CAS 環境에서 媒介變數 槪念과 因數分解 一般化에 대한 質的 硏究

다국어 초록 (Multilingual Abstract)

The purpose of the study was to analyze the process of students’ understanding in technique-oriented ways and to investigate which roles CAS plays epistemically in this process. The concept of the parameter and factorization’ generalization were selected as math domain for this study, and the two qualitative researches were implemented. The theoretical framework consisted of the cognitive tools, the instrumental approach, the anthropological approach, and the concept of the parameter and factorization’ generalization.
The first research of six high school students was conducted to investigate how students understand the concept of the parameter for solving the algebra problem in a CAS calculator environment. After analyzing the process of students’ understanding on the concept of parameter in technique-oriented ways, it was found that students’ mathematics learning using the CAS techniques changed their epistemologies. The CAS techniques’ potential and opportunity gave rise to new meanings, and CAS’ constraints often caused an obstacle in students’ mathematics learning. The CAS technique help students justify mathematical property and concept in their own words, and various combinations of the paper-and-pencil techniques and the CAS techniques enriched their problem solving strategies.
The characteristics on the process of students’ understanding the concept of parameters in a CAS calculator environment are as follows. In understanding the parameter as placeholder, it was characterized that there were the tendency to choose a parameter value if it is unknown and to move between specific parameter values and a generic parameter. In understanding the parameter as changing quantity, it was characterized that there were the misperception of meta-variables’ change, the dominance of the changing quantity view of parameter and the formulation of the dynamics in natural language. In understanding the parameter as unknown, it was characterized that there were renaming flexibly variables and parameters, and perception of the parameter as unknown with respect to which an equation is solved. In understanding the parameter as generalizer, it was characterized that there were making an example of generalization and pattern recognition, solving a parametric general solution, and exploring the graph of function-family. Also, this research observed the hierarchy of the parameters. The results indicate that a CAS environment offers opportunities for activities that may support the development of algebraic understanding of the concept of parameter.
The Second research of six high school students was conducted to investigate how students generalize the factoring of in technique-oriented ways in a CAS application environment and the role of CAS in this process. This research had designed the Task-Technique-Theorization(T-T-T) frame adapted from Chevallard’s anthropological approach of Didactics. Students’ mathematical theorization is classified and analyzed according to ‘generalization of the expansion’, ‘the complete factorization of ()’, ‘the conjecture and its elaboration’ and ‘proving’. This research found that Task-Technique-Theorization activates mutual interaction between paper-and-pencil techniques and the CAS techniques. This interaction led to the students’ theoretical reflection and conceptual understanding. In this process, three epistemic roles of CAS were identified : the role of checking the result(CR), the role of cognitive stimulation(CS) and the role of extending thinking(ET). Therefore CAS played epistemic roles of checking the result of a task, stimulating the students’ cognition, and extending their thinking as well as a pragmatic role of producing the result of an input.
In addition, this study pointed out two suggestions of mathematics education in the school setting. The first suggestion is to introduce the concept of parameter in a new textbook and to teach it systematically. It is necessary to help students understand that the parameters work as meta-variables unlike variables. The other suggestion is to change the teaching system focusing on memorizing formulas about factoring polynomials. Students must be taught in the way of understanding and learning mathematical principles and concept of factoring polynomials. Through this study, the ‘finding-common-factor’ technique and the ‘adding-eliminable-terms’ technique were suggested as alternatives.
In conclusion, two achievements were made through the two qualitative research studies. The first one is that the importance of the technique in math learning was perceived. In the first research the students experienced changes in their epistemology through the CAS technique; in the second research the students led the technique-oriented mathematical theorization. The second achievement is that the epistemic roles of CAS were identified in the algebra learning. As a spoon plays a pragmatic role while eating but an epistemic role while measuring the amount of soy sauce to season soup, CAS played three epistemic roles as well as a pragmatic role. Therefore, CAS functions as a math laboratory to be with student for doing mathematics. When students use CAS as spontaneous math laboratory regardless of time and place, they can be self-directed learners. To educate students in information-oriented society, teachers’ consistent efforts in order to apply hand-held technology including CAS are needed.

목차 (Table of Contents)

• Ⅰ. 서론 1
• 1. 연구의 필요성 및 목적 1
• 2. 연구문제 10
• 3. 용어 정의 11
• Ⅱ. 이론적 배경 12
• 1. 인지적 도구 12
• 1) CAS 도구의 진화 12
• 2) 인지적 테크놀로지 17
• 3) White box/Black box 22
• 4) 실재화 25
• 2. 도구적 접근 27
• 1) 도구적 발생 28
• 2) 도구적 오케스트레이션 31
• 3. 인류학적 접근 33
• 1) 과제-기법-이론화 34
• 2) 실용주의적 역할과 인식론적 역할 36
• 3) 실험 수학 37
• 4) 교수학적 변환 39
• 4. 도구적 접근과 인류학적 접근의 통합 41
• 5. 매개변수 개념과 의 인수분해 일반화 46
• 1) 변수의 개념 47
• 2) 매개변수의 개념 49
• 3) 다항식의 인수분해 54
• 4) 의 인수분해 일반화 63
• Ⅲ. CAS 환경에서 개념적 이해에 관한 질적 연구 66
• 연구 1. 매개변수 개념에 대한 학생의 이해 66
• 1. 서론 66
• 1) 연구의 필요성 및 목적 66
• 2) 연구문제 69
• 2. 연구 방법 70
• 1) 참여자 70
• 2) 연구 절차 71
• 3) 자료 수집 및 분석 72
• (1) 매개변수 개념의 이해 과정에 나타난 기법 73
• (2) 질적 분석을 통한 매개변수 개념의 이해 74
• 3. 연구 결과 및 분석 75
• 1) 매개변수 개념의 이해 과정에 나타난 기법의 분석 75
• (1) <문항 1>에 나타난 기법의 분석 76
• (2) <문항 2>에 나타난 기법의 분석 77
• (3) <문항 3>에 나타난 기법의 분석 80
• (4) <문항 4>에 나타난 기법의 분석 82
• (5) <문항 5>에 나타난 기법의 분석 83
• (6) <문항 6>에 나타난 기법의 분석 84
• 2) 질적 분석을 통한 매개변수 개념의 이해 분석 87
• (1) 자리지기로서의 매개변수 87
• (2) 변하는 양으로서의 매개변수 90
• (3) 미지수로서의 매개변수 94
• (4) 일반화로서의 매개변수 97
• 4. 결론 103
• 연구 2. 의 인수분해 일반화에 대한 학생의 이해 107
• 1. 서론 107
• 1) 연구의 필요성 및 목적 107
• 2) 연구문제 110
• 2. 연구 방법 110
• 1) 참여자 111
• 2) 연구 절차 112
• 3) 자료 수집 및 분석 113
• (1) 의 인수분해 일반화 114
• (2) 의 인수분해 일반화에서 CAS의 역할 114
• 3. 연구 결과 및 분석 115
• 1) 의 인수분해 일반화 115
• (1) 전개의 일반화 116
• (2) (단, 인 자연수)의 완전한 인수분해 118
• (3) 추측과 추측의 수정 124
• (4) 증명 131
• 2) 의 인수분해 일반화 과정에서 CAS의 역할 140
• (1) 결과 확인의 역할 141
• (2) 인지 자극의 역할 143
• (3) 사고 확장의 역할 146
• 4. 결론 154
• Ⅳ. 요약 및 결론 157
• 1. 연구의 요약 157
• 1) 매개변수 개념에 대한 학생의 이해 157
• 2) 의 인수분해 일반화에 대한 학생의 이해 160
• 2. 결론 및 제언 163
• 참고문헌 168
• 부록 189
• <부록 1> 연구 1-사전 설문 문항 189
• <부록 2> 연구 1-교수실험 문항 190
• <부록 3> 연구 2-탐구과제 활동지 193
• ABSTRACT 199
• 감사의 글 203