Mathematics Education in the Early Years: Results from the POEM4 Conference, 2018

Mathematics Education in the Early Years
	Introduction
	Children’s Mathematical Reasoning
	Early Years Mathematics Teaching
	Parents’ Role in Children’s Mathematical Development
	Contents
	About the Editors
	Chapter 1: Early Years Mathematics: Semiotic and Cultural Mediation
		1.1 Introduction
			1.1.1 What Does Early Years Mathematics Mean?
			1.1.2 My Personal Trajectory
		1.2 Some Outcomes of the Preschool Program
			1.2.1 The Giant Slavonic Abacus
				1.2.1.1 Task 1: The First Impact (The Narrator’s Voice)
				1.2.1.2 Task 2: The Structure of the Artifact (The Constructor’s Voice)
				1.2.1.3 Task 3: The Use of the Artifact (The User’s Voice)
				1.2.1.4 Task 4: The Justification for Use (The Theoretician’s Voice)
				1.2.1.5 Task 5: New Problems (The Problem Poser and Solver’s Voice)
			1.2.2 The Time Tube
			1.2.3 Semiotic Mediation at the Preschool Level
		1.3 The Shift to Toddler Centers (and to Prelinguistic Kids)
			1.3.1 From Semiotic Mediation to Cultural Mediation
			1.3.2 The Newborn Experience in a Cultural Context
			1.3.3 Vygotsky’s Hints to Non-Verbal Mediation: The Inner Visual Field
			1.3.4 Vygotsky’s Analysis of the Pointing Gesture
			1.3.5 A Replica of a Classical Experiment: Learning to Drink from a Cup
			1.3.6 The Effects of Social Relationships
		1.4 Concluding Remarks
		References
	Chapter 2: Young Children’s Early Mathematical Competencies: The Role of Mathematical Focusing Tendencies
		2.1 Introduction
		2.2 SFON
		2.3 SFONS
		2.4 SFOR
		2.5 SFOP
		2.6 Conclusion and Discussion
			2.6.1 Conceptual Issues
			2.6.2 Measurement Issues
			2.6.3 Developmental Issues
			2.6.4 Promotion of Mathematical Focusing Tendencies
		References
	Chapter 3: Play and the Production of Subjectivities in Preschool
		3.1 Introduction
		3.2 The Production of Individuals In and Through Play
		3.3 Semiotic Systems of Cultural Signification
		3.4 Being, Becoming, and Subjectivity
		3.5 Playing a Mathematical Game
		3.6 The Inscription of the Children in the Social World
			3.6.1 The Role of Rules
			3.6.2 The Role of the Mathematical Content
			3.6.3 The Teacher as the Embodiment of an Ideal Form
		3.7 Synthesis and Concluding Remarks
		References
	Part I: Children’s Mathematical Reasoning
	Chapter 4: Copying and Comparing Repeating Patterns: Children’s Strategies and Descriptions
		4.1 Introduction
		4.2 Duplicating, Copying, and Comparing Patterns
		4.3 Methodology
			4.3.1 Data Analysis
		4.4 Findings
			4.4.1 Copying and Comparing AB-Structured Necklaces
			4.4.2 Copying and Comparing AAB-Structured Necklaces
			4.4.3 Comparing the AB and AAB Patterns
		4.5 Discussion
		References
	Chapter 5: Patterning as a Mathematical Activity: An Analysis of Young Children’s Strategies When Working with Repeating Patterns
		5.1 Introduction
		5.2 Repeating Patterns and Mathematics
		5.3 Patterning Strategies
		5.4 Method
			5.4.1 Setting and Participants
			5.4.2 Tasks and Materials
			5.4.3 Data Collection and Analysis
		5.5 Results
		5.6 Discussion
		References
	Chapter 6: Preschoolers’ Ways of Using Fingers in Numerical Reasoning
		6.1 Background and Aim
		6.2 Research Review
		6.3 The Study
		6.4 Analysis
		6.5 Results
			6.5.1 Fingers as an Image of Numbers
			6.5.2 Fingers to Create Numbers of Single Units
			6.5.3 Fingers to Visualize the Structure of Numbers
		6.6 Conclusions
		References
	Chapter 7: Young Children’s Cross-Domain Mapping of Numerosity in Path Navigation
		7.1 Introduction and Background
		7.2 Theoretical Framework
		7.3 Methodology
			7.3.1 The Intervention and Case Selection
			7.3.2 Procedure of the Navigation Task
			7.3.3 Operationalisation and Data Analysis
		7.4 Results
			7.4.1 General Results
			7.4.2 Rich Description of Three Groups of Task Solutions
		7.5 Discussion
		7.6 Summary and Concluding Remarks
		References
	Chapter 8: Mathematical Problem-Solving Visualised in Outdoor Activities
		8.1 Introduction
		8.2 Theoretical Frameworks About Mathematical Problem-Solving
		8.3 Methodology
		8.4 Data
			8.4.1 A Child-Initiated, Problem-Solving Activity
		8.5 Analysis and Discussion
			8.5.1 An Analysis and Discussion of the Measurement Understanding
			8.5.2 An Analysis and Discussion Using Copley’s Characteristics of an Effective Problem Solver
			8.5.3 An Analysis and Discussion Using Polya’s Problem-Solving Stages
			8.5.4 An Analysis and Discussion Using the Didaktic Space
		8.6 Implication for the Kindergarten Teacher’s Role
		8.7 Conclusion
		References
	Chapter 9: Making Choices and Explaining Them: An Experiment with Strategy Games in Kindergarten
		9.1 Introduction
		9.2 Theoretical Framework
		9.3 The Teaching Experiment
			9.3.1 The Thirteen Buttons Game
			9.3.2 The Didactical Choices
		9.4 Data Analysis
			9.4.1 Phase 1: Introduction to the “Thirteen Buttons Game”
			9.4.2 Phase 2. The Discovery of the “Magic Buttons”
			9.4.3 Phase 3: The Dinosaur Goes on Holiday
		9.5 Conclusion
		References
	Chapter 10: Measuring Rice in Early-Childhood Education Activities: A Bridge Across Discrete and Continuous Magnitudes
		10.1 Introduction
		10.2 Theoretical Background of the Study
		10.3 Methodology
		10.4 Analyses
			10.4.1 Strategies Before Introduction of the Artifacts
				10.4.1.1 Evaluation of Numerosity
				10.4.1.2 Evaluation of Surface
				10.4.1.3 Evaluation of Volume
			10.4.2 Strategies with Artifacts
		10.5 Conclusions and Future Perspectives
		References
	Chapter 11: The Semiotic Resources Children Use in Their Explanations of Hypothetical Situations
		11.1 Introduction
		11.2 Semiotic Resources and Explanations
		11.3 Methodology
		11.4 Results
			11.4.1 Introducing an Imaginary Layout
			11.4.2 Adapting Gestures
			11.4.3 Clarifying What Is Discussed
			11.4.4 Alternative Support for Deductive Explanations
			11.4.5 Explicit Explanation
		11.5 Discussion
		11.6 Conclusion
		References
	Chapter 12: Drawings, Gestures and Discourses: A Case Study with Kindergarten Students Discovering Lego Bricks
		12.1 Introduction
		12.2 Theoretical Framework
		12.3 Research Methodology
		12.4 Overview of the Teaching Sequence
		12.5 Analysis and Preliminary Results
			12.5.1 Different Points of View of Matilde, Laura and Elio
				12.5.1.1 Episode 1
				12.5.1.2 Episode 2
				12.5.1.3 Episode 3
		12.6 Conclusions
		References
	Chapter 13: Perceiving and Using Structures When Determining the Cardinality of Sets: A Child’s Learning Story
		13.1 Introduction
		13.2 Role of Structures for Numerical and Arithmetical Development
		13.3 Research Question
		13.4 Design of the Study
			13.4.1 Tasks
			13.4.2 Implementation
			13.4.3 Aspects of Data Analysis
			13.4.4 Data Analysis by the Use of Eye-Tracking Technology
			13.4.5 Example from the Data Analysis
		13.5 Results of the Learning Story of “Luca”
		13.6 Summary and Discussion
		References
	Chapter 14: “A Triangle Is Like a Tent”: Children’s Conception of Geometric Shapes
		14.1 Introduction
		14.2 Theoretical Background
			14.2.1 Explaining Shapes
			14.2.2 Drawing Shapes
			14.2.3 Identifying Shapes
			14.2.4 Coherence
		14.3 Research Question
		14.4 Design of the Study
			14.4.1 Tasks
			14.4.2 Analysis
		14.5 Results
			14.5.1 Explaining Shapes
			14.5.2 Drawing Triangles
			14.5.3 Coherence Between the Children’s Explanations and Drawings
			14.5.4 Identifying Circles, Squares, and Triangles
			14.5.5 Coherence Between Children’s Explanations and Identifications
		14.6 Summary and Discussion
		14.7 Conclusion
		References
	Chapter 15: Framing Mathematics Teaching with Narratives: The Ambiguity of Goldilocks
		15.1 Introduction
		15.2 Narratives as Pedagogical Tools
		15.3 Methodology
		15.4 Two Empirical Examples
			15.4.1 Example 1
			15.4.2 Example 2
		15.5 Content Analysis of the Goldilocks Story
			15.5.1 What Mathematical Challenges Does the Story Impose?
		15.6 Discussion
		Appendix: The Story of Goldilocks and the Three Bears
		References
	Chapter 16: Kindergarten Teacher’s Knowledge to Support a Mathematical Discussion with Pupils on Measurement Strategies and Procedures
		16.1 Introduction
		16.2 Theoretical Framework
		16.3 Context and Method
		16.4 Analysis and Discussion
			16.4.1 Episode 1: Ordering by Length Vs. Height
			16.4.2 Episode 2: The Magnitude of Length
			16.4.3 Final Comments
		References
	Chapter 17: The Materialisation of Children’s Mathematical Thinking Through Organisation of Turn-Taking in Small Group Interactions in Kindergarten
		17.1 Introduction
		17.2 Theoretical Framework
		17.3 Methodology
		17.4 Results
			17.4.1 Segment 1, from Kindergarten 1 (K1)
			17.4.2 Segment 2.1, from Kindergarten 2 (K2)
			17.4.3 Segment 2.2, from Kindergarten 2 (K2)
		17.5 Discussion
			17.5.1 The Characterisation of Children’s Turn-Taking: Similarities and Differences in the Three Segments
			17.5.2 The Role of Children’s Organisation of Turn-Taking for the Materialisation of Children’s Mathematical Thinking in the Joint Activity
		References
	Chapter 18: Mathematical Activity in Early Childhood and the Role of Generalization
		18.1 Introduction
		18.2 Analysis of Mathematical Activity in Early Years
		18.3 Development of Mathematical Ideas and Generalization
		18.4 A Teaching Approach Encouraging Generalization
		18.5 Closing
		References
	Part II: Early Years Mathematics Teaching
	Chapter 19: Situational Perception in Mathematics (SPiM)—Results of a Cross-Country Study in Austria and Norway
		19.1 Introduction
		19.2 Situation-Specific Skills: An Essential Part of Kindergarten Teachers’ Competence
		19.3 Kindergarten Teachers’ Situational Perception of Mathematics
		19.4 Research Questions
		19.5 Methods
			19.5.1 Sample
			19.5.2 Data Collection
			19.5.3 Data Analysis and Analytical Process
		19.6 Identified Categories from the Analysed Data
		19.7 Discussion
		References
	Chapter 20: How to Support Kindergarten Children in Spontaneous Measuring Activities
		20.1 Background
			20.1.1 Measuring
			20.1.2 The Knowledge Quartet
			20.1.3 The Purpose of Education: Biesta
		20.2 Methodology
			20.2.1 Limitations of the Study
		20.3 Analysis
			20.3.1 The Quantitative Data
			20.3.2 The Qualitative Data
			20.3.3 Relative Heights—Dorothea’s Observation
			20.3.4 Who Is Tallest?—Turid’s Observation
			20.3.5 Selling Water—Astrid’s Observation
			20.3.6 An Empty Cardboard Box—Lina’s Observation
			20.3.7 Santa Hats—Lars’s Observation
		20.4 Discussion
			20.4.1 Reflecting on Practise with the Quartet Knowledge
			20.4.2 The Active-Passive Dilemma
		20.5 Conclusion
		References
	Chapter 21: Kindergarten Teachers’ Stories About Young Children’s Problem Posing and Problem Solving
		21.1 Introduction
		21.2 Methodology
		21.3 Results
			21.3.1 Goldilocks and the Three Bears
			21.3.2 Train Crash
			21.3.3 Counting Orange Boats
			21.3.4 Packing the Police Car
		21.4 Discussion
		21.5 Conclusion
		References
	Chapter 22: ‘You Are Already Bigger Than the Giraffe!’—The Use of Adjectives in Measurement Activities in Kindergarten
		22.1 Introduction
		22.2 Measuring Length and Weight
		22.3 Learning Opportunities: Adjectives
		22.4 Research Design and Analysing Methods
		22.5 Analysis and Interpretation of Empirical Data
		22.6 Empirical Examples for Learning Opportunities
		22.7 Conclusion
		References
	Chapter 23: The Complexity of Teaching Mathematics in Kindergarten: A Case Study and Conceptualization
		23.1 Introduction
		23.2 Theoretical Background
		23.3 Design of the Study
		23.4 Tasks of Teaching Mathematics in Kindergarten
			23.4.1 Initiating Mathematical Discussions
			23.4.2 Responding to Unexpected Questions
			23.4.3 Dealing with Wrong Answers
			23.4.4 Using Representations
			23.4.5 Positioning Children as Valuable Contributors
			23.4.6 Asking Productive Mathematical Questions
		23.5 Discussion
		23.6 Conclusion
		References
	Part III: Parents’ Role in Children’s Mathematical Development
	Chapter 24: Parents’ Valuing of Mathematics for Young Children
		24.1 Introduction
		24.2 Theoretical Perspective
			24.2.1 Narrative Approach to the Learning of Mathematics
		24.3 Methodology
			24.3.1 Analysis of the Data
		24.4 Results
			24.4.1 Yahtzee and Ludo, the Value of Learning Numbers
			24.4.2 Everyday Activities, the Value of Learning Counting Skills
			24.4.3 Length, Volume, Time, and the Value of Learning Measurement Skills
			24.4.4 Money and the Valuing of Equivalence
		24.5 Discussion
		24.6 Conclusion
		References
	Chapter 25: “Pedagogical” Mathematics During Play at Home: An Exploratory Study
		25.1 Introduction
		25.2 Theoretical Framework
		25.3 Background Literature
		25.4 Method
			25.4.1 Data Sources
		25.5 Results
			25.5.1 Establishing the Mathematics Goal
			25.5.2 Sustaining the Mathematics Goal
			25.5.3 Addressing Each Child’s Role in Play-Based Mathematics
			25.5.4 Beyond Pedagogical Moves: Mother-as-Teacher and Funds of Knowledge
		25.6 Conclusion
		Appendix
		References
	Index