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.
When we begin our lessons and units with meaningful word problems we provide a context within which mathematical ideas can be seen. When the word problems that we use to begin our lessons can be solved using the math we will be teaching, this allows students to learn the mathematics in the setting in which it will be used. Notice that we are saying that the word problem can be solved using the mathematics that the student will be learning. The exposure to a word problem comes before instruction in the skill needed to solve it. This is counterintuitive for most people, but it is, in fact, the most effective way that we currently know of to help students develop understanding of mathematical concepts.
At the very start of our lessons we present a word problem for which the students have not yet learned a procedure. We ask the students to solve this problem in whatever way makes sense to them. It is important for learners to realize (and for teachers to have faith) that there can be any number of different ways to solve any given problem. When we ask learners to solve a problem in whatever way makes sense to them, we are communicating that there are many possible ways to go about it. We are also communicating that the method that a student uses can and should make sense. In a classroom setting we are also opening the door to many different solution methods, because of the diversity of students within our classrooms. Students from varying cultural backgrounds will very likely see problems in different ways from each other. Likewise, students with different learning styles will approach problems differently. Students who differ in their preference for different intelligences (Gardner, 1983) will approach problems differently. One student may easily construct a visual way to represent a problem. Another student may solve the same problem by using an organized list or table. Yet another child may use objects such as base-10 blocks or interlocking cubes to represent the problem. The point is that the students are encouraged to find their own ways to solve the problem. This same principle holds for the reader of this text. The reader is always encouraged to be creative and find a way or ways of solving that make sense personally.
Following the initial solution, students are asked to share with the class the methods that they found for solving the problem. Several different students are asked to share. This gives the whole class an opportunity to hear and see a variety of methods of solving a given problem. We work hard to create a safe and comfortable classroom environment in which students can explain their thinking to one another. The goal is for each child to fully understand the methods presented by each of his or her classmates. In the case of users of this text, we recommend that math coaches or faculty development leaders establish a community of inquiry among teachers in order to facilitate a dynamic exchange of ideas. The goal of this dynamic exchange of ideas is to deepen understanding of the powerful ideas.
After classmates have presented a variety of different methods, the teacher finally offers a standard algorithm, or the more formal mathematical way of solving the problem. By this point in the lesson the students are equipped to see this more formal mathematical solution as simply one more way to approach the problem. Since they have already seen the problem solved in a number of different ways, and since these various ways of solving have been sensible and meaningful to them, they can expect to see the formal mathematical method in the same light. They expect it to make sense, and they work to understand how it works. This method of beginning with problems, asking students to find their own ways of solving, having students share their methods with one another, and following this with instruction from the teacher is what we mean when we say that we are teaching mathematics in the context of real-world problems. This is what we refer to as problem-focused teaching.
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Excerpt from Elementary Mathematics Pedagogical Content Knowledge: Powerful Ideas for Teachers, by J.E. Schwartz, 2008 edition, p. 8-9.
© ______ 2008, Allyn & Bacon, an imprint of Pearson Education Inc. Used by permission. All rights reserved. The reproduction, duplication, or distribution of this material by any means including but not limited to email and blogs is strictly prohibited without the explicit permission of the publisher.
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