Teaching and Learning Mathematics Online

Description

Online education has become a major component of higher education worldwide. In mathematics and statistics courses, there exists a number of challenges that are unique to the teaching and learning of mathematics and statistics in an online environment. These challenges are deeply connected to already existing difficulties related to math anxiety, conceptual understanding of mathematical ideas, communicating mathematically, and the appropriate use of technology. Teaching and Learning Mathematics Online bridges these issues by presenting meaningful and practical solutions for teaching mathematics and statistics online. It focuses on the problems observed by mathematics instructors currently working in the field who strive to hone their craft and share best practices with our professional community. The book provides a set of standard practices, improving the quality of online teaching and the learning of mathematics. Instructors will benefit from learning new techniques and approaches to delivering content. Features Based on the experiences of working educators in the field Assimilates the latest technology developments for interactive distance education Focuses on mathematical education for developing early mathematics courses Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface Acknowledgments Editors Contributors Part 1 Course Design Chapter 1 Teaching Cross-Listed Mathematics Courses Online 1.1 Introduction 1.2 The Courses 1.2.1 Groups and Geometry 1.2.2 Advanced Linear Algebra 1.2.3 Mathematical Modeling 1.3 Adapting the Courses to Online Delivery 1.3.1 Groups and Geometry 1.3.2 Advanced Linear Algebra 1.3.3 Mathematical Modeling 1.4 Technical Issues 1.4.1 Creating the Lectures 1.4.2 Online Interactions with Students 1.4.2.1 Asynchronous Tools 1.4.2.2 Synchronous Tools: Office Hours 1.4.3 Assessment 1.4.4 Ensuring Rigor 1.5 Conclusions Bibliography Chapter 2 What Do We Know about Student Learning from Online Mathematics Homework? 2.1 Online Homework and Student Achievement 2.1.1 Exam Scores 2.1.2 Course Grades 2.1.3 Homework Completion Rates and Homework Grades 2.1.4 Homework Grades and Exam Grades 2.1.5 Findings about Online Homework for Various Sub-Populations 2.2 Students’ Perceptions of Online Homework 2.2.1 Online Homework and Students’ Beliefs about Learning Mathematics and the Nature of Mathematics 2.2.2 What Students Like about Online Homework 2.2.3 What Students Dislike about Online Homework 2.3 How Do Students Engage with Online Homework? 2.3.1 Multiple Attempts per Problem 2.3.2 Resources 2.4 The Importance of Homework Content 2.5 Research-Based, Effective Practices for Utilizing Online Homework 2.5.1 Instructors’ Attitudes Influence Students’ Attitudes 2.5.2 Online Homework Has the Same Effects on Student Achievement, or Increases Achievement Slightly, as Compared to Paper-and-Pencil Homework 2.5.3 Students Complete Online Homework at Higher Rates Than They Complete Paper-and-Pencil Homework 2.5.4 Students Like Having Both Online and Paper-and-Pencil Homework, and Aspects of Both May Be Important for Student Success 2.5.5 If Employing Both Online and Paper-and-Pencil Homework, Be Wary of the Total Workload 2.5.6 Be Intentional about Due Dates, Particularly If Employing Both Online and Written Homework Components 2.5.7 Students Become Rapidly Frustrated with Online Homework Platforms That Make It Difficult to Format Answers, or Are Inconsistent in Formatting 2.5.8 Students Do More Homework When It Counts toward Their Course Grade 2.5.9 Offering Unlimited Attempts per Problem Is Linked to Student Persistence in Repeating Assignments/Problems to Obtain High Scores, and It May Also Increase Students’ Confidence in Their Ability to Do Math 2.5.10 Offering Unlimited/Multiple Attempts Is Linked Both to Students Employing Those Attempts to Make Sense of the Mathematics, and in Employing Those Attempts to Guess Answers 2.5.11 When Students Can Submit Each Part of a Question Individually, They Often Employ Feedback on One Part to Guide Their Work on the Other Parts of the Question 2.5.12 Students Like “Help” Features. We Do Not Know Much about How They Employ These Tools 2.5.13 Content Is Incredibly, Incredibly Important 2.6 Closing Remarks References Chapter 3 Designing Mathematics Hybrid Classrooms in High School: The Case of Valeria 3.1 Introduction 3.2 Conceptual Framework 3.3 Methodology 3.3.1 The Context of the Research 3.3.2 Participants: The Teachers and Their Students 3.3.3 Data Collection 3.4 The Case of Valeria 3.4.1 The Teaching Context 3.4.2 Lesson Image 3.4.2.1 Class A 3.4.2.2 Class B 3.4.3 In-the-Moment Decision Making 3.5 Discussion References Chapter 4 Designing Mathematics Hybrid Classrooms in High School: The Cases of Nicoletta and Lorenza 4.1 Introduction 4.2 The Case of Nicoletta 4.2.1 The Teaching Context 4.2.2 Lesson Image 4.2.3 In-the-Moment Decision Making 4.4 The Case of Lorenza 4.4.1 The Teaching Context 4.4.2 Lesson Image 4.4.3 In-the-Moment Decision Making 4.5 Discussion 4.6 Conclusions References Chapter 5 Upper-Level Mathematics and Statistics Courses Shared across Campuses 5.1 Introduction 5.2 Case Study: Graduate Real Analysis (Garcia) 5.2.1 Scheduling 5.2.2 Equipment 5.2.3 Recording 5.2.4 File Transfer 5.2.5 Video Delivery 5.2.6 Homework 5.2.7 Examinations 5.2.8 Summary 5.3 Synchronous Versus Asynchronous Instruction (Miller) 5.3.1 Introduction 5.3.2 Asynchronous Content 5.3.3 Synchronous Content 5.3.4 Summary 5.4 Creating and Fostering an Online Learning Community (Hu) 5.4.1 Introduction 5.4.2 Self-Introduction Posts 5.4.3 Project Introduction Video Posts 5.4.4 A Learning Environment for Reading a Research Paper 5.4.5 A Learning Environment for Guest Introductory Videos 5.4.6 Summary 5.5 Future Work Chapter 6 Online Statistics Teaching and Learning 6.1 Introduction 6.2 Description of the Online Courses 6.2.1 An Online Course in Exploratory Data Analysis 6.2.2 A Bayesian Statistics Course for Cross-Campus Share 6.2.3 A Five-Course MOOC Specialization: Statistics with R 6.3 Online Course Design 6.3.1 Presentation of Content 6.3.2 Course Material in Video Form 6.3.3 Learning Objectives 6.3.4 Suggested Readings and Practice 6.4 Use of Technology 6.4.1 R Package 6.4.2 EDA Blog, YouTube Videos 6.4.3 Shiny Activities 6.4.4 Use of Technology in Lectures 6.4.5 Computing Lab 6.5 Assessments and Engagement 6.5.1 Weekly Data Analysis Assignments 6.5.2 Final Project 6.5.3 Homework 6.5.4 Case Studies 6.5.5 Quizzes 6.6 Interaction 6.7 Challenges 6.8 Concluding Comments 6.8.1 Presentation of Content 6.8.2 Interaction 6.8.3 Assessment 6.8.4 Using Software References Chapter 7 Statistics for Engineers 7.1 Introduction 7.2 Project Components 7.3 Common Mistakes 7.4 Performance 7.5 Conclusion Bibliography Part 2 Student Interaction Chapter 8 Encouraging Higher-Order Thinking in Online and Hybrid Mathematics and Statistics Courses 8.1 What Is This Chapter About? 8.2 What Is HOT? 8.3 Why Teach HOT, and Why Not? 8.4 How to Teach HOT? 8.4.1 Analyze 8.4.2 Evaluate 8.4.3 Create/Synthesize 8.4.4 Comfort with Multiplicity and Ambiguity 8.4.5 Self-Awareness 8.4.6 Questioning 8.4.7 Independence 8.4.8 Risk-Taking 8.4.9 Communicating 8.5 Can HOT be taught online? 8.6 Might the O2S Forum Allow Teaching HOT? 8.6.1 Challenges and Responses 8.6.2 Critiques 8.6.3 Dynamics 8.6.4 More Examples 8.7 Does O2S Work? 8.8 How Can O2S Be Used Effectively? 8.9 Is O2S Available? 8.10 What Can We Conclude? References Chapter 9 Tools for Communication and Interaction in Online Mathematics Teaching and Learning 9.1 Tracking Data on Students Found in the Learning Management System (LMS) 9.2 How to Engage Students in an Online Environment 9.3 Communicating Using Online Tools 9.4 Set Times for Communication 9.5 Communicating out of Sync 9.6 Communication through the Course Content 9.7 Using Local Supports 9.7.1 Schools 9.7.2 Parents 9.8 Using Online Tools 9.9 GeoGebra 9.10 A Standardized Method of Delivery 9.11 Conclusion References Chapter 10 Managing Students’ Mathematics Anxiety in the Context of Online Learning Environments 10.1 Anxiety in the Teaching and Learning of Mathematics 10.2 A Genetic Account of Mathematics Anxiety 10.2.1 Emotions and the Role of Cognition in Their Elicitation 10.2.2 Mathematics Anxiety as a Prospect-Based Emotion 10.2.2.1 Goal Structures 10.2.2.2 Identity 10.2.2.3 Intensity of Emotional Experiences 10.2.2.4 Somatic Markers 10.3 Strategies for Managing Students’ Mathematics Anxiety in Online Learning Environments 10.3.1 Assist Students in Establishing Meaningful Learning Goals 10.3.2 Enhance Students’ Appraisal of Their Psychological Resources 10.3.3 Support Students’ Construction of Quantitative Meanings for Mathematical Symbols, Expressions, and Equations 10.3.3.1 Figurative Versus Operative Modes of Thought 10.3.3.2 Piagetian Abstraction as the Mechanism for the Construction of Operative Structures 10.4 Concluding Thoughts References Chapter 11 A Face-to-Face Program of Support for Students in a Hybrid Online Developmental Mathematics Course 11.1 Introduction 11.2 Literature Review 11.2.1 Students and Computer-Supported Instruction 11.2.2 Peer-Mentoring Supports 11.2.3 Theoretical Framework 11.3 Institutional Setting 11.3.1 A Course before College Algebra 11.3.2 Peer-Mentoring Program Description 11.4 Methods 11.5 Results 11.5.1 Student Success 11.5.1.1 Gender Subgroups 11.5.1.2 STEM-Intending Subgroups 11.5.1.3 First-Generation Students 11.5.2 Measured Anxiety 11.6 Conclusions 11.7 Recommendations for Practitioners References Chapter 12 A Practical Guide to Discussions in Online Mathematics Courses 12.1 Introduction 12.2 The Value of Discussions in an Online Course 12.3 Mathematicians Communicate and Collaborate 12.4 Online Discussions and Learning 12.5 Theoretical Model 12.6 Synthesis and Analysis 12.7 Solving Content Problems 12.8 Applications of Mathematics 12.9 Social and Emotion Prompts and Study Tips 12.10 Adaptive Reasoning 12.11 Conclusion and Recommendations Bibliography Part 3 Using Technology Chapter 13 Cognitive Load Theory and Mathematics Instruction through MOOCs 13.1 Introduction 13.2 MOOCs in Mathematics 13.3 Cognitive Load Theory and Mathematics Learning 13.4 Methodology 13.5 Results and Discussion 13.6 Discussion 13.7 Conclusions 13.8 Recommendations 13.9 Limitations of the Study References Chapter 14 Technological Pedagogical Content Knowledge for Meaningful Learning and Instrumental Orchestrations: A Case Study of a Cross Product Exploration Using CalcPlot3D 14.1 Introduction 14.2 Cross Product 14.3 Theoretical Framework and Research Questions 14.3.1 Instrumental Orchestration 14.3.2 TPACK 14.3.3 Research Questions 14.4 Research Context 14.5 Method 14.5.1 Self-Study 14.5.2 Data Collection and Analysis 14.6 Results 14.6.1 Instrumental Orchestrations of the Cross Product Exploration 14.6.2 TPACK for Meaningful Learning with ICT of the Developer and Instructors 14.7 Discussion and Conclusion 14.7.1 Further Development of Online Resources 14.7.2 Synergy and Limitations of the Theoretical Frameworks References Chapter 15 Enhancement of Mathematics Learning through Novel Online Tools 15.1 Introduction 15.2 Pedagogical Approaches 15.2.1 Integrating Technology in Teaching Mathematics 15.2.2 Backward Design Module 1: Algebraic Manipulations 15.2.3 Educated Assessments 15.2.4 Promotion of Active Learning 15.2.5 Promotion of Creativity 15.3 Online Calculus Modules: Design Objectives and Organization 15.3.1 Module Components 15.3.1.1 Modules 1–8 15.3.1.2 Modules 9–12 15.3.2 Implementation 15.3.3 Online Mathematics Preparedness Course—2016 15.3.4 Challenges 15.4 Math In Action Journal 15.4.1 Survey Results References Appendix Chapter 16 Making Online Mathematics Method Courses Interactive and Effective with OER 16.1 Introduction 16.2 Challenges and Issues 16.3 Open Education Resource (OER) 16.4 OER’s Utilization 16.4.1 Virtual Manipulative 16.4.2 The National Library of Virtual Manipulatives (NLVM) 16.4.3 Illumination (NCTM) 16.4.4 Mathematics Software 16.4.5 Mathematical Apps 16.4.6 Mathematics Teaching Video 16.5 Conclusion References Chapter 17 Developing Interactive Demonstrations for the Online Mathematics Classroom: Interactive Diagrams 17.1 Introduction 17.2 Incorporating Features of an Effective Mathematics Classroom in a Hybrid Setting 17.3 Interactions, Communication, and Assessments in an Online Mathematics Classroom 17.4 Research Method 17.5 Exploration-Based Activities: Mysterious Geometrical Transformation 17.6 Dragging as a Conceptual Tool: Interplay between Dragging and Variation 17.7 Diagrammatic Argumentation Using IDs 17.8 Constructing a Golden Rectangle 17.9 Conclusion Acknowledgment References Part 4 Teacher Education Chapter 18 MOOCs for Mathematics Teacher Education: New Environments for Professional Development 18.1 Introduction 18.2 Theoretical Framework 18.3 Math MOOC UniTo and Its Methodological Choices 18.4 MOOCs’ Materials and Trainees’ Interactions 18.5 Examples from MOOC Geometria 18.5.1 Difference between Perpendicular and Vertical 18.5.2 Difference between Arc and Angle 18.6 Examples from MOOC Numeri 18.6.1 MERLO Methodology 18.6.2 Arithmetic and Algebra 18.7 Results 18.8 Discussion and Conclusion 18.9 Limitation of the Study and Future Research Acknowledgment Bibliography Chapter 19 Online Mathematics “Self-Help Kiosks” to Support Pre-Service Teachers 19.1 Introduction 19.1.1 Numeracy and Schooling 19.1.2 Numeracy and Teacher Education 19.2 Self-Help Kiosks 19.2.1 The Monash University Master of Teaching Program 19.2.2 Rationale 19.2.3 Structure 19.2.4 Details of One SHK: Representing and Interpreting Data 19.2.4.1 Part A: Introduction 19.2.4.2 Part B: Key Mathematical Topics 19.2.4.3 Part C: Issues and Misconceptions 19.2.4.4 Part D: Examples for Teaching and Learning 19.2.4.5 Part E: Self-Check Quiz 19.3 Students’ Reactions to and Use of the SHKs 19.3.1 LANTITE Questionnaire and Interviews 19.3.2 SHKs Questionnaire 19.3.3 Moodle Usage Data 19.4 Conclusions Acknowledgments References Part 5 Commentary Chapter 20 Online Mathematics Education, the Good, the Bad, and the General Overview 20.1 Introduction 20.2 Benefits of Teaching Mathematics Online 20.3 Difficulties of Teaching Mathematics Online 20.3.1 Cheating 20.3.2 Isolation 20.3.3 Typing Math Online 20.4 Favorite Aspect of Teaching Math Online 20.5 Least Favorite Aspect of Teaching Math Online 20.6 Mathematics Courses Best and Poorest Suited for Online Study 20.7 Conclusion References Index