A computer-assisted learning system for exploring geometry ...

Kong, S.C., Ogata, H., Arnseth, H.C., Chan, C.K.K., Hirashima, T., Klett, F., Lee, J.H.M., Liu, C.C., Looi, C.K., Milrad, M., Mitrovic, A., Nakabayashi, K., Wong, S.L., Yang, S.J.H. (eds.) (2009). Proceedings of the 17th International Conference on Computers in Education [CDROM] . Hong Kong: Asia-Pacific Society for Computers in Education.

A computer-assisted learning system for exploring geometry conjectures online Wing-Kwong WONGa, Chang-Zhe YANGb, Sheng-Kai YINc a Department of Electronic Engineering b Institute of Computer Science and Information Engineering c Graduate School of Engineering Science & Technology National Yunlin University of Science & Technology, Douliou, Yunlin, Taiwan {wongwk, g9617727, g9310811}@yuntech.edu.tw Abstract: We believe it would be fun and meaningful for students to make geometric conjectures and experience the joy of mathematical inquiry by manipulating dynamic geometry figures in a system we have designed. In order to facilitate the students to explore geometric conjectures, we have designed tools for constructing geometric objects, measurement tools, and tools to construct geometric conjectures. In an experiment, two classes of high school students were compared. One class used our system while the other used a JavaSketchpad environment. The study indicated students produced more and better conjectures with our system. Keywords: Math exploration, conjecture generation, dynamic geometry environment, Geometer’s Sketchpad, JavaSketchpad

1. Introduction According to some researchers (e.g., Hanna, 2000), geometry proofs serve a number of purposes in math education: verification, explanation, systematization, discovery, communication, construction, and exploration. Geometry theorem proving is highly regarded as an important skill. In traditional geometry education in Taiwan, learning is often passive, recitative, and repetitive (Lin & Tsao, 1999). When students encounter a novel problem, they often fail to solve it. Instead of following traditional instruction approach, instructors should focus more on students’ development of skills in discovering conjectures and facing intellectual challenges. We believe that self-guided problem solving and conjecture exploration will facilitate student learning in geometry theorem proving. Lakatos (1976) proposed that the first step of making discoveries in math is to produce conjectures. Conjectures make interesting connections between geometric concepts and properties. The interleaving steps of conjecturing, justifying, proving, and modifying would enhance students’ problem solving skills. Therefore, it is very important to encourage students to make conjectures systematically. According to Hoyles and Healy (1999), exploration of geometric properties in a dynamic geometry environment (DGE) could motivate students to explain their empirical conjectures by constructing a formal proof. In a DGE, students can easily be convinced of the general validity of a conjecture by seeing its invariance in graphic display while geometric objects undergo continuous transformation (de Villiers, 2003). We have designed and developed a DGE that encourages students to explore geometric properties associated with given geometry theorems. In order to facilitate the exploration of geometry conjectures, the system provides a set of object construction tools, such as midpoint and angle bisector. To evaluate the system, we compared two classes of high school students in an experiment. One class used our system while the other used a popular dynamic geometry environment called JavaSketchpad (JSP, 167

Kong, S.C., Ogata, H., Arnseth, H.C., Chan, C.K.K., Hirashima, T., Klett, F., Lee, J.H.M., Liu, C.C., Looi, C.K., Milrad, M., Mitrovic, A., Nakabayashi, K., Wong, S.L., Yang, S.J.H. (eds.) (2009). Proceedings of the 17th International Conference on Computers in Education [CDROM] . Hong Kong: Asia-Pacific Society for Computers in Education.

http://www.keypress.com/sketchpad/). We want to find out whether the tools of our system would facilitate better student performance in making geometry conjectures. 2. User Interface This section describes the user-interface and tools of the web-based dynamic geometry conjecture system that we have designed. With the system’s friendly user-interface, the user can take measurements of geometric objects in order to confirm his suspicion of relationship between the objects. If the user wants to add auxiliary lines, she can do it with a system tool. After enough exploration, if the user has convinced herself of the validity of a proposition, she can use a tool to write down the proposition and enter it as a conjecture. All these activities help students reflect on the meaning of the conjecture and might even help them to prove the conjecture, since some or all the propositions needed by the conjecture might have already been considered by the user. Figure 1 shows the user interface of the system.

Figure 1. User interface 3. Experiment of Conjecture Generation Two high school classes (about 15 ~ 16 years old) participated in an experiment to compare how well students did in generating conjectures in our system and in a JSP environment. The former class of 39 students was the experimental group while the latter class of 42 students was the control group. The math scores of the students in school indicated that the two classes had similar competence levels in math. The experiment took place in a classroom with personal computers. Each class spent 50 minutes for the experiment. There were two geometry problems used in the experiment. In a textbook, the first problem was solved with a proof that involved the use of auxiliary lines but the second problem was proved without adding any auxiliary line. However, this experiment did not ask the students to prove the problem. For the first problem of the experiment, the students were asked to find propositions that might be needed for proving the theorem. For the second problem, the students were asked to find all conjectures, including those that were not needed for a proof. The experimental group constructed auxiliary lines, made measurements, and produced conjectures in our system, all of which were recorded 168

Kong, S.C., Ogata, H., Arnseth, H.C., Chan, C.K.K., Hirashima, T., Klett, F., Lee, J.H.M., Liu, C.C., Looi, C.K., Milrad, M., Mitrovic, A., Nakabayashi, K., Wong, S.L., Yang, S.J.H. (eds.) (2009). Proceedings of the 17th International Conference on Computers in Education [CDROM] . Hong Kong: Asia-Pacific Society for Computers in Education.

automatically in a MySQL database. The control group constructed auxiliary lines and produced conjectures on paper. According to the experimental data, the experimental group produced more conjectures than the control group and the proportion of conjectures that were valid propositions was also higher. In fact, the experimental group produced about twice as many auxiliary lines as the control group. 4. Conclusion Most researchers would agree that a dynamic geometry environment is a good tool to demonstrate the invariance of geometric properties in a figure that is dynamically changing. But there are several problems in using DGE as a tool for students to explore and produce geometric conjectures. First of all, instructors need to construct DGE with a script of with the graphical user interface of software such as GSP. Moreover, no tools, among those we have surveyed, allow the user to add auxiliary lines freely in the process of exploration. We propose to address these problems in designing a system for students to explore geometric conjectures in the context of geometry proofs. Finally, few tools can automatically record the user’s actions in a DGE. Our system provides tools for a user to construct auxiliary lines to explore various conjectures with a web browser. Measurements can also be freely made by the user in order to confirm the relationship she suspects to hold between two geometric objects, such as whether two segments have the same length. The user can also construct conjectures with menu-based interface. Such features are seldom implemented in any Web-based environments for conjecture exploration. An empirical experiment was conducted to compare the experimental group that used our system with the control group. For the control group, students could play with a dynamic geometry figure a Web-based plain JSP environment without online tools for adding auxiliary lines, making measurements, or constructing conjectures. Results indicated that the experimental group produced more conjectures and produced conjectures with a greater proportion of validity. This might mean that our system was successful in motivating students to make more useful conjectures. Whether this would lead to their success in proving the conjectured propositions remain an important question, which we intend to explore further in future work. Acknowledgements This study is supported 95-2520-S-224-001-MY3).

by

the

National

Science

Council,

Taiwan

(NSC

References [1] [2] [3] [4] [5]

Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1), 5-23. Lin, F. L. & Tsao, L. C. (1999). Exam Mathsre-examined. In C. Hoyles, C. Morgan & G. Woodhouse (Eds.), Rethinking the mathematics curriculum (pp. 228-239). London: Falmer Press. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press. Hoyles, C. & Healy, L., 1999, Linking Information Argumentation with Formal Proof Through Computer-Integrated Teaching Experiences. In Zaslavsky (ed. ), Proceedings of the 23nd conference of the International Group for the Psychology of Mathematics Education, Haifa, Israel. De Villiers, M. (2003). Rethinking Proof with Sketchpad, Key Curriculum Press, Emeryville: USA.

169