		Conclusion
		Cross-References
		References
	Deductive Reasoning in Mathematics Education
		Definition
		The Importance of Deductive Reasoning
		Different Theoretical Perspectives on What a Deductive Inference Is
		The Role of Axioms in Deductive Reasoning
		Purposes of Deductive Reasoning and Proof
		Students´ Proving Abilities, Conceptions, and Perceptions
			What Types of Evidence Provide Students with Mathematical Certainty?
			Do Students Justify with the Aim of Obtaining Certainty?
			What Do Students and Teachers Think the Role of Proof Is?
		Open Research Questions with Regard to Deductive Reasoning and Proof
			What Factors Prevent Deductive Reasoning from Playing a Prominent Role in K-12 Mathematics Classrooms?
			How Can We Have Proof Serve the Functions of Providing Insight, Promoting Discovery, and Illustrating Methods?
			What Strategies and Heuristics Can Help Students Learn to Prove?
		References
	22q11.2 Deletion Syndrome, Special Needs, and Mathematics Learning
		Characteristics
		Cross-References
		References
	Design Research in Mathematics Education
		Definition
		The Origins and Need for Design Research
		Characterizing Design-Based Research
			Creative and Visionary
			Ecologically Valid
			Interventionist and Iterative
			Theory-Driven
		Some Issues and Challenges
		Cross-References
		References
	Dialogic Teaching and Learning in Mathematics Education
		Definition
		Characteristics
			Sources of Inspiration
			The Inquiry Cooperation Model
			New Qualities in Teaching and Learning
		Cross-References
		References
	Didactic Contract in Mathematics Education
		Introduction
		Definition
		Characteristics
			Background: Illustrative Examples
			Didactical and Ethical Responsibility
			Paradoxes of the Didactical Contract
			Observations of Reactions of Teachers to Difficulties
			Predicting and Explaining Certain Long-Term Effects
			Extensions
		Cross-References
		References
	Didactic Engineering in Mathematics Education
		Definition
		History
		DE as a Research Methodology
		Realizations
		Challenges and Perspectives
		Cross-References
		References
	Didactic Situations in Mathematics Education
		Didactical Situation
			Components
			Teaching Methods
		Origin and Necessity of the Concept of ``Didactical Mathematical Situation´´
			The Reform of the Foundations (1907-1980)
			Learning Mathematics by Doing It Reverses the Classic Pedagogical Order
			The Project of a Mathematical Science: Didactique
		Mathematical Situations
			Definition
			Examples
			Types of Mathematical Knowledge, Reference Knowledge (Savoirs)
			Examples of Milieux
		Types of Mathematical Situations Characteristic of Activities, of Pieces of Knowledge, and of Pieces of Mathematical Learning
		The Processes
		Some of the Results of Research on Didactical Situation
		Research Perspectives
		Cross-References
		References
	Didactic Transposition in Mathematics Education
		Definition
		Scope
		An Emancipatory Tool
		Enlargement of the Object of Study
		The Need for Researchers´ Own Epistemological Models
		From Didactic Transposition to the Anthropological Approach
		Cross-References
		References
	Didactical Phenomenology (Freudenthal)
		What Is Meant by Didactical Phenomenology?
		Merit of a Didactical Phenomenology for Mathematics Education
		Relation with Realistic Mathematics Education
		Examples of Didactical Phenomenology
		The Method
		Mathematics-Related Analyses Constituting the Didactics of Mathematics
		Cross-References
		References
	Differential Equations Teaching and Learning
		Definition
		Differential Equations Curriculum
		Learning
		Teaching
		For Further Research
		Cross-References
		References
	Discourse Analytic Approaches in Mathematics Education
		Introduction
		Critical Discourse Analysis
		Poststructural Approaches
		Mathematical Discourse, Thinking, and Learning
		Cross-References
		References
	Discrete Mathematics Teaching and Learning
		Definition
		What Is Discrete Mathematics?
		Why Teach and Learn Discrete Mathematics? New Context, Concepts, and Ways of Reasoning: A New Realm of Experience for the Clas...
			To Integrate Discrete Mathematics into the School Curriculum: A Current Challenge
			Benefits from Teaching and Learning Discrete Mathematics: Some Examples
				Accessible Problems and Concepts
				Richness of Discrete Concepts: A Way to Deal with the Construction of Axiomatic Theory
				Several Ways of Questioning, Proving, and Modeling
				A Mathematical Experience
		Interesting Perspectives for Research in Mathematics Education
		Cross-References
		References
	Discursive Approaches to Learning Mathematics
		Definition
		Background
		Foundations
		Strands
		Methods
		Cross-References
		References
	Documentational Approach to Didactics
		Introduction
		Sources of the Approach
		The Documentational Approach to Didactics: A Holistic Approach to Teachers´ Work
		Deepening the Approach: Schemes and Systems
		Reflexive Investigation: A Developing Methodological Construct
		Perspectives for Further Evolutions
		Cross-References
		References
	Down Syndrome, Special Needs, and Mathematics Learning
		Characteristics
		Cross-References
		References
E
	Early Algebra Teaching and Learning
		What Is Early Algebra?
		What Is Algebraic Reasoning?
			Operating on Unknowns
			Thinking About Variables
			Acknowledging Algebraic Structure
		Approaches to Early Algebra Instruction
			Reasoning About Physical Quantities and Measures
			The Properties of Number Systems: Generalized Arithmetic
			Functions Approaches to Early Algebra
		In Summary
		References
	Early Childhood Mathematics Education
		What Is Meant by Early Childhood Mathematics Education?
		History
		Recent Interest in Early Childhood Mathematics Education
		Mathematics Taught in Early Childhood
		Ways of Teaching Mathematics to Young Children
			Play
			Picture Books
		Information and Communication Technology
		Future Perspectives in Early Childhood Mathematics Education
		Cross-References
		References
	Education and Professional Development of Teacher Educators
	Education of Facilitators (for Educators of Practicing Teachers)
	Education of Mathematics Teacher Educators
		Synonyms
		Definition
		Background
		Mathematics Teacher Educators´ Knowledge, Skills, and Practice
		Education and Development of Mathematics Teacher Educators
		Further Research
		Cross-References
		References
	Education of Professional Development Providers (for Educators of Practicing Teachers)
	Education of Teachers of Teachers
	Elkonin and Davydov Curriculum in Mathematics Education
		Definition
		Characteristics and Origin
		Implementation and Adaptation
		Cross-References
		References
	Embodied Cognition
		Definition
		Some Definitional Differences in Mathematics Education
		Embodied Cognition in Mathematics Education
		Cognitive Science of Embodied Cognition
		Dienes´ Contributions to Embodied Mathematics
		Cross-References
		References
	Enactivist Theories
		Definition
		Characteristics
		Cross-References
		References
	Engagement with Mathematics
		Definition
		Behavioral Engagement
		Cognitive Engagement
		Affective Engagement
		Social Engagement and Opportunities to Engage
		Intertwined Constructs
		Cross-References
		References
	Epistemological Obstacles in Mathematics Education
		References
	Equity and Access in Mathematics Education
		Definition
		Characteristics
		Cross-References
		References
	Ethnomathematics
		Introduction
		Conceptual Foundations of Ethnomathematics
		Current Work in Ethnomathematics
		Cross-References
		References
	External Assessment in Mathematics Education
		Characteristics
		Design-Based Assessment
		References
F
	Fieldwork/Practicum in Mathematics Education
		Definition
		Features
			Research on the Field Experience
		Cross-References
		References
	Frameworks for Conceptualizing Mathematics Teacher Knowledge
		Introduction
		Lee Shulman
		Mathematical Knowledge for Teaching
		The Knowledge Quartet
		Culture: A Caveat
		References
	Functions Learning and Teaching
		Definition and Brief History
		Challenges to the Teaching and Learning of Function
		Overcoming Learning Difficulties
		Future Research
		Cross-References
		References
G
	Gender in Mathematics Education
		Issues of Definition
		Historical Overview of Gender and Mathematics Education Research
		Theoretical Considerations
		Methodological Considerations
		Recent Developments
		Cross-References
		References
	Gestures in Mathematics Education
		Introduction to the Literature on Gesture
		Definition of Gesture in Mathematics Education
		Cross-References
		References
	Giftedness and High Ability in Mathematics
		Introduction
		Mathematical Giftedness, Ability, Potential, and Talent
		Definition of Mathematical Giftedness and Its Relativity
			Mathematical Expertise and Mathematical Giftedness
			Mathematical Creativity, Insight, and Mathematical Giftedness
		Cognitive and Neurocognitive Characteristics Associated with Mathematical Giftedness
		Affect and Personality
		Development of Mathematical Ability
		Concluding Comment
		Cross-References
		References
H
	Heuristics and Biases
		Definition
		Cross-References
		References
	Heuristics in Mathematics Education
		Definition
		Characteristics
		Cross-References
		References
	History of Mathematics and Education
		History and Epistemology of Mathematics: An Ever-Increasing Interest for New Conceptions and Practices in Mathematics Education
		Epistemological Contributions
		Cultural Contributions
		History of Mathematics in Teachers´ Training
		History as an Instrument for an Interdisciplinary Approach in Teaching
		History of Mathematics in the Classroom
		Current Concerns and Emergent Questions in the Field
		Cross-References
		References
	History of Mathematics Teaching and Learning
		Introduction
		General Education and Professional Training in Mathematics
		Mathematics in Primary Schooling
		Mathematics in Secondary Schooling
		Curriculum
		Textbooks for Mathematics
		Mathematics Learning for Girls
		Teachers of Mathematics
		Mathematics in the Global World
		Research into History of Mathematics Education as a Field
		Cross-References
		References
	History of Research in Mathematics Education
		Definition
		Main Text
		Mathematics Education as an Academic Field
		Comparative Studies of School Mathematics
		Becoming Scientific
		Studying the Teaching of Mathematics
		A Flourishing Academic Enterprise
		Cross-References
		References
	Hypothetical Learning Trajectories in Mathematics Education
		Definition
		Explanation of the Construct
		Cross-References
		References
I
	Immigrant Students in Mathematics Education
		Introduction
		Educational Policy and Immigration
		Different Forms of Mathematics
		Teacher Education in an Immigration Context
		Multilingualism and Mathematics Teaching and Learning
		Immigrant Parents´ Perceptions of Mathematics Education
		Some Implications
		Cross-References
		References
	Immigrant Teachers in Mathematics Education
		References
	Inclusive Mathematics Classrooms
		Definition
		Characteristics
		Classroom Practice
		Language Use in Classrooms
		Mathematics
		Cross-References
		References
	Indigenous Students in Mathematics Education
		Introduction
		Gaining Western Mathematics
		Using Cultural Knowledge and Artifacts
		Ensuring Access and Control of the Decision Making
		Cross-References
		References
	Informal Learning in Mathematics Education
		What is Informal Learning?
		From Terminology to Conceptualization: Consciousness, Status, and Knowledge Transfer
			Consciousness of Concepts Versus Non-consciousness Concepts
			Transfer of Knowledge and Abilities
			The Question of Context
			Status of Knowledge
		Informal Learning in Mathematics Education
		About the Paradigms of Research on Informal Learning in Mathematics Education
			Ways of Approaching the Issue and Methods
			Models of Formal Knowledge and Informal Learning in Mathematics
			Model of Daily Learning and Informal Learning in Mathematics
		An Alternative Approach
		Cross-References
		References
			Further Reading
	Information and Communication Technology (ICT) Affordances in Mathematics Education
		Definition
		Characteristics
		Cross-References
		References
	Inquiry-Based Mathematics Education
		Related Terms (and Acronyms)
		Definition
		Sociopolitical Background
		Historical Background
		IBME and Mathematics Education Research
		Evidences from Research of IBME Benefits
		Research on Teachers´ Practices Regarding IBME
		Cross-References
		References
	Instrumental and Relational Understanding in Mathematics Education
		Characteristics
		Cross-References
		References
	Instrumentalization in Mathematics Education
		Introduction
		Mediated Action as a Source of Tensions
		Instrumentation/Instrumentalization Dialectic
		Instrumentalization as a Key Feature of the Instrumental Approach to Didactics
		Rethinking the Concept of Instrumentalization
			Taking into Account the Evolution of Artifacts
			Taking into Account the Teacher´ Role for Designing the Artifacts Environment
			Rethinking Instrumentalization from the Teacher´s Perspective
		Conclusion
		Cross-References
	Acknowledgment
		References
	Instrumentation in Mathematics Education
		Introduction
		Instrumentation and Instruction
		The Critical Notion of Affordance
		The Dialectical Relationships Between Artifact and Instrument
		An Essential Dialectic Relationship between Instrumentation and Instrumentalization
		From a Set of Artifacts to a System of Instruments: The Crucial Notion of ``Orchestration´´
		Conclusion: From Student Instrumentation to Teacher Instrumentation
		Cross-References
	Acknowledgment
		References
	Interactionist and Ethnomethodological Approaches in Mathematics Education
		Definition
		Reception of Symbolic Interactionism and Ethnomethodology in Mathematics Education
		Perspectives for Future Research
		Sociological Aspects of a Theory of Mathematics Learning
		Combination and Expansion of Symbolic Interactionism and Ethnomethodology with Other Theoretical Approaches
		Cross-References
		References
	Interdisciplinary Approaches in Mathematics Education
		Introduction
		Definition
		Interdisciplinarity and Mathematics (Education)
		Perspectives
		Cross-References
		References
	International Comparative Studies in Mathematics: An Overview
		Characteristics
		What Are ICS?
		Some Examples of ICS
		Why Are ICS Important?
		Some Issues with ICS
		The Way Forward
		References
	Interpretative Knowledge
		Introduction
		Interpretative Knowledge
		Main Implications for Teacher Education
		Cross-References
		References
	Intuition in Mathematics Education
		Definition
		Characteristics, Approaches, and Role in Mathematics Education
		Cross-References
		References
J
	Joint Action Theory in Didactics (JATD)
		Introduction
		The Didactic Game
		Learning Games, the Double Dialectics Reticence-Expression/Contract-Milieu, the Equilibration Process
		Epistemic Games
		Cooperative Engineering
		Cross-References
		References
L
	Language Background in Mathematics Education
		Introduction
		Theoretical Perspectives on Language Background in Mathematics Education
		Language Background and Attainment in Mathematics
		Language Background and Learning and Teaching Mathematics
			Second Language Education Settings
			Bilingual Education Settings
			Plurilingual Societies
			Immersion Education Settings
		Language Background in Mathematics Teacher Education
		Language Background and Researchers in Mathematics Education
		Future Directions
		Cross-References
		References
	Language Disorders, Special Needs and Mathematics Learning
		Definition
		Characteristics
		Cross-References
		References
	Learner-Centered Teaching in Mathematics Education
		Definition
		Characteristics
			Intellectual Autonomy
			History
			Characteristics of Student-Centered Classrooms
			Problem Solving
			Classroom Environment
			Collaboration
			Mathematical Discourse
			Tools/Manipulatives
			Critiques
		Cross-References
		References
	Learning Difficulties, Special Needs, and Mathematics Learning
		Introduction
		Two Sorts of Mathematical Skills to Be Learned
		The Specificity of Mathematics Learning Difficulties
		Further Research
		Cross-References
		References
	Learning Environments in Mathematics Education
		Computer Scaffolded Learning
		Computer-Supported Collaborative Learning
		Cross-References
		References
	Learning Framework in Number, The
		Description
		Origin
		Applications
		Cross-References
		References
	Learning Practices in Digital Environments
		Definition
		Characteristics
		Cross-References
		References
	Learning Study in Mathematics Education
		Characteristics
		The Cyclic Process and the Object of Inquiry
		Guided by a Learning Theory
		Exploring the Object of Learning
		Teachers´ Learning and Students´ Learning: A Parallel Process
		References
	Lesson Study in Mathematics Education
		Definition
		Historical Developments and Contexts
		Key Elements
		The Role of Outside Experts
		Lesson Study Adopted as a Model of Professional Development in Other Countries
		Cross-References
		References
	Linear Algebra Teaching and Learning
		Introduction
		Preliminary Work
		Different Approaches to the Learning and Teaching of Linear Algebra Concepts
		The Use of Realistic Problems and Models in the Teaching and Learning of Linear Algebra
		Use of Technology
		Cross-References
		References
	Logic in Mathematics Education
		Definition
		The Role of Logic in Mathematics Education
		Logic and Mathematical Language
		The Role of Logic in Mathematical Proof, Argumentation, and Reasoning
		Conclusion
		Cross-References
		References
	Logic in University Mathematics Education
		Definition
		Characteristics
		Asking and Answering Logical Questions
		Instruction in Mathematical Logic
		Cross-References
		References
M
	Manipulatives in Mathematics Education
		Definition
		Characteristics
			Manipulatives and Mathematics Education
			Critical Issues
			A Comprehensive Theoretical Approach to Manipulatives: Semiotic Mediation After a Vygotskian Approach
			Some Examples of Manipulatives and Tasks
			Counting Stick
			Pascaline
			Pair of Compasses and Other Curve Drawers
			Pantographs
			Open Questions
		Cross-References
		References
	Mathematical Ability
		Definitions
		Characteristics
			The Evolvement of Mathematical Abilities
			Characterizing Different Students on the Spectrum of Mathematical Abilities
			Mathematical Abilities and General Intelligence
			Measuring and Evaluating Students´ Mathematical Ability
		Cross-References
		References
	Mathematical Approaches
		Introduction
		Definition
		Developments
		Characteristics
		Stoffdidaktik
		An Epistemological Program
		Realistic Mathematics Education
		Mathematical Knowledge for Teaching
		Some Further Aspects of Mathematical Approaches
		Unresolved Issues
		Cross-References
		References
	Mathematical Cognition: In Secondary Years [13-18] Part 1
		Definition/Introduction
		Slope and Proportional Reasoning
		Rate of Change
		Covariance
		Functions
		Generalizing
		Types of Mathematical Generalization
		Strategies in Pattern Generalizing
		Cognitive Difficulties in Generalizing Mathematical Ideas
		Justification of Mathematical Generalization
		Summary
		Cross-References
		References
	Mathematical Cognition: In Secondary Years [13-18] Part 2
		Definition/Introduction
		Geometry
		Theoretical Models Guiding Research on Geometry Learning
		Proofs
		Genres of Investigation
		Cross-References
		References
	Mathematical Cognition: In the Elementary Years [6-12]
		Introduction
		Cognition in Elementary Years: Geometric Thinking
			Piagetian Views
			Non-Piagetian Views: Van Hiele Levels of Geometric Thinking
		Cognition in Elementary Years: Enumeration and (Whole) Number Operations
			Piagetian Views
			Non-Piagetian Views: Cognitively Guided Instruction
		References
	Mathematical Games in Learning and Teaching
		Definition
		Characteristics
		Cross-References
		References
	Mathematical Language
		Introduction: What Is Mathematical Language?
		Characteristics of Mathematical Language
		Variations in Language and Thinking Mathematically
		Cross-References
		References
	Mathematical Learning Difficulties and Dyscalculia
		Introduction
		The Mathematics Education Perspectives on MLD: The Issues of Prevention and Remediation
		Communicating Across Fields
		Cross-References
		References
	Mathematical Literacy
		Definition
		Characteristics and Delimitation
		Motivations for Introducing Mathematical Literacy
		Critique and Further Research
		Cross-References
		References
	Mathematical Modelling and Applications in Education
		Characteristics
		Theoretical Debate on Mathematical Modelling: Historical Development and Current State
		The Modelling Process as Key Feature of Modelling Activities
		Detailed Description of One Modelling Cycle Based on the Lighthouse Example
		Westerhever Lighthouse
		Development of a Real-World Model
		Development of a Mathematical Model
		Interpretation and Validation
		Further Explorations and Extensions
		Modelling Competencies and Their Promotion
		Results of Empirical Studies on the Implementation of Mathematical Modelling in School
		Cross-References
		References
	Mathematical Proof, Argumentation, and Reasoning
		Definition
		Characteristics
		Mathematical Proof
		Reasoning and Proof
		Argumentation and Proof
		Practical Classroom Approaches
		Trends in Proof
		Cross-References
		References
	Mathematical Representations
		Definitions
		Characteristics
		Research
		Cross-References
		References
	Mathematics Classroom Assessment
		Definition
		Background
		A Social-Constructivist Approach to Classroom Assessment
		Eliciting Evidence of Student Learning
		Interpreting Evidence of Student Learning
		Acting on Evidence of Student Learning
		Issues in Classroom Assessment
		Cross-References
		References
	Mathematics Curriculum Evaluation
		Definition
		Notions and Meanings
		Curriculum
		Curriculum Evaluation
		Models of Mathematics Curriculum Evaluation
		Mathematics Curriculum Evaluation and Large-Scale Reforms
		Cross-References
		References
	Mathematics Learner Identity
		Definition
		Cross-References
		References
	Mathematics Teacher as Learner
		Rationale
		Sources and Strategies
		Teacher Learning Theory
		Cross-References
		References
	Mathematics Teacher Education Organization, Curriculum, and Outcomes
		Structure and Characteristics
		Institutions
		Study Programs
		Who Teaches Future Teachers?
		Who Enters the Profession?
		Professional Teacher´s Competences
		Cross-References
		References
			Further Readings
	Mathematics Teacher Educator as Learner
		Mathematics Teacher Educators: Definition
		Mathematics Teacher Educators´ Learning Through Research
		Mathematics Teacher Educators´ Learning Through Action Research and Intervention Research
		Cross-References
		References
	Mathematics Teacher Identity
		Definition
		A Brief History
		Identity Research in Mathematics Education and MTI
		Clusters of Research in MTI
		Concluding Comments
		Cross-References
		References
	Mathematics Teachers and Curricula
		Definition and Historical Background
		Different Cultures Shaping Different Forms of Interaction Between Teachers and Curricula
		Teachers and Curricula Within a Collaborative Perspective
		Open Questions
		Cross-References
		References
	Mathematization as Social Process
		Definition
		Characteristics
		Mathematization and Demathematization
		There Is Mathematics Everywhere
		Mathematics in Action
		New Challenges
		Cross-References
		References
	Metacognition
		Definition
		Characteristics
		Cross-References
		References
	Meta-didactical Transposition
		Definition
		Why Do We Need a Dynamic Framework Relating to Teachers´ Practices?
		Meta-Didactical Transposition
			The Institutional Aspects
			The Meta-Didactical Praxeologies
			The Dynamics Between Internal and External Components
			The Role of the Broker
			The Double Dialectic
			Applications, Integration, and Evolution of the Meta-Didactical Transposition Framework
		References
	Metaphors in Mathematics Education
		Definition
		Introduction
		Characteristics
			Metaphors for Metaphor
		Metaphor and Reification
		Metaphor, Embodied Cognition, and Gestures
		Metaphors for Teaching and Learning
		Educational Metaphors
		Metaphoring (Metaphorical Thinking) in Mathematics Education
		Examples of Metaphors for Multiplication
		On the Metaphorical Nature of Mathematics
		Open Ends and Questions
		Cross-References
		References
	Misconceptions and Alternative Conceptions in Mathematics Education
		Definition
		Characteristics
		Cross-References
		References
	Models of In-Service Mathematics Teacher Education Professional Development
		Cross-References
		References
	Models of Preservice Mathematics Teacher Education
		Definition
		Characteristics
		Cross-References
		References
	Motivation in Mathematics Learning
		Definition
		Characteristics and Findings from Various Theoretical Perspectives
		Attribution Theory
		Goal Theory
		Intrinsic Motivation and Interest
		Self-Regulated Learning
		Social Motivation
		Affect
		Cross-References
		References
N
	Noticing of Mathematics Teachers
		Definition of Mathematics Teacher Noticing
		Studying Mathematics Teacher Noticing
		Supporting the Development of Expertise in Mathematics Teacher Noticing
		Contributions of Mathematics Teacher Noticing
		Cross-References
		References
	Number Lines in Mathematics Education
		Characteristics
		The Empty Number Line
		Flexible Solution Strategies
		Further Elaboration
		An Alternative Approach
		Imagery
		Effect Studies
		The Double Number Line
		Cross-References
		References
	Number Teaching and Learning
		Introduction
		Natural Numbers
			Natural Numbers, Operations, and Estimation
			Arithmetic and Algebra
			Numbers and Word Problems
			Number Sense
		Rational Numbers
			Epistemological and Cognitive View of Rational Numbers Learning
			Teaching of Rational Numbers
		Negative Numbers
		Irrational Numbers
		Cross-References
		References
P
	Pedagogical Content Knowledge Within ``Mathematical Knowledge for Teaching´´
		Definition
		Characteristics
		Theorizations of PCK
		Measuring PCK
		Using PCK to Support the Development of Pedagogic Practice
		Emerging Directions
		Cross-References
		References
	Policy Debates in Mathematics Education
		Definition
		History
		Analytical Framework
		``Math Wars´´
		Defusing the Debates
		Cross-References
		References
	Political Perspectives in Mathematics Education
		Definition
		History
		Weak Political Perspectives
		Strong Political Perspectives
		Recent Overviews
		Cross-References
		References
	Poststructuralist and Psychoanalytic Approaches in Mathematics Education
		Definition
		Characteristics
		Poststructuralist Approaches
		Psychoanalytic Approaches
		Emancipatory Possibilities
		Cross-References
		References
	Preparation and Professional Development of University Mathematics Teachers
		Introduction
		What Is Included in Preparation and Professional Development?
		University Provision for Both New and Experienced Teachers of Mathematics
		Learning from Research and Scholarship in Mathematics Education
		Teachers´ Learning through Engagement in Research and Development Projects
		Innovation in Teaching
		Professional Development Activity Influencing Practice
		Mathematics Support
		Conclusion
		Cross-References
		References
	Probabilistic and Statistical Thinking
		Definition
		Probabilistic and Statistical Thinking
		Probabilistic Thinking in Probability Education
		Statistical Thinking in Statistics Education
		The Data Deluge
		Supplantation
		Cross-References
		References
	Probability Teaching and Learning
		Different Meanings of Probability
		Probability in the School Curriculum
		Intuitions and Misconceptions
		Challenges in Teaching Probability
		Cross-References
		References
	Problem-Solving in Mathematics Education
		Introduction
		Focusing of Problems in the Development of Mathematics
		Mathematics Education and Problem-Solving Developments
			Curriculum Proposals and Instruction
		Regional Problem-Solving Developments and the Use of Digital Tools
			Digital Technologies and Mathematical Problem-Solving
		Directions for Future Research
		Cross-References
		References
	Professional Learning Communities in Mathematics Education
		Characteristics
		Data-Informed and Knowledge-Based Enquiry
		Leadership
		Impact and Research
		References
	Psychological Approaches in Mathematics Education
		Characteristics
		The Constructivist Approach
		The Piagetian Approach: Research on Conceptions and Conceptual Change
		Departing from Piaget: From Research on Concept Formation to Teaching Experiments
		Sociocultural Approaches
		Open Issues
		Cross-References
		References
Q
	Quasi-empirical Reasoning (Lakatos)
		Definition
		Characteristics
		Cross-References
		References
	Questioning in Mathematics Education
		Definition
		Introduction
		Mathematical Perspectives
		Asking as Telling
		Asking as Enquiring
		Intention and Effect
		Classroom Ethos
		Questions as Typifying What Mathematics Is About
		Internalizing Questions
		Students Asking ``Good Questions´´
		Further Investigation
		Cross-References
		References
R
	Realistic Mathematics Education
		What is Realistic Mathematics Education?
		The Onset of RME
		Freudenthal´s Guiding Ideas About Mathematics and Mathematics Education
		The Core Teaching Principles of RME
		Various Local Instruction Theories
		Implementation and Impact
		A Long-Term and Ongoing Process of Development
		Cross-References
		References
	Recontextualization in Mathematics Education
		Characteristics
		Cross-References
		References
	Reflective Practitioner in Mathematics Education
		References
	Registers of Semiotic Representation
		Introduction
		Cognitive Requirements for Understanding Conversions
		Cognitive Requirements for Understanding Treatments
		Knowledge and Recognition in Mathematics Understanding and Learning
		Cross-References
		References
	Risk and Decision Making: Fundamental Aspects
		Introduction
		Meanings of Risk
			Probability and a First Approach to Risk
			Definitions of Risk
			Risk and Hazard
			Knight´s Conception of Risk and Uncertainty
			Types of Situations and Analytical Investigation of a Situation under Risk
		Risk and Decisions
			Rational Approach
			Behavioristic Approach
			Mathematical Approach
			Paradigmatic Contexts for Decisions under Uncertainty
			Risk with Small Probabilities
		Conclusions
		Cross-References
		References
	Risk and Decision Making: Psychological and Educational Aspects
		Introduction
		Psychological Aspects of Risk
			Probability and Utility
			Inconsistencies: Psychologically and Otherwise
			Heuristics and Biases: The Approach by Kahneman and Tversky
			Ecological Approach: Gigerenzer and Todd
		Aspects of Risk in Education in Probability and Statistics
			Twin Concepts of Probability and Risk
			Probabilistic Literacy Is Risk Literacy
			Tools to Foster Risk Literacy
		Conclusions
		Cross-References
		References
	Risk Education
		Definition
		Classroom Approaches
		Pedagogy of Risk
		Future Research about the Understanding of Risk
		Cross-References
		References
	Rural and Remote Mathematics Education
		Definition(s)
		Differences in Student Performance
		Opportunities in Rural and Remote Settings
		Cross-References
		References
S
	Scaffolding in Mathematics Education
		Definition
		Characteristics
			Tutoring Learning
		Cross-References
		References
	Secondary-Tertiary Transition in Mathematics Education
		Introduction
		Students´ Difficulties and Their Origins
			Difficulties Linked with Specific Mathematical Subjects
			The Proof Practices of Professional Mathematicians
			Different Institutional Cultures
		Supporting the Transition
			Providing Help to the Students
			Innovative Teaching Practices
			Professional Development and Secondary-Tertiary Transition
		Cross-References
		References
	Semiotics in Mathematics Education
		Definitions and Background
		Semiotic Lenses and Their Uses
		Questions for Research on Semiosis in Learning and Teaching Mathematics
		Cross-References
		References
	Service-Courses in University Mathematics Education
		Definition
		Importance of Mathematics for other Disciplines
		Main Problematic Issue of Service-Courses
		Studies on the Vocational Relevance of Service-Course Mathematics
		Students´ Practices
		Measures Supporting Students Learning in Service-Courses
		Final Remark
		Cross-References
		References
	Shape and Space: Geometry Teaching and Learning
		Definition and Teaching Situation
		Theories Concerning Geometry Teaching and Learning
			Piaget
			van Hiele
		Geometry in Preschool and Elementary School
			Freedom in Selecting Geometrical Context, Content, and Teaching/Learning Paradigms
			Some Comments on Difficulties and Relevant Research
		High-School Geometry or Shapes in Space as Ingredients for Constructing a Theory
		Cross-References
		References
	Single-Sex Mathematics Classrooms
		Introduction
		Characteristics
			Mathematics Classes in Single-Sex Schools
			Mathematics Classes in Sex-Segregated Classes in Coeducational Schools
		Cross-References
		References
	Situated Cognition in Mathematics Education
		Introduction
		Characteristics
			History
			Mathematics Education
			Knowing
			Research Methodologies
			Implication for Teaching
			Critiques of Situated Cognition
		Cross-References
		References
	Socioeconomic Class and Socioeconomic Status in Mathematics Education
		Definition
		Overview
		Statistical Evidence
		Sociopolitical Analyses
		Explanatory and Analytical Frameworks
		Research on Action or Intervention
		Concluding Remarks
		References
	Socioepistemology in Mathematics Education
		Introduction
		Socioepistemology: Origins and Evolution of the Theoretical Approach
		Socioepistemology: Theoretical Foundations, Main Concepts, and Specific Methodologies
			Example of the Problematization of Knowledge in Use: Successive Variation
		Main Research Questions and Results
		Cross-References
		References
	Sociological Approaches in Mathematics Education
		Topic and History
		Issues of Research
		Cross-References
		References
	Sociomathematical Norms in Mathematics Education
		Definition
		Characteristics
			Social Norms
			Sociomathematical Norms
			Origin of the Term
			Growth of the Concept
			Common Issues
		Cross-References
		References
	Stoffdidaktik in Mathematics Education
		Definition
		History
		Characteristics
		Some Examples
		Recent Development
		Cross-References
		References
	Structure of the Observed Learning Outcome (SOLO) Model
		Introduction
		Modes and Multimodal Functioning
		SOLO Levels
		Cycles Within Modes
		Conclusion
		Cross-References
		References
	Students´ Attitude in Mathematics Education
		Definition
		The Origin of the Construct
		The Problematic Relationship Between Attitude and Achievement
		Changes of Perspective in Research into Attitude in Mathematics Education
		The TMA Model: A Definition of Attitude Grounded on Students´ Narratives
		Cross-References
		References
	Subject Matter Knowledge Within ``Mathematical Knowledge for Teaching´´
		Definition
		Characteristics
		Defining, Theorizing, and Measuring SMK
		Other Categorizations and Related Research
		Emerging Directions
		Cross-References
		References
T
	Task„-Based„ Interviews in Mathematics Education
		Definition
			The Clinical Interview
			Task„-Based„ Interview
			Methodology
			Techniques and Resources
			Task„-Based„ Interviews for Assessment
			The Teaching Experiment
			Significance
		Cross-References
		References
	Teacher as Researcher in Mathematics Education
		Cross-References
		References
	Teacher Beliefs, Attitudes, and „Self„-Efficacy„ in Mathematics Education
		References
	Teacher Education Development „Study„-Mathematics„ „(TEDS-„M„)
		Definition
		Institutions
		Curriculum
		The Characteristics of Future Primary and Secondary Teachers Who Are Expected to Teach Mathematics
		The Outcomes of Teacher Education: Mathematics Professional Knowledge and Beliefs
			Mathematics Content and Mathematics Pedagogy Content Knowledge for Teaching
		Beliefs
		Cross-References
		References
			Further Reading
	Teacher Supply and Retention in Mathematics Education
		Definition
		Characteristics
		Cross-References
		References
	Teacher„-Centered„ Teaching in Mathematics Education
		Definition
		Intellectual Heteronomy
		Contrast with „Student„-Centered„ Instruction
		Characteristics
			Role of the Instructor
			Direct Instruction Lesson Design
			Further Areas of Research
		Cross-References
		References
	Teaching Practices at University Level
		Introduction
		Studies of „Lecture„-Oriented„ Instruction
		Studies of Alternatives or Complements to Lecture
		Studies of University Mathematics Teachers’ Pedagogical Perspectives
		Taking Stock and Looking Forward
		Cross-References
		References
	Teaching Practices in Digital Environments
		Definition
		Characteristics
			Teachers’ Changing Uses of Digital Technologies
			Teacher Practice and Technology: Theoretical Perspectives
			Central Considerations for the Future
		Cross-References
		References
	Technology and Curricula in Mathematics Education
		Definition
		Antecedents and New Developments
		Theoretical Perspectives
		The Presence and Uses of Technology in the Mathematics Curriculum
		The Influence of Technology Advances
		Cross-References
		References
	Technology Design in Mathematics Education
		Introduction and Historical Background
		Example of a Microworld
		Virtual Learning Communities in Mathematics Education
		Applications and Task Design with Technology
		New Paths in Research and Practice with Technology Design
			Mobile Learning Design
			Combining Multiple Platforms
			Design of Games for Teaching and Learning Mathematics
			New Trends in the Design and Implementation of Virtual Manipulatives
		Cross-References
		References
	Theories of Learning Mathematics
		Definition„
		Characteristics
			The History and Nature of Theory Development
			Curricular Standards and Mathematics Education Research
			Theories Versus Models
			The Complexity of Models in Mathematics Education
		Cross-References
		References„
	Types of Technology in Mathematics Education
		Terms and Definitions
		Historical Background
		First Computers and Their Use in Education
		E„-learning„: Web 2.0 Tools and Their Use in Mathematics
		M„-Learning„: Anytime, Anywhere with Laptops and Other Handheld Devices
		New Technologies and Literacies for the Twenty„-First„-Century„ Classroom: Robotics, 3D Printing, Virtual Reality, Minecraft, Scratch, and’More
		Cross-References
		References
U
	University Mathematics Education
		Introduction
		Communities, Theories, and Methodologies in RUME
		Courses and Curricula
			From Preuniversity Mathematics Education to UME
			UME Content Areas
			UME Curricula
			From UME to Professions
		Student, Teacher, and Institutional Perspectives
			The Student Perspective
			The Teacher Perspective
			The Institutional Perspective
		Intervention Research and Impact
			Design- and Intervention-Based RUME
			Impact
		Cross-References
		References
	Urban Mathematics Education
		Definition
			Suburban School/Urban School Binary
			Research in Urban Mathematics Education
		Cross-References
		References
V
	Values in Mathematics Education
		Introduction
			Values as Personal Constructs
			Values as Sociocultural Constructs
			Educating Values and Developing Mathematics Education
		Cross-References
		References
	van Hiele Theory, The
		Definition
		Van Hiele Levels
		Features of the Van Hiele Levels
		Van Hiele Phases
		Summary
		Cross-References
		References
	Visualization and Learning in Mathematics Education
		Definitions and Background
			Preference for Visualization in Mathematics
			Theoretical Lenses
			Interaction of Visualization Styles in Learning and Teaching
			Difficulties and Affordances of Use of Visualization in Mathematics
			Questions for Research on Visualization and Learning in Mathematics
		Cross-References
		References
W
	Wait Time
		Definition
		Introduction
			Changes in Student Behavior
			Changes in Teacher Behavior
			Changing Practice
		Conclusion
		Cross-References
		References
	Word Problems in Mathematics Education
		Definition and Function of Word Problems
		Characteristics
			Research Perspectives on Word Problem Solving
			Phases and Components of Competent Word Problem Solving
			Solving Word Problems Versus Problems in the Real World
			Teaching Word Problem Solving
		References
Z
	Zone of Proximal Development in Mathematics Education
		Definition
		Zone of Proximal Development: An Example
		Recent Critical Reworking of the Notion
		Summary
		Cross-References
		References