Browse > Article

Gifted Middle School Students' Genetic Decomposition of Congruent Transformation in Dynamic Geometry Environments  

Yang, Eun Kyung (Graduate School, Korea National University of Education)
Shin, Jaehong (Korea National University of Education)
Publication Information
Journal of Educational Research in Mathematics / v.25, no.4, 2015 , pp. 499-524 More about this Journal
Abstract
In the present study, we propose four participating $8^{th}$ grade students' genetic decomposition of congruent transformation and investigate the role of their dragging activities while understanding the concept of congruent transformation in GSP(Geometer's Sketchpad). The students began to use two major schema, 'single-point movement' and 'identification of transformation' simultaneously in their transformation activities, but they were inclined to rely on the single-point movement schema when dealing with relatively difficult tasks. Through dragging activities, they could expand the domain and range of transformation to every point on a plane, not confined to relevant geometric figures. Dragging activities also helped the students recognize the role of a vector, a center of rotation, and an axis of symmetry.
Keywords
congruent transformation; APOS theory; dragging activity;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
연도 인용수 순위
1 교육과학기술부(2011). 수학과 교육과정. (교육과학기술부 고시 제 2011-361호 [별책8]).
2 김자경 (2005). van Hieles의 기하 학습 사고 수준 이론을 적용한 도형 학습이 합동 변환의 이해력과 기하 수준 변화에 미치는 영향, 이화여자대학교 석사학위논문.
3 김정학(1971). 도형의 함수적인 고찰, 논문집, 4, 111-128.
4 박연정(2006). 고등학교 교육과정에서 도형의 변환에 대한 연구, 서울시립대학교 석사학위논문.
5 박혜숙, 김서령, 김완순(2005). 수학적 개념의 발생적 분해의 적용에 대하여 - 추상대수학에서의 $Z_n$의 경우 -. 수학교육, 44(4), 547-563.
6 백용배(1995). 기하학개론. 서울: 교학연구사.
7 손홍찬(2011). 우리나라 수학교육에서 공학 활용의 역사와 현황. 학교수학, 13(3), 525-542.
8 송석준, 박종국(1998). 고등학교 교육과정에서 도형의 변환에 대한 지도내용 분석 및 개선 방안, 과학교육, 15, 151-168.
9 안웅용(1995). 도형의 변환지도에 관한 연구. 단국대학교 석사학위논문.
10 양은경, 신재홍(2014a). 개방형 기하 문제에서 학생의 드래깅 활동을 통해 나타난 수학적 추론 분석. 수학교육학연구, 24(1), 1-27.
11 양은경, 신재홍(2014b). 작도 접근 방식에 따른 중학생의 기하학적 특성 인식 및 정당화. 수학교육학연구, 24(4), 507-528.
12 우정호(1998). 학교수학의 교육적 기초, 서울: 서울대학교 출판부.
13 최종렬(1992). 중등교육과정에서의 도형의 변환에 대한 연구, 경성대학교 석사학위논문.
14 Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2013). APOS theory: A framework for research and curriculum development in mathematics education. Springer Science & Business Media.
15 Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. Zentralblatt fur Didaktik der Mathematik, 34, 66-72.   DOI
16 Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1997). A framework for research and curriculum development in undergraduate mathematics education. MAA NOTES, 37-54.
17 Baccaglini-Frank, A. (2010). Conjecturing in dynamic geometry: A model for conjecture-Generation through maintaining dragging, Doctoral dissertation, University of New Hampshire, Durham, NH, USA. ISBN: 9781124301969.
18 Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall(Ed.), Advanced mathematical thinking. Dordrecht : Kluwer Academic Publishers, 류희찬, 조완영, 김인수 (공역) (2002), 고등수학적 사고. 서울:경문사.
19 Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247-285.   DOI
20 Dixon, J. K. (1997). Computer use and visualization in students' construction of reflection and rotation concepts. School Science and Mathematics, 97(7), 352-358.   DOI
21 Guven, B. (2012). Using dynamic geometry software to improve eight grade students' understanding of transformation geometry. Australasian Journal of Educational Technology, 28(2), 364-382.
22 Hollebrands, K. F. (2003). High school students' understandings of geometric transformations in the context of a technological environment. Journal of Mathematical Behavior, 22(1), 55-72.   DOI
23 Hollebrands, K. F. (2004). High school students' intuitive understandings of geometric transformations. The Mathematics Teacher, 207-214.
24 Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 164-192.
25 Klein, F. (1932). Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis (Vol. 1). Courier Corporation.
26 Law, C. K. (1991). A genetic decomposition of geometric transformations (Unpublished doctoral dissertation). Purdue University, Indiana, U.S.
27 Martin, G. E. (1982). Transformation geometry: An introduction to symmetry. Springer Science & Business Media.
28 Leung, A. (2012). Discernment and Reasoning in Dynamic Geometry Environments. www.icme12.org/upload/submission/1961_F.pdf.
29 Leung, A., & Lopez-Real, F. (2002). Theorem justification and acquisition in dynamic geometry: A case of proof by contradiction, International Journal of Computers for Mathematical Learning, 7, 145-165.   DOI
30 Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry environments. International Journal of Mathematical Education in Science and Technology, 37(6), 665-679.   DOI
31 Maxwell, J. A. (2012). Qualitative research design: An interactive approach (Vol. 41). Sage.
32 Meagher, M., Cooley, L., Martin, B., Vidakovuc, D., & Loch, S. (2006). The learning of linear algebra from an APOS perspective. In S. Alatorre, J. L. Cortina, M. Saiz, & A. Mendez (Eds.), Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Mexico: Universidad Pedagogica Nacional.
33 Monaghan, J., Sun, S., & Tall, D. O. (1994). Construction of the limit concept with a computer algebra system. Proceedings of PME, 18, 279-286.
34 Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. Handbook of research design in mathematics and science education, 267-306.
35 Vidakovic, D.(1996). Learning the concept of inverse function. Journal of computers in Mathematics and Science Teaching, 15, 295-318.
36 Zazkis, R., Dubinsky, E., & Dautermann, J. (1996). Coordinating Visual and Analytic Strategies: A study of students' understanding of the group D 4. Journal for research in Mathematics Education, 435-457.
37 Weller, K., Clark, J., Dubinsky, E., Loch, S., McDonald, M., & Merkovsky, R.(2003). Student performance and attitudes in courses based on APOS Theory and the ACE Teaching Cycle. In A Selden, E Dubinsky, G Harel & F Hitt (eds). Research in Collegiate Mathematics Education V. Providence, RI: American Mathematical Socie.
38 Wesslen, M., & Fernandez, S. (2005). Transformation geometry. Mathematics Teaching, 191, 27-29.