Exploring Mathematical Modeling with Young Learners (Early Mathematics Learning and Development)

✍ Scribed by Jennifer M. Suh (editor), Megan H. Wickstrom (editor), Lyn D. English (editor)

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 440
• Category: Library

✦ Synopsis

This book conceptualizes the nature of mathematical modeling in the early grades from both teaching and learning perspectives. Mathematical modeling provides a unique opportunity to engage elementary students in the creative process of mathematizing their world.
A diverse community of internationally known researchers and practitioners share studies that advance the field with respect to the following themes:

• The Nature of Mathematical Modeling in the Early Grades

• Content Knowledge and Pedagogy for Mathematical Modeling

• Student Experiences as Modelers

• Teacher Education and Professional Development in Modeling

Experts in the field provide commentaries that extend and connect ideas presented across chapters. This book is an invaluable resource in illustrating what all young children can achieve with mathematical modeling and how we can support teachers and families in this important work.

✦ Table of Contents

Exploring Mathematical
Modeling with Young
Learners
Acknowledgments
Contents
Contributors
Part I: The Nature of Mathematical Modeling in the Early Grades
Chapter 1: Mathematical and Interdisciplinary Modeling in Optimizing Young Children’s Learning
1.1 Early Modeling
1.2 Defining Modeling for the Elementary Grades
1.3 Model-Eliciting Activities
1.4 Data Modeling
1.5 Modeling Within Cultural and Community Contexts
1.6 STEM-Based Modeling
1.7 Examples of Modeling Problems
1.7.1 Model-Eliciting Activity
1.7.1.1 The Sweets Pantry (Part A)
1.7.2 STEM-Based Modeling Problem
1.8 Concluding Points
References
Chapter 2: Synthesizing Research of Mathematical Modeling in Early Grades
2.1 Introduction
2.2 Purpose of the Chapter
2.3 Selection Criteria
2.4 Results
2.4.1 Summary of Studies Published in Research Journals
2.4.2 Summary of Studies Published in TCM
2.5 Discussion and Future Direction
References
Chapter 3: Mathematical Modeling Thinking: A Construct for Developing Mathematical Modeling Proficiency
3.1 Introduction
3.2 Unpacking the Meaning of Proficiency in Mathematical Modeling
3.2.1 Representation and Communication as Central to Mathematical Modeling
3.3 Conceptualization of Mathematical Modeling Thinking
3.3.1 Specialized Ways of Thinking Required for Mathematical Modeling Proficiency
3.3.2 Characterization of Mathematical Modeling Thinking
3.4 Practices that Target Mathematical Modeling Thinking
3.4.1 Recognize Assumptions
3.4.2 Approximate and Estimate to Reason Quantitatively
3.4.3 Prioritize Factors that Affect the Solution as a Means of Simplifying the Problem
3.4.4 Use Multiple Representations to Express Mathematical Ideas
3.4.5 Reflect on a Solution, Meaning, and Reasonableness Within the Original Context
3.4.6 Reconsider, Revise, and Refine the Solution
3.5 Situating Our Work with Teachers as Modelers
3.5.1 Snapshots of Mathematical Modeling Thinking
3.6 Discussion
3.6.1 Implications for Mathematics Education: Developing Mathematical Modeling Thinking
3.7 Conclusion
References
Chapter 4: Data Modelling and Informal Inferential Reasoning: Instances of Early Mathematical Modelling
4.1 Introduction
4.1.1 Theoretical Perspective
4.1.1.1 Definition of Informal Inferential Reasoning (IIR)
4.1.1.2 Reasoning About Aggregates
4.1.1.3 Data Modelling Environments and Informal Inferential Reasoning (IIR)
4.2 Research Approach
4.2.1 Participants
4.2.2 Method
4.2.2.1 Framing the Lesson
4.2.2.2 Task Design
4.2.2.3 Presenting the Tasks
4.2.2.4 Role of the Pre-service Teacher
4.2.2.5 Analysis of the Data
4.3 Findings
4.3.1 Principle 1: Generalisations, Including Predictions and Conclusions, that Extend Beyond Describing the Given Data
4.3.1.1 Using the Sequence of Counting Numbers as a Mechanism to Generate an Estimation
4.3.1.2 Moving Beyond the Sequence of Counting Numbers to Make Generalisations
4.3.2 Principle 2: The Use of Data as Evidence for Those Generalisations
4.3.3 Principle 3: Employment of Probabilistic Language in Describing the Generalisation, Including Informal Reference to Levels of Certainty About the Conclusions Drawn
4.3.4 Reasoning About Aggregates
4.3.4.1 A Case Value Lens Was Limited to Specific Scenarios
4.3.4.2 Awareness of Trends in the Data Provides Evidence for the Presence of a Pre-aggregate Lens
4.4 Conclusions
Appendix
References
Chapter 5: Development in Mathematical Modeling
5.1 Three Traditions of Modeling-Related Research
5.2 Contrasts Between the Traditions
5.3 Grounds for Complementarity
5.4 Considering Development
5.5 Children’s Capabilities as Modelers
5.6 A Modeling Competencies Lens and Its Entailments
5.7 Perspectives on Modeling Competencies from Chapters in This Section
5.8 Directions for Future Research
5.9 Conclusion
References
Part II: Identifying the Knowledge of Content and Pedagogy Needed for Mathematical Modeling in the Elementary Grades
Chapter 6: Elementary Teachers’ Enactment of the Core Practices in Problem Formulation through Situational Contexts in Mathematical Modeling
6.1 Introduction
6.2 Identifying Practices that Support Problem Formulation in Mathematical Modeling
6.3 Research Questions
6.4 Context for Our Study
6.5 Data Sources and Analysis
6.6 Results
6.6.1 Core Practice of Posing a Modeling Problem While Engaging Students in the Process
6.7 Collective Activities that Supported Students as They Mathematized, Made Assumptions, and Defined Variables Related to the Problem Posed
6.8 Vignettes to Illustrate the Interplay Between Teacher and Students in Problem Formulation
6.9 Discussion
References
Chapter 7: Teaching Practices to Support Early Mathematical Modeling
7.1 The Modeling Cycle
7.2 Teaching Practices and Mathematical Modeling
7.2.1 Developing Competency in Modeling
7.2.2 Developing Teachers of Modeling
7.3 The Current Study
7.3.1 Participants and Setting
7.3.2 Data and Analysis
7.4 Results
7.4.1 Working with Student Ideas
7.4.2 Interacting with the Real-World Context
7.4.3 Monitoring and Supporting Mathematics Learning
7.4.4 Linking Student Ideas and the Mathematical Solution
7.5 Discussion and Conclusion
References
Chapter 8: Teachers’ Use of Students’ Mathematical Ideas in Mathematical Modeling
8.1 Introduction
8.2 Literature
8.2.1 Modeling Cycle and Mathematics’ Role in Modeling
8.2.2 Teachers’ Interactions with Student’s Thinking
8.2.3 Teaching Mathematical Modeling
8.2.4 Research Question
8.2.5 Theoretical Perspectives
8.3 Methodology
8.3.1 Setting and Participants
8.3.2 Data Collection
8.3.3 Data Analysis
8.4 Results
8.4.1 Nature of the Modeling Tasks
8.4.1.1 Rebecca’s Task
8.4.1.2 Amy’s Task
8.4.2 Organizing Students to Introduce and Use Mathematical Ideas
8.4.3 Teachers’ Interaction with Student’s Ideas
8.4.3.1 Monitor
8.4.3.2 Moving from Monitor to Regroup
8.4.3.3 Regrouping
Justify and Clarify
Using Students Ideas to Guide and Teach
Connecting Ideas
Not Pursuing all Mathematical Ideas
8.4.4 Modeling Instructional Practices
8.4.4.1 Teaching to Elicit Student’s Mathematical Ideas
8.4.4.2 Teaching to Illuminate Mathematical Processes and Content
8.4.4.3 Coordinating Student’s Mathematical Ideas
8.4.4.4 Maintaining High Cognitive Demand Tasks in the Midst of Emergent Mathematics
8.4.4.5 Relationship Between Categories
8.5 Discussion
References
Chapter 9: Teaching and Facilitating Mathematical Modeling: Teaching, Teaching Practices, and Innovation
9.1 Teaching and Facilitating Mathematical Modeling
9.2 Contributions of Chapters
9.3 Teaching Practices in a Broader Field
9.4 Effective, Equitable, and High-Leverage Teaching Practice
9.5 Innovations in Teaching
9.6 Framing Teaching Practices for Mathematical Modeling
9.7 To Be Continued
9.8 Concluding Thoughts
References
Part III: Mathematical Modeling and Student Experiences as Modelers
Chapter 10: Mathematical Modeling: Analyzing Elementary Students’ Perceptions of What It Means to Know and Do Mathematics
10.1 Benefits of Mathematical Modeling
10.2 The Child’s Voice
10.3 Theoretical Framework: Mathematical Identity and Construction of the Discipline
10.4 Methodology
10.4.1 Setting and Participants
10.4.2 The Modeling Task
10.4.3 Data Collection
10.4.4 Data Analysis
10.5 Results
10.6 Interview 1
10.6.1 Students Descriptions of Mathematics
10.6.2 Doing Mathematics
10.6.3 Students’ Beliefs About Their Mathematical Ability
10.6.4 Students’ Beliefs About Learning New Mathematics
10.6.5 Mathematics in Relation to Other Disciplines
10.7 Interview 2
10.7.1 Students’ Descriptions of Modeling
10.7.2 Challenges and Affordances of Modeling in Relation to Mathematics
10.7.3 Mathematical Modeling and Student Identity
10.8 Discussion
10.9 Conclusion
References
Chapter 11: Upcycling Plastic Bags to Make Jump Ropes: Elementary Students Leverage Experiences and Knowledge as They Engage in a Relevant, Community-Oriented Mathematical Modeling Task
11.1 Mathematical Modeling in the Elementary Grades
11.2 Mathematical Modeling with Cultural and Community Contexts
11.3 The Mathematical Modeling Process
11.3.1 Phase 1: Make Sense of a Situation or Problem
11.3.2 Phase 2: Construct a Model
11.3.3 Phase 3: Operate on Model
11.3.4 Phase 4: Interpret/Analyze Solutions and Refine Model
11.3.5 Phase 5: Validate and Generalize Model
11.3.6 Movement Across the Phases
11.4 Context of the Study
11.4.1 M2C3 Teacher Professional Development
11.5 Methods
11.5.1 Case Study Design
11.5.2 Participants
11.5.3 Data Sources
11.5.4 Data Analysis and Analytical Framework
11.6 Findings
11.6.1 Phase 1: Making Sense of the Situation or Problem
11.6.1.1 Students Connecting to Family Practices with Plastic Shopping Bags
11.6.1.2 Students Connecting to Experiences with Play
11.6.2 Phase 2: Constructing a Model
11.6.2.1 Reasoning About the Length of Jump Ropes
11.6.2.2 Reasoning About What Constitutes a Set of Jump Ropes (How Many and What Kinds)
11.6.3 Phase 3: Operating on a Model
11.6.4 Phase 4: Interpret/Analyze Solutions and Refine Model
11.6.5 Phase 5: Generalizing the Model to Other Contexts
11.7 Discussion
11.8 Implications
11.9 Conclusion
References
Chapter 12: A Window into Mathematical Modeling in Kindergarten
12.1 Introduction
12.2 The IMMERSION Project
12.2.1 Our Perspective on Modeling
12.2.2 Teachers Professional Development Cycle
12.3 Theoretical Framing of Mathematical Modeling in the Classroom
12.4 The Case Study
12.4.1 Methodology
12.4.2 A Perspective of Enacted Classroom Lessons
12.4.3 Classroom Context
12.4.4 Mathematical Modeling Lesson Sequence
12.4.5 Vignette 1: Mouse Count
12.4.6 Our Discussion of Vignette 1
12.4.7 Vignette 2: Cupcakes for a Party
12.4.8 Our Discussion of Vignette 2
12.4.9 Vignette 3: Two Ways to Count to Ten: A Liberian Folktale
12.4.10 Our Discussion of Vignette 3
12.5 Discussion: The Work of Teaching Mathematical Modeling in Kindergarten
12.6 Revisiting Mathematical Modeling Frameworks
12.7 Final Remarks
References
Chapter 13: The Genesis of Modeling in Kindergarten
13.1 Conceptual Framework for Mathematical Modeling
13.2 Research on Teachers’ Knowledge and Perceptions of Modeling
13.2.1 Teachers’ Knowledge of Mathematical Models and Modeling
13.2.2 Teachers’ Beliefs and Perceptions About Modeling
13.3 Present Study
13.4 Method
13.4.1 Participants
13.4.2 The Ideal Classroom Activity
13.4.3 Researcher and Teacher Roles
13.4.4 Data Sources
13.5 Findings
13.5.1 Teachers’ Conceptions of Modeling
13.5.2 Mathematical Knowledge for Teaching
13.5.3 Constraints to Incorporating Modeling
13.6 Discussion
References
Chapter 14: Mathematical Modeling with Young Learners: A Commentary
14.1 Why Modeling?
14.2 What Kinds of Modeling Tasks Were Used in These Elementary Classrooms?
14.3 What Were the Consequences for Students of Engaging in These Tasks?
14.4 What Were the Consequences for Teachers of Engaging in These Tasks?
14.5 What Can We Learn from These Studies About Supporting Teachers?
14.6 What Implications Do These Studies Suggest for Future Work?
References
Part IV: Interdisciplinary and Community-Based Modeling
Chapter 15: Convergent Nature of Modeling Principles Across the STEM Fields: A Case Study of Preservice Teacher Engagement
15.1 Introduction: Issues Facing Elementary Teachers in STEM
15.2 Literature Review
15.3 Conceptual Framework
15.3.1 Goals for the Chapter
15.4 Methods
15.5 Findings
15.5.1 Analysis of STEM Modeling Instruction Through Lesson Vignettes
15.5.2 Use of Mathematical Modeling to Help Make Decisions, Describe, Predict, and Explain Phenomena About Science, Engineering, and Technology
15.6 Discussion
15.6.1 Converging STEM Practices as a Means to Articulate “STEM Modeling”
15.6.2 The Value of Modeling for Elementary Teaching Practice
15.7 Conclusions
References
Chapter 16: Supporting Students’ Critical Literacy: Mathematical Modeling and Economic Decisions
16.1 Introduction
16.2 Why Mathematical Modeling
16.3 Economics and Mathematical Modeling
16.4 Theoretical Framework
16.5 Our Context
16.5.1 Identifying the Problem
16.5.2 Making Assumptions and Identifying Variables
16.5.3 Doing Math
16.5.4 Implementing the Model
16.5.5 Implications for Teacher Education
16.6 Effects of Opportunity Costs: Mathematical Modeling with Third Graders
16.6.1 Identifying the Problem
16.6.2 Making Assumptions and Identifying Variables
16.6.3 Doing Math
16.6.4 Implementing the Model
16.7 Discussion
References
Chapter 17: Culturally Relevant Pedagogy and Mathematical Modeling in an Elementary Education Geometry Course
17.1 Overview of the Literature
17.1.1 Culturally Relevant Pedagogy
17.1.2 Culturally Relevant Pedagogy and Mathematical Modeling
17.1.3 Using Community Knowledge to Build Social Justice Tasks
17.1.4 Mathematical Modeling and Critical Consciousness
17.1.5 Conceptual Framework of Mathematics Modeling for Social Justice
17.2 Goals of the Chapter
17.3 Methods
17.4 Findings
17.4.1 Course Goals and Outcomes
17.4.2 Assignments
17.4.3 Instructor Notes
17.4.3.1 Task Design
17.4.3.2 CRCD Analysis
17.5 Discussion
17.5.1 Tensions Between Content, Pedagogy, and Critical Pedagogy
17.5.2 Interdisciplinary Nature of Modeling and Grade-Level Appropriateness
17.5.2.1 Setting Constraints and Open-Ended Problems
17.5.3 Tension in Authenticity in Planning a Curriculum That Is Static
References
Chapter 18: Learning from Mothers as They Engage in Mathematical Modeling
18.1 Review of Literature on Mathematical Modeling and Funds of Knowledge
18.2 Context and Methods
18.3 The Paper Flowers Task
18.3.1 The Work of Lidia’s Group
18.3.2 The Work of Sandra’s Group
18.3.3 Elements of Mathematical Modeling
18.4 Co-developed Modeling Tasks
18.4.1 The Case of the Cupcake Task
18.4.2 Commentary on the Cupcake Task
18.5 Some Closing Considerations
References
Chapter 19: Insights Regarding the Professional Development of Teachers of Young Learners of Mathematical Modeling
19.1 Articulating Values in Order to Make Decisions
19.2 Emphasizing Agency for Modelers
19.3 Engaging in the Practice of Modeling in Anticipation of Teaching Others to Model
19.4 Learning to Develop Modeling Tasks
19.5 Does Professional Development for Pre-service Audiences Differ from That for In-service Audiences?
19.6 Conclusion
References