8학년 '삼각형의 성질' 증명 지도를 위한 IDEAL 전략별 단계적 발문 = Phased Questions in Each Step of The IDEAL Strategy for The Proof Instruction of 'The Properties of Triangle' at The Eighth Grade

다국어 초록 (Multilingual Abstract)

The proof in mathematics has been essential for the mathematical consideration with problem-solving since Euclid Principles and it has significant meanings for students to develop deductive inference ability. However, it is true that both teachers and students have some troubles with instructing and learning about the proof. The teachers have a tendency to explain proof contents at eye height of teacher oneself and it is distant from student's activities. Because the students are just trying to memorize the proof without any thinking process, it is easy for them to lose an interest in learning about the proof. To make the proof take back the original meaning for the development of the mathematical thinking and reasoning ability, it is a better way to learn by passing the systematic process focused on students in detail than just by memorizing automatically like a machine. Also the teachers would help the student to be able to learn proofs by themselves, providing phased questions during this process.
On the point of view like this, this study has proposed the plan which applies phased questions in each step of the IDEAL strategy for the proof instruction of 'the properties of triangle' at the eighth grade where the deductive proof is initially introduced.
The background theories of this study are like this: the contents and the real conditions and the way to improve the proof instruction in the geometry area, the IDEAL strategy, the phased questions proposed by Yi, Yong-Ryool(1997a, 1997b) to improve the thinking ability.
The results of this study are following below.
First, this study has investigated all the detail elements of the proof instruction at the eighth grade in 'the properties of triangle' unit and then analyzed the characteristics of the proof instruction.
Second, this study has tried to develop systematic phased questions in each step of the IDEAL strategy for the proof instruction. For the proof instruction, this study has specified the detail elements of the proof instruction in the IDEAL strategy. And then, to apply the phased questions in the IDEAL strategy, it has investigated by connecting the IDEAL strategy with the 4-step-process for problem-solving (recognizing and formulating problems→ building rough plans to solve problems→ executing the plans→ reviewing logically and developing into more advanced way) proposed by Yi, Yong-Ryool(1997b) and Pyun Dong Joong Nam. After that, it presented "phased questions in the IDEAL strategy" as the IDEAL strategy and phased questions(which are proposed by Yi, Yong-Ryool(1997b) in his problem solving 4-step-process) are connected, in the proof instruction. Phased questions are composed of the following three questions making the order (questionⅠ→ questionⅡ→ questionⅢ) at each problem solving process step: questions(questionⅠ) that help to have the mathematical attitudes, questions(questionⅡ) that make to pop up the mathematical thoughts (methods or ideas), questions(questionⅢ) that suggest the mathematical knowledges and functions.
Third, this study has drawn up a development section which applies phased questions in each step of the IDEAL strategy for the proof instruction. This development section was prepared as an example for proof instruction regarding 'the meaning and characteristics of triangle's incenter'. Through considering each step of the IDEAL strategy, this study specified the meaning of each step and phased questions which applies at each step in the proof instruction. Then, this study showed an example of the activity result of the students who finished each step.
The following suggestions can be given on the basis of the results of this study.
First, we need to develop various examples which applies phased questions in each step of the IDEAL strategy in the proof instruction for the students.
Second, we need to verify the real effect of phased questions in each step of the IDEAL strategy through tests before and after.
Third, IDEAL strategy is generalizable to all curriculum areas and to a variety of problems, whether they are well-defined problems(e.g., the word problems typically seen in math curricula) or ill-defined problems with no single answer, with many solution paths. Not only for the proof instruction but also for the general mathematics instruction, we need to develop various instruction plans which applies phased questions in each step of the IDEAL strategy.

목차 (Table of Contents)

• Ⅰ.서론 1
• 1.연구의 필요성 및 목적 1
• 2.연구의 방법과 제한점 3
• Ⅱ.이론적 배경 5
• 1.도형 영역의 증명 지도 5
• 2.IDEAL 전략 11
• 3.단계적 발문 20
• Ⅲ.8학년 ‘삼각형의 성질’ 증명 지도를 위한 IDEAL 전략별 단계적 발문 30
• 1.8-나 삼각형의 성질 증명 지도의 세부 요소 30
• 2.증명 지도를 위한 IDEAL 전략별 단계적 발문 32
• 3.IDEAL 전략별 단계적 발문을 활용한 증명 수업의 전개안 42
• Ⅳ.결론 및 제언 53
• 1.결론 53
• 2.제언 54
• 참고문헌 55
• 영문초록 57
• 부록 60