How many of us remember "word problems" as the most difficult part of learning mathematics? Over the years, most mathematics texts have presented word problems at the end of each chapter. The theory was that students would first learn the basic procedure of a specific chapter and then they would be expected to apply that procedure in selected word problems. This seemed to work fine for each chapter; students appeared to be successful. (What the students knew, however, and apparently the teachers didn't, was that you didn't really need to think about the word problems in order to be successful. If you had just learned a new procedure, such as subtraction, then you could safely assume that the word problems at the end of the chapter could be solved by subtraction.) The difficulty came when you had a test that included several different kinds of word problems. Then you actually had to think about what was being asked in order to know what procedure to apply. That's when things really got difficult. Does this describe your experience? This difficulty was related to an overemphasis on teaching the procedures of mathematics and an underemphasis on teaching the concepts of mathematics. As you use this text, you will find yourself faced primarily with word problems (which we will refer to simply as problems). The reason for this is that word problems are the most effective way to help you develop the conceptual understandings that you need.
One of the most significant changes in mathematics teaching in recent years has actually been one of the simplest: we have moved word problems from the end of the chapters and lessons to the beginning. Now, instead of starting by teaching mathematics as if it were some naked, abstract, symbol manipulation, we are teaching mathematics as it really occurs in our lives. We are teaching mathematics in context. Real-world problems have become the settings in which mathematics lessons are presented. Skill in mathematics arises from context, rather than the other way around. This simple change has been a part of the broader change moving toward a greater emphasis on conceptual understanding.
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