A Study of Multi-Representation of Geometry Problem Solving with Virtual Manipulatives and Whiteboard System

Article excerpt


Geometry is one of fundamental methods which people use to understand and to explain the physical environment by measuring length, surface area and volume. For this reason, enhancing geometric thinking is very important for high-level mathematical thinking, and it should be developed with spatial interaction and manipulation in daily life (Clements & Battista, 1992; Tan, 1994). However, in traditional classrooms, geometry learning is usually conducted only through the description of text, 2D graphs and mathematical formulas on whiteboards or paper. In some important topics, such as measuring the area and volume of 2D or 3D objects, traditional teaching methods often focus too heavily on the application of mathematical formulas, and lack opportunities for students to manipulate the objects under study. Consequently, many students can memorize the formulas and even appear to succeed in their course work without fully understanding the physical meaning of the math formulas or geometry concepts (Tan, 1994).

Tan (1994) suggested that the development of understanding of concepts such as the measurement of area and volume should come from the experience of covering and stacking manipulations, so that when formal mathematical concepts and formulas are introduced or applied, children would actually understand the formulas and their meanings. This implies that the construction of the geometry knowledge should be acquired via manipulating spatial objects (concrete experience), brainstorming (imagery concept) and writing symbolic solutions (abstract representation) (Battista & Clements, 1991, 1996).

Therefore, to provide an environment to facilitate such deep, rich learning, researchers employed both computer 3D graphics and simulations to create a multi representative construction model, offering learners more flexible ways to organize their thinking with manipulation (like coordinating, restructuring and comparing operations) and symbolic terms, such as text, graphics and speech. Researchers incorporated translucent multimedia whiteboards into a 3D virtual space, combining Virtual Manipulatives and a Multimedia Whiteboard to facilitate geometry problem solving (Figure 1), to create a new tool called the Virtual Manipulatives and Whiteboard, or VMW. In the VMW system, learners can solve geometry problems by manipulating virtual objects or exploring the problems from various viewpoints in 3D space. Then, learners can choose appropriate viewpoints in the 3D space and generate their own translucent whiteboards atop their images, to write down math equations or textual explanations. By providing students with an easy way to move back and forth from the concrete to the symbolic, the tool facilitates children's thinking in geometry problem solving as per the pedagogical theory which states: Children should construct their geometry concept from multiple representations like mapping the concrete items to abstract ideas through physical or mental manipulation.

VMW system also recorded user's manipulation into a database. Analysis of physical manipulation in a 3D scene and symbol expressions of the same on whiteboard provides insight into the thinking of each learner, such as strategies used, and misconceptions held. Thus the VMW system also provides teachers with valuable information, which can be used to guide the development of subsequent lessons.

Perceived acceptation of the VMW system was investigated using the Technology Acceptance Model-based questionnaire. The obtained results show that VMW students found the tool to be useful and easy to use.. The combination of learners' manipulations and their solving content on the whiteboard was also analyzed. Varieties of solving strategies were found and some important insights into effective teaching practice were also acquired. In the future, researchers aim to extend multi representative construction model to various domain knowledge in addition to the learning of Mathematics. …