• School Mathematics
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• v.18 no.1
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• pp.215-229
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• 2016

This study aims to identify Archimedes' idea used while proving proposition 23 in 'Quadrature of the Parabola' and to provide an alternative way for finding the sum of geometric series without applying the concept of limit by extending the idea though metaphor. This metaphorical model is characterized as static and thus can be complimentary to the dynamic aspect of limit concept adopted in Korean high school mathematics textbooks. In addition, middle school students can understand $0.999{\cdots}=1$ with this model in a structural way differently from the operative one suggested in Korean middle school mathematics textbooks. In this respect, I argue that the metaphorical model can be an useful educational tool for Korean secondary students to overcome epistemological obstacles inherent in the concepts of infinity and limit by making it possible to transfer from geometrical context to algebraic context.

• Communications of Mathematical Education
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• v.23 no.4
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• pp.1079-1092
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• 2009

The purpose of this study is to investigate Newton's binomial theorem that was on epistemological basis of the emergent background and developmental course of infinite series and power series. Through this investigation, it will be examined how finding the approximate of square root of given numbers, the method of the inverse method of fluxions by Newton, and Gregory and Mercator series were developed in the course of history of mathematics. As the result of this study pedagogical analysis and discussion of the history of mathematics on Newton's binomial theorem will be presented.

• Journal of Educational Research in Mathematics
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• v.25 no.3
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• pp.263-279
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• 2015

The introduction of real numbers is one of the most difficult steps in the teaching of school mathematics since the mathematical justification of the extension from rational to real numbers requires the completeness property. The author elucidated what questions about real numbers can be unanswered as the "institutional didactic void" in school mathematics defining real numbers as the union of the rational and irrational numbers. The pre-service teachers' explanations on the extension from rational to real numbers and the raison d'$\hat{e}$tre of arbitrary non-recurring decimals showed the superficial and fragmentary understanding of real numbers. Connecting school mathematics to university mathematics via the didactic void, the author discussed how pre-service teachers could break the illusion of transparency about the real number.

• School Mathematics
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• v.16 no.2
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• pp.317-334
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• 2014

The purpose of this study is to grasp pre-service secondary teachers' understanding-realities for Taylor series convergence conceptions and to examine a teaching way using GeoGebra based on the understanding-realities. In this study, most pre-service teachers have abilities to calculate the Taylor series and radius of convergence, but they are vulnerable to conceptual problems which give meaning of the equality between a given function and its Taylor series at any point. Also they have some weakness in determining the change of radius of convergence according to the change of Taylor series' center. To improve their weakness, we explore a teaching way using dynamic and CAS functionality of GeoGebra. This study is expected to improve the pedagogical content knowledge of pre-service secondary mathematics teachers for infinite series treated in high school mathematics.

• Journal of Educational Research in Mathematics
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• v.27 no.3
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• pp.375-389
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• 2017

The purpose of this study is to analyze the difference and inter-connectivity between the definition of continuity in school mathematics and the definition of academic mathematics in four perspectives. These difference and inter-connectivity have not analyzed in previous papers. According to this study, the definition of 'continuity and discontinuity at one point' in school mathematics depends on the limit processing but in academic mathematics it depends on the topology of the domain. The target function of the continuous function in school mathematics is a function whose domain is limited to an interval or a union of intervals, but the target function of the continuous function in academic mathematics is all functions. Based on these results, the following two opinions are given in relation to the concept of continuity in school mathematics. First, since the notion of local continuity in school mathematics is based on limit processing, the contents of 2009-revised textbooks that deal with discontinuity at special point not belonging to the domain is appropriate. Here the discontinuity appears as types of infinite discontinuity, removable discontinuity, and step discontinuity. Second, the definition of a general continuous function is proposed to "if there is no discontinuity point in the domain of a function y = f(x), we call the function f a continuous function." This definition allows the discontinuity at special point in non-domain, but is consistent with the definition in academic mathematics.

• Journal of Educational Research in Mathematics
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• v.11 no.2
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• pp.341-349
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• 2001

Infinity is one of the important concept in mathematics, science, philosophy etc. In history of mathematics, potential infinity concept conflicts with actual infinity concept. Reason that mathematicians refuse actual infinity concept during long period is because that actual infinity concept causes difficulty in our perceptions. This phenomenon is called epistemological obstacle by Brousseau. Potential infinity concept causes difficulty like history of development of infinity concept in mathematics learning. Even though students team about actual infinity concept, they use potential infinity concept in problem solving process. Therefore, we must make clear epistemological obstacles of infinity concept and must overcome them in learning of infinity concept. For this, it is useful to experience visualization about infinity concept. Also, it is to develop meta-cognition ability that students analyze and control their problem solving process. Conclusively, students must adjust potential infinity concept, and understand actual infinity concept that is defined in formal mathematics system.

• Journal for History of Mathematics
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• v.18 no.3
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• pp.55-66
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• 2005

In our school mathmatics, the irrational numbers and the real numbers are defined and instructed on the basis of decimal fractions. In relation to this fact, we identified the essences of the real number and the irrational number defined by the decimal fractions through the historical analysis. It is revealed that the formation of real numbers means the numerical measurements of all magnitudes and the formation of irrational numbers means the numerical measurements of incommensurable magnitudes. Finally, we suggest instructional plan for the meaninful understanding of the real number concept.

• Journal for History of Mathematics
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• v.14 no.1
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• pp.41-46
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• 2001

In the western culture, the definition of infinity and finiteness developed into various features, being combined with classical Hebraism and Hellenism, and after modernism, it has been paralleled with natural sciences within the circle of its tradition, whereas the definition of infinity has been showed in Buddhism, the traditional religion, which has been handed down without scientific consideration, in the eastern culture. This is the most notable characteristics between two hemispheres. In this paper, I will show sameness and differences of these two cultures in terms of defining infinity. Also in terms of examining the definition of infinity and finiteness in the Buddhist principles which has been handed down since 600∼500 B.C. I will show how the definition of Buddhist infinity connect with the mathematical definition of today.

• The Mathematical Education
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• v.42 no.3
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• pp.303-325
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• 2003

This study sought to provide an explanation of university students' concept understanding on the infinity and infinite process and utilized a psychological constructivist perspective to examine the differences in transitions that students make from static concept of limit to actualized infinity stage in context of problems. Open-ended questions were used to gather data that were used to develop an explanation concerning student understanding. 47 university students answered individually and were asked to solve 16 tasks developed by Petty(1996). Microgenetic method with two cases from the expert-novice perspective were used to develop and substantiate an explanation regarding students' transitions from static concept of limit to actualized infinity stage. The protocols were analyzed to document student conceptions. Cifarelli(1988)'s levels of reflective abstraction and Robert(1982) and Sierpinska(1985)'s three-stage concept development model of infinity and infinite process provided a framework for this explanation. Students who completed a transition to actualized infinity operated higher levels of reflective abstraction than students who was unable to complete such a transition. Developing this ability was found to be critical in achieving about understanding the concept of infinity and infinite process.

• Journal of Educational Research in Mathematics
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• v.15 no.3
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• pp.287-313
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• 2005

The decimal fraction concept plays an important role in understanding the real number which is one of the major concepts in school mathematics. In the school mathematics of Korea, the decimal fraction is treated merely as a sort of name of the common fraction, while many other important aspects of the decimal fraction concept are ignored. In consequence students fail to understand the decimal fraction concept properly, and merely consider it as a kind of number for formal computation. Preceding studies also identified students' narrow understanding of the decimal fraction concept. But none of them succeeded in clarifying the essences of the decimal fraction concept, which are crucial for discussing the didactical problems of it. In this study we attempted a didactical analysis of the decimal fraction concept and disclosed the roots of didactical problems and presented measures for its improvement. First, we attempted a phenomenological analysis of the decimal fraction concept and extracted 9 elements of the decimal fraction concept. Second, we has analyzed of the essence of the decimal fraction concept more clearly by relating it to the situations where it functions and its representations. For this we tried to construct the conceptual field of the decimal fraction. Third, we categorized he developmental levels of the decimal fraction concept from the aspect of external manifestation of the internal order. On the basis of these results, we attempted hierarchical structuring of the elements of the decimal fraction concept. And using the results of such a didactical analysis on the decimal number concept we analyzed the mathematics curriculum and textbooks of our country, investigated levels of students' understanding of the decimal fraction concept, and disclosed related problems. Finally we suggested directions and measures for the improvement of teaching decimal fraction concept.