Title: Problem Posing: An Opportunity for Increasing Student Responsibility.
Abstract: INTRODUCTION This article examines the benefits and teaching procedures resulting from a classroom activity in student problem posing. The benefits included an enhancement of student reasoning and reflection and a heightened level of engagement. Principles and Standards for School Mathematics [3] indicates that one of the critical requirements for successful problem solving is a productive disposition to problem pose. To develop such a disposition, students must be provided with more than explicit instruction. When students, rather than the teacher, formulate new problems, it can foster the kind of responsibility that students need to take for constructing their own knowledge. BACKGROUND Many students fail to experience problem posing in their study of mathematics [4]. Often, students are exposed to mathematics instruction which requires little student input, much as in the Instrumental or Platonist perspective [1]. The Instrumentalist perspective emphasizes the mastery of skills, rules, and procedures that become ends in themselves, rather than devices for understanding, representation, and reasoning. From this perspective, the student's role is to master what the teacher says. The Platonist perspective also emphasizes narration, and students are considered to be receptors. When mathematics instruction mimics these narrative perspectives, the textbook or teacher is continually relied upon for problem generation and for explicit instructions on how to solve problems. Students serve mostly as listeners and have little responsibility for constructing their own knowledge. Problem posing can be used to challenge over-reliance on the teacher and the textbook, and give the student an improved sense of ownership and engagement in their education. Moreover, extending problems with problem posing offers other potential benefits. As part of the critical look back process of problem solving, it can enhance student reasoning and reflection needed for a deep understanding of mathematics. Also, student-generated connections between mathematics and the real world often spring from such creative experiences. PROCEDURES The following activity utilizes a common problem involving automobile depreciation, which could be labeled the Jetta Problem. This problem was assigned to a class of community college students enrolled in a precalculus course. They worked on the problem in small groups of 3 or 4 students and had graphing calculators at their disposal. Jetta Problem: A new 2000 Volkswagon Jetta GL-Sedan has a base price of $15,230 and will depreciate 44% over the first 5 years. a) Model the Jetta's depreciation using a linear function. b) Model the Jetta's depreciation using an exponential function. c) If you owned the Jetta and wished to trade it in after 4 years, which method of depreciation would be most beneficial to you as the owner? To encourage problem posing by the students, the following was added to the problem. d) Extend this problem by writing your own question and solution using or modifying the information given in the problem. EXAMPLE SOLUTIONS AND STUDENT RESPONSES Presented below are sample solutions to parts a, b, and c, along with seven examples of student-generated problems resulting from part d of the Jetta Problem. The solutions to Parts a, b and c serve merely as a point of reference for the reflective and creative problems posed, as well as the related discussions prompted by Part d. Although all of the groups found a solution for part a, most needed help with Part b. However, once part b was completed, all of the groups managed to complete Part c and to generate a problem and a solution for Part d. Solution to Part c: Students entered function y^sub 1^ and their choice for y^sub 2^ into the calculator. In Figure 1 a printout of the calculator displays shows the functions, window, and graphs used. Because of the values of y^sub 1^ and y^sub 2^ at x = 4 are very close together, some students examined the table of values for y^sub 1^ and y^sub 2^, and a printout of the table forms the last display in Figure 1. …
Publication Year: 2004
Publication Date: 2004-01-01
Language: en
Type: article
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Cited By Count: 40
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