Research Questions and Background
Elementary school students typically do not have too much difficulty understanding the geometry of circles, squares, and rectangles. The majority, however, recognize only equilateral and isosceles triangles as triangles. The shapes of triangles are diverse, and include obtuse triangles, acute triangles, and right-angled triangles. Although most students can recite area formulas, they cannot explain the origins of these formulas clearly, which shows that their perception of the area involves merely rote memory, rather than an understanding of the derivation process. This often results in students using formulas incorrectly when solving problems (Tan, 1998). Furthermore, in traditional lecture instruction, the student is often considered a passive receiver of knowledge; with such unidirectional communication, it is difficult to achieve real learning progress. By contrast, constructivism emphasizes the active role of the student. Education is currently inclined toward free constructivist learning approaches, in which new information is linked to existing knowledge, thereby corresponding to developed archetypes (Resnick, 1983). Therefore, students must actively participate in activities geared toward different learning modes to stimulate and strengthen cognitive growth (Larkin & Chabay, 1989). The microcosm of the physical world that can be represented on computers enables students to explore, experiment, and operate artificial systems or models with actual scientific effects (Dede, 1987).
In the IT Whitepaper in K-12 Education, the Ministry of Education (MOE, 2008) stated that schools must develop diverse digital teaching resources, and enhance the IT equipment and network services in classrooms, so that in the future, the incorporation of IT into the teaching field will no longer be restricted to computer laboratories. Software associated with mathematics includes SAS for statistics, Maple and Mathematica for algebra, and Excel for numerical computations. However, it is difficult for math teachers to master these programs and apply them to teaching (Kaput, 1992). Numerous researchers and teachers are currently studying dynamic geometry, but the majority are focusing on problem-solving. Rarely have they concentrated on the process of cognitive psychology. Jackiw (1991) created the Geometer’s Sketchpad (GSP), a tool that enables users to construct and manipulate basic geometric shapes. Phonguttha, Tayraukhamh, and Nuangchalerm (2009) compared the effectiveness of GSP as a teaching medium against traditional textbook teaching. Their results revealed that the GSP can improve mathematics achievements and the analytical thinking of students effectively. The GSP allows the teaching of mathematics to begin from dynamic shapes, enabling students to follow an intuitive process of learning. It provides a geometric construction, continuous dynamic transformation, structure retention, a view of the specific as general, and the recording of the geometric construction process. These functions provide a simple teaching environment in which to discuss the properties of different geometrical shapes. Students can thus observe, experiment, conjecture, and generalize the constant properties of continuously changing dynamic shapes. Kurz, Middleton, and Yanik (2005) stated that the GSP enables students to test their hypotheses regarding geometrical shapes, relations, and transformations, and other researchers have agreed that using dynamic visual teaching methods helps deepen students’ understanding of mathematical concepts (Fallstrom & Walter, 2009; Teoh & Fong, 2005).
Van Hiele and van Hiele-Geldof (1958)
divided the development of geometric thought into five levels: (a) visualization, (b) analysis, (c) informal deduction, (d) formal deduction, and (e) rigor. These five levels must be advanced in order, and students must comprehend the concepts and strategies of one level fully before they can proceed to the next. Progress is not correlated with age, but rather to previous teaching and learning experiences. For example, most younger elementary school students are at the first level (i.e., visualization; Liu, 1998); therefore, they require a manipulation, observation, description, and comparison of physical objects as well as multiple concrete experiences before they can progress toward a higher level of learning. The purpose of this study was to integrate the GSP with the five levels of geometric thought proposed by van Hiele (1986), to develop a novel computer micro-world concerning the area of triangles. This system uses constructivist learning and computer visualization to create an environment in which users can manipulate icons freely, thereby learning of the area of triangles. This learning model considers both the process of cognitive psychology and the historical development of mathematics, and encourages students to actively explore new concepts.