• 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.

• The Mathematical Education
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• v.45 no.2 s.113
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• pp.177-189
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• 2006

This study focuses on the analysis of prospective elementary teachers' representation of trapezoid area and teacher educator's reflecting in the context of a mathematics course. In this study, I use my own teaching and classroom of prospective elementary teachers as the site for investigation. 1 examine the ways in which my own pedagogical content knowledge as a teacher educator influence and influenced by my work with students. Data for the study is provided by audiotape of class proceeding. Episode describes the ways in which the mathematics was presented with respect to the development and use of representation, and centers around trapezoid area. The episode deals with my gaining a deeper understanding of different types of representations-symbolic, visual, and language. In conclusion, I present two major finding of this study. First, Each representation influences mutually. Prospective elementary teachers reasoned visual representation from symbolic and language. And converse is true. Second, Teacher educator should be prepared proper mathematical language through teaching and learning with his students.

• East Asian mathematical journal
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• v.31 no.2
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• pp.167-188
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• 2015

Formula for the area of a trapezoid is an educational material that can handle algebraic and geometric perspectives simultaneously. In this note, we will make up the expression equivalent algebraically to the formula for the area of a trapezoid, and deal with the conversion of a geometric point of view, in algebraic terms of translating and interpreting the expression geometrically. As a result, the geometric conversion model, the first algebraic model, the second algebraic model are obtained. Therefore, this problem is a good material to understand the advantages and disadvantages of the algebraic and geometric perspectives and to improve the mathematical insight through complementary activity. In addition, these activities can be used as material for enrichment and gifted education, because it helps cultivate a rich perspective on diverse and creative thinking and mathematical concepts.

• Journal of Educational Research in Mathematics
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• v.23 no.2
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• pp.253-267
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• 2013

The newly revised mathematics curriculum of 2007 speaks of ultimate goal to develop ability to think and communicate mathematically, in order to develop ability to rationally deal with problems arising from the life around, which puts emphasize on mathematical communication. In this study, analysis on mathematical communication and analogical thinking process of group of students with similar level of academic achievement and that with different level, and thus analyzed if such communication has affected analogical thinking process in any way. This study contains following subjects: 1. Forms of mathematical communication took placed at the two groups based on achievement level were analyzed. 2. Analogical thinking process was observed through trapezoid's area obtaining activity and analyzed if communication within groups has affected such process anyhow. A framework to analyze analogical thinking process was developed with reference of problem solving procedure based on analogy, suggested by Rattermann(1997). 15 from 24 students of year 5 form of N elementary school at Gunpo Uiwang, Syeonggi-do, were selected and 3 groups (group A, B and C) of students sharing the same achievement level and 2 groups (group D and E) of different level were made. The students were led to obtain areas of parallelogram and trapezoid for twice, and communication process and analogical thinking process was observed, recorded and analyzed. The results of this study are as follow: 1. The more significant mathematical communication was observed at groups sharing medium and low level of achievement than other groups. 2. Despite of individual and group differences, there is overall improvement in students' analogical thinking: activities of obtaining areas of parallelogram and trapezoid showed that discussion within subgroups could induce analogical thinking thus expand students' analogical thinking stage.

• Journal of Elementary Mathematics Education in Korea
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• v.15 no.2
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• pp.361-383
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• 2011

This study is on teaching about the areas of plane figures through open instruction, which aims to discover the pedagogical meanings and implications in the application of open methods to math classes by running the Math A & B classes regarding the areas of parallelogram, triangle, trapezoid and rhombus for fifth graders of elementary school through open instruction method and analyzing the educational process. This study led to the following results. First, it is most important to choose proper open-end questions for classes on open instruction methods. Teachers should focus on the roles of educational assistants and mediators in the communication among students. Second, teachers need to make lists of anticipated responses from students to lead them to discuss and focus on more valuable methods. Third, it is efficient to provide more individual tutoring sessions for the students of low educational level as the classes on open instruction methods are carried on. Fourth, students sometimes figured out more advanced solutions by justifying their solutions with explanations through discussions in the group sessions and regular classes. Fifth, most of students were found out to be much interested in the process of thinking and figuring out solutions through presentations and questions in classes and find it difficult to describe their thoughts.

• Education of Primary School Mathematics
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• v.19 no.2
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• pp.113-125
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• 2016

The concept and measuring activities of the height of figures are essential to find the areas or volumes of the corresponding figures. For plane figures, the height of a triangle is defined to be the line segment from a vertex that is perpendicular to the opposite side of the triangle, whereas the height of a parallelogram(trapezoid) is defined to be the distance between two parallel sides. For the solid figures, the height of a prism is defined to be the distance of two parallel bases, whereas the height of a pyramid is defined to be the perpendicular distance from the apex to the base. In addition, the height of a cone is defined to be the length of the line segment from the apex that is perpendicular to the base and the height of a cylinder is defined to be the length of the line segment that is perpendicular to two parallel bases. In this study, we discuss some pedagogical problems on the concepts and measuring activities of the height of figures to provide alternative activities and suggest their educational implications from a teaching and learning point of view.

• Journal of the Korean School Mathematics Society
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• v.15 no.1
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• pp.27-51
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• 2012

This paper analyzed the pedagogical content knowledge (PCK) presented in three novice teachers' mathematics instruction. PCK was analyzed in terms of the knowledge of mathematics content, the knowledge of students' understanding, and the knowledge of teaching methods. Teacher A executed a concept-oriented instruction with manipulative materials because she had difficulties in learning mathematics during her childhood. Teacher B attempted to implement an inquiry-centered instruction in the lesson of looking for the area of a trapezoid. Teacher C focused on the real-life connection to mathematics instruction. There were substantial differences among the teachers' PCK revealed in mathematics teaching, depending on their instructional goals. The detailed analyses of three teachers' teaching in terms of their PCK will give rise to the issues and suggestions of professional development for beginning elementary school teachers in mathematics teaching.

• School Mathematics
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• v.8 no.3
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• pp.327-339
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• 2006

In this study, a teaching units for teaching and learning of secondary preservice teachers' mathematising is designed, focusing on reinvention of Bretschneider's formula. preservice teachers can obtain the following through this teaching units. First, preservice teachers can experience mathematising which invent a noumenon which organize a phenomenon, They can make an experience to invent Bretscheider's formula as if they invent mathematics really. Second, preservice teachers can understand one of the mechanisms of mathematics knowledge invention. As they reinvent Brahmagupta's formula and Bretschneider's formula, they understand a mechanism that new knowledge is invented Iron already known knowledge by analogy. Third, preservice teachers can understand connection between school mathematics and academic mathematics. They can understand how the school mathematics can connect academic mathematics, through the relation between the school mathematics like formulas for calculating areas of rectangle, square, rhombus, parallelogram, trapezoid and Heron's formula, and academic mathematics like Brahmagupta's formula and Bretschneider's formula.