Let’s begin this book by exploring five general principles that will help improve your math skills.
Having a broad knowledge of different math topics lays a solid foundation for solving math problems. In middle school and high school, you will gain general knowledge in number theory, counting, algebra, probability, and geometry. This book will review some key concepts in these math topics, but it is necessary to have a firm grasp of these concepts so you can tackle math problems.
Your problem-solving skills also depend on the breadth of your knowledge. A basic understanding of a broad range of topics often helps you analyze problems, explore alternative approaches, and come up with efficient solutions.
As avid readers ourselves, we found reading skills to be helpful in solving math problems. Many of the problems involve the translation of words into equations or geometric figures. You need to have excellent reading skills to make sure that you efficiently understand a problem and quickly and properly formulate it before answering it. Furthermore, active reading and writing also improve one’s analytical skills and the ability to communicate one’s ideas, which in turn help develop problem-solving skills.
When you solve math problems, we suggest you focus on the concepts, not the answers. In classroom teaching, many teachers focus on teaching the answers. Understanding the concepts is a much deeper level of understanding. Developing insights helps build problem-solving skills. Memorizing the theorems and specific rules enables you to solve problems where such theorems and rules directly apply. When you need to use a similar concept in a different setting or combine multiple concepts to solve a problem, the insights behind the theorems and rules are far more powerful than the theorems and the rules themselves.
In many Asian countries, the education philosophy is to gain proficiency through frequent practice, which often means performing similar or the same tasks again and again to achieve proficiency. Such an approach helps build solid math knowledge, but taken to the extreme, such an approach may stifle creativity and dampen one’s interests in math. In American schools, since preschool, teachers have always encouraged students to read (improve through practice). However, schools often do not actively encourage students to practice math problems. There is even a cultural bias against students who practice math. We believe that practice is a significant component in building math skills. The goal is to enforce one’s understanding of the underlying concepts instead of memorizing solutions. The right amount of practice allows you to build a solid understanding of a key concept so you can easily apply the concept when you need to use it to solve problems.
Practice does not mean mechanical repetition, especially not a repeat of what you already know. In order to learn and improve, you need to step outside of your comfort zone, which means focusing on what you do not know. Learning unfamiliar concepts and practicing until you feel comfortable is the fastest way to improve.