# A study of multi-representation of geometry problem solving with virtual manipulatives and whiteboard system.

Introduction

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.

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Literature Review

Mathematical Geometry Learning

Enhancing geometrical thinking is important, and this occurs naturally through by spatial interaction with real objects and problems in everyday life. However, in traditional education environments, geometry is most commonly taught using text, 2D images and mathematical formulas. For some important topics, such as measuring distance, area and volume, some studies have shown that such teaching method are not highly effective. Consequently, many children do not understand the physical meaning of the formulas found in their textbooks (Tan, 1994). Schoenfeld (1992) stated that "mathematics relies on logic rather than observation as its standard of truth, yet employs observation, simulation, measurement and even experimentation as a means of discovering truth". The statement reflects a growing understanding of mathematics as an empirical discipline, one in which mathematical practitioners gather "data" in the same ways that scientists do. Moreover, the tools of mathematics can support abstraction, symbolic representation, and symbolic manipulation.

As for learning the concepts of volume and surface area by children, Tan (1994) found that most children used visual perception to intuitively compare the size of objects. They usually considered one dimension for comparison even though the geometry problems concerned multi-dimensions. For example, in comparing the area of one triangle and one square; it is not easy to distinguish which one is bigger using intuition or by comparing only one dimension. Therefore, some manipulations are first needed to restructure the two shapes into one similar shape whereupon accurate comparisons can be easily made. If only mathematical formulas are used to make comparison of the areas of these two shapes, students have no experiential basis by which to understand the physical meaning of math concepts and formulas. Therefore, students develop their understanding of geometry through first decomposing, structuring, comparing and coordinating, and formalize that understanding by learning and applying mathematical formulas that are both meaningful and useful.

Virtual Reality and Manipulatives

Some studies have concentrated on conceptual learning and virtual reality (Roussos et al., 1999; Kirner et al., 2001; Moustakas, Nikolakis, Tzovaras, & Strintzis, 2005). Most of these studies are based on the constructivism approach. For example, the NICE project (Roussos et al., 1999) was one of the immersive, multi-user environments designed specifically for collaborative learning. The main activity in NICE requires students to collaboratively construct and to tend a healthy virtual garden. The NICE project employed the social constructivism approach and helped children to build virtual gardens collaboratively by learning and discussing the relationships between plant growth, sunlight, and water.

Kirner et al. (2001) developed the CVE-VM (Collaborative Virtual Environment), which provided tools to help children to build their own virtual world. It also supported collaborative learning according to the constructivism approach. Moustakas et al. (2005) also designed collaborative augmented reality as a medium for teaching by using 3D dynamic geometry to facilitate mathematics and geometry education.

Many studies used Virtual Reality and Constructivism to enhance collaborative learning, but the major difference between the VMW system and past studies is that VMW system combined Virtual Manipulatives and multimedia whiteboard to propose a multi representative construction model. In this model, knowledge is constructed through transformation of concrete imagery and abstracted representations through physical or mental manipulation. In VMW, the manipulation was defined as Virtual Manipulatives which was first proposed by Moyer, Bolyard and Spikell (2002). Virtual Manipulatives is an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge. Virtual Manipulatives promised to have great assets for learning because computer materials have portability, safety, cost-efficiency, minimization of error, amplification or reduction of temporal and spatial dimensions, and flexible, rapid, and dynamic data displays features. Regarding learning performance, Triona and Klahr (2003) also showed that the performance in concept learning was almost the same when taught with either virtual or physical materials.

Multimedia Whiteboard

Another feature of the proposed system is to incorporate translucent multimedia whiteboard into virtual 3D environments. We know the whiteboard or chalkboard plays an important role in knowledge construction process. For thousands of years, people have communicated about objects in the physical world using 3D models or two-dimensional drawings (Bimber & Stork, 2000). For convenience and ease of use, most people usually use 2D symbols to represent 3D physical objects in everyday life, requiring access to a 2D-sketch tool like a paper, chalkboard or whiteboard to show their ideas. In schools, teachers and students need these tools to show their ideas by writing texts and drawing graphs. Teachers generally still used traditional chalkboard or white boards to teach mathematical reasoning and calculating skills. However, use of the physical boards is constrained by the limited physical space available so that frequent erasing is required. Hwang et al. (2007) proposed using a multimedia whiteboard system without physical space limitations to improve mathematical problem solving with multiple representations. The proposed multimedia whiteboard allows students to express their thoughts with text, images or the spoken word. While using multimedia whiteboards to support multiple representations, students start with multi-model solving processes in mind and then translate their thoughts into multiple representations. The results showed most students were satisfied with the usefulness and ease of use of the multimedia whiteboard system.

The use of language to explain mathematical thinking is significantly related to learning performance and Weitz, Wachsmuth and Mirliss (2006) employed tablet PC as an electronic whiteboard and studied its usefulness to support learning activities. They evaluated faculty applications of tablet PCs apropos their contribution to teaching and learning. The result showed approximately two-thirds of the responding faculty used their tablets as a whiteboard, and approximately 90% felt that their use of a tablet for writing mathematics and drawing diagrams, charts and/or graphs had a positive impact on learning. In this study, the multimedia whiteboard allowed for abstract representations of real objects, facilitation expression of ideas, as well as annotation, mathematical reasoning, and peer communication. With help of Java Web Start technology, the client user interface of VMW system was integrated with asynchronous discussion threads (reference Figure 4). Teachers can use the whiteboard to start a new discussion thread with an initial question and then students use the whiteboard to respond or reply to messages already posted, engaging in a potentially rich opportunity to share and discuss ideas on the internet.

Technology Acceptance Model

To evaluate functionality of VMW system, the valid measurement scales were needed to predict user acceptance of the system. Therefore, the Technology Acceptance Model (TAM) was used in this study. The TAM was developed by Davis (1989) and is based on the Theory of Reasoned Action (TRA) (Ajzen & Fishbein, 1975). In TAM, there are two beliefs focused on information system acceptance, which is perceived usefulness and perceived ease of use. Perceived usefulness was defined as the degree to which a person believes that using a particular system would enhance his or her job performance. The point of perceived usefulness is that if the users think a system might help some way, the attitudes they express will be positive. Perceived ease of use was defined as the degree to which a person believes that using a particular system would be free of effort, the point being that if users think the system is easy to use, then their attitude towards it will be positive, and increase the likelihood of continued use. For high reliable and valid measures, Davis used the step-by-step process to refine and streamline it, resulting in two six-item scales. In the context of education, TAM has been applied to educational technologies such as e-learning systems used with children (Liu, W., Cheok, A. D., Mei-Ling, C. L., & Theng, Y. L, 2000) (Shayo, C., Olfman, L., & Guthrie, R, 2000)(Hwang, W. Y., Chen, N. S., & Hsu, R. L, 2006). In this study, the proposed system was designed for upper intermediate elementary school students and their teachers.

Supporting Theories for Design

According to the literature review in the previous section, two pedagogical concepts, constructivism and multiple representation transformation, were employed to support the VMW design. For mathematics problems, especially for geometry, these two theories were considered together and employed to support our design for help students to build their geometric knowledge. Constructivism provides the pedagogical support to design VMW for students to construct their own knowledge, while multiple representation transformation gives the support of cognitive symbol translations to facilitate students' learning by transforming various forms of construction.

Constructivism

From the previous literature review, we know the emerging media holds promise to improve classroom learning activities, and the geometry concepts can result from experience in covering and stacking manipulations. The use of computer-based simulations has been recognized as a powerful tool to stimulate students to engage in the learning activities and to construct meaningful knowledge. As Whiteside (1986) stated Computer simulation-based instruction is useful to reach the analysis, synthesis, and evaluation--hierarchical levels in Bloom's taxonomy. Now from the constructivist viewpoint, using computer 3D virtual reality and its manipulation to support mathematical geometry learning activity is beneficial. Constructivists claim that individuals learn through a direct experience of the world, through a process of knowledge construction that takes place when learners are intellectually engaged in personally meaningful tasks (Conceicao-Runlee & Daley, 1998). Chittaro and Serra (2004) claimed that Constructivism is the fundamental theory that motivates educational uses of Virtual environments as follows:

"Our type of experience is a firstperson one, that is a direct, non-reflective and, possibly, even unconscious type of experience. On the contrary, third-person experiences, that result from the interaction through an intermediate interface. In many cases, interaction in VEs (Virtual Environments) can be a valuable substitute for a real experience, providing a first person experience and allowing for a spontaneous knowledge acquisition that requires less cognitive effort than traditional educational practices."

Winn (1993) also claimed virtual environments can provide three kinds of knowledge-building experience that are not available in the real world; they are concepts of size, transduction and reification, which have invaluable potential for education. In this study, 3D computer simulation was employed to model real-world geometry problems, and learners can be deeply engaged in task-related manipulation to solve them.

Multi-Representation Transformation

In cognitive psychology, representations refer to hypothetical internal cognitive symbols that represent external reality. Lesh, Post & Behr (1987) pointed out five outer representations used in mathematics education including real world object representation, concrete representation, arithmetic symbol representation, spoken-language representation, and picture or graphic representation. Among them, the last three are more abstract and higher level of representations for mathematical problem solving (Johanna, n.d.; Kaput, 1987; Lesh et al., 1987; Milrad, 2002; Zhang, 1997).

Some learners favor visual or concrete representations, while others favor symbolic or abstract representations. Normally, students with high problem solving abilities are those who can skillfully manipulate the translation of language representation (verb or vocal), picture representation (picture, graphic) and formal representation (sentence, phrase, rule and formula). On the other hand, students with low problem solving abilities generally have difficulty in the translation of different representations in problem solving. Furthermore, since students have different learning styles, it is useful to provide various learning strategies and media to allow students to explore multiple representations in class, thereby enhancing their learning performance (Hwang et al., 2007).

To understand how the VMW system supports multi-representation transformation, we can imagine a fictitious instructional space in the classroom. The teacher and all students take their transparent whiteboards and stand around a table, where several kinds of geometric objects are placed to allow the students to study geometrical problems and concepts. Students can manipulate geometric objects or observe others manipulating geometric objects, so some ideas for problem solving or concepts underlying the questions may appear in their mind (imagery). This kind of representation transformation is from concrete manipulation or observation to imagery concept in the mind (as shown in the upper part of Figure 2). Afterward, the students choose appropriate viewpoints behind the whiteboard to view the 3D theme and write down their solutions on the whiteboard with symbol or texts, moving from imagery concepts in the mind to abstract symbols on the whiteboard.

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Students can also share their ideas via the virtual whiteboard. When students study others' solutions in the whiteboard, they may try to understand the equation and symbolic explanation (abstract to imagery) of others. Furthermore, students can validate or refute others' thinking by manipulating geometric objects (imagery to concrete). This reversed transformation was shown in the lower part of Figure 2. (abstraction -> imagery -> concrete)

Research Design

Measurement

An 18 item questionnaire with five-point Likert-type scale was used to investigate perceived acceptance of the VMW system. The perceived satisfaction and perceived improvement in geometry problem solving with the VMW system were also surveyed. The questionnaire's design was based on the Technology Acceptance Model, and involved some aspects of multi-representation usage (multimedia whiteboard and Virtual Manipulatives). Previous study of multimedia whiteboard questionnaire (Hwang et al., 2006) was referenced to develop the new questionnaire. This study follows TAM's definition, with perceived usefulness being defined as the degree to which a user believes that using a VMW system would enhance his or her task performance for solving geometry problem. In contrast, perceived ease of use refers to the degree to which a person believes that using a VMW system would be free of effort. The scale items are worded in reference to Virtual Manipulatives, whiteboard usage and geometry problem solving issues.

The factors of questionnaire items are summarized in Table 4. Notice the table implies each item is corresponding to distinct factor. For perceived usefulness, corresponding factors include task performance, useful, effectively, make task easier and enjoyment. They all come from the TAM, with the exception of enjoyment. The major difference between the current study and the original TAM questionnaire design is that in the original TAM questionnaire, questions covered the entire system without looking at functional details. In order to help young students understand the meaning and intent of the questions, each item was described with a detailed description of the function of the VMW system. The factor of enjoyment was added based upon the belief that will enjoy working with a well designed system (Norman, 2004). With regards to perceived ease of use, the corresponding factors were made explicit and included controllability, ease of learning, ease of skill development and ease of use, all of which come from TAM.

The questionnaire was originally in Chinese, and was reviewed by an elementary Math teacher to make sure that students could understand the meaning of each item. All the items in questionnaire were framed in positive ways, based on the authors' review of related journal studies (Selim, 2003) (Hwang et al., 2006). In addition, the contents of students' solutions were analyzed, and their solving strategies classified and quantified into different classes.

Subjects

Twenty three 6th grade elementary students participated in this research. The evaluated period was one and half months. The students followed their Math teacher's instruction to solve eight geometry problems. The perceived acceptance of the proposed VMW system and its influences on geometry concept learning were investigated. Interviews the subjects and their content analysis were carried out to further investigate the causes underlying some interesting phenomena.

Activity Design

Since the proposed VMW system can be accessed from a website, teachers and students can easily engage themselves in problem solving anywhere and anytime if Internet access is available. In the beginning, teachers organized and stacked geometric objects to build geometry problems in the 3D space and gave problem descriptions on a virtual whiteboard. Students then started to solve problem and to express their solutions on their own virtual whiteboards. When students completed their answers, a peer review session was started. In this session, students were asked to review the answers of others and to critique their work using the virtual whiteboard. Students could therefore continually revise their answers, affirm or refute the work of others and engage in discussions of the work until the next problem was posed.

Research Tool: VMW and its Implementation

The VMW system is a collaborative tool for geometry problem posing and solving. We designed this system based on work in the literature review above. The geometry knowledge construction should be based upon covering and stacking activities. The VMW system provided virtual 31) geometric solids as a form of concrete representation and users could pose and solve geometry problems by manipulating them. These geometric solids included cube, pyramid and sphere and so on. Users could select the solids and place them into the virtual space. The operations of stacking, partition, comparison and measuring operations are explained as following:

1. Stacking: the stacking operation involved movement of one block at at time into configurations with other blocks. Figure 3 shows the user how to use the stacking operation to move one cube on the top of a stair. Stacking blocks was one of the most useful activities in lesson. For example, teacher could stack many different kinds of blocks into a larger one, and ask students to determine the surface area or volume. Students could use stacking operation to rearrange the blocks to help them to find solutions to the questions asked.

2. Partition: Sometimes it was necessary to move a group of blocks or to partition one large block into many small pieces. In Figure 3, we can see the partition operation pushed all the blue blocks forward and the original blocks were divided into two groups. The partition mechanism was made it possible to break large blocks into different layers, turning a large problem into smaller and simpler components which the students could solve in stages.

3. Measurement: Measurement was employed to figure out the distance between two points. It is useful to find the width and height of multiple blocks to help in calculating their surface area or volume. For example, the width of stair blocks was measured as 5 units in Figure 3.

4. Comparison: Comparing was employed to find the difference between two distinct stacks of blocks. The VMW system can generate shadow blocks from the existing blocks. The shadow blocks are just one pseudo-figure of the original blocks, and it cannot be stacked and partitioned. Afterwards it was possible to change the transparency of shadow blocks so that students could explore the structure of stacked blocks and make some comparisons. The shadow blocks are useful to teachers, as they can use this feature to pose stacked blocks as shadow blocks. Then students used the above operations to manipulate their own blocks to compare their constructions with posed shadow blocks.

To model the 3D geometric solids in the VMW system, the open XML-enabled 3D standard-X3D was employed, which was proposed by the Web3D consortium. To help and to realize the X3D application development, the Web3D Consortium also promoted the open Java source project Xj3D. The open source project Xj3D provides Java API and toolkit for X3D application development and SAI (Scene Access Interface) could be used to access X3D objects with Java programming languages. In this study, we used X3D to describe the virtual 3D models in the VMW system and employed Xj3D API to implement and to access X3D objects.

For the VMW system delivery and maintenance mechanism, the Java Network Launching Protocol (JNLP) was employed to maintain the consistency of the Java module or library between the clients and the server. When users login and launch the VWM system using JNLP on the web page, the newest Java module, of the VWM system (if available), was automatically downloaded, installed and executed. After launch, the client program communicated with the server by Simple Object Access Protocol (SOAP).

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To support collaborative work for geometry problem solving among students and teachers, a discussion forum was employed, shown in the upper part of Figure 4. In the beginning, teachers first posed a new geometry problem with the VWM system, in which a 3D theme was built, with a problem description given in the whiteboard. All students studied the new geometry problem given by teachers and gave their answers by clicking a circular icon in the upper part of Figure 4 to launch the VWM client program. Students could also revise their solutions many times by replying their previous answers or by giving comments to others solution via others' discussion thread. The VMW system, combined with discussion thread, provides an asynchronous collaboration model. Its implementation is easier than a synchronous model. From the viewpoint of learning, Won et al. (2003) and Hwang et al. (2006) indicated that the asynchronous collaboration offers flexibility to the students. With the assistance of a semi-automatic administration system, students can study the materials of the course, take exams, as well as being assessed. Following the social constructivism and the scaffolding theory, this type of system gives students the support needed to acquire and consolidate new learning.

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Students can study the problem by manipulating 3D objects, and then choose appropriate viewpoints to create their own translucent whiteboards. The whiteboards can help them to find out the solutions via observation of the manipulated 3D theme behind of the whiteboards (the lower part of Figure 4). Moreover, after the teacher initiates a new question, students can collaboratively discuss the question by generating their own translucent whiteboards combined with 3D scene behind. In Figure 5, there were three whiteboards that were created asynchronously. One was created by the teacher and two were created by students. Teacher used one whiteboard to describe geometry problems, while on the students' whiteboards they discussed how to solve the problem together.

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Results and Discussion

Questionnaires

The questionnaires were sent to the 23 subjects and the exclusion of responses from incomplete questionnaires resulted in a total of 20 valid questionnaires. Table 1 shows the reliability of the questionnaires. It was found that the value of Cronbach alpha in the two dimensions, namely perceived ease of use and perceived usefulness, were both higher than 0.8 which is deemed acceptable. Table 2 and 3 show the results of perceived usefulness and perceived ease of use in the questionnaires, respectively. The subjects agreed the VMW system was very useful and the values were higher than 4.0 out of 5.0 on average. Furthermore, it was found that most subjects agreed that the VMW can help them to use different representations for solving geometry problems as well as facilitating and broadening their thinking from different viewpoints in the 31) theme. Meanwhile, they think the VMW can help them to show their solutions more completely.

Significantly, the high scores on the perceived usefulness questionnaire showed the proposed system and the multi-representation transformation mechanism were considered to provide some help in geometry problem solving by students. Most questions for assessing usefulness of the system were related to multi-representation, such as using 3D block as the concrete representation and using the multimedia whiteboard for expressing ideas with abstract symbols.

As for ease of use, most subjects had a positive attitude toward the VMW system. However, some perceived ease of use items revealed that the children had some trouble using the VMW system. Teachers found that although the whiteboard was integrated in a virtual 3D space, some students preferred using pen and paper for reasoning when they were exploring the 3D scene. The whiteboards were therefore only used for writing down their final solutions. Based on evidence gleaned from interviews, we know this was because that some students felt more comfortable when they used pen and paper for Math reasoning work. To promote students' use of the whiteboard tool in addition to pen and paper, researchers have decided to implement some calculating widgets to help with Math reasoning in a future release of the VMW system.

Importantly, according to the observations, the students were highly motivated by the interesting, fresh, new 3D manipulative software. Such an effect may have caused students to put more effort than usual into using the tools to explore geometry problems. Therefore a longer study is needed to determine if student enthusiasm and interest stemming from the novelty of the new tool was more responsible for improved performance than the tool itself.

Content Analysis

The Math teacher posed eight geometry questions in this study. They were all area and volume reasoning problems. The eight questions were divided into two categories, irregular and regular shapes stacking with blocks. In the irregular category, some blocks were employed to build irregular shapes. Since these shapes were not regular, it was neither easy nor straightforward to find the math rules needed to get the answers. Sometime students could re-stack cubes into regular or near regular shape to find an efficient way to get the answer. In regular stacking questions, blocks were stacked in specific ways so students could use the stacking rule for the solutions. However, if the number of blocks was huge, and if students used a straightforward counting method to get the answers, a great deal of time was required and there was a high probability of mistakes being made.

There were two questions, one regular and one irregular, chosen for content and statistical analysis, which we called question A and B respectively (see Figure 6 and Figure 7). The students' whiteboard contents were analyzed to find their solving strategies (not including revised solutions from the peer review session). To show solving strategies clearly, the description of their strategies and the statistical distribution of correct answers and wrong ones were given. Their solving strategies of question A and B were described in details as follows.

Question A: How can we find the volume and surface area of irregular stacking blocks? In this question, the Math teacher used 19 blocks to create an irregular, three tiered object. Although the stack had no specific rule, students were expected to use a more efficient way to figure out the answers rather than merely counting.

To determine the volume, more than half students used a re-stacking method to simplify the problem, as shown in Table 4. One of those students got the wrong answer due to mistakes in calculation. The remaining students simply counted the blocks. The first group of students re-stacked the blocks into one-level or multi-level rectangular solids. Four students first calculated the volume of cuboids, and then added the remaining cubes or subtracted missing ones to get their answers, as shown in Figure 6A. Three other students re-stacked all cubes into several one-level rectangles of different sizes and added up the volume of all the new rectangles to get the answer. Five students reorganized the object into one single-layer rectangle and calculated the answer.

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The surface area reasoning was more difficult than volume reasoning and most students used some kind of rule to get their answer. As shown in Table 5, there were four students who got wrong answers because of a misconception or errors in calculation. The students who got the right answer efficiently applied a rule related to classifying six surfaces of irregular blocks: six surfaces were divided into three pairs, two surfaces in each pair have the same area. Therefore, the sum of three different area surfaces' areas was first calculated and multiplied by 2 to get the answers. Only five students used a straightforward counting method to determine the surface area of the stack.

Question B: How do we find the volume and surface area of pyramid? To study how students figured out the rules to find the volume and surface area of a regular stacking shape, the Math teacher stacked a seven-level pyramid. Then she asked students to find its volume and surface area.

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Most students used partition strategies to find the rule and simplify the computation to determine the volume, as shown in Table 6. Students using a horizontal partition strategy divided the pyramid into seven square levels and easily calculated the volume of each one. The total volume was then determined by adding the volume of the levels together. However, one student employed a vertical partition strategy, decomposing the pyramid into vertical slices, and then adding up the volume of all the slices to get the answer (in Figure 7). The results showed the versatility of the VMW system, as it provided the students with the flexibility to explore new ideas, and gave the teacher ways to understand and promote the way the students' thinking in a geometrical problem solving context.

In surface area portion of the problem only half of the students got the correct solution, as shown in Table 7. Seven students found the rule that the top and bottom surface belonged to the same dimension and their surface areas were equal (bottom surface area multiplied by 2) and four other faces' surface areas (the front, back, right and left side of the pyramid) were equal (one face surface area multiplied by 4). The others used a method similar to the one used to answer question A to figure out the surface area. Even pyramid is regular in shape. But question B seemed to be more difficult than the question A, because the pyramid of the question B had a large number of blocks. Almost half students did not get the correct answer for volume and surface area.

Asynchronous Collaboration

During the activity, teachers did not only ask students to write their own solutions but also encouraged them to comment on the solutions provided by their classmates. As stated earlier, with the VMW system, the multimedia whiteboard can be used to represent abstract ideas in Virtual 3D scene, and allows teachers and students to communicate and share ideas through n asynchronous discussion board. Some examples of asynchronous collaboration between students are described below.

Figure 8 shows one example regarding mathematical discussion. Although student A gave correct answers to the volume and surface area questions, students B and C gave critiqued the work and asked student A to find a way to simplify the Math calculation and to describe the physical meaning of the solution. In Figure 9, student A gave the incorrect answer to the problem of surface area, and student B responded by pointing out what was wrong, after which student A revised his answer. Figure 10 gives another example of the way students critiqued each other's work and revised their own answers. Comments by Student B reminded student A that he had made some mistakes in calculating the volume, whereupon Student A corrected his answer and gave a clearer description of how he solved the question.

The ability to share their ideas, and to affirm, refute and respond to the work of their peers, as facilitated by the VMW system, helped the students to clarify their thinking about mathematical problem solving. With the asynchronous discussion board and teacher's encouragement, most students were willing to use the system to describe their ideas and solutions clearly and thoroughly. Through this kind of interactive communication, more correct answers and meaningful responses to others' comments or queries were derived.

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Conclusions

In this paper, the Virtual Manipulatives and Multimedia Whiteboard system was proposed to promote a multi-representative construction model to help solve geometry problems. The proposed system was also evaluated with using a perceived acceptance questionnaire, and the results showed that most of subjects considered the system to be useful and easy to use, the researchers having used questionnaires to assess the usability and friendly of proposed system. Making the system even more user-friendly will be a major improvement in the next generation of the system.

Students' solutions were analyzed to determine how they used our tools to solve geometry problems in VMW system. The result showed that more than half of the subjects relied on partition and stacking methods to find the solutions. Meanwhile, versatile approaches to solve geometry problems were also found in this study, indicating that the VMW system could provide more flexible thinking than paper and pencil activities or even manipulation of actual physical objects to allow students to reach their full potential in understanding and solving geometry problems. As previously mentioned, recent literature shows that 3D simulations have invaluable potential for education and promising new directions for study were found as our study sought to enhance geometrical learning by integrating the multimedia whiteboard and Virtual Manipulatives in the VMW system.

For future research, the researchers will include the technology of pattern recognition to classify students' manipulation automatically or semi-automatically and try to find its relationship with the solving strategies, or perhaps study other issues such as the role of gender in problem solving. Although it is difficult to probe mysterious depths of the human mind, it is possible to discover the processes involved in cognitive operations by observing and analyzing the behaviours and verbal expressions of people engaged in intellectual activity (Ashcraft, 2001). Therefore, determining correlations between manipulation and problem solving strategies is helpful in the investigation of how children learn geometry concepts.

Moreover, the authors consider the multi-representative construction model to be a generalized concept. The combination of Virtual Manipulatives and Multimedia Whiteboard design put multi-representation and Constructivism into practice. It can be applied not only to geometry problem solving but also to other domain of learning. Since the VMW system provided a forum for discussion, it allowed teachers and students to pose and solve questions and to share ideas, actively engaging the students in the learning process. We can extend such a model to Internet users so they could use the whiteboard to share ideas and to work collaboratively with 3D model manipulation. It could also be useful in the promotion of product design, game playing and advertising, because useful suggestions from users around the world can be collected, analyzed and used in a variety of ways. Researchers plan to design more 3D models and manipulation widgets to support such endeavor, and to extend their design into various applications and to verify its effectiveness in future studies.

Acknowledgements

This research is partially financial supported by National Science Council, Taiwan, R.O.C. The contract number is NSC96-2524-S-008-002.

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Wu-Yuin Hwang (1), Jia-Han Su (2), Yueh-Min Huanga (3) * and Jian-Jie Dong (2)

(1) Network Learning Technology Institute, Graduate School of Network Learning Technology, National Central University, Taiwan

(2) Department of Computer Science & Information Engineering, National Central University, Taiwan

(3) Department of Engineering Science, National Cheng Kung University, Tainan City, Taiwan // huang@mail.ncku.edu.tw

* Contact author
```Table 1. Reliability analysis for perceived acceptance
questionnaires

Part   Questionnaire           Cronbach alpha

1.     Perceived usefulness    0.90
2.     Perceived ease of use   0.82

Table 2. Perceived usefulness

NO   Perceived usefulness             SD   D   U   A    SA   Ave.
1    2   3   4    5

l.   Teacher uses voice recording     0    1   4   8    7    4.05
and 3D blocks to organize
and pose geometry question
is clearer  than using text
description only.

2.   I find it useful to have free    0    0   3   6    11   4.04
viewpoints to solve geometry
problem.

3.   Using text writing,              0    4   8   8    4    3.80
free-sketching and voice
recording which provided
by multimedia whiteboard
meets my requirement to
solve geometry problem.

4.   Using 3D blocks to design        0    0   3   10   7    4.20
geometry problem promotes
me to think from wide angle.

5.   I find it useful to use          0    0   5   9    6    4.05
VMW system to show my
solution completely.

6.   Using VMW system gives           0    1   6   5    8    4.00
enjoyment to me for
solving geometry problem.

7.   I find it useful to place        0    0   1   12   7    4.30
multimedia whiteboard in
different viewpoints for
solving geometry problem.

8.   Using VMW system helps           0    0   5   7    8    4.15
me to learn geometry
problem solving.

9.   Using VMW system gives           0    1   7   8    4    3.75
clues to me for discovering
solving strategy

SD: Strongly disagree, D: Disagree, U: Unsure,
A: Agree, SA: Strongly agree, Ave: Average

Table 3. Perceived ease of use

NO.   Perceived ease of use        SD   D   U   A    SA   Ave
1    2   3   4    5

1.    I find it easy to use view   0    3   7   9    1    3.40
3D virtual scene.

2.    I find it easy to            0    4   8   7    1    3.25
manipulate 3D blocks
in 3D virtual scene.
3.    I find it easy to            1    2   9   6    2    3.30
change 3D blocks'
color and transparency.

4.    It takes a short time        0    1   8   9    2    3.60
to learn to use VMW
system.

5.    I find it easy to use        0    3   2   9    6    3.90
text writing function
of multimedia
whiteboard.
6.    I find it easy to use        0    4   7   7    2    3.35
free-sketching function
of multimedia whiteboard.

7.    It takes a short time        0    1   6   10   3    3.75
to learn to use
multimedia whiteboard.

8.    Overall, I find it easy      0    6   9   5    0    2.95
to get VMW system to do
what I want it to do.

9.    Overall, learning to         1    3   7   7    2    3.30
operate VMW system is
easy for me.

SD: Strongly disagree, D: Disagree, U: Unsure,
A: Agree, SA: Strongly agree, Ave: Average

Table 4. Corresponded factors of perceived
usefulness and ease of use items

NO.           Factor

Usefulness

1.            Performance for posing/solving geometry question
2.            3D viewpoint is useful to figure out solution.
3.            Whiteboard presentation enhances effectiveness.
4.            3D presentation enhances effectiveness.
5.            VMW system is useful to show my solution.
6.            Enjoyment.
7.            Free whiteboard placement is useful to figure out
solution.
8.            VMW system is useful to figure out solution.

Ease of Use

2.            Virtual Manipulation is controllable.
3.            Changing block's attribute is controllable.
4.            Easy to learn VMW system.
5.            Text writing function of whiteboard is controllable.
6.            Free-sketching function is controllable.
7.            Easy to learn whiteboard function.
8.            Easy to become skillful with VMW system.
9.            Easy to use VMW system.

Table 5. The distribution of volume solution methods in question A

Method                                                 Correct   Wrong

1. Straightforward Counting method                     10        0
2. Re-stacking Method                                  12        1
2.1 Multilayer Stack                                 7         0
2.1.1. Stack into rectangles with variable sizes   3         0
2.1.2. Stack into cuboids                          4         0
2.2. Single-layer Stack                            5         1
Total                                                  22        1

Table 6. The distribution of surface area
solution methods in question A

Method                                                  Correct   Wrong

1. Straightforward Counting method                      3         2
2. Calculating three surfaces area and multiplying 2    16        1
3. Misconception                                        0         1
Total                                                   19        4

Table 7. The solution distribution of volume problems in question B

Method                                          Correct   Wrong

1. Give up                                      0         4
2. Horizontal partition                         15        2
2.1. Summing all levels                       12        2
2.2. Specifying extra blocks in next level    3         0
3. Vertical partition (Outward extension)       1         1
Total                                           16        7

Table 8. The solution distribution of surface area in question B

Method                                          Correct   Wrong
1. Give up                                      0         5
2. Misconception                                0         6
3. Specifying each faces' surface area          11        1
3.1. Bottom surface area multiplied by 2+     7         1
one side surface area multiplied by 4
3.2. Summation of top, left and front         4         0
surface area multiplied by 2
Total                                           11        12
```
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