• Journal of the Korean School Mathematics Society
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• v.20 no.2
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• pp.119-139
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• 2017

The construction in the ancient Greek era had more meanings than a construction in the present education. Based on this fact, this study examines the meaning of the current textbook. In contrast, we have extracted the meaning of the constructions in Euclid Elements. In addition, we have been thinking about what benefits can come up if the meaning of the construction in Euclid Elements was reflected in current education, and suggested a way to exploit that advantage. As results, it was confirmed that the construction in the current textbook was merely a means for introducing and understanding the congruent conditions of the triangle. On the other hand, the construction had four meanings in Euclid Elements; Abstract activities that have been validated by the postulates, a mean of demonstrating the existence of figures and obtaining validity for the introduction of auxiliary lines, refraining from intervening in the argument except for the introduction of auxiliary lines, a mean of dealing with numbers and algebra. Finally we discussed the advantages of using the constructions as a means of ensuring the validity of the introduction of the auxiliary line to the argument. And we proposed a viewpoint of construction by intervention of virtual tools for auxiliary lines which can not be constructed with Euclid tool.

• Journal for History of Mathematics
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• v.34 no.1
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• pp.1-20
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• 2021

Logic and intuition are considered as the opposite extremes of teaching geometry, and any teaching method of geometry is to be placed between these extremes. The purpose of this study is to identify the characteristics of logical and intuitive approaches for teaching geometry and to derive didactical implications by taking Euclid's Elements and Clairaut's Elements respectively representing the extremes. To this end, comparing the composition and contents of each book, we analyze which propositions Clairaut chose from Euclid's Elements, how their approaches differ in definitions, proofs, and geometrical constructions, and what unique approaches Clairaut took. The results reveal that Clairaut mainly chose propositions from Euclid's books 1, 3, 6, 11, and 12 to provide the contexts that show why such ideas were needed, rather than the sudden appearance of abstract and formal propositions, and omitted or modified the process of justification according to learners' levels. These propose a variety of intuitive strategies in line with trends of teaching geometry towards emphasis on conceptual understanding and different levels of justification. Specifically, such as the general principle of similarity and the infinite geometric approach shown in Clairaut's Elements, we could confirm that intuition-based geometry does not necessarily aim for tasks with low cognitive demand, but must be taught in a way that learners can understand.

• Journal for History of Mathematics
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• v.33 no.1
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• pp.33-53
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• 2020

Euclid's Elements has been considered as the stereotype of logical and deductive approach to mathematics in the history of mathematics. Nonetheless, it has been criticized by its dryness and difficulties for learning. It is worthwhile to noticing mathematicians' struggle for providing some alternatives to Euclid's Elements. One of these alternatives was written by a French scientist, Pardies who called it 'Elemens de Geometrie ou par une methode courte & aisee l'on peut apprendre ce qu'il faut scavoir d'Euclide, d'Archimede, d'Apllonius & les plus belles inventions des anciens & des nouveaux Geometres.' A precedent research presented its historical meaning in traditional mathematics of China and Joseon as well as its didactical meaning in mathematics education with the overview of this book. However, it has a limitation that there isn't elaborate comparison between Euclid's and Pardies'in the aspects of contents as well as the approaching method. This evokes the curiosity enough to encourage this research. So, this research aims to compare Pardies' Elements and Euclid's Elements. Which propositions Pardies selected from Euclid's Elements? How were they restructured in Pardies' Elements? Responding these questions, the researcher confirmed his easy method of learning geometry intended by Pardies.

• Education of Primary School Mathematics
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• v.20 no.1
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• pp.117-130
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• 2017

Euclid's elements have been recognized as a significant textbook in mathematics and mathematics education because of importance of its contents and methodology. This study discussed how the elements is connected with understanding of math textbooks in elementary school, trying to reveal the value for teacher training. First, when details in elementary textbooks were considered in aspect of elements, the importance of elements was illustrated with the purpose of understanding contents of elementary school by examining educational implications. In addition, the study discussed the value of the elements as the place for teachers and would-be teachers to experience math system.

• Journal for History of Mathematics
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• v.17 no.3
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• pp.23-32
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• 2004

In this paper, Ive investigated algebraic concepts which are contained in Euclid's Elements. In the Books II, V, and VII∼X of Elements, there are concepts of quadratic equation, ratio, irrational numbers, and so on. We also analyzed them for mathematical meaning with modem symbols and terms. From this, we can find the essence of the genesis of algebra, and the implications for students' mathematization through the experience of the situation where mathematics was made at first.

• Journal of Educational Research in Mathematics
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• v.25 no.3
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• pp.367-379
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• 2015

The existence of mathematical objects was considered through diorism which was used in ancient Greece as conditions for the existence of the solution of the problem. Proposition I-22 of Euclid Elements has diorism for the existence of triangle. By discussing the diorism in Elements, ancient Greek mathematician proved the existence of defined object by postulates or theorems. Therefore, the existence of mathematical object is verifiability in the axiom system. From this perspective, construction is the main method to guarantee the existence in the Elements. Furthermore, we suggest some implications about the existence of mathematical objects in school mathematics.

• Journal of Educational Research in Mathematics
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• v.23 no.2
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• pp.173-192
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• 2013

It may be replacement proofs with understanding and explaining geometrical properties that was a remarkable change in school geometry of 2009 revised national curriculum for mathematics. That comes from the difficulties which students have experienced in learning proofs. This study focuses on one of those difficulties which are caused by the forms of proofs: using letters for designating some sides or angles in writing proofs and understanding some long sentences of proofs. To overcome it, this study aims to investigate the applicability of Byrne's method which uses coloured diagrams instead of letters. For this purpose, the proofs of three geometrical properties were taught to middle school students by Byrne's visual method using the original source, dynamic representations, and the teacher's manual drawing, respectively. Consequently, the applicability of Byrne's method was discussed based on its strengths and its weaknesses by analysing the results of students' worksheets and interviews and their teacher's interview. This analysis shows that Byrne's method may be helpful for students' understanding of given geometrical proofs rather than writing proofs.

• Journal for History of Mathematics
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• v.20 no.2
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• pp.33-46
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• 2007

In this Paper the change of trends over historical research of mathematics, that has been developed since 1970, is inquired. Most of all it deals with the controversy concerning so-called 'geometrical algebra'. It covers the contents of Euclid' work II. And the relation of the controversy with the change of direction over historical research of mathematics is examined.

• Journal for History of Mathematics
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• v.19 no.2
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• pp.101-114
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• 2006

The content and method of geometry taught in secondary school is rooted in 'Elements' by Euclid. On the other hand, however, there are differences between the content and structure of the current textbook and the 'Elements'. The gaps are resulted from attempts to develop the geometry education. Specially, the content and method for the proportional relations of geometric figures has been varied. In this study, we reviewed the changes of the proportional relations of geometric figures with pedagogical point of view. The conclusion that we came to is that the proportional relations in incommensurable case Is omitted in secondary school. Teacher's understanding about the proportional relations of geometric figures is needed for meaningful geometry education.

• Journal of Educational Research in Mathematics
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• v.13 no.3
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• pp.351-364
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• 2003

by A.C. Clairaut is the first geometry textbook based on the historico-genetic principle against the logico-deduction method of Euclid's This paper aims to recognize Clairaut's historico-genetic principle by inquiring into this book and to search for its applications to school mathematics. For this purpose, we induce the following five characteristics that result from his principle and give some suggestions for school geometry in relation to these characteristics respectively : 1. The appearance of geometry is due to the necessity. 2. He approaches to the geometry through solving real-world problems.- the application of mathematics 3. He adopts natural methods for beginners.-the harmony of intuition and logic 4. He makes beginners to grasp the principles. 5. The activity principle is embodied. In addition, we analyze the two useful propositions that may prove these characteristics properly.