• Journal for History of Mathematics
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• v.36 no.2
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• pp.21-34
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• 2023

Spherical trigonometry refers to the geometry related to spherical triangles. It has been an important tool for studying astronomy since ancient times. In trigonometry, concepts such as trigonometric functions naturally emerge from the relationship between arcs and chords of a circle. In this paper, we briefly examine the origin of spherical trigonometry. To introduce the basics of spherical trigonometry, we present fundamental and important theorems such as Menelaus's theorem, the law of sines and the law of cosines on a sphere, along with their proofs. Furthermore, we discuss the educational value and potential applications of spherical trigonometry.

• Communications of Mathematical Education
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• v.34 no.3
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• pp.393-419
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• 2020

Trigonometry allows us to recognize the usefulness of mathematics through connection with real life and other disciplines, and lays the foundation for the concept of higher mathematics through connection with trigonometric functions. Since international comparisons on the trigonometry area of textbooks can give implications to trigonometry teaching and learning in Korea, this study attempted to compare trigonometry in textbooks in Korea, Australia and Finland. In this study, through the horizontal and vertical analysis presented by Charalambous et al.(2010), the objectives of the curriculum, content system, achievement standards, learning timing of trigonometry content, learning paths, and context of problems were analyzed. The order of learning in which the three countries expanded size of angle was similar, and there was a difference in the introduction of trigonometric functions and the continuity of grades dealing with trigonometry. In the learning path of textbooks on the definition method of trigonometric ratios, the unit circle method was developed from the triangle method to the trigonometric function. However, in Korea, after the explanation using the quadrant in middle school, the general angle and trigonometric functions were studied without expanding the angle. As a result of analyzing the context of the problem, the proportion of problems without context was the highest in all three countries, and the rate of camouflage context problem was twice as high in Korea as in Australia or Finland. Through this, the author suggest to include the unit circle method in the learning path in Korea, to present a problem that can emphasize the real-life context, to utilize technological tools, and to reconsider the ways and areas of the curriculum that deal with trigonometry.

• The Mathematical Education
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• v.56 no.1
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• pp.101-118
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• 2017

In this article we study a reconstruction of mathematical knowledge on trigonometry by the method of ascent from the abstract to the concrete from the pedagogical viewpoint of dialectic. The direction of education is shifting in a way that emphasizes the active constitution of knowledge by the learning subjects from the perspective that knowledge is transferred from the teacher to the student. In mathematics education, active discussions on the construction of mathematical knowledge by learners have been going on since the late 1990s. In Korea, concepts and aspects of constructivism such as operational constructivism, radical constructivism, and social constructivism were introduced. However, examples of practical construction according to the direction of construction of mathematical knowledge are very hard to find. In this study, we discuss the direction of the actual construction of mathematical knowledge and suggest a concrete example of the actual construction of trigonometry knowledge from a constructivist point of view. In particular, we discuss the process of the construction of theoretical knowledge, the ascent from the abstract to the concrete, based on the literature study from the pedagogical viewpoint of dialectic, and show how to construct the mathematical knowledge on trigonometry by the method of ascent from the abstract to the concrete. Through this study, it is expected to introduce the new direction and new method of knowledge construction as 'the ascent from the abstract to the concrete', and to present the possibility of applying dialectic concepts to mathematics education.

• Journal for History of Mathematics
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• v.35 no.3
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• pp.73-113
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• 2022

In this paper we survey on the geometric development in the history of Joseon mathematics. We have relatively many research papers on the history of equations in Joseon but the history of geometry is limited to that of trigonometry (gugosul). We survey on the results on the whole geometry including the introduction of western geometry in Joseon. Joseon mathematics developed differently during several different periods. We investigate how geometric theories developed during those periods and the meaning behind them. We do not claim that our survey is anywhere close to a complete one. This is rather a preliminary attempt to collect research results to plan our research following those of our predecessors.

• The Pure and Applied Mathematics
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• v.19 no.4
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• pp.337-347
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• 2012

The taxicab distance and Chinese-checker distance in the plane are practical distance notions with a wide range of applications compared to the Euclidean distance. The ${\alpha}$-distance was introduced as a generalization of these two distance functions. In this paper, we study alpha circle, trigonometry, and the area of a triangle in the plane with ${\alpha}$-distance.

• Journal of the Korean Mathmatical Society
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• v.4 no.1
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• pp.18-24
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• 1967

• Research in Mathematical Education
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• v.13 no.2
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• pp.151-170
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• 2009

This study reports a research which was conducted on how frequently and where the students use the unit circle method while dealing with trigonometric functions in solving the trigonometry questions. Moreover, the reasons behind the choice of the methods, which could be the unit circle method, the ratio method, or the use of trigonometric identities, are also investigated to get an insight about their understanding. In this study, the relationship between the students' choices of methods in solving questions is examined in terms of instrumental or relational understanding. This is a multi-method research which involves a range of research strategies. The research techniques used in this study are test, verbal protocol (think aloud), and interview. The test has been applied to ten tenth grade students of a public school to get students' solution processes on the paper. Later on, verbal protocol has been performed with three students of these ten who were of the upper, middle and lower sets in terms of their performance in the test. The aim was to get much deeper data on the students' thinking and reasoning. Finally, interview questions have been asked both these three students and other three from the initial ten students to question the reasons behind their answers to the trigonometry questions. Findings in general suggest that students voluntarily choose to learn instrumentally whose reasons include teachers' and students' preference for the easier option and the anxiety resulting from the external exam pressure.

• Journal of Educational Research in Mathematics
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• v.24 no.4
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• pp.607-622
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• 2014

The purpose of this paper is that it analyzes the historical development of the concept of trigonometric functions and discuss some didactical implications. The results of the study are as follows. First, the concept of trigonometric functions is developed from line segments measuring ratios to numbers representing the ratios. Geometry, arithmetic, algebra and analysis has been integrated in this process. Secondly, as a result of developing from practical calculation to theoretical function, periodicity is formalized, but 'trigonometry' is overlooked. Third, it must be taught trigonometry relationally and structurally by the principle of similarity. Fourth, the conceptual generalization of trigonometric functions must be recognized as epistemological obstacle, and it should be improved to emphasize the integration revealed in history. The results of these studies provide some useful suggestions to teaching and learning of trigonometry.

• School Mathematics
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• v.6 no.1
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• pp.21-35
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• 2004

This article was to develop a tool and apply it to the high school classrooms for the performance assessment in trigonometry Bloom(1956)'s cognitive domain and holistic rubric and analytic rubric(NCTM, 1999) were used to guide the development of 12 problems. To find validity and credibility of this developed tool, Cronbach n and Rasch's BIGSTEPS were used with the samples of high students, 208, using SPSS 10.0K. The results from the investigation, indicated that the tool was very worth assessing students' achievement and there was no difference between the areas where students lived, but were differences between genders as well as between a specialized high school and preparatory high schools.

• Research in Mathematical Education
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• v.7 no.2
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• pp.101-117
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• 2003

In this paper, we give examples of models we have created for use in university mathematics courses. We explain the concept of linear transformation, investigate the roles of each component of 2 ${\times}$ 2 and 3 ${\times}$ 3 transformation matrices, consider the relation between sound and trigonometry, visualize the Riemann sum, the volume of surfaces of revolution and the area of unit circle. This paper illustrates how one can use Mathematica to visualize abstract mathematical concepts, thus enabling students to understand mathematics problems effectively in class. Development of these kinds of teaching and learning models can stimulate the students' curiosity about mathematics and increase their interest.