• Communications of Mathematical Education
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• v.21 no.2 s.30
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• pp.177-197
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• 2007

This study analyzes the theoretical background concerning problem solving, mathematization and real-life problems. Further it examines how middle school mathematics teachers and high school students of first grade recognize the real-life problems provides in textbooks concerning the area of geometry. Following those results found from this analysis, this paper reveals the issues and problems that we noticed through the analysis of real-life problems from textbooks, level 8 and level 9, Also we suggest the application of them along with the development of real-life problems for students' uplifting problem solving skills.

• The Mathematical Education
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• v.55 no.1
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• pp.21-39
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• 2016

This research examined the Korean and American $6^{th}$ grade students' mathematical problem solving ability and methods via an intuitive thinking. For this, the survey research was used. The researcher developed the questionnaire which consists of problems with intuitive and algorithmic problem solving in number and operation, figure and measurement areas. 57 Korean $6^{th}$ grade students and 60 American $6^{th}$ grade students participated. The result of the analysis showed that Korean students revealed a higher percentage than American students in correct answers. But it was higher in the rate of Korean students attempted to use the algorithm. Two countries' students revealed higher rates in that they tried to solve the problems using intuitive thinking in geometry and measurement areas. Students in both countries showed the lower percentages of correct answer in problem solving to identify the impact of counterintuitive thinking. They were affected by potential infinity concept and the character of intuition in the problem solving process regardless of the educational environments and cultures.

• The Mathematical Education
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• v.49 no.3
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• pp.313-328
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• 2010

Mathematical modeling has been observed in the way of a possibility to contribute in improving students' problem solving abilities. One of the important views of real life problem context could be described such as a useful ways to interpret the real life leading to children's abstraction process. The problem contexts for the grade 6 with mathematical modeling perspectives were developed by reviewing the current 7th National Mathematics Curriculum of Korea. Those include the 5 content areas such as number & operation, geometry, measurement, probability & statistics, and pattern & problem solving. One of problem contexts, "Space", specially designed for pattern & problem solving area, was applied to the grade 6 students and analyzed in detail to understand student's mathematical modeling progress.

• East Asian mathematical journal
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• v.30 no.2
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• pp.173-195
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• 2014

This study, utilizing several features about plane figures covered in the current secondary curriculum of mathematics and reviewing two solutions to cube duplication problem presented by Menaechmus, proving the solution by Nicomedes and visualizing solutions based on Apollonius' 'Conics' by medieval Islam geometricians such as Ab$\bar{u}$ Bakr al-Haraw$\bar{i}$, AbAb$\bar{u}$ J$\acute{a}$far al-Kh$\bar{a}$zin, Nas$\bar{i}$r al-D$\bar{i}$n al-T$\bar{u}s\bar{i}$, Y$\bar{u}$suf al-Mu'taman ibn H$\bar{u}$d, introduce to teachers and students in the field where the question of cube duplication problem comes from and which solving method has developed it and suggests new methods for visualization using dynamic geometry program as well so that the contents reviewed can be used in the filed. The solving methods to cube duplication problem in this paper are very creative and increase the practicality, efficiency and value of Mathematics, and provide students and teachers with the opportunities to reconfirm the importance and beauty of basic knowledge in the secondary geometry in the process of visualization of drawing figures using dynamic geometry program.

• Korean Journal of Computational Design and Engineering
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• v.20 no.1
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• pp.44-54
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• 2015

Solving partial differential equations (PDEs) on a manifold setting is frequently faced problem in CAD, CAM and CAE. However, unlikely to a regular grid, solutions for those problems on a triangular mesh are not available in general, as there are no well-established intrinsic differential operators. Considering that a triangular mesh is a powerful tool for representing a highly-complicated geometry, this problem must be tackled for improving the capabilities of many geometry processing algorithms. In this paper, we introduce mathematically well-defined differential operators on a triangular mesh setup, and show some examples of their applications. Through this, it is expected that many CAD/CAM/CAE application will be benefited, as it provides a mathematically rigorous solution for a PDE problem which was not available before.

• The Mathematical Education
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• v.60 no.2
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• pp.133-157
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• 2021

Ill-structured problems have drawn attention in that they can enhance problem-solving skills, which are essential in future societies. The purpose of this study is to analyze and evaluate students' spatial reasoning(Intrinsic-Static, Intrinsic-Dynamic, Extrinsic-Static, and Extrinsic-Dynamic reasoning) and problem solving abilities(understanding problems and exploring strategies, executing plans and reflecting, collaborative problem-solving, mathematical modeling) that appear in ill-structured problem-solving. To solve the research questions, two ill-structured problems based on the geometry domain were created and 11 lessons were given. The results are as follows. First, spatial reasoning ability of sixth-graders was mainly distributed at the mid-upper level. Students solved the extrinsic reasoning activities more easily than the intrinsic reasoning activities. Also, more analytical and higher level of spatial reasoning are shown when students applied functions of other mathematical domains, such as computation and measurement. This shows that geometric learning with high connectivity is valuable. Second, the 'problem-solving ability' was mainly distributed at the median level. A number of errors were found in the strategy exploration and the reflection processes. Also, students exchanged there opinion well, but the decision making was not. There were differences in participation and quality of interaction depending on the face-to-face and web-based environment. Furthermore, mathematical modeling element was generally performed successfully.

• Journal of Educational Research in Mathematics
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• v.27 no.1
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• pp.89-112
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• 2017

In this study we have analyzed processes of generalization in which students have geometrically solved cubic equation $x^3+ax=b$, regarding geometrical solution of cubic equation $x^3+4x=32$ as examples. The result of this research indicate that students could especially re-interpret the geometric solution of the given cubic equation via dynamically understanding the variables in dynamic geometry environment. Furthermore, participants could simultaneously re-interpret the given geometric solution and then present a different geometric solutions of $x^3+ax=b$, so that the level of generalization could be improved. In conclusion, the study could provide useful pedagogical implications in school mathematics that the dynamic geometry environment performs significant function as a means of students-centered exploration when understanding variables and enhancing the level of generalization in problem solving.

• The Mathematical Education
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• v.56 no.1
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• pp.63-80
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• 2017

By analysing the examination papers from third grade high school students, we classified the errors occurred in the problem solving process of high school 'Geometry and Vectors' into several types. There are five main types - (A)Insufficient Content Knowledge, (B)Wrong Method, (C)Logical Invalidity, (D)Unskilled Expression and (E)Interference.. Type A and B lead to an incorrect answer, and type C and D cannot be distinguished by multiple-choice or closed answer questions. Some of these types are classified into subtypes - (B1)Incompletion, (B2)Omitted Condition, (B3)Incorrect Calculation, (C1)Non-reasoning, (C2)Insufficient Reasoning, (C3)Illogical Process, (D1)Arbitrary Symbol, (D2)Using a Character Without Explanation, (D3) Visual Dependence, (D4)Symbol Incorrectly Used, (D5)Ambiguous Expression. Based on the these types of errors, answers of each problem was analysed in detail, and proper ways to correct or prevent these errors were suggested case by case. When problems that were used in the periodical test were given again in descriptive forms, 67% of the students tried to answer, and 14% described flawlessly, despite that the percentage of correct answers were higher than 40% when given in multiple-choice form. 34% of the students who tried to answer have failed to have logical validity. 37% of the students who tried to answer didn't have enough skill to express. In lessons on curves of secondary degree, teachers should be aware of several issues. Students are easily confused between 'focus' and 'vertex', and between 'components of a vector' and 'coordinates of a point'. Students often use an undefined expression when mentioning a parallel translation. When using a character, students have to make sure to define it precisely, to prevent the students from making errors and to make them express in correct ways.

• Journal of Computational Design and Engineering
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• v.3 no.2
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• pp.161-177
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• 2016

In history based parametric CAD modeling systems, persistent identification of the topological entities after design modification is mandatory to keep the design intent by recording model creation history and modification history. Persistent identification of geometric and topological entities is necessary in the product design phase as well as in the re-evaluation stage. For the identification, entities should be named first according to the methodology which will be applicable for all the entities unconditionally. After successive feature operations on a part body, topology based persistent identification mechanism generates ambiguity problem that usually stems from topology splitting and topology merging. Solving the ambiguity problem needs a complex method which is a combination of topology and geometry. Topology is used to assign the basic name to the entities. And geometry is used for the ambiguity solving between the entities. In the macro parametrics approach of iCAD lab of KAIST a topology based persistent identification mechanism is applied which will solve the ambiguity problem arising from topology splitting and also in case of topology merging. Here, a method is proposed where no geometry comparison is necessary for topology merging. The present research is focused on the enhancement of the persistent identification schema for the support of ambiguity problem especially of topology splitting problem and topology merging problem. It also focused on basic naming of pattern features.

• Journal of the Korean School Mathematics Society
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• v.13 no.2
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• pp.225-241
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• 2010

Even though geometry is an important part basic to mathematics, studies on the program designs and teaching strategies of geometry are insufficient. The aims of this study are to propose the model of program design for autonomous learners taking their characteristics of the mathematically gifted into consideration. The core of teaching materials are analytic geometry and projective geometry. And the new teaching strategy will introduce three steps ; a draft strategies step(problem presentation, problem solving), a supportive strategies step(abstraction of a mathematical concept, mathematical induction, and extension), a transference strategies step to teaching strategy suitable for mathematically gifted. As a result, this study will suggest the effective methods of geometry teaching for the mathematically gifted.