• School Mathematics
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• v.12 no.3
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• pp.439-454
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• 2010

This study presents how two eighth grade students generated their own algorithms in the context of fraction measurement division situations by modifications of unit-segmenting schemes. Teaching experiment was adopted as a research methodology and part of data from a year-long teaching experiment were used for this report. The present study indicates that the two participating students' construction of reciprocal relationship between the referent whole [one] and the divisor by using their unit- segmenting schemes and its strategic use finally led the students to establish an algorithm for fraction measurement division problems, which was on par with the traditional invert-and-multi- ply algorithm for fraction division. The results of the study imply that teachers' instruction based on understanding student-generated algorithms needs to be accounted as one of the crucial characteristics of good mathematics teaching.

• The Mathematical Education
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• v.60 no.1
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• pp.1-19
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• 2021

This study investigates students' conceptions of fractions from a measurement approach while providing a technological environment designed to support students' understanding of the relationships between quantities and adjustable units. 13 third-graders participated in this study and they were involved in a series of measurement tasks through task-based interviews. The tasks were devised to investigate the relationship between units and quantity through manipulations. Screencasting videos were collected including verbal explanations and manipulations. Drawing upon the theory of semiotic mediation, students' constructed concepts during interviews were coded as mathematical words and visual mediators to identify conceptual profiles using a fine-grained analysis. Two students changed their strategies to solve the tasks were selected as a representative case of the two profiles: from guessing to recursive partitioning; from using random units to making a relation to the given unit. Dragging mathematical objects plays a critical role to mediate and formulate fraction understandings such as unitizing and partitioning. In addition, static and dynamic representations influence the development of unit concepts in measurement situations. The findings will contribute to the field's understanding of how students come to understand the concept of fraction as measure and the role of technology, which result in a theory-driven, empirically-tested set of tasks that can be used to introduce fractions as an alternative way.

• Journal of Science Education
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• v.46 no.1
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• pp.109-120
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• 2022

Fraction division is the content that is important but difficult to learn because it includes the process of finding a numerical expression in the real-world context, the process of making a context that matches a numerical expression, how to solve division, and the justification of standard algorithm. This study analyzes the word problems and problem solving methods about fraction division which elementary pre-service teachers represented. Pre-service teachers have more difficulty in making word problem where the dividend is less than the divisor and they also show typical errors in making the word problems. There were differences in the methods presented according to the contexts of division in problem solving. Through this study, it is necessary to rethink the teaching methods for fraction division instruction in the curriculum for pre-service teachers and analyze the formation process of 'knowledge for content and teaching' because of the differences in responses between grades.

• Strategy21
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• s.43
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• pp.273-302
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• 2018

1980년대 북 핵개발을 처음 발견 이후, 미국은 북한의 비핵화라는 정책목적달성을 위해 다양한 정책을 사용해 왔지만 현재까지 실패하였다. 미국의 대북 정책 실패의 결과는 북핵 문제의 고착화 속에서, 평양의 핵무기 개발 가속화 야기로 한반도 및 미국을 핵위협 속에 놓이게 하였다. 특히, 지난 해, 북미간의 가열된 공격적 수사와 행동에 의한 한반도 위기설은 절정에 달하였다. 해결의 실마리가 보이지 않던 한반도의 갈등 및 위기는, 지난 4월에 열린 남북 정상회담을 통해 25년간의 핵위협의 굴레를 벗어날 기회를 다시 한번 맞이하게 되었다. 남북 정상회담 이후 이어질 북미 정상회담 등 향후 미국의 정책은 한반도 비핵화를 위한 중요한 분수령에 다시 한번 서있다. 하지만, 과거의 25년간의 역사는 다시 맞이한 '한반도의 봄'에 대한 낙관적 희망만을 주지 않는다. 과거, 양자적, 다자적 협상을 이룸에도 불구하고, 북핵 문제는 다시 위기에 접어드는 반복된 패턴과 사이클 속에 악화 되어 왔기 때문이다. 비핵화의 분수령에 있는 미 정부는 다시 한번 과거의 정책을 뒤돌아 보고, 남북 정상회담을 통해 어렵게 맞이한 기회를 결실로 이룰 수 있도록 어느 때 보다 신중한 노력이 필요하다. 최근 몇 달간 북핵 문제는 경이로운 속도로 진전을 보였지만, 한순간의 정책의 실패는 최근 보여진 진전의 속도 이상의 속도로 문제를 악화 시킬 수 있으며, 그 결과는 작년 여름과 겨울의 위기보다 더욱 심각 할 수 있음을 명심해야 한다. 이러한 점에서 이 보고서는 과거의 역사 및 이론적 분석을 통해 과거 미국의 북핵정책 실패 원인을 분석하고 정책을 제언하는데 그 목적이 있다. 과거 미 북핵 실패의 원인은 크게 3가지로 보인다. 먼저, 포괄적인 그리고 북한 정권의 특성에서 비롯된 북핵 개발의 모티브를 정확히 이해하는데 실패하여, 북한의 정책적 계산을 변화 시키는데 실패 하였다. 둘째, 북한 문제를 둘러싼 외부적 복잡성이 미북핵 정책실패를 야기하였다. 한반도 문제는 과거부터 다양한 국가들의 이해관계에 둘러 싸여 왔다. 북핵 문제도 남북 및 미국을 비롯 중국 등 주변국의 복잡성이 불확실성을 가중시켜 문제를 더욱 복잡하게 하였으며, 미국의 대북 협상의 영향력을 약화 시켰다. 셋째, 과거 누적된 두 국가간의 불신은 협상 이후 상대의 신뢰 있는 이행에 대한 불신을 야기하여 미국의 정책의 효과성을 저해하였다. 미국은 북핵 개발 모티브에 대한 포괄적 이해와 한국 및 중국과의 다자외교로 과거의 실패를 극복하고 25년간의 북핵문제의 고리를 끊어야 할 것이다.

• Journal of Gifted/Talented Education
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• v.24 no.1
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• pp.17-43
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• 2014

On the subjects of elementary 3rd grade three child prodigies who had learned the four fundamental arithmetic operations and primary concepts of fraction, this study conducted a qualitative case research to examine how they composed schema of addition and multiplication of fractions and transformed schema through recognition of precise concepts and linking of concepts with addition and multiplication of fractions as the contents. That is to say, this study investigates what schema and transformed schema child prodigies form through composition of primary mathematic concepts to succeed in relational understanding of addition and multiplication of fractions, how they use their own formed schema and transformed schema for themselves to approach solutions to problems with addition and multiplication of fractions, and how the subjects' concept formation and schema in their problem solving competence proceed to carry out transformations. As a result, we can tell that precise recognition of primary concepts, schema, and transformed schema work as crucial factors when addition of fractions is associated with multiplication of fractions, and then that the schema and transformed schema that result from the connection among primary mathematic concepts and the precise recognition of the primary concepts play more important roles than any other factors in creative problem solving with respect to addition and multiplication of fractions.

• School Mathematics
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• v.18 no.3
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• pp.647-666
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• 2016

This study investigates pre-service teacher's understanding of the concept and representations of irrational numbers. We classified the representations of irrational numbers into six categories; non-fraction, decimal, symbolic, geometric, point on a number line, approximation representation. The results of this study are as follows. First, pre-service teachers couldn't relate non-fractional definition and incommensurability of irrational numbers. Secondly, we observed the centralization tendency on symbolic representation and the little attention to other representations. Thirdly, pre-service teachers had more difficulty moving between symbolic representation and point on a number line representation of ${\pi}$ than that of $\sqrt{5}$ We suggested the concept of irrational numbers should be learned in relation to various representations of irrational numbers.

• Journal of Gifted/Talented Education
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• v.24 no.3
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• pp.339-358
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• 2014

On the subjects of elementary 3-rd grade three child prodigies who learned primary concept of division, this study explored how they could compose schema and transformed schema through recognition of precise concepts and linking with the contents of division of fraction. That is to say, this study examined in depth what schema and transformed schema as primary concept of division they composed to get relational understanding of division of fraction, and how they used the schema and transformed schema composed by themselves to approach problem solving as well as how they transformed the schema in their concept composition and problem solving competence. As a result, it was found that learning of primary concept of division played a key role of composing schema and transformed schema needed for coping with division of fraction, and that at this time, composition of the transformed schema and transformed schema derived from the recognition of primary concept of division could play the inevitable role of problem solving for division of fraction.

• Journal of Educational Research in Mathematics
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• v.14 no.2
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• pp.207-219
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• 2004

There have been studies reporting the increase in student confidence in mathematics when using technology. However, past studies indicating a positive correlation between technology and confidence in mathematics do not explain why they see this positive outcome. With increased availability and easy access to the Internet in schools and the development of free online virtual manipulatives, this research was interested in how the use of virtual manipulatives in mathematics can affect students confidence in their mathematical abilities. Our hypothesis was that the classes using virtual manipulatives which allows students to connecting dynamic visual image with abstract symbols will help students gain a deeper conceptual understanding of math concept thus increasing their confidence and ability in mathematics. The participants in this study were 46 fifth-grade students in three ability groups: one high, one middle and one low. During a two-week unit on fractions, students in three groups interacted with several virtual manipulative applets in a computer lab. Data sources in the project included a pre and posttest of students mathematics content knowledge, Confidence in Learning Mathematics Scale, field notes and student interviews, and classroom videotapes. Our aim was to find evidence for increased level of confidence in mathematics as students strengthened their understanding of fraction concepts. Results from the achievement score indicated an overall main effect showing significant improvement for all ability groups following the treatment and an increase in the confidence level from the preassessment of the Confidence in Learning Mathematics Scale in the middle and high ability groups. An interesting finding was that the confidence level for the low ability group students who had the highest confidence level in the beginning did not change much in the final confidence scale assessment. In the middle and high ability groups, the confidence level did increase according to the improvement of the contest posttest. Through interviews, students expressed how the virtual manipulatives assisted their understanding by verifying their answers as they worked and facilitated their ability to figure out math concept in their mind and visually.

• Education of Primary School Mathematics
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• v.25 no.2
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• pp.179-195
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• 2022

Mathematical communication competency, one of the six mathematical competencies emphasized in the latest mathematics curriculum, plays an important role both as a means and as a goal for students to learn mathematics. Therefore, it is meaningful to find instructional methods to improve students' mathematical communication competency and analyze their communication competency in detail. Given this background, this study analyzed 64 sixth graders' mathematical communication competency after they participated in the lessons of fraction division emphasizing mathematical communication. A written assessment for this study was developed with a focus on the four sub-elements of mathematical communication (i.e., understanding mathematical representations, developing and transforming mathematical representations, representing one's ideas, and understanding others' ideas). The results of this study showed that students could understand and represent the principle of fraction division in various mathematical representations. The students were more proficient in representing their ideas with mathematical expressions and solving them than doing with visual models. They could use appropriate mathematical terms and symbols in representing their ideas and understanding others' ideas. This paper closes with some implications on how to foster students' mathematical communication competency while teaching elementary mathematics.

• Journal of the Korean School Mathematics Society
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• v.13 no.3
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• pp.483-498
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• 2010

The purpose of this paper is to investigate the dispositions of university students' understanding and reasoning about rational number concept. For this, we surveyed for the subject groups of prospective math teachers(33), engineering major students(35), American engineering and science major students(28). The questionnaire consists of four problems related to understanding of rational number concept and three problems related to rational number operation reasoning. We asked multi-answers for the front four problem and the order of favorite algorithms for the back three problems. As a result, we found that university students don't understand exactly the facets of rational number and prefer the mechanic approaches rather than conceptual one. Furthermore, they reasoned illogically in many situations related to fraction, ratio, proportion, rational number and don't recognize exactly the connection between them, and confuse about rational number concept.