دانلود کتاب Encyclopedia of Mathematics Education

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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