• Journal for History of Mathematics
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• v.23 no.2
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• pp.101-121
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• 2010

This article includes an introduction, a history of Pick's theorem on lattice polyhedron and its proof, Reeve's theorem on 3-dimensional lattice polyhedrons extended from the Pick's theorem and Ehrhart polynomial generalized as an n-dimensional lattice polyhedron, and then shows the relationship between the volume of 3-dimensional polyhedron and the number of its lattice points by means of Reeve's theorem. It is aimed to apply the relationship to the visualization of sums in sequences.

• School Mathematics
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• v.13 no.1
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• pp.175-187
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• 2011

The purpose of this study is to develop and utilize various kinds of mathematics teaching materials for gifted class in elementary school by utilizing polyominoes and a what-if-not strategy. Blokus is used to let students understand the characteristics of polyominoes, and omok is utilized to let them grasp interior point. Thus, the activities that utilized the new materials, blokus and omok, are developed to teach Pick's theorem. Besides, recreation activities were additionally prepared to provide education in an easy, intriguing and creative manner. The findings of the study is as follows: First, each of the materials was utilized in a different manner when the students engaged in basic and enrichment learning. Second, the mathematically gifted students were able to discover Pick's theorem in the course of utilizing the materials that contained recreational elements. Third, the students were taught to foster their problem-solving skills about area, girth and interior point by making use of the materials that were designed to be linked to each other. Fourth, existing programs were just designed to attain particular objects, to be conducted at a fixed time and to cater to particular graders. Fifth, when the students made problems by making use of the what if (not) strategy and the materials, they responded in diverse ways and were able to apply them.

• Journal of Educational Research in Mathematics
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• v.23 no.4
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• pp.553-565
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• 2013

The purpose of this study was to examine how the Lakatos method works in the elementary teacher education program. Elementary preservice teachers were given a task in which they examined the Pick's theorem. The finding revealed that Lakatos method was usable in the elementary teacher education. They produced initial conjecture and found counterexamples, and finally made improved conjectures. These experience encourage them to change their belief of teaching and learning mathematics and to find alternative ways of teaching mathematics.

• Journal of Educational Research in Mathematics
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• v.17 no.3
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• pp.221-232
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• 2007

In school mathematics, activities to make particular convex polygons by attaching edgewise some pieces of tangram are introduced. This paper focus on deepening these activities. In this paper, by using Pick's Theorem and 和 草's method, all the convex polygons by attaching edgewise seven pieces of tangram, Sei Shonagon(淸少納言)'s tangram, and Pythagoras puzzle are found out respectively. By using Pick's Theorem to the square seven-piece puzzles satisfying conditions of the length of edge, it is showed that the number of convex polygons by attaching edgewise seven pieces of them can not exceed 20. And same result is obtained by generalizing 和 草's method. The number of convex polygons by attaching edgewise seven pieces of tangram, Sei Shonagon's tangram, and Pythagoras puzzle are 13, 16, and 12 respectively.

• School Mathematics
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• v.18 no.1
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• pp.85-103
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• 2016

This study was to find characteristics of argumentation derived from a discourse in a math-gifted 5 grade class, which was held for finding a Pick's formula using Geoboard function of TI-73 calculator. For the analysis, a video record of the class, transcript of its voice record, and activity paper were used as data and Toulmin's argument schemes were applied as analysis standard. As a result of the study, we found that the graphing calculator helped the students to create an experimental environment that graphing a grid-polygon and figuring out its area. Furthermore, it also provided a basic demonstration through 'data->claim' composition and reasoning activities which consisted of guarantee, warrant, backing, qualifier and refutal for justifying. The basic argumentation during the process of deriving the Pick's theorem by the numbers of boundary points and inner points was developed into a 'collective argumentation' while a teacher took a role of a conductor of the argumentation and an authorizer on the knowledge at the same time.

• Journal of Elementary Mathematics Education in Korea
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• v.19 no.4
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• pp.589-608
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• 2015

The inquiry subject of this paper is the number of convex polygons one can form by attaching the seven pieces of a tangram. This was identified by two mathematical proofs. One is by using Pick's Theorem and the other is 和々草's method, but they are difficult for elementary students because they are part of the middle school curriculum. This paper suggests new methods, by using unit area and the minimum area which can be applied at the elementary level. Development of programs for the mathematically gifted elementary students can be composed of 4 class times to see if they can prove it by using new methods. Five mathematically gifted 5th grade students, who belonged to the gifted class in an elementary school participated in this program. The research results showed that the students can justify the number of convex polygons by attaching edgewise seven pieces of tangrams.