• 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.30 no.5
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• pp.289-304
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• 2017

The interdisciplinary study explores the discussion of actual infinity between Georg Cantor and Roman Catholic Church. Regarding the actual infinity, we first trace the theological background of Cantor by interpreting his correspondence and major works including ${\ddot{U}}ber$ die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche(1885) and Mitteilungen zur Lehre vom Transfiniten (1887), and then investigate his argumentation for two points at issue: (1) pantheism and (2) inconsistency of the necessity with freedom of God. In terms of mathematics and theology, Cantor defined the actual infinity(aphorismenon) as characterized by (1) the transfinite infinity(Transfinitum) and (2) the absolute infinity(Absolutum). Transfinitum is conceptualized here in mathematical terms as a multipliable actual infinity, whereas Absolutum is not as a multipliable actual infinity. The results imply that Cantor's own concept of Transfinitum and Absolutum is adequate for Roman Catholic theology as well as mathematics including the reflection principle.

• Journal for History of Mathematics
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• v.16 no.3
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• pp.57-68
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• 2003

Infinity is very important concept in mathematics. In history of mathematics, potential infinity concept conflicts with actual infinity concept for a long time. It is reason that actual infinity concept causes difficulty in our perceptions. This phenomenon is called epistemological obstacle by Brousseau. So, in this paper, we examine the infinity in terms of mathematical didactics. First, we examine the history of development of infinity and reveal the similarity between the history of debate about infinity and episternological obstacle of students. Next, we investigate obstacle of students about infinity and the contents of curriculum which treat the infinity Finally, we suggest the methods for overcoming obstacle in learning of infinity concept.

• Journal of Educational Research in Mathematics
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• v.18 no.4
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• pp.469-482
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• 2008

This study analysed difficult points on the understanding of infinity when the concept is considered as actual infinity or as potential infinity. And I consider examples that the concept of actual infinity is used in texts of elementary and middle school mathematics. For understanding of modem mathematics, the concept of actual infinity is required necessarily, and the intuition of potential infinity is an epistemological obstacle to get over. Even so, it might be an excessive requirement to make such epistemological rupture from the early school mathematics, since the concept of actual infinity is not intuitive, derives many paradoxes, and cannot offer any proper metaphor.

• Journal for History of Mathematics
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• v.17 no.2
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• pp.1-8
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• 2004

In this paper we try to research in the influences which the concepts of infinity have made to our life, and how they have led the trend of the times through studying on the process of changes of concepts of infinity. Also we intend to make a research in how the shift of paradigm on the view of life have been changing under the circumstances in which the concepts of infinity have been accepted as an actual meaning.

• Journal of Educational Research in Mathematics
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• v.20 no.3
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• pp.293-303
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• 2010

The purpose of this study is to classify a three-fold role of the history of mathematics through epistemological analysis. Based on the history of infinity and limit, the "potential infinity" and "actual infinity" discourses are described using four different historical epistemologies. The interdependence between the mathematical concepts is also addressed. By using these analyses, three different uses of the history of mathematical concepts, infinity and limit, are discussed: past, present, and future use.

• The Mathematical Education
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• v.49 no.2
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• pp.259-265
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• 2010

Even though Sanhak has a long history, it has disappeared from the stage of modern mathematics. What happened to Sanhak? This article tries to answer the question. In fact, the authors argue that the oriental perception toward to infinity has played an important role in such situation. The authors claim that actual infinity and virtual infinity have resulted in quite different types of mathematics, respectively.

• Journal for History of Mathematics
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• v.21 no.3
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• pp.31-52
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• 2008

The concept of infinity, as with other scientific concepts, has a history of evolution. In the present work we intend to discuss the subject matter with regard to Bolzano since he is considered to be the first to accept the idea of actual infinity not just from a metaphysical perspective but from a mathematical one. Like modem platonists, Bolzano defended the infinite set itself regardless of the construction process; this is based on the principal of comprehension and unicity of denotation regarding all concepts. In addition, instead of considering as paradoxical the fact that a one-to-one correspondence existed between an infinite set and its parts, he regarded it in a positive way as a special characteristic. While the Greek era recognized the existence of only one infinity, Balzano acknowledged the existence of various types of infinity and formulated a logical definition for it. The question of infinity is a touchstone of constructive method which holds an increasingly important role in mathematics. The present study stops with just a brief reference to the subject matter and we will leave further in-depth investigation for later.

• Journal of the Korean School Mathematics Society
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• v.10 no.1
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• pp.91-111
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• 2007

This study investigated how high school students predict the rule, the sum of sequence for the concept of sequence, for the given patterns based on inductive approach using computers that provide dynamic functions and materials that are visual. Students for themselves were able to induce the formula without using the given formula in the textbook. Furthermore, this study examined how these technology and materials affect students' understanding of the concept of actual infinity for those who have the concept of the potential infinity which is the misconception of infinity in a infinity series. This study shows that students made a progress from the concept of potential infinity to that of actual infinity with technology and materials used I this study. Students also became interested in the use of computer and the visualized materials, further there was a change in their attitude toward mathematics.

• The Mathematical Education
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• v.47 no.4
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• pp.447-465
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• 2008

The purpose of this study is to suggest effective teaching methods on the concept of infinity for students to obtain the right concept in the middle school curriculum. Many people have thought that infinity is something vouge and unapproachable. But, nowadays it is rather something with a precise definition that lies at the core of modern mathematics. To understand mathematics and science very well, it is necessary to comprehend the concept of infinity. But students tend to figure out the properties of infinite objects and limit concepts only through their experience closely related to finite process, and so they are apt to have their spontaneous intuition and misconception about it. Since most of them have cognitive obstacles in studying the infinite concepts and misconception, mathematics teachers need to help them overcome the obstacles and establish the right secondary intuition for the concepts through good examples and appropriate explanation. In this study, we consider the developing process of the concept of infinity in human history and give some comments and suggestions in teaching methods relative to that concept with new suitable examples.