A Problem-Solving Model of Quadratic Min Values Using Computer

Article excerpt

INTRODUCTION

There are two facets in the direction of reforming mathematics education. First, it is a trend that many educators in mathematics education have paid attention to problem solving since 1980's. The National Council of Teachers of Mathematics' (1980) recommendations to make problem solving the focus of school mathematics posed fundamental questions about the nature of school. The aspects of mathematics education proposed by the NCTM (1989) emphasized fostering students' problem-solving ability as one of faculties students should develop. After students graduate from schools and enter job markets, they face highly developed scientific industry in the era of information and should be able to apply what they learned through various activities for problem solving in schools. Obviously, it is necessary that mathematics instruction should be designed so that the students can have a sufficient experience of problem solving in school mathematics. So, mathematics educators are interested in operation and creative activities related to problem solving such as exploration, investigation and observation.

Second, there has been a classroom renovation with advanced multi-media facility including a computer in Korea. Also, the development of industry related to the computer changes contents and methods in teaching and learning mathematics in classrooms. Through this diffusion of the advanced computer in every area of modern society, broad and profound mathematical knowledge is required. In past, mathematical knowledge was only for a few elite of society and scholars, but nowadays, administrators in a company need manpower well prepared with more mathematical knowledge than ever before. Since computer software provides a convenient tool in teaching mathematics, various teaching methods need to be developed. Mathematical properties and principles, which used to be hardly explained and observed, can be visualized at ease, as well as complicated calculations time-consuming by paper-and-pencil can be done at once on the monitor of computer. This changing circumstance calls for reforming school mathematics.

There has been some research (cf. Roman, 1975; Roman & Laudato, 1974) using computer software for drill and practice in computer assisted instruction (CAI), however, few research (cf. Bobango, 1988; Schwartz & Yerushalmy, 1987) focused students problem-solving performance using dynamic software. As mentioned earlier, mathematics classrooms in Korea have been equipped with a pentium computer, a 43-inch TV projector, and other high-tech educational materials. However, existing software such as Geometer's Sketchpad and Mathematica written in English have difficulties being used in Korean secondary-school classrooms due to a large number of students. In addition, using computer software as a tool, teachers lack the knowledge on how to use computers and how to integrate it into regular math programs. Thus, a new computer program needs to put the computer between teachers and students for problem solving as an instructional goal and helps to lessen these difficulties. The purpose of the study is to introduce a problem-solving model using computer as a tool and discuss how effectively it integrates mathematics instruction of problem solving into a regular math program of quadratic minimum value for fostering students' problem solving ability.

BACKGROUND

Visualization

Zimmermann and Cunningham (1991) stated that visualization is the process of producing or using geometrical or graphical representations of mathematical concepts, principles or problems, whether hand drawn or computer generated (p. 1). Reinforcing the visual and intuitive side of mathematics opens a new possibility for mathematical work, especially now that visualizing figures related to a problem has enough power and resolution to support it with accurate representations of problems and their solutions. The benefits of visualization include the ability to focus on specific components and details of very complex problems, to show the dynamics of systems and processes, and to increase intuition and understanding of mathematical problems and processes (Cunningham, 1991). …