• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1573-1594
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• 2017

This paper deals with the existence and energy estimates of solutions for a class of degenerate nonlocal problems involving sub-linear nonlinearities, while the nonlinear part of the problem admits some hypotheses on the behavior at origin or perturbation property. In particular, for a precise localization of the parameter, the existence of a non-zero solution is established requiring the sublinearity of nonlinear part at origin and infinity. We also consider the existence of solutions for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz. In what follows, by combining two algebraic conditions on the nonlinear term which guarantees the existence of two solutions as well as applying the mountain pass theorem given by Pucci and Serrin, we establish the existence of the third solution for our problem. Moreover, concrete examples of applications are provided.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.307-321
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• 2004

In this paper, the unsteady Stokes problem is considered and also the pressure-correction method for the problem is described. At a fixed time level, we reduce the problem to two symmetric positive definite problems which depend on a time step parameter. Linear systems that arise from the problems are large, sparse, symmetric, positive definite and ill-conditioned as the time step tends to zero. Preconditioned problems based on an additive Schwarz method for solving the symmetric positive definite problems are derived and preconditioners are defined implicitly. It will be shown that the rate of convergence is independent of the mesh parameters as well as the time step size.

• The Mathematical Education
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• v.46 no.1 s.116
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• pp.69-79
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• 2007

Students experience many problem solving activities in school mathematics. These activities have focused on finding the solution whose existence was known, and then again conjecture about existence of solution or posing of problems has been neglected. It needs to put more emphasis on conjecture and problem posing activities in school mathematics. To do this, a model and examples of designing mathematical activities centered on conjecture and problem posing are needed. In this article, we introduce some examples of designing such activities (from the pythagorean theorem, the determination condition of triangle, and existing solved-problems in textbook) and examine suggestions for mathematics education. Our examples can be used as instructional materials for mathematically able students at middle school.

• Education of Primary School Mathematics
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• v.20 no.3
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• pp.193-204
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• 2017

The purpose of the study was to investigate and develop a practical approach to integrating student-driven mathematical problems posing in mathematics instruction. A problem posing activity was performed during regular mathematics instruction. A total of 540 mathematical problems generated by students were recorded and analysed using systemic procedures and criteria. Of the problems, 81% were mathematically solvable problem and 18% were classified as error type problems. The Mathematically solvable problem were analysed and categorized according to the complexity level; 13% were of a high-level, 30% mid-level and 57% low-level. The error-type problem were classified as such within three categories: non-mathematical problem, statement or mathematically unsolvable problem. The error-type problem category was distributed variously according to the leaning theme and accomplishment level. The study has important implications in that it used systemic procedures and criteria to analyse problem generated by students and provided the way for integrating mathematical instruction and problem posing activity.

• Journal of the Korean School Mathematics Society
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• v.22 no.3
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• pp.329-350
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• 2019

Productive struggle is a student's persevering effort to understand mathematical concepts and solve challenging problems that are not easily solved, but the problem can lead to curiosity. Productive struggle is a key component of students' learning mathematics with a conceptual understanding, and supporting it in learning mathematics is one of the most effective mathematics teaching practices. In comparison to research on students' productive struggles, there is little research on preservice mathematics teachers' productive struggles. Thus, this study focused on the productive struggles that preservice mathematics teachers face in solving a non-routine mathematics problem. Polya's four-step problem-solving process was used to analyze the collected data. Examples of preservice teachers' productive struggles were analyzed in terms of each stage of the problem-solving process. The analysis showed that limited prior knowledge of the preservice teachers caused productive struggle in the stages of understanding, planning, and carrying out, and it had a significant influence on the problem-solving process overall. Moreover, preservice teachers' experiences of the pleasure of learning by going through productive struggle in solving problems encouraged them to support the use of productive struggle for effective mathematics learning for students, in the future. Therefore, the study's results are expected to help preservice teachers develop their professional expertise by taking the opportunity to engage in learning mathematics through productive struggle.

• Journal of the Korean School Mathematics Society
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• v.13 no.4
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• pp.655-678
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• 2010

In the elementary mathematics, geometric education emphasize spatial sense and understandings of figures through development of intuitions in space. Especially space visualization is one of the factors which try conclusion with geometric problem solving. But studies about space visualization are limited to middle school geometric education, studies in elementary level haven't been done until now. Namely, discussions about elementary students' space visualization process and methods in plane or space figures is deficient in relation to geometric problem solving. This paper examines these aspects, especially in relation to plane and space problem solving in elementary levels. First, we investigate visualization methods for plane problem solving and space problem solving respectively, and analyse in diagram form how progress understanding of figures and visualization process. Next, we derive constituent factor on visualization process, and make a check errors which represented by difficulties in visualization process. Through these analysis, this paper aims at deriving an influence of visualization on geometric problem solving in the elementary mathematics.

• Education of Primary School Mathematics
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• v.4 no.1
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• pp.51-61
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• 2000

Mathematical Problem solving has been the focus of a considerable amount of research over past 30 years. But nowadays problem solving is being beginning to be of less interest to mathematics education researchers. Moreover, mathematics teachers have an urgent need to be provided with well-documented informations about "teaching of(expecially, via) problem solving" though following research issues ：ⅰ) the role of the teacher in a problem-centered classroom, ⅱ) what actually takes place in problem-centered classrooms, and iii) groups and whole classes' problem solving rather than individuals. This paper intends to give some informations about practice of teaching mathematical problem solving in elementary school.ry school.

• The Mathematical Education
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• v.54 no.1
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• pp.65-81
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• 2015

The purpose of this study is to investigate the international trends of how the core competencies are reflected in mathematics curriculum, and to find the implications for the revision of Korean mathematics curriculum. For this purpose, the curriculum of the 9 countries including the U.S., Canada(Ontario), England, Australia, Poland, Singapore, China, Taiwan, and Hong Kong were thoroughly reviewed. It was found that a variety of core competencies were reflected in mathematics curricula in the 9 countries such as problem solving, reasoning, communication, mathematical knowledge and skills, selection and use of tools, critical thinking, connection, modelling, application of strategies, mathematical thinking, representation, creativity, utilization of information, and reflection etc. Especially the four most common core competencies (problem solving, reasoning, communication, and creativity) were further analyzed to identify their sub components. Consequently, it was recommended that new mathematics curriculum should consider reflecting various core competencies beyond problem solving, reasoning, and communication, and these core competencies are supposed to combine with mathematics contents to increase their feasibility. Finally considering the fact that software education is getting greater attention in the new curriculum, it is necessary to incorporate computational thinking into mathematics curriculum.

• Research in Mathematical Education
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• v.19 no.1
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• pp.19-41
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• 2015

Previous studies indicated that students tended to hold less satisfactory beliefs about the discipline of mathematics, beliefs about themselves as learners of mathematics, and beliefs about mathematics teaching and learning. However, only a few studies had developed curricular interventions to change students' beliefs. This study aimed to examine the effect of a problem-solving curriculum (i.e., Mathematical Problem Solving for Everyone, MProSE) on Singaporean Grade 7 students' beliefs about mathematical problem solving (MPS). Four classes (n =142) were engaged in ten lessons with each comprising four stages: understand the problem, devise a plan, carry out the plan, and look back. Heuristics and metacognitive control were emphasized during students' problem solving activities. Results indicated that the MProSE curriculum enabled some students to develop more satisfactory beliefs about MPS. Further path analysis showed that students' attitudes towards the MProSE curriculum are important predictors for their beliefs.

• Journal of applied mathematics & informatics
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• v.7 no.3
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• pp.787-800
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• 2000

In this paper we consider an optimal control system described by n-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem. We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.