• Education of Primary School Mathematics
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• v.27 no.3
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• pp.281-301
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• 2024

Many students have experienced difficulties due to the discontinuity in instruction between arithmetic and algebra, and in the field of elementary education, algebra is often treated somewhat implicitly. However, algebra must be learned as algebraic thinking in accordance with the developmental stage at the elementary level through the expansion of numerical systems, principles, and thinking. In this study, algebraic thinking-based classes were developed and conducted for 6th graders in elementary school, and the effect on the ability to solve word-problems in fraction division was analyzed. During the 11 instructional sessions, the students generalized the solution by exploring the relationship between the dividend and the divisor, and further explored generalized representations applicable to all cases. The results of the study confirmed that algebraic thinking-based classes have positive effects on their ability to solve fractional division word-problems. In the problem-solving process, algebraic thinking elements such as symbolization, generalization, reasoning, and justification appeared, with students discovering various mathematical ideas and structures, and using them to solve problems Based on the research results, we induced some implications for early algebraic guidance in elementary school mathematics.

• The Mathematical Education
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• v.62 no.1
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• pp.35-55
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• 2023

The purpose of this study is to examine not only students' cognition in the mathematical error-finding activity of the concept of irrational numbers, but also the students' learning stance regarding the use of errors and a teacher's questioning strategies that lead to changes in the level of mathematical discourse. To this end, error-finding individual activities, group activities, and additional interviews were conducted with 133 middle school students, and students' cognition and the teacher's questioning strategies for changes in students' learning stance and levels of mathematical discourse were analyzed. As a result of the study, students' cognition focuses on the symbolic representation of irrational numbers and the representation of decimal numbers, and they recognize the existence of irrational numbers on a number line, but tend to have difficulty expressing a number line using figures. In addition, the importance of the teacher's leading and exploring questioning strategy was observed to promote changes in students' learning stance and levels of mathematical discourse. This study is valuable in that it specified the method of using errors in mathematics teaching and learning and elaborated the teacher's questioning strategies in finding mathematical errors.

• 기호학연구
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• no.54
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• pp.7-36
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• 2018

Michael Polanyi's idea of tacit knowledge came from the realization that scientific objectivity and critical philosophy had become too restrictive for philosophy, especially in the realm of meaning, which is beyond positivistic proof and contains more non-critical elements than critical ones. In social life, people still share certain kinds of knowledge and beliefs which they obtain without making or learning those explicitly. Contemplating the role and significance of tacit knowledge, he called for a post-critical philosophy that integrates the realm of meaning and thereby appreciates the intertwined nature of tacit and explicit knowledge. Polanyi's position towards skepticism and doubt shows similarities with Charles S. Peirce's thinking about the relationship between belief and doubt. Although Peirce's semeiotics stands firmly in the tradition of critical philosophy, he affirms that doubt cannot be a constant state of mind and only belief can form a basis for a specific way of life. Polanyi's approach differs from Peirce's by focusing on the impossibility of scientific knowledge based solely on principles and precision, and his emphasis on the crucial role of the community of scientists. Nevertheless, the deeper implications of Peirce's contemplations on belief and doubt have myriad ramifications on the philosophy of science as well as the sociology of science.

• Education of Primary School Mathematics
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• v.24 no.4
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• pp.247-264
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• 2021

The area of a trapezoid is an important concept to develop mathematical thinking and competency, but many students tend to understand the formula for the area of a trapezoid instrumentally. A clue to solving these problems could be found in Dienes' theory of learning mathematics and Watson and Mason' concept of example spaces. The purpose of this study is to obtain implications for the teaching and learning of the area of the trapezoid. This study analyzed the example spaces constructed by students in learning the area of a trapezoid based on Dienes' theory of learning mathematics. As a result of the analysis, the example spaces for each stage of math learning constructed by the students were a trapezoidal variation example spaces in the play stage, a common representation example spaces in the comparison-representation stage, and a trapezoidal area formula example spaces in the symbolization-formalization stage. The type, generation, extent, and relevance of examples constituting example spaces were analyzed, and the structure of the example spaces was presented as a map. This study also analyzed general examples, special examples, conventional examples of example spaces, and discussed how to utilize examples and example spaces in teaching and learning the area of a trapezoid. Through this study, it was found that it is appropriate to apply Dienes' theory of learning mathematics to learning the are of a trapezoid, and this study can be a model for learning the area of the trapezoid.

• Communications for Statistical Applications and Methods
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• v.15 no.2
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• pp.245-254
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• 2008

According to the $7^{th}$ reform of high school education curriculum, contents on probability and statistics in mathematics of high school curriculum have been expanded compared to the previous curriculum. Thus if the curriculum contains the contents to achieve the goals for probability and statistics, more efficient education on statistics is expected to meet the needs of information age. In this thesis, we studied through network analysis if contents on probability and statistics in mathematics of high school curriculum are composed to achieve the goals. We reviewed contents on probability and statistics in mathematics of the $7^{th}$ reform of high school curriculum whether they are conformed the purpose and direction of the reform or not. As a result, the concept of probability distribution and statistical estimation with a random variable was described clearly. But, census and sample survey were not connected with other items. In a part, there were expressional mistakes.

• School Mathematics
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• v.5 no.3
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• pp.361-384
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• 2003

The purpose of this study was to investigate students' problem solving process based on the model of IDEAL if they learn to solve word problems of simultaneous linear equations through structure-representation instruction. The problem solving model of IDEAL is followed by stages; identifying problems(I), defining problems(D), exploring alternative approaches(E), acting on a plan(A). 160 second-grade students of middle schools participated in a study was classified into those of (a) a control group receiving no explicit instruction of structure-representation in word problem solving, and (b) a group receiving structure-representation instruction followed by IDEAL. As a result of this study, a structure-representation instruction improved word-problem solving performance and the students taught by the structure-representation approach discriminate more sharply equivalent problem, isomorphic problem and similar problem than the students of a control group. Also, students of the group instructed by structure-representation approach have less errors in understanding contexts and using data, in transferring mathematical symbol from internal learning relation of word problem and in setting up an equation than the students of a control group. Especially, this study shows that the model of direct transformation and the model of structure-schema in students' problem solving process of I and D stages.

• School Mathematics
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• v.9 no.3
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• pp.375-396
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• 2007

The purpose of this study is to analyze how a mathematics teacher and a high school student make the meaning in a graph and how aspects of the interpretation of a graph are interacted during the signification process, and to suggest considerations for teaching and learning of a graph. The findings of a case study have led to conclusions as follows: All of them have a difficulty in making the meaning in a graph and construct the meaning as a nested signification model. In the process which they make the meaning, they interrelate cognitive, contextual, and affective aspects and construct interpretants. In this process, a teacher focuses on cognitive aspect, based on a qualitative approach. But a student considers contextual aspect more, based on a quantitative approach. This study suggests three considerations for teaching and learning of a graph.

• The Mathematical Education
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• v.57 no.4
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• pp.433-452
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• 2018

The purpose of this study is to provide implications for improvement of mathematics textbook based on discursive approach to textbook analysis that complementarily combines a communicational approach to cognition and social semiotics. For this purpose, we reconstructed an analytic framework for discursive approach to written discourses of Korean textbook and CMP, and applied it to our analysis. Results show that several characteristics in meanings were developed by the use of words and visual mediators. First, in the case of ideational meaning, there were qualitative and quantitative differences between vocabularies used and between information addressed by visual mediators. Second, in the case of structural meaning, an offer and application of procedure was emphasized in a Korean textbook, whereas expectation and selection experiences of diverse possibilities for problem solving was underlined in CMP. In the case interpersonal meaning of student-author, imperative instructions were paid attentions in a Korean textbook. In contrast, students' interdependence and active participation were stressed in CMP. Therefore, this study addressed ideas about how to analyze mathematics textbooks based on integrated meanings developed by the use of words and visual mediators. In addition, it distributes implications for improvements of Korean mathematics textbooks through the analytic framework of both mathematical meanings and interpersonal meanings of student-author.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.4
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• pp.583-599
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• 2016

In this paper, aspects of equalities with monomial left-hand side presented in Korean elementary school mathematics textbooks are analyzed focusing on the component of expressions. According to this analysis, the textbooks deal with equalities with monomial left-hand side as though the students already know them, rather than to introduce and deal with them systematically. In this paper, the following four suggestions based on this analysis are proposed as conclusions. First, A-type equalities (with one kinds of calculation symbols and two or more numbers, variables, denominative numbers in the right-hans side) and B-type equalities (with two or more kinds of calculation symbols and two or more numbers, variables, denominative numbers in the right-hans side) may need to be introduced by the explicit description. Second, it is necessary to establish clearly the order of dealing with numeric expressions, expressions with ${\Box}$(blank) expression, expressions with words, expressions with ${\Box}$(variable), expressions with variables. Third, it needs to be noted that equalities with monomial left-hand side cab be used with a variety of meanings. Fourth, it is necessary to widen the range of the number constituting equalities with monomial left-hand side to the natural number 0 and as well as fractions, decimals.

• Communications of Mathematical Education
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• v.24 no.2
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• pp.475-502
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• 2010

In this paper, the ability to use mathematical representations in solving math problem was analyzed according to student assessment levels using 113 first-year high school students, and the characteristics of their representation usage according to student assessment levels were also examined. For this purpose, problems were presented that could be solved using various mathematical representations, and the students were asked to solve them using a maximum of three different methods. Also, based on the comparative analysis results of a paper evaluation, six students were selected and interviewed, and the reasons for their representation usage differences were analyzed according to their student assessment levels. The results of the analysis show that over 50% of high ranking students used two or more representations in all questions to solve problems, but with middle ranking students, there were deviations depending on the difficulty of the questions. Low ranking students failed to use representation in diverse ways when solving problems. As for characteristics of symbol usage, high ranking students preferred using formulas and used mathematical representations efficiently while solving problems. In contrast, middle and low ranking students mostly used tables or pictures. Even when using the same representations, high ranking students' representations were expressed in a more structurally refined manner than those by middle and low ranking students.