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Drawing Dynamic Geometry Figures with Natural Language

Wing-Kwong Wonga, Sheng-Kai Yinb, Chang-Zhe Yangc

a Department of Electronic Engineeringb Graduate School of Engineering Science & Technology

c Institute of Computer Science and Information EngineeringNational Yunlin University of Science & Technology

Douliou, Yunlin, Taiwan{wongwk, g9310811, g9617727}@yuntech.edu.tw

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Introduction (1/2)

The NCTM publication: Principles and Standards for School Mathematics (2000), which focused more on the skill of writing formal proofs of geometry (Knuth, 2002).

The ability in writing geometry proofs involve important skills that are difficult to learn (Koedinger, 1998; Whiteley, 1999).

DGEs are used in teaching geometry in school, such as Geometer's Sketchpad (GSP), Cabri Geometry II, Geometry Expert, and Cinderella’s Café.

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Introduction (2/2) Though researches have found DGE has many

advantages, there are two common problems in teaching activities:

(1) Teachers and students have to learn to use the DG software and the learning process can be difficult.

(2) There are few data on how students manipulate the dynamic figures to discover conjectures or help write their proofs.

To address the problems, we need a dynamic geometry system that both can “understand” geometry problems and collect data of students’ interactions with the dynamic geometry figures.

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Background (1/3)

Problems of using DGE in class:

(1) Students need both basic geometry knowledge and the knowledge to work in DGE. This can be an obstacle to some students, reducing the effectiveness of learning in DGE (Talmon & Yerushalmy, 2004).

(2) Despite the potential benefits in using DGE in geometry classes, DGE is not available to some high schools in Taiwan: funding, over-emphasis on public exams, time-consuming

To make DGE more accessible is very important.

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Background (2/3) Natural language understanding for computer-assist

ed learning:

(1) Lees & Cowie (1996) proposed an enquiry system for training students to learn UNIX commands.

(2) Li & Chen (1988) proposed a Chinese enquiry system about fundamental knowledge of computer.

(3) Lu, Wu et al. (2005) proposed a model to simulate procedural knowledge of basic arithmetic operations.

(4) Wong, Hsu et al. (2007) proposed a system LIM-G (Learners’ Initiated Model for Geometry), which is used to understand geometry word problems and help elementary school student comprehend geometry word problems.

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Background (3/3)

Can users construct dynamic geometry figures on a webpage by inputting a geometry problem in natural language?

We present a tool for drawing dynamic geometry figures by understanding the texts of geometry problems.

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System architecture & user interface

Dynamic geometry figure

User information

Problem text for input

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InfoMap: knowledge engineering A knowledge engineering tool provided by the Intelli

gent Agent System Lab, Institute of Information Science, Academia Sinica. It is an ontology-based system for knowledge representation and template matching (Hsu et al., 2001).

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Knowledge base of geometry concepts Before the understanding task, we need to

build knowledge base of geometry concepts first. There are more than 50 concept nodes of geometry in the KB.

Lexical node

Linguistic knowledge node (template)

Concept node

“Point A is the midpoint of segment BC”

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JavaSketchpad: Dynamic geometry A computer program with which authors publish

dynamic geometry figures of Geometer’s Sketchpad as a Java applet embedded in a HTML file so that users can interact with the figures with a web browser on the internet.

The JavaSketchpad script

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Text understanding

Text understanding: an example “(1) Consider parallelogram ABCD. (2) Point E is the midpoint of segment AB. (3) F is the midpoint of segment CD. (4) Prove the length of segment DE is equal to the length of segment FB.”

Parsing all sentences of a text at one time (Wong et al., 2007) is not flexible.

Parsing a sentence at one time and integrating the results of all sentences is better.

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Script generation:(1) To map the concepts of a sentence into one or

more JavaSketchpad commands to draw the concepts.

(2) JavaSketchpad cannot draw some basic geometric objects directly, such as equilateral triangle, isosceles triangle, trapezoid, parallelogram, angle bisector and arc.

To address the problem, some scripts are designed by imitating the compass and straightedge constructions.

Generation of JavaSketchpad Script

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Core technologies of the system (7/7) An example of constructing an angle bisector:

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Experiment on understanding theorems and proofs From textbooks and reference books for junior high

school, pick 34 theorems and proofs on quadrilaterals and 27 problems on triangles.

Experimental Results:

Type Quadrilaterals Triangles Total

Correct 31 24 55

Incorrect 3 3 6

Total 34 27 61

Rate of correctness 92% 89% 90%

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Conclusion

Dynamic geometry environment such as Geometer's Sketchpad is recognized as a tool with great potential educational value.

We propose to mechanically draw dynamic figures from input problem texts.

Empirical experiments indicate that about 90% of problems from textbooks for junior high school can be comprehended to produce correct dynamic geometry figures.