Abstract

Problem solving is fundamental not only to the learning and application of mathematics, but to all walks of life. Many people consider mathematics and problem solving to be synonymous. However, there are many mathematicians who do not solve problems or who do more than solve problems. Some work to build new theories or advance the language of mathematics. Others unify or explain previous results, sometimes from many fields of mathematics. Yet others consider the very nature and philosophy of mathematics as a discipline. In modern society, mathematics teaching at all levels seeks to develop students' abilities to effectively address a wide variety of mathematics problems such as: proving theorems; reducing new problems to previously solved problems; formulating and solving both real-life and abstract word problems; finding and creating patterns; interpreting figures, graphs and data; developing geometric constructions; and doing appropriate computations or simulations, often with computers or calculators. Problem-solving is also an instructional approach in which students actively learn fundamental concepts through their contextualization within problems rather than from a passive lecture. What fundamentally connects these activities, beyond the mathematics techniques and skills necessary to solve them, is the notion of "how to think." Not only must students have the tools and techniques at their disposal through a solid education in the fundamentals, they must be able to analyze the characteristics and requirements of a problem in order to decide which tools to apply, or to know that they do not have the appropriate tool at their disposal. Further, students must practice with these mathematical tools in order to become skilled and flexible problem solvers, in the same way athletes or craftsmen practice their trades. As Hungarian mathematician George Polya expressed, "If you wish to become a problem solver, you have to solve problems." This extends to the notion that problem solving is by its nature cyclic and dynamic. In many cases, the solution to a problem results in one or more new problems, or opens the path to solving an older problem for which a solution has previously proven to be elusive. Sometimes mathematics problems have real and immediate applications, and many new mathematical disciplines, like operations research or statistical quality control, have developed from these sorts of problems. In contrast, there are many issues in theoretical mathematics that do not appear to have any immediate benefit to society. In some cases people question the need to explore such abstract problems when there are more immediate needs to be met. Often, these abstract problems turn out to have very concrete applications decades or even centuries after their initial introduction. Even if that is not the case, theoretical problem-solving adds to the growing body of mathematics knowledge and, just as importantly, shows people yet another way to think about the world.