• 대한수학교육학회지:수학교육학연구
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• 제13권1호
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• pp.25-55
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• 2003

In the school mathematics, the negative numbers have been instructed by means of intuitive models(concrete situation models, number line model, colour counter model), inductive-extrapolation approach, and the formal approach using the inverse operation relations. These instructions on the negative numbers have caused students to have the difficulty in understanding especially why the rules of signs hold. It is due to the fact that those models are complicated, inconsistent, and incomplete. So, students usually should memorize the sign rules. In this study we studied on the didactical phenomenology of the negative numbers as a foundational study for the improvement of teaching negative numbers. First, we analysed the formal nature of the negative numbers and the cognitive obstructions which have showed up in the historic-genetic process of them. Second, we investigated what the middle school students know about the negative numbers and their operations, which they have learned according to the current national curriculum. The results showed that the degree they understand the reasons why the sign rules hold was low Third, we instructed the middle school students about the negative number and its operations using the formal approach as Freudenthal suggest ed. And we investigated whether students understand the formal approach or not. And we analysed the validity of the new teaching method of the negative numbers. The results showed that students didn't understand the formal approach well. And finally we discussed the directions for improving the instruction of the negative numbers on the ground of these didactical phenomenological analysis.

• 한국수학교육학회지시리즈A:수학교육
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• 제45권4호
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• pp.439-457
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• 2006

This research was to understand the features of mathematization and didactical phenomenology, in a way that was not a routine calculation of equation, rather a complete comprehension by the reinventing historical principles of the equation. To achieve the purpose of this study, one-mate middle school student participated in the study. Interview and observation were used for collecting data during the student's performance. The results of research were: First, the student understood the mathematical concepts from a real life and developed the abstract concepts from it, which were very intimately related with his life. Second, the skill and formula definition were accomplished with the accompanying predicted and consequently derived mathematical concepts. Third, through the approach of using the history of mathematics, he became more interested in what he was doing and took lessons with confidence. Forth, the student performed his learning based on the historical reinventing principle under the proper guidance of a teacher.

• 한국수학교육학회지시리즈A:수학교육
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• 제61권3호
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• pp.419-439
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• 2022

본 연구의 목적은 프로이덴탈의 수학화 과정과 일차함수 개념의 교수학적 현상학의 분석을 통하여 학생들이 변화율이 일정한 현상을 표, 그래프, 식으로 표현하는 과정과 일차함수 개념의 심상을 구성하는 과정, 본질을 구성하는 과정을 서술하고 분석하는 것이다. 연구 결과, 학생들은 변화율이 일정한 현상을 표로 표현할 때 합성단위로서의 비를 사용하고, 그래프로 표현할 때 한 학생을 제외하고 직선으로 표현하였다. 식으로 표현할 때 학생별로 주어진 상황, 공변 관점, 대응 규칙을 이용하는 수준의 차이가 있었다. 학생들은 두 변화량 사이의 관계를 곱셈적으로 비교하였고, 그 비율이 하나의 상수가 된다는 것을 교사의 안내에 따라 구성하였다. 특히 시간의 변화량과 거리의 변화량이 하나의 값, 속력이 되는 상황을 통하여 변화율이 일정하다는 심상을 구성하였다. 단, 변화율을 직선의 기울기와 연결하는 데에는 어려움을 겪었으나, 변화율이 일정하다는 심상과 그래프가 직선이며 식의 모양이 y=ax+b (a는 변화율, b는 절편)라는 심상을 정리하고 조직하여 일차함수의 본질(개념)을 구성하였다.

• 대한수학교육학회지:수학교육학연구
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• 제9권1호
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• pp.81-109
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• 1999

This study aims to reflect the basic principles and teaching-teaming principles of Realistic Mathematics Education in order to suppose an way in which mathematics as an activity is carried out in primary school. The development of what is known as RME started almost thirty years ago. It is founded by Freudenthal and his colleagues at the former IOWO. Freudenthal stressed the idea of matheamatics as a human activity. According to him, the key principles of RME are as follows: guided reinvention and progressive mathematisation, level theory, and didactical phenomenology. This means that children have guided opportunities to reinvent mathematics by doing it and so the focal point should not be on mathematics as a closed system but on the process of mathematisation. There are different levels in learning process. One should let children make the transition from one level to the next level in the progress of mathematisation in realistic contexts. Here, contexts means that domain of reality, which in some particular learning process is disclosed to the learner in order to be mathematised. And the word of 'realistic' is related not just with the real world, but is related to the emphasis that RME puts on offering the students problem situations which they can imagine. Under the background of these principles, RME supposes the following five instruction principles: phenomenological exploration, bridging by vertical instruments, pupils' own constructions and productions, interactivity, and interwining of learning strands. In order to reflect how to realize these principles in practice, the teaming process of algorithms is illustrated. In this process, children follow a learning route that takes its inspiration from the history of mathematics or from their own informal knowledge and strategies. Considering long division, the first levee is associated with real-life activities such as sharing sweets among children. Here, children use their own strategies to solve context problems. The second level is entered when the same sweet problems is presented and a model of the situation is created. Then it is focused on finding shortcomings. Finally, the schema of division becomes a subject of investigation. Comparing realistic mathematics education with constructivistic mathematics education, there interaction, reflective thinking, conflict situation are many similarities but there are alsodifferences. They share the characteristics such as mathematics as a human activity, active learner, etc. But in RME, it is focused on the delicate balance between the spontaneity of children and the authority of teachers, and the development of long-term loaming process which is structured but flexible. In this respect two forms of mathematics education are different. Here, we learn how to develop mathematics curriculum that respects the theory of children on reality and at the same time the theory of mathematics experts. In order to connect the informal mathematics of children and formal mathematics, we need more teachers as researchers and more researchers as observers who try to find the mathematical informal notions of children and anticipate routes of children's learning through thought-experiment continuously.