Awesome Math

Teaching Mathematics with Problem Based Learning

Titu Andreescu, Kathy Cordeiro, Alina Andreescu

• Publisher: Jossey-Bass
• Year: 2019
• Print ISBN: 9781119575733
• eBook ISBN: 9781119575702
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

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.

SECTION II
Teaching Problem Solving

In this section, we will provide teachers with the strategies and tools for guiding innovative problem solvers who can work independently and think deeply. And, as you may have guessed, this won't require complete conformity (e.g., “do exactly as I say”) to a program. Rather, we encourage you to expand on the ideas presented to create a curriculum that works for your style and the students who make up your classes. Teaching is all about having a plan; however, anyone who has taught knows that lessons rarely go exactly as intended. Flexibility and creativity are required when working with students of varied levels and abilities, so this section seeks to provide a well-conceived plan, along with ideas from our teaching and learning community to inspire new ideas and methods. As detailed in Section 1, it's critical to create a kind and supportive environment that respects the intellectual abilities of all participants along with interesting problems, like the one below, that are scalable for students of different mathematical skill levels.

---

We have a large pile of identical 20 gram 1 × 1 square pieces. For some positive integers m and n the following displays use the same number of pieces:

• an m × (n + 2) rectangle with a 8 × 8 square removed from its interior;
• an (m − 2) × n rectangle with a 4 × 4 square removed from its interior;
• an (m + 4) × (n − 6) rectangle with a 2 × 2 square removed from its interior.

Prove that each of the three displays weighs 2020 g.

Solutions

Because the first two displays use the same number of pieces, we have m(n + 2) − 8² = (m − 2)n − 4², yielding m = 24 − n. Then, because the last two displays use the same number of pieces, we have

In other words, we have m(n + 2) − 8² = (m − 2)n − 4², yielding m = 24 − n.

Then (24 − n − 2)n − 4² = (24 − n + 4)(n − 6), yielding n = 13 and m = 11.

Hence, each display has 101 pieces and weighs 2020 g.

---

This section is about helping educators be their better selves, as well and providing the information necessary to successfully launch a problem-based curriculum.

You will be provided with:

• Five Steps to Problem-Based Learning
• The Three Cs: Competitions, Collaboration, Community
• Mini-Units

CHAPTER 6
Five Steps to Problem-Based Learning

When creating a problem-based learning (PBL) program you need the problems, tools, management, and support to effectively guide each student toward the primary goal of creating innovative problem solvers. With the five steps outlined below, you will be well on your way to launch PBL in your classroom, utilizing the provided set of mini-units (10–15-minute lessons) in Chapter 8 or full units (45–55-minute lessons) in Section III.

1. Start with meaningful problems.
2. Utilize teacher resources.
3. Provide an active learning environment.
4. Understand the value of mistakes.
5. Recognize that everyone is good at math!

Start with Meaningful Problems

In this section, we will cover the following:

• What are good problems?
• Show relentless curiosity.
• Don't forget grace.

WHAT ARE ...

Table of contents

• Cover
• Table of Contents
• Acknowledgments
• About the Authors
• Introduction
• SECTION I: Why Problem Solving?
• SECTION II: Teaching Problem Solving
• SECTION III: Full Units
• Index
• End User License Agreement