• Communications of Mathematical Education
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• v.32 no.3
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• pp.257-273
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• 2018

In this paper, we deal with the isoprimetric problem of polygons from the point of view of learning materials for elementary gifted students. The isoperimetric problem of the polygon of odd degree can be solved by E-transformation(see Figure III-1) and M-transformation(see Figure III-3). But in the case of even degree's polygon, it is quite difficult to solve the problem because of the connected components of diagonals (here we consider the diagonals forming triangle with two adjacent sides of polygon). The primary purpose of this paper is to give an idea to solve the isoperimetric problem of polygons of even degree using the properties of ellipse. This idea is derived from the programs of the Institute of Science Education for Gifted Students in the Jeju National University.

• School Mathematics
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• v.11 no.2
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• pp.227-241
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• 2009

The isoperimetric problem has been known from the time of antiquity. But the problem was not rigorously solved until Steiner published several proofs in 1841. At the time it stood at the center of controversy between analytic and geometric methods. The geometric approach give us more productive thinking (insight, structural understanding) than the analytic method (using Calculus). The purpose of this paper is to analysis and then to construct the isoperimetric problem which can be applied to the mathematically gifted students in the elementary school. The theoretical backgrounds of our analysis about our problem are based on the Gestalt psychology and mathematical reasoning. Our active program about the isoperimetric problem constructed by the Gestalt's view will contribute to improving a mathematical reasoning and to serving structural (relational) understanding of geometric figures.

• School Mathematics
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• v.16 no.4
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• pp.677-690
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• 2014

The main goals of learning geometry include developing spatial ability and concepts on geometric objects based on understanding the attributes and relationships of them. While the instructions on geometric objects follow the concept development models, the ones on spatial ability are designed from the perspective of geometric transformation. However, there is a need for instructional materials to emphasizing the relationships among geometric concepts. This study hypothesizes that spatial ability stems from the intuitive understanding of geometric objects and the relational understanding on concepts, and it considers the isoperimetric problems as instructional materials to foster spatial ability.

• East Asian mathematical journal
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• v.31 no.4
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• pp.445-458
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• 2015

In this paper, we provide a geometrical solving method about the equiangular problem appeared in the solving process of the isoperimetric problem of polygon. In fact we deal with the following problem in the view of the productive thinking centered on the circle: Let B and G be fixed points, and let $\bar{AB}=\bar{AP_1}=\bar{DP_1}=\bar{DP_2}=\bar{FP_2}=\bar{FP_3}=\bar{HP_{n-1}}=\bar{HG}$. Then find the position of moving points $P_i(1{\leq}i{\leq}n)$ to maximize the sum of areas of the triangles that lie on the line segment $\bar{BG}$.

• The Mathematical Education
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• v.50 no.4
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• pp.441-466
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• 2011

In this article, we study on the teaching design, focused on the geometric methods, of 2-D isoperimetric problem for the elementary mathematically gifted students. For our teaching design, we discussed the ideals of Zenodorus's polygon proof, Steiner's four-hinge proof, Steiner's mean boundary proof, Steiner's snowball-packing proof, Edler's finite existence proof and Lawlor's dissection proof, and then the ideals achieved were modified with the theoretical backgrounds-the theory of Freudenthal's mathematisation, the method of analysis-synthesis. We expect that this article would contribute to the elementary mathematically gifted students to acquire and to improve spatial sense.

• Communications of Mathematical Education
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• v.31 no.2
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• pp.223-239
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• 2017

In this paper, it is aimed to design the teaching units 'Inquiry into the isoperimetric problem of triangle and quadrilateral' to give elementary gifted students experience of mathematization. For this purpose, the teacher and the class observer (researcher) made a discussion about the design of the teaching unit through the analysis of the class based on the thought processes appearing during the problem solving process of each group of students. The following is a summary of the discussions that can give educational implications. First, it is necessary to use mathematical materials to reduce students' cognitive gap. Second, it is necessary to deeply study the relationship between the concept of side, which is an attribute of the triangle, and the abstract concept of height, which is not an attribute of the triangle. Third, we need a low-level deductive logic to justify reasoning, starting from inductive reasoning. Finally, there is a need to examine conceptual images related to geometric figure.

• East Asian mathematical journal
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• v.36 no.2
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• pp.131-146
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• 2020

The isoperimetric problem is a well-known optimization problem from ancient Greek. Among plane figures with the same perimeter, which is the largest area surrounded? The answer to the question is circle. Zenodorus and Steiner's pure geometric proofs, which left a lot of achievements in this matter, looked beautiful with ideas at that time. But there was a fatal flaw in the proof. The weakness is related to the existence of the solution. In this paper, from a view of the existence of the solution, we investigate proofs of Zenodorus and Steiner and get educational implications.