Awesome Math Teaching Mathematics with Problem Based Learning

Help your students to think critically and creatively through team-based problem solving instead of focusing on testing and outcomes. 

Professionals throughout the education system are recognizing that standardized testing is holding students back. Schools tend to view children as outcomes rather than as individuals who require guidance on thinking critically and creatively. Awesome Math focuses on team-based problem solving to teach discrete mathematics, a subject essential for success in the STEM careers of the future. Built on the increasingly popular growth mindset, this timely book emphasizes a problem-solving approach for developing the skills necessary to think critically, creatively, and collaboratively.

In its current form, math education is a series of exercises: straightforward problems with easily-obtained answers. Problem solving, however, involves multiple creative approaches to solving meaningful and interesting problems. The authors, co-founders of the multi-layered educational organization AwesomeMath, have developed an innovative approach to teaching mathematics that will enable educators to:

• Move their students beyond the calculus trap to study the areas of mathematics most of them will need in the modern world
• Show students how problem solving will help them achieve their educational and career goals and form lifelong communities of support and collaboration
• Encourage and reinforce curiosity, critical thinking, and creativity in their students
• Get students into the growth mindset, coach math teams, and make math fun again
• Create lesson plans built on problem based learning and identify and develop educational resources in their schools

Awesome Math: Teaching Mathematics with Problem Based Learning is a must-have resource for general education teachers and math specialists in grades 6 to 12, and resource specialists, special education teachers, elementary educators, and other primary education professionals.

Dr. Titu Andreescu is an Associate Professor at the University of Texas at Dallas in the Science and Mathematics Education department. As a mathematics educator and leader, he has developed math camps, competitions, and curricula that have received international attention. Titu was a long-time head coach of Team USA participating in the International Math Olympiad. He is cofounder of AwesomeMath, a premiere mathematics camp held across the United States.

Kathy Cordeiro, MBA, is a leading developer of education initiatives for business and academia. She founded Eudaimonia Academy, where she coached mathematics teams and worked with highly and profoundly gifted students. Other roles include facilitator for the Metroplex Math Circle as well as Director of Marketing and Communications for AwesomeMath, where she was able to work with the finest mathematical minds from across the globe. She continues to speak about maximizing mathematics education for parents, schools, and gifted organizations.

Alina Andreescu, MA, is the co-founder and Operations Director of AwesomeMath and founder of XYZ Press. She fosters an international community of staff, students, and instructors that values critical thinking, creativity, passionate problem solving, and lifetime mathematical learning.

Acknowledgments xi

About the Authors xiii

Introduction xvii

I. Why Problem Solving?

Chapter 1: Rewards for Problem-Based Approach: Range, Rigor, and Resilience 5

Range Ignites Curiosity 5

Rigor Taps Critical Thinking 9

Resilience is Born Through Creativity 10

Chapter 2: Maximize Learning: Relevance, Authenticity, and Usefulness 13

Student Relevance 13

Mathematical Relevance 14

Mathematical Relevance: The Math Circle Example 16

Curriculum Relevance 18

Authenticity: The Cargo Cult Science Trap 21

Authenticity in Learning 22

Usefulness 25

Chapter 3: Creating a Math Learning Environment 27

Know Yourself: Ego and Grace 27

Know Your Students 30

Know Your Approach 35

Chapter 4: What is the Telos? 47

Autonomy to Solve Your Problems 47

Mastery Through Inquiry 48

Purpose with Competitions 50

Quadrants of Success 52

Chapter 5: Gains and Pains with a Problem-Based Curriculum 57

Teachers 58

Students 61

Parents 67

II. Teaching Problem Solving

Chapter 6: Five Steps to Problem-Based Learning 75

Start with Meaningful Problems 75

Utilize Teacher Resources 79

Provide an Active Learning Environment 91

Understand the Value of Mistakes 97

Recognize That Everyone is Good at Math 99

Chapter 7: The Three Cs: Competitions, Collaboration, Community 103

Competitions 103

Collaboration 107

Community 117

Aspire to Inspire: Stories from Awesome Educators 121

Chapter 8: Mini-Units 147

Relate/Reflect/Revise Questions 147

Roman Numeral Problems 148

Cryptarithmetic 151

Squaring Numbers: Mental Mathematics 155

The Number of Elements of a Finite Set 157

Magic Squares 159

Toothpicks Math 163

Pick’s Theorem 165

Equilateral versus Equiangular 168

Math and Chess 170

Area and Volume of a Sphere 172

III. Full Units

Chapter 9: Angles and Triangles 177

Learning Objectives 177

Definitions 177

Angles and Parallel Lines 177

Summary 180

Chapter 10: Consecutive Numbers 185

Learning Objectives 185

Definitions 185

Chapter 11: Factorials! 191

Learning Objectives 191

Definitions 191

Chapter 12: Triangular Numbers 199

Learning Objectives 199

Definitions 199

Chapter 13: Polygonal Numbers 205

Learning Objectives 205

Definitions 205

Chapter 14: Pythagorean Theorem Revisited 213

Learning Objectives 213

Definitions 213

Pythagorean Theorem 214

Rectangular Boxes 214

Euler Bricks 216

Assessment Problems 219

Chapter 15: Sequences 221

Learning Objectives 221

Definitions 221

Introduce a Geometric Progression 222

Chapter 16: Pigeonhole Principle 227

Learning Objectives 227

Definitions 227

Chapter 17: Viviani’s Theorems 235

Learning Objectives 235

Definition 235

Chapter 18: Dissection Time 239

Learning Objectives 239

Definitions 239

Chapter 19: Pascal’s Triangle 245

Learning Objective 245

Summary 249

Chapter 20: Nice Numbers 255

Learning Objectives 255

Definitions 255

Index 259

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