• Journal of Educational Research in Mathematics
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• v.9 no.2
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• pp.473-491
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• 1999

The learning of mathematics happens in some situations. It is natural that students should learn mathematics in more appropriate situations. But, so far It has been hardly studied about concrete situation and milieu where math can be successfully taught. In today's math education, the situation of education as a external circumstance become realized more and more importantly with influence of open education. But they don't embody situation as an internal circumstance where the intrinsic concept of mathematics can be obtained. We started this thesis from this tried to answer it on the basis of Brousseau's question, have theory about the didactical situations. One of the purpose of this study is to understand the theory of didactical situations, which focuses on how we can elaborate situations which really make a mathematical notion function. In this study, It is attempted clarify some concepts of the theory of didactical situations. The other is to discuss about what the theory of didactical situations suggests us in math zeducation. The method of math teaching and learning and the teacher's role were discussed in the viewpoint of Brousseau's theory. Finally, We elaborated and presented some didactical situations which make the notion of the area of rectangle.

• School Mathematics
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• v.1 no.2
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• pp.417-431
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• 1999

In this study, I consider Brousseau's theory of didactical situation focused on 'the development process of situations', and analyze some examples of didactical situation related to instruction of 'decimal' concept. To elaborate situations which really make a mathematical notion function, we have to analyze the essence of the notion, and to construct the situation which can be developed to situations of 'action-formulation-validation - institutionalization'. From this view, it can be said that the instruction of decimal concept in our country mainly lies in the situations of 'action' and 'institutionalization'. we have to provide more situations of 'formulation' and 'institutionalization' which can connect 'action' and 'institutionalization'.

• The Mathematical Education
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• v.41 no.3
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• pp.329-339
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• 2002

This study analyzes the epistemological meaning of‘social construction’in mathematical instruction. The perspective that consider the cognition of mathematical concept as a social construction is explained by a cyclic scheme of an academic context and a school context. Both of the contexts require a public procedure, social conversation. However, there is a considerable difference that in the academic context it is Lakatos' ‘logic of mathematical discovery’In the school context, it is Vygotsky's‘instructional and learning interaction’. In the situation of mathematics education, the‘society’which has an influence on learner's cognition does not only mean‘collective members’, but‘form of life’which is constituted by the activity with purposes, language, discourse, etc. Teachers have to play a central role that guide and coordinate the educational process involving interactions with learners in this context. We can get useful suggestions to mathematics education through this consideration of the social contexts and levels to form didactical situations of mathematics.

• Journal of Educational Research in Mathematics
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• v.13 no.1
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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.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.23-48
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• 2007

This study begins from posing a problem, 'formal introduction of mathematical induction in school mathematics'. Most students may learn the mathematical induction at the level of instrumental understanding without meaningful understanding about its meaning and structure. To improve this didactical situation, we research on the historical progress of mathematical induction from implicit use in greek mathematics to formalization by Pascal and Fermat. And we identify various types of thinking included in the developmental process: recursion, regression, analytic thinking, synthetic thinking. In special, we focused on the role of regression in mathematical induction, and then from that role we induce the implications for teaching mathematical induction in school mathematics.

• Journal of Educational Research in Mathematics
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• v.15 no.2
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• pp.131-145
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• 2005

This study intends to analyze didactically on triangle-determining conditions and triangle-congruence conditions. The result of this study revealed the followings: Firstly, many pre-service mathematics teachers and secondary school students have insufficient understanding or misunderstanding on triangle-determining conditions and triangle-congruence conditions. Secondly, the term segment instead of edge may show well the concern of triangle-determining conditions. Thirdly, when students learn the method of finding six elements of triangle using the law of sines and cosines in high school, they should be given the opportunity to reflect the relation and the difference between triangle-determining situation and the situation of finding six elements of triangle. Fourthly, accepting some conditions like SSA-obtuse as a triangle-determining condition or not is not just a logical problem. It depends on the specific contexts investigating triangle-determining conditions. Fifthly, textbooks and classroom teaching need to guide students to discover triangle-deter-mining conditions in the process of inquiry from SSS, SSA, SAS, SAA, ASS, ASA, AAS, AAA to SSS, SAS, ASA, SAA. Sixthly, it is necessary to have students know the significance of 'correspondence' in congruence conditions. Finally, there are some problems of using the term 'correspondent' in describing triangle-congruence conditions.

• East Asian mathematical journal
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• v.31 no.4
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• pp.569-585
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• 2015

Many mathematics teachers in characterization high schools have been troubled to teach students because most of the students have weak interests in mathematics and they are also lack of preliminary mathematical knowledges. Currently many of mathematics teachers in such schools teach students using worksheets owing to the situation that proper textbooks for the students are not available. In this study, we referred to Chevallard's didactic transposition theory based on Brousseau's theory of didactical situations for mathematical teaching and learning. Our lessons utilizing worksheets necessarily entail encouragement of students' self-directed activities, active interactions, and checking the degree of accomplishment of the goal for each class. Through this study, we recognized that the elaborate worksheets considering students' level, follow-up auxiliary materials that help students learn new mathematical notions through simple repetition if necessary, continuous interactions in class, and students' mathematical activities in realistic situations were all very important factors for effective mathematical teaching and learning.

• Journal of Elementary Mathematics Education in Korea
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• v.19 no.4
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• pp.563-588
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• 2015

In elementary school, didactical transposition is inevitable due to several reasons. In mathematics, addition and multiplication are taught as binary operations, subtraction and division are taught as unary operations. But in elementary school, we try to teach all the four operations as binary operations by didactical transposition. In 'Mastering' the concepts of the four operations, the way of concept introduction is dealt importantantly. So it is different from understanding the four operations. In this study, we analyzed the four operations of natural numbers and fractions from two perspectives: concept understanding (how to introduce concepts and how to choose an operation) and connection between the operations. As a result, following implications were obtained. In division of fractions, students attempted a connection with multiplication of fractions right away without choosing an operation, based on the situation. Also, to understand division of fractions itself, integrate division of fractions presented from the second semester of the fifth grade to the first semester of the sixth grade are needed. In addition, this result can be useful in the future textbook development.

• Journal of Educational Research in Mathematics
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• v.9 no.1
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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.

• Education of Primary School Mathematics
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• v.23 no.3
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• pp.117-134
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• 2020

Based on the current curriculum, students learn the concept of fraction in the 3rd grade for the first time. At that time, fraction is introduced as whole-part relationship. But as the idea of fraction expands to improper fraction and so on, fraction as measurement would be naturally appeared. In that situation where fraction as whole-part relationship and fraction as measurement are dealt together, it is necessary for students to get experiences of understanding and exploring unit and whole adequately in order to fully understand the concept of fractions. Therefore, the purpose of this study is to analyze how to deal with unit fractions, how to implement activities to find the standard of reference from the part, and what visual representations were used to help students to understand the concept of fractions in elementary mathematics textbooks from the 7th to the 2015 revised curriculum. And we analyzed 60 3rd graders' understanding of finding and drawing the whole by looking at the part. Several didactical implications for teaching the concept of fractions were derived from the discussion according to the analysis results.