• Journal of the Korean School Mathematics Society
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• v.14 no.3
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• pp.355-375
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• 2011

This paper was designed for the purpose of helping the functional comprehension on the concept of a circumcenter and an incenter of triangle and offering the help for teaching-learning process on their definitions. We analysed the characteristic of the definition on a circumcenter and an incenter of triangle and studied the context, mean and purpose on the definition. The definition focusing on the construction is the definition stressed on the consistency of the concept through the fact that it is possible to draw figure of the concept. And this definition is the thing that consider the extend of the concept from triangle to polygon. Meanwhile this definition can be confused because the concept is not connected with the terminology. The definition focusing on the meaning is easy to memorize the concept because the concept is connected with the terminology but is difficult to search for the concept truth. And this definition is the thing that has the grounds on the occurrence but is taught in a made-knowledge. The definition focusing on both the construction and meaning is the definition that the starting point is vague in the logical proof process. We hope that the results are used to improve the understanding the concept of a circumcenter and an incenter of triangle in the field of mathematical education.

• Journal of the Korean School Mathematics Society
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• v.12 no.1
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• pp.47-70
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• 2009

The purpose of this research was to confirm one of constructivists' assumptions that even children 조o are with low reasoning ability can make reflective abstracting ability and cognitive structures by this ability can make generation ability of new knowledge by themselves. To investigate the assumption, learner-centered instruction were implemented to 2nd grade classroom located in Suseong Gu, DaeGu City and with lesson plans which initially were developed by Burns and corrected by the researchers. Recordings videoed using 2 video cameras, observations, instructions, children's activity worksheets, instruction journals were analyzed using multiple tests for qualitative analysis. Some conclusions are drawn from the results. First, even children with low reasoning ability can construct mathematical knowledge on multiplication in their own. ways, Thus, teachers should not compel them to learn a learning lesson's goals which is demanded in traditional instruction, with having belief they have reasoning ability. Second, teachers need to have the perspectives of respects out of each child in their classroom and provide some materials which can provoke children's cognitive conflict and promote thinking with the recognition of effectiveness of learner-centered instruction. Third, students try to develop their ability of reflective and therefore establish cognitive structures such as webs, not isolated and fragmental ones.

• Journal of the Korean School Mathematics Society
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• v.16 no.2
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• pp.383-407
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• 2013

As an alternative of making students active and independent under the passive learning conditions in school math classes, many researchers have paid much attention to problem posing and done a lot of research on it. Above all, Brown and Walter proposed What I f Not strategy as a means of problem posing. In this strategy, during the process of posing problems, the transformation of their attributes is inevitably made, and so after problem posing, the process is finished by explaining the problem. But only the simple transformation of attributes could pose wrong problems. It suggests that it is very important to recognize the relationship which leads to organic connection between attributes in order to pose the right problem. However, many other studies of problem posing haven't focused on this fact. Thus, this study tried to design a model of problem posing to help recognize inherent knowledge in the problem and then pose the right problem by adding an activity of meaning analysis. We concretely showed a model of problem posing emphasizing the analysis of meaning by means of an example, thereby examining the meaning of the model. This study expects students to have the chance to understand the true meaning of problem posing and to be active learners after all.

• Journal of the Korean School Mathematics Society
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• v.16 no.4
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• pp.797-820
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• 2013

This study intended on analyzing the error patterns of mathematic problem posing sentences by the 100 elementary pre-teachers and discussing about the solutions. The results showed that the problem posing sentences have five error patterns: phonological error patterns, word error patterns, sentence error patterns, meaning error patterns, and notation error patterns. Divided into fourteen specific error patterns, they are as in the following. 1) Phonological error patterns are consisted of the 'ㄹ' addition error pattern and the abbreviated word error pattern. 2) Words error patterns are divided with the inappropriate usage of word error pattern and the inadequate abbreviation error pattern, which are formulized four subgroups such as the case maker, ending of the word, inappropriate usage of word, and inadequate abbreviation of article or word error pattern in detail. 3) Sentence error patterns are assumed four kinds of forms: the reference, ellipsis of sentence component, word order, and incomplete sentence error pattern. 4) Meaning error patterns are composed the logical contradiction and the ambiguous meaning. 5) Notation error patterns are formed four patterns as the spacing, punctuation, orthography of Hangul, and spelling rules of foreign words in Korean. Furthermore, the solutions for these error patterns were discussed: First, it has to be perceived the differences between spoken and written language. Second, it has to be rejected the spoken expressions in written contexts. Third, it should be focused on the learning of the basic sentence patterns during the class. Forth, it is suggested that the word meaning should have the logical development perception based on what it means. Finally, it is proposed that the system of spelling of Korean has to be learned. In addition to these suggestions, a new understanding is necessary regarding writing education for college students.

• The Mathematical Education
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• v.57 no.3
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• pp.197-213
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• 2018

Descriptive problems can be used to grow student's ability of thinking logically and creatively, because it shows if the students had a reasonable way of thinking. Rate of descriptive problems is increasing in middle and high school exams. However, students in middle and high schools are generally used to answering multiple-choice or short-answer questions rather than describing the solving process. The purpose of this paper is to gain a theoretic ground to increase the rate of descriptive problems. In this study, students were to solve some multiple-choice problems, and after a few weeks, to solve the problems of same contents in the form of descriptive problems which requires the students to write the solving process. The difference of the scores were measured for each problems to each students, and students were asked what they think the reason for rise or fall of the score is. The result is as follows: First, average scores of 7 of 8 problems used in this study had fallen when it was in descriptive form, and for 5 of them in the rate of 11.2%~16.8%. Second, the main reason of falling is that the students have actual troubles of describing the solving process. Third, in the case of rising, the main reason was that partial scores were given in the descriptive problems. Last, there seems a possibility gender difference in the reason of falling. From these results, followings are suggested to advance the learning, teaching and evaluation in mathematics education: First, it has to be emphasized enough to describe the solving process when solving a problem. Second, increasing the rate of descriptive problems can be supported as a way to advance the evaluation. Third, descriptive problems have to be easier to solve than multiple-choice ones and it is convenient for the students to describe the solving process. Last, multiple-choice problems have to be carefully reviewed that the possibility of students' choosing incorrect answer with a small mistake is minimal.

• Journal of Educational Research in Mathematics
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• v.17 no.2
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• pp.179-199
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• 2007

The purpose of this study was to discover differences between mathematically gifted students (MGS) and non-gifted students (NGS) when making probability judgments. For this purpose, the following research questions were selected: 1. How do MGS differ from NGS when making probability judgments(answer correctness, answer confidence)? 2. When tackling probability problems, what effect do differences in probability judgment factors have? To solve these research questions, this study employed a survey and interview type investigation. A probability test program was developed to investigate the first research question, and the second research question was addressed by interviews regarding the Program. Analysis of collected data revealed the following results. First, both MGS and NGS justified their answers using six probability judgment factors: mathematical knowledge, use of logical reasoning, experience, phenomenon of chance, intuition, and problem understanding ability. Second, MGS produced more correct answers than NGS, and MGS also had higher confidence that answers were right. Third, in case of MGS, mathematical knowledge and logical reasoning usage were the main factors of probability judgment, but the main factors for NGS were use of logical reasoning, phenomenon of chance and intuition. From findings the following conclusions were obtained. First, MGS employ different factors from NGS when making probability judgments. This suggests that MGS may be more intellectual than NGS, because MGS could easily adopt probability subject matter, something not learnt until later in school, into their mathematical schemata. Second, probability learning could be taught earlier than the current elementary curriculum requires. Lastly, NGS need reassurance from educators that they can understand and accumulate mathematical reasoning.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.1
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• pp.81-102
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• 2010

The purpose of this study was to analyze the content and design of the seven math camp programs for students of the education centers for the elementary gifted students. The analysis focused on the goals, content, and evaluations utilized in the math camp programs. The results of the study were as follows. First, there was no big difference between the goals set for each camp, and they mainly focused on the goals in affective domain. Second, the content of math camp programs was focused on enrichment rather than acceleration. Most of the programs were focused on geometry, whereas fewer programs were focused on measurement, probability and statistics. Based on the Analysis, we found that only nine out of 27 programs applied level-wised or individual exercise programs. Third, all centers for the mathematically gifted carried out evaluations of their math camp programs. However, a specific evaluation plan was not established for the math camp program plans. We suggested the direction of math camp programs as follows. First, the goals should reflect on the intended outcomes of the math camp programs. Also, the goals of math camp programs need to be distinctive from general education goals. Second, the programs should contain harmonious contents with enrichment and acceleration and must include various reactions and task commitment. The math camp programs need to include references and an appropriate information for the gifted students to encourage self-directed learning. Third, a more specific evaluation plan for math camp programs needs to be developed for effective education for the gifted students.

• Journal of Educational Research in Mathematics
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• v.22 no.1
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• pp.53-68
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• 2012

This paper investigated the conceptual schemes four children constructed as they related division number sentences to various types of fraction: Proper fractions, improper fractions, and mixed numbers in both contextual and abstract symbolic forms. Methods followed those of the constructivist teaching experiment. Four fifth-grade students from an inner city school in the southwest United States were interviewed eight times: Pre-test clinical interview, six teaching / semi-structured interviews, and a final post-test clinical interview. Results showed that for equal sharing situations, children conceptualized division in two ways: For mixed numbers, division generated a whole number portion of quotient and a fractional portion of quotient. This provided the conceptual basis to see improper fractions as quotients. For proper fractions, they tended to see the quotient as an instance of the multiplicative structure: $a{\times}b=c$ ; $a{\div}c=\frac{1}{b}$ ; $b{\div}c=\frac{1}{a}$. Results suggest that first, facility in recall of multiplication and division fact families and understanding the multiplicative structure must be emphasized before learning fraction division. Second, to facilitate understanding of the multiplicative structure children must be fluent in representing division in the form of number sentences for equal sharing word problems. If not, their reliance on long division hampers their use of syntax and their understanding of divisor and dividend and their relation to the concepts of numerator and denominator.

• Journal of Educational Research in Mathematics
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• v.17 no.4
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• pp.395-418
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• 2007

The purpose of this study was to analyze the influence of mathematical tasks on mathematical communication. Mathematical tasks were classified into four different levels according to cognitive demands, such as memorization, procedure, concept, and exploration. For this study, 24 students were selected from the 5th grade of an elementary school located in Seoul. They were randomly assigned into six groups to control the effects of extraneous variables on the main study. Mathematical tasks for this study were developed on the basis of cognitive demands and then two different tasks were randomly assigned to each group. Before the experiment began, students were trained for effective communication for two months. All the procedures of students' learning were videotaped and transcripted. Both quantitative and qualitative methods were applied to analyze the data. The findings of this study point out that the levels of mathematical tasks were positively correlated to students' participation in mathematical communication, meaning that tasks with higher cognitive demands tend to promote students' active participation in communication with inquiry-based questions. Secondly, the result of this study indicated that the level of students' mathematical justification was influenced by mathematical tasks. That is, the forms of justification changed toward mathematical logic from authorities such as textbooks or teachers according to the levels of tasks. Thirdly, it found out that tasks with higher cognitive demands promoted various negotiation processes. The results of this study implies that cognitively complex tasks should be offered in the classroom to promote students' active mathematical communication, various mathematical tasks and the diverse teaching models should be developed, and teacher education should be enhanced to improve teachers' awareness of mathematical tasks.

• Journal of Elementary Mathematics Education in Korea
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• v.14 no.3
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• pp.937-957
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• 2010

The purpose of this study is to discover the features of symbol sense. This study tries to sum up the meaning and elements of symbol sense and the measures to improve them through documents. Also based on this, it analyzes the learning conditions about symbol sense for 6th grade mathematically able students and suggests the method that activates symbol sense in the math of elementary schools. Considering various studies on symbol sense, symbol sense means the exact knowledge and essential understanding in a comprehensive way. Symbol sense is an intuition about symbols that grasps the meaning of symbols, understands the situation of question, and realizes the usefulness of symbols in resolving a process. Considering all other scholars' opinions, this study sums up 5 elements of the symbol sense. (The recognition of needs to introduce symbol, ability to read the meaning of symbols, choice of suitable symbols according to the context, pattern guess through visualization, recognize the role of symbols in other context) This study draws the following conclusions after applying the symbol questionnaires targeting 6th grade mathematically able students : First, although they are math talents, there are some differences in terms of the symbol sense level. Second, 5 elements of the symbol sense are not completely separated. They are rather closely related in terms of mainly the symbol understanding, thereby several elements are combined.