극한 개념 학습 과정에 나타나는 메타인지적 수행과 내적 인지 표상의 특성 및 관계에 관한 연구

Abstract

The conclusions from the above findings are as follows. First, in this study, the frequency of direction setting and guidance (O) was the highest in metacognitive performance in the extreme concept learning process. In particular, 'information and condition analysis' was the highest frequency among meta-cognitive tasks of orientation and guidance (O). Speaking ’showed the lowest frequency. To understand a given task, the students analyzed the information in small units and tried to find the key concepts that could be derived from each information. This could be understood in the same context as Silver's (1982a) study of searching for content to find the "key word" of a condition in the process of understanding the concept. And repetitive reading; The lowest frequency of speaking was in contrast to Baker's (1979) study, in which college students used a variety of remedial strategies to recognize the meaning of tasks when they read them repeatedly and to solve related problems. On the other hand, the lowest percentage of organizational (G) among metacognitive performances in this study is that the eye patch (O) and organizational (G) during metacognitive performance will be minimized for the tasks that are lower than students' cognitive level. This is in contrast to Garofalo & Lester(1985) studies, which showed that (E) and justification (V) are mainly present. In addition, the results of Lester and Garofalo(1987) study showed that the justification of metacognitive performance (V) is reflected in the student's self-confidence of mathematics competency, self-assessment and belief of learning task. Second, metacognitive performance in the extreme concept learning process in this study was found to be simultaneous and complex for each metacognitive task. This suggests that meta-cognitive tasks are complex in the students' abstract understanding process, as Schroeder and Thomas (1993) found that mathematical problem solving interacts with various cognitive activities in terms of mathematical connectivity. When these metacognitive tasks were coordinated with each other, it was confirmed that students could carry out their own metacognitive strategies. In addition, the complex relationship between metacognitive performance described by Garofalo & Lester(1985) in problem solving can be confirmed in the extreme concept learning through student interaction. Third, the characteristics of metacognitive performance in the extreme concept learning process appeared in various ways. Among them, six meta-components of Sternberg (1982) in that they perform and organize mainly local planning in organizing (G) and execution (E), and pay more attention to processes than goals in justification (V). We found that metacognitive tasks were mainly performed through the selection of sub-elements and the combination strategy of sub-elements. In the process of understanding extreme concepts, students' meta-cognitive task of justification (V) focusing on 'process' rather than' goal 'is especially important for each mathematical analysis process. It was understood in the same context as Silver's (1982b) claim to focus on action. In addition, justification (V) was found to play a role in developing students' new knowledge structure by linking their mathematical cognition with metacognitive structure, which is suggested by Garofalo & Lester(1985). It seems to be the result of combining Polya's (1957) problem solving and metacognitive activities. Fourth, verbal representation (IV) was the highest among internal cognitive representations in the extreme concept learning process. In particular, among the representational activities of linguistic representation (IV), 'mathematical language' was the most frequent, followed by 'self-language'. This is in line with Goldin and Shteingold's (2001) study that learning tasks that are already refined representations do not make students feel the need for original expression or sign development. In contrast, Goldin and Kaput (1996) opposed the development of the mathematical representation of the three stages (the stage of the symbol invention—the stage of expansion—the stage of autonomy). On the other hand, when the ratio of representational activities among students was large, Ozgun-Koca (2001) found that students' preference for representation also exists in the study subjects, and the representation was verbal representation (IV). Fifth, according to Hitt (1998), individual representation does not appear as a one-off in the process of elaboration through student interaction, and the representation of a certain stage does not reveal the mathematical thinking process of students. In this study, as a result of checking the overlapping frequency of representational activities in the internal cognitive representation in the extreme concept learning process, the internal cognitive representation was also simultaneous and complex like metacognitive performance. This was supported by Goldin's (2002a, 2003) study of the multiplicity of cognitive representation activities used in a mixture of two or more. In particular, it was confirmed that the representational activities of internal cognitive representation through students' interactions are simultaneous and complex as in Cobb's (1991, 2000). In addition, Kim Min-kyeong and Kwon Hyuk-jin (2010) explained that more than 50% of the top-level students solved the problem by using two or more representations in all questions. Sixth, there were various features of internal cognitive representation in the extreme concept learning process. Among them, the representation activity 'specific explanation' played a positive role in the student's learning of extreme concepts. This is enough to examine Gee (2011) seven of the building tasks for which mathematical language the student considers important and what kinds of student activities focus on the mathematical language deemed important. It was. On the other hand, in this study, the subjective expression of the learner showed little that Dufour-Janvier (1987) plays an important role in representation development. This can be interpreted in the same context as the three phases of Goldin and Kaput (1996) did not appear in the student's representation activities (the phase of sign invention-the phase of expansion-the level of autonomy). In addition, Vinner and Hershkowitz's (1980) study confirmed that a simple geometric image is a common path for developing mathematical conceptual images. However, students do not believe that they can prove complete logic alone, and the visual conceptual image was performed only to help symbolic representation (F) and verifiable representation (C). However, in the process of performing symbolic representations (F), there was no formal manipulation of symbols that were focused solely on procedures that Silver (1982a) was concerned about. Students' understanding of signs was accompanied by representational activities that included not only the interpretation of signs but also the meaning of signs and an understanding of the role of each condition in a given definition. Verification representation (C), on the other hand, is a representation that plays a crucial role in understanding a given task or identifying errors. It could be understood in the same context as it claimed to help form a mathematical structure. Verbal representations based on local monitoring appear throughout metacognitive performance in support of Lester (1978) 's assertion that metacognition is not limited to the last stage of Polya's fourth stage of problem solving. Schoenfeld (1984) also explains self-awareness and belief systems that can affect individual behavior with metacognitive knowledge, and Goldin (1998) understands and solves problems in mathematics learning. Emotional representation (A) has been shown to help students verify and monitor their thinking processes, as the emphasis is on expressing students' inner feelings in the process of solving problems. Seventh, internal cognitive representation was found to influence metacognitive performance. Cobb (1995, 2000) emphasized that the various representational activities negotiated through student interaction elaborate the concept of representation so that the representation of the task and the representation best suited to the student can be achieved. Garofalo & Lester(1985) found that the various activities of students in mathematical practice directly affect metacognitive activities. This study also confirmed that the various activities in mathematical performance described by Garofalo & Lester(1985), among others, were directly related to metacognitive performance.

The purpose of this study is to analyze the characteristics and relations of metacognitive performance and internal cognitive representation in the process of first-year high school students' learning of rigorous and abstract extreme concepts through student interaction. . For this purpose, the student discourses and student performances were collected and analyzed. For a meaningful quantitative analysis of the frequency of representational activities and the relationship between them based on metacognitive tasks and internal cognitive representations for metacognitive performance, data from three semesters with different classes were analyzed together. Therefore, it is necessary to improve the research method according to the data collection method for the reliable study on the characteristics of metacognitive performance and internal cognitive representation in the extreme concept learning process. In addition, 13 students who participated in the experiment class Ⅲ confirmed that their attitude toward mathematics was improved through the experiment class. Accordingly, mathematicians and curriculum development staff need to encourage students to actively participate in classes and to provide lessons that enable the learners' metacognitive performance and internal cognitive representation through discussion activities. In other words, in the follow-up study, a teaching and learning method was developed in consideration of the characteristics of metacognitive performance and internal cognitive representation obtained through this study, and the specific guidance plan was prepared through the process of verifying the effect by applying it to school sites. In the meantime, 39 high school first grade students participated in the discussion activities on the basis of self-directed extreme concepts. Through this, students' internal cognitive representation had a positive effect on metacognitive performance, and it was confirmed that metacognitive performance and internal cognitive representation were simultaneously and complex in this process. Therefore, if there are studies that can identify the structure of metacognitive performance and internal cognitive representation based on this, it will be possible to analyze the characteristics and relationships of metacognitive performance and internal cognitive representation more closely. Finally, this study studied the characteristics and relationships of metacognitive performance and internal cognitive representation in the process of understanding the definition, axiom, and property, which are the most basic units of mathematics, through interaction among students. will be. However, from the 4th curriculum to the 2015 revised curriculum, school mathematics has emphasized problem-solving ability, and many previous studies show that students are actively involved in metacognitive and representational activities in solving unstructured problems. Therefore, in the process of solving real-life and atypical problems, a study that analyzes the characteristics and relationships of metacognitive performance and internal cognitive representation will be meaningful. In addition, this study has limitations in that students who have been researched have high interest and interest in mathematics and their academic level is above a certain level. Should be. In particular, the study of metacognitive performance and internal cognitive representation for the gifted students of mathematics will be helpful in understanding the multifaceted aspects of metacognitive performance and internal cognitive representation.;이상의 연구결과에서 얻은 결론은 다음과 같다. 첫째, 학생들은 주어진 과제를 이해하기 위해 주어진 정보를 작은 단위로 분석하고, 각 정보에서 얻을 수 있는 핵심적인 개념을 찾기 위해 노력하였다. 이것은 개념을 이해하는 과정에서 조건의‘핵심어(Key word)’를 찾기 위해 내용을 탐색한다는 Silver(1982a)의 연구와 같은 맥락으로 이해할 수 있었다. 그리고 ‘반복적 읽기 ㆍ 말하기’가 가장 낮은 빈도를 보인 것은 대학생들이 과제를 반복적으로 읽을 때 과제의 의미를 알아차리고 연관된 문제들을 해결하는 다양한 수정 전략을 구사한다는 Baker(1979)의 연구와는 상반되는 결과였다. 둘째, 본 연구에서 극한 개념 학습 과정에 나타나는 메타인지적 수행은 각 메타인지 과업별로 동시다발적이고 복합적으로 나타났다. 이는 수학적 연결성 측면에서 수학적 문제해결이 다양한 인지적 활동들과 상호작용을 한다는 Schroeder(1993)의 연구처럼, 학생들의 추상적인 개념 이해 과정에서도 메타인지 과업이 전체적으로 복합적으로 나타남을 확인할 수 있었으며, 이러한 메타인지 과업이 서로 조율이 될 때, 학생 나름의 메타인지적 전략을 수행할 수 있음을 확인할 수 있었다. 또한 문제해결에서 Garofalo와 Lester(1985)가 설명한 메타인지적 수행의 복합적인 관계를 학생들 간 상호작용을 통한 극한의 개념 학습에서도 확인할 수 있었다. 셋째, 메타인지적 수행의 정당화(V)는 학생들의 수학적인 인지를 메타인지 구조로 연계시켜 새로운 지식 구조로 발전시켜나가는 역할을 하는 것으로 확인되었는데, 이것은 Garofalo와 Lester(1985)가 제시한 ‘인지-메타인지 구조’가 Polya(1957)의 문제해결 과정과 메타인지활동을 결합한 결과로 보인다. 넷째, 본 연구에서 극한 개념 학습 과정에 나타나는 내적 인지 표상은 이미 정제된 표상으로 이루어진 학습 과제는 학생 개인의 독창적인 표현이나 기호 개발의 필요성을 못 느끼게 한다는 Goldin과 Shteingold(2001)의 연구결과와 같았다. 반면 Goldin과 Kaput(1996)의 3단계(기호 발명의 단계-확장의 단계-자율의 단계)의 수학적 표상의 개발 과정과는 상반되었다. 다섯째, Hitt(1998)에 따르면 개개인의 표상은 학생들 간 상호작용을 통하여 정교화 되는 과정에서 일회성으로 나타나지 않으며, 특정 단계에서 나타나는 표상으로는 학생의 수학적 사고 과정을 알아낼 수 없다고 하였다. 이에 본 연구에서 극한 개념 학습 과정에 나타나는 내적 인지 표상 내 표상활동의 중복 빈도를 확인해 본 결과, 내적 인지 표상 역시 메타인지적 수행과 마찬가지로 동시다발적이고 복합적으로 나타났다. 여섯째, 본 연구에서 극한 개념 학습 과정에 나타나는 내적 인지 표상의 특징은 다양하게 나타났다. 그 중 표상활동‘구체적인 설명’은 학생이 극한 개념을 학습하는데 있어 긍정적인 역할을 하였다. 이는 Gee(2011) 7가지 학습 과업(building tasks) 중 학생이 중요하다고 생각하는 수학적 언어가 무엇인지, 그리고 중요하다고 여긴 수학적 언어에 중점을 두는 학생들의 활동에는 어떠한 종류가 있는지를 살펴보기에 충분한 결과였다. 한편 검증적 표상(C)은 주어진 과제를 핵심적으로 이해하거나 오류를 파악하는데 결정적인 역할을 하는 표상이었는데, 이는 Bruner(2010)가 정당화 과정을 수행할 수 있는 학생들은 수학적 수준이 높고, 정당화 과정이 직접 수학적 구조를 형성하는 것에 도움을 준다고 주장한 것과 같은 맥락으로 이해할 수 있었다. 또한 국소적인 모니터링을 바탕으로 한 검증적 표상이 메타인지적 수행 과정에 지속적으로 나타나는 것은 Lester(1978)가 메타인지는 Polya의 문제해결 4단계의 마지막 단계로 제한되지 않는다고 주장한 것을 뒷받침하는 결과였다. 일곱째, 내적 인지 표상은 메타인지적 수행에 영향을 주는 것으로 나타났다. Cobb(1995, 2000)은 학생 상호작용을 통하여 협상된 다양한 표상활동은 표상의 개념을 정교화시켜 과제의 특성과 학생에게 맞는 가장 적합한 표상을 할 수 있게 한다고 강조하였다. Garofalo와 Lester(1985)는 수학적 수행에서 나타난 학생들의 다양한 활동은 메타인지활동에 직접적인 영향을 준다고 하였다. 이는 본 연구에서도 확인되는 바, Garofalo와 Lester(1985)에서 설명한 수학적 수행 내 다양한 활동, 그중에서도 학생들 간 상호작용을 통한 내적 인지 표상이 메타인지적 수행과 직접적으로 연관되어 있음을 확인할 수 있었다. 본 연구의 목적은 고등학교 1학년 학생들이 교육과정에서 더 나아가 엄밀하고 추상적인 극한 개념을 학생들 간 상호작용을 통해 학습하는 과정에서 나타나는 메타인지적 수행과 내적 인지 표상의 특성 및 관계를 분석하는 것이다. 이를 위해 실험 수업을 통해 학생 담화와 학생 수행 결과물을 수집하고, 분석하였다. 메타인지적 수행을 위한 메타인지 과업 및 내적 인지 표상을 바탕으로 한 표상활동의 빈도 및 이들 간 관계에 대한 의미 있는 양적 분석을 위하여 수업 시기가 다른 총 3학기의 자료를 함께 분석하였다. 따라서 극한 개념 학습 과정에 나타나는 메타인지적 수행과 내적 인지 표상의 특징에 대한 신뢰도 높은 연구를 위하여 자료의 수집 방식에 따른 연구 방법 개선이 필요하다. 또한 실험 수업 Ⅲ에 참여한 13명의 학생들은 해당 실험 수업을 통해서 수학에 대한 태도가 신장되었음을 확인할 수 있었다. 이에 수학교사 및 교육과정 개발 담당자들은 학생들의 적극적인 수업 참여를 장려하고, 토론 활동을 통한 학습자 개인의 메타인지적 수행 및 내적 인지 표상을 활성화시킬 수 있는 수업을 제공할 수 있도록 노력할 필요가 있다. 즉, 후속연구에서는 본 연구를 통하여 얻은 메타인지적 수행 및 내적 인지 표상의 특성을 고려하여 교수ㆍ학습 방법을 개발하고, 이를 학교 현장에 적용하여 그 효과를 검증하는 과정을 통해 구체적인 지도 방안을 마련할 수 있을 것으로 판단된다. 한편 본 연구에 참여한 고등학교 1학년 학생 39명은 개별적으로 자기주도학습 한 극한 개념을 바탕으로 하여 학생들 간 토론 활동을 수행하였다. 이를 통해 학생의 내적 인지 표상이 메타인지적 수행에 긍정적인 영향을 주었으며 이 과정에서 메타인지적 수행 및 내적 인지 표상이 동시다발적이고 복합적으로 나타남을 확인할 수 있었다. 따라서 이를 바탕으로 메타인지적 수행 및 내적 인지 표상의 구조를 파악할 수 있는 연구가 있다면 보다 면밀히 메타인지적 수행과 내적 인지 표상의 특성 및 관계를 분석할 수 있을 것이다. 마지막으로, 본 연구는 연구 대상이었던 학생들이 수학에 흥미와 관심이 높고 학업 수준이 일정 수준 이상이었다는 점에서 제한점이 있으므로 연구 결과를 보다 일반화하기 위해서는 대상 학생의 범주를 넓혀 다양한 학생들을 대상으로 한 연구가 진행되어야 할 것이다. 특히 후속 연구로 수학 영재학생들에 대한 메타인지적 수행 및 내적 인지 표상에 대하여 연구한다면 메타인지적 수행과 내적 인지 표상의 특성 및 관계에 대한 다각적인 측면을 이해하는데 도움이 될 것이다.