• The Mathematical Education
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• v.56 no.1
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• pp.101-118
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• 2017

In this article we study a reconstruction of mathematical knowledge on trigonometry by the method of ascent from the abstract to the concrete from the pedagogical viewpoint of dialectic. The direction of education is shifting in a way that emphasizes the active constitution of knowledge by the learning subjects from the perspective that knowledge is transferred from the teacher to the student. In mathematics education, active discussions on the construction of mathematical knowledge by learners have been going on since the late 1990s. In Korea, concepts and aspects of constructivism such as operational constructivism, radical constructivism, and social constructivism were introduced. However, examples of practical construction according to the direction of construction of mathematical knowledge are very hard to find. In this study, we discuss the direction of the actual construction of mathematical knowledge and suggest a concrete example of the actual construction of trigonometry knowledge from a constructivist point of view. In particular, we discuss the process of the construction of theoretical knowledge, the ascent from the abstract to the concrete, based on the literature study from the pedagogical viewpoint of dialectic, and show how to construct the mathematical knowledge on trigonometry by the method of ascent from the abstract to the concrete. Through this study, it is expected to introduce the new direction and new method of knowledge construction as 'the ascent from the abstract to the concrete', and to present the possibility of applying dialectic concepts to mathematics education.

• Research in Mathematical Education
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• v.5 no.1
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• pp.13-24
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• 2001

Using computer technology in teaching school mathematics creates new instructional environments. The emphases on the use of computer technology in the classrooms and in particular the use of computer-based exploration as a context of mathematics instruction have been reflected in the recommendation of the NCTM (Curriculum and Evaluation Standards for School Mathematics, 1989). Although the power of using computer technology in the exploration of mathematical problems has been recognized and stressed by many educators, we do not have many research studies on mathematics in computer-based explorations. Especially research has failed to clarify how computer technology can contribute to the construction of procedural and conceptual knowledge of mathematics. Up to now most researches on procedural and conceptual knowledge in computer environments have only focused on classifying programming languages which program language has more random access and rich interrelationship characteristic in relation to conceptual knowledge in humans, and which computer language has more characteristic flavor of procedural knowledge. How computer-based explorations affect the knowledge construction of mathematics, therefore, emerges as an issue of research on teacher education program for theoretical framework. This situation leads to do research on the effectiveness of using computer explorations in pre-service teacher education in terms of procedural and conceptual knowledge construction.

• Journal of Educational Research in Mathematics
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• v.18 no.1
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• pp.1-24
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• 2008

The purpose of this study is to uncover weaknesses in the constructivism in mathematics education and to search for ways to complement these deficiencies. We contemplate the relationship between the capability of construction and the performance of it, with the view of the 'Twofold-Structure of Mind.' From this, it is claimed that the construction of mathematical knowledge should be to experience and reveal the upper layer of Mind, the Reality. Based on the examination on the conflict and relation between the structuralism and the constructivism, with reference to the 'theory of principle' and the 'theory of material force' in Neo-Confucianist theory, it is asserted that the construction of mathematical knowledge must be the construction of the structure of mathematical knowledge. To comprehend the processes involved in the construction of the structure of mathematical knowledge, the epistemology of Michael Polanyi is studied. And also, the theory of mathematization, the historico-genetic principle, and the theory on the levels of mathematical thinking are reinterpreted. Finally, on the basis of the theory of twofold-structure, the roles and attitudes of teachers and students are discussed.

• East Asian mathematical journal
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• v.38 no.1
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• pp.85-94
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• 2022

Recently, an LWE-based commitment scheme is proposed. Their construction is statistically hiding as well as computationally binding. On the other hand, the construction of related zero-knowledge protocols is left as an open problem. In this paper, we present zero-knowledge protocols with hardness based on the LWE problem. we show how to instantiate efficient zero-knowledge protocols that can be used to prove linear and sum relations among these commitments. In addition, we show how the variant of LWE, spLWE problem, can be used to instantiate efficient zero-knowledge protocols.

• Journal of Educational Research in Mathematics
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• v.19 no.3
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• pp.355-369
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• 2009

Though there is no agreement on the definition of analogical reasoning, there is no doubt that analogical reasoning is the means of mathematical knowledge construction. Mathematicians generally have a tendency or desire to find similarities between new and existing Ideas, and new and existing representations. They construct appropriate links to new ideas or new representations by focusing on common relational structures of mathematical situations rather than on superficial details. This focus is analogical reasoning at work in the construction of mathematical knowledge. Since analogical reasoning is the means by which mathematicians do mathematics and is close]y linked to measures of intelligence, it should be considered important in mathematics education. This study investigates how mathematicians used analogical reasoning, what role did it flay when they construct new concept or problem solving strategy.

• Research in Mathematical Education
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• v.19 no.2
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• pp.101-115
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• 2015

In the present paper, we argue any research concerning human knowledge construction, components, or types needs to clarify its epistemological stance regarding 'knowledge' in that such viewpoint might have much influence on the nature of knowledge the researcher sees and the way in which evidence for knowledge development is gathered. Thus, we suggest two alternative research groups who conducted their studies on mathematical knowledge for teaching with an explicit epistemological standpoint. We finalize our discussion by reviewing concrete examples in the previous literature on teacher knowledge of fraction conducted by the two groups.

• Research in Mathematical Education
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• v.8 no.2
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• pp.81-93
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• 2004

This study investigated three preservice teachers' mathematical problem solving among hand-in-write-ups and final projects for each subject. All participants' activities and computer explorations were observed and video taped. If it was possible, an open-ended individual interview was performed before, during, and after each exploration. The method of data collection was observation, interviewing, field notes, students' written assignments, computer works, and audio and videotapes of preservice teachers' mathematical problem solving activities. At the beginning of the mathematical problem solving activities, all participants did not have strong procedural and conceptual knowledge of the graph, making a model by using data, and general concept of a sine function, but they built strong procedural and conceptual knowledge and connected them appropriately through mathematical problem solving activities by using the computer technology.

• Journal of Educational Research in Mathematics
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• v.14 no.1
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• pp.129-142
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• 2004

These days, constructivism has become a central theory in mathematics education. A essential concept in constructivism is 'construction' and the meaning of construction of mathematical knowledge is a core issue in mathematics educational field. In the basis of Dewey's epistemology, this article is trying to explicate the meaning of construction of mathematical knowledge. Dewey, Kant and Piaget coincide in construction of knowledge from the viewpoint of the interaction between mind and environment. However, unlike Dewey's concept, Kant and Piaget are still in the line of traditional realistic epistemology. Dewey's concept of construction logically implies teaching-learn learning principles. This can be named as a principle of genetic construction and a principle of progressive consciousness.

• Journal of Educational Research in Mathematics
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• v.20 no.1
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• pp.57-71
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• 2010

Similarity between concept and concept, principle and principle, theory and theory is known as a strong motivation to mathematical knowledge construction. Metaphor and analogy are reasoning skills based on similarity. These two reasoning skills have been introduced as useful not only for mathematicians but also for students to make meaningful conjectures, by which mathematical knowledge is constructed. However, there has been lack of researches connecting the two reasoning skills. In particular, no research focused on the interplay between the two in didactic transposition. This study investigated the process of knowledge construction by metaphor and analogy and their roles in didactic transposition. In conclusion, three kinds of models using metaphor and analogy in didactic transposition were elaborated.

• Journal for History of Mathematics
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• v.17 no.3
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• pp.109-120
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• 2004

This study investigated three pre-service teachers' mathematical problem solving among hand-in-write-ups and final projects for each subject. All participants' activities and computer explorations were observed and video taped. If it was possible, an open-ended individual interview was performed before, during, and after each exploration. The method of data collection was observation, interviewing, field notes, students' written assignments, computer works, and audio and videotapes of pre- service teachers' mathematical problem solving activities. At the beginning of the mathematical problem solving activities, all participants did not have strong procedural and conceptual knowledge of the graph, making a model by using data, and general concept of a sine function, but they built strong procedural and conceptual knowledge and connected them appropriately through mathematical problem solving activities by using the computer technology.