Objectives
Teacher Objectives
This unit has objectives for the teacher and for the students. First, I will discuss objectives for the teachers. In Ma's book (1999), the author compares math teachers in China and in the United States. There are striking differences in the knowledge base of the teachers. The Chinese teachers are more knowledgeable than American teachers in their own math skills and methods for solving equations. The book then analyzes reasons why. One main reason is summarized well with the analogy of the taxi driver. She compares the way the Chinese teachers understand math to a proficient taxi driver who knows all of the roads and many ways to get to the same place. Ma associates the American teachers to a newcomer in an area. A newcomer may know a few roads and one way to get to a location. Ma states, "A teacher's map of school mathematics must be more complicated and flexible" (Ma, 1999, p. 123).
A major objective of the seminar that led to the creation of this unit is for teachers to analyze and categorize word problems based on similarities and differences. It is this in depth analysis that really begins to help the teacher become a more proficient "taxi driver". At first glance, categorizing the word problems seems to be an easy task, but comparing similarities and differences goes beyond grouping the problems based on the operation involved. It looks more closely at the components and the structure of the problems the students are solving. There are several categories within each operation. Below the categories are defined and an example of each type of problem is given. These categories are from Children's Mathematics Cognitively Guided Instruction (Carpenter, et al., 1999). Additional insight has been added based on the article (Sowder, 1995).
Student Objectives
This unit clearly meets national and state standards since the four operations are such an integral part of elementary mathematics. In addition to simply learning the four operations, this unit looks at the relationships shared by the various operations. The inverse relationship of addition and subtraction, the inverse relationship of multiplication and division, and the relationship that addition and multiplication share are all explored through problem solving. By making connections between related word problems and examining different ways to solve the same word problems, students should develop a better understanding of the relationships between the four operations. As stated earlier, developing an understanding of the connectedness that the operations share is crucial to student learning.
Going beyond the basic calculations, students need to think critically and to develop problem-solving skills. This unit not only provides opportunities to develop these skills, but also teaches new strategies and approaches for dealing with word problems. The suites of problems should help make students aware of the connections that closely related problems share, and thus help them to develop a greater mathematical understanding. The expectation is that if students have a better understanding of the four operations they can use this knowledge to closely examine word problems and become better problem solvers.
As the strategies and activities of this unit unfold, the students will learn the Singapore bar model approach. This is just one strategy to help students solve word problems. It is visual and a more abstract "manipulative" to help students solve problems. This method will provide the students with one more approach or option when they face a word problem.
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