Browse > Article
http://dx.doi.org/10.14477/jhm.2021.34.1.001

Revisiting Logic and Intuition in Teaching Geometry: Comparing Euclid's Elements and Clairaut's Elements  

Chang, Hyewon (Dept. of Math. Edu., Seoul National Univ. of Edu.)
Publication Information
Journal for History of Mathematics / v.34, no.1, 2021 , pp. 1-20 More about this Journal
Abstract
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.
Keywords
Euclid's Elements of geometry; Clairaut's Elements of geometry; logic; intuition;
Citations & Related Records
연도 인용수 순위
  • Reference
1 O. Byrne, The first six books of the elements of Euclid: In which coloured diagrams and symbols are used instead of letters for the greater ease of learners, London: William Pickering, 1847.
2 J. Casey, The first six books of the elements of Euclid, and propositions I-XXI of book XI, and an appendix on the cylinder, sphere, cone, etc, 1885. https://www.maths.tcd.ie/-wilkins/Courses/MA232A/MA232A_Mich2015/EuclidResources/Euclid-Casey__ProjectGutenberg-21076-pdf.pdf
3 Chang H., A Study on geometrical construction in middle school mathematics, The Journal of Educational Research in Mathematics 7(2) (1997), 327-336.
4 Chang H., A study on the historico-genetic principle revealed in Clairaut's Elements of Geometry, The Journal of Educational Research in Mathematics 13(3) (2003), 351-364.
5 Chang H., A comparative study on Euclid's Elements and Pardies' Elements, Journal for History of Mathematics 33(1) (2020), 33-53.   DOI
6 Chang H., B. Reys, If only Clairaut had dynamic geometric tools, Mathematics Teaching in the Middle School 19(5) (2013), 280-287.   DOI
7 A. C. Clairaut, Elemens de geometrie, Gauthier-Billars et Cle, Editeurs, 1741/1920. (translated by Chang, H. 2018). 장혜원(역), 클레로 기하학원론, 서울: 지오아카데미, 2018.
8 J. Dewey, The psychology and the logical in teaching geometry, 1903. In J. A. Boydston, (Ed.) John Dewey: the middle works, 1899-1924, Carbondale: Southern Illinois University Press, 1977.
9 E. Fischbein, Intuition in science and mathematics, Kluwer Academic Publishers, 1987. (translated by Woo, J. et al., 2006). 우정호 외(역), 수학과학학습과 직관, 서울: 경문사, 2006.
10 T. Fujita, K. Jones, S. Yamamoto, The role of intuition in geometry education: learning from the teaching practice in the early 20th century, 2004. https://eprints.soton.ac.uk/14300/1/Fujita-Jones-Yamamoto_ICME10_TSG29_2004.pdf
11 T. L. Heath, The thirteen books of Euclid's Elements, Translated from the text of Heiberb with introduction and commentary, Vol. I, II, III, Cambridge: at the University Press, 1908.
12 K. Jones, Deductive and intuitive approaches to solving geometrical problems, 1998, 78-83. In C. Mammana, V. Villani(Eds.), Perspectives on the teaching of geometry for the 21st century, Dordrecht: Kluwer Academic Publishers.
13 Lee M. H.(trans.), Euclid's Elements of geometry, 이무현(역), 기하학원론, 서울: 교우사, 1997.
14 Ministry of Education, Mathematics 6-2, 2020.
15 H. Poincare, Intuition and logic in mathematics, The Mathematics Teacher 62(3) (1969a), 205-212. http://www.jstor.org/stable/27958100.   DOI
16 Yoo J. G., Park M. H., A Study on the meaning of similarity in school mathematics, The Journal of Educational Research in Mathematics 29(2) (2019), 283-299.   DOI
17 H. Poincare, Mathematical definitions and teaching, The Mathematics Teacher 62(4) (1969b), 295-304. http://www.jstor.org/stable/27958128.   DOI
18 E. Schuberth, Geometry lessons in the Waldorf school, Mannheim: AWSANS publications, 2004.