Volume 10 · Number 3 · Pages 338–347
Elementary Students’ Construction of Geometric Transformation Reasoning in a Dynamic Animation Environment

Nicole Panorkou & Alan Maloney

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Context: Technology has not only changed the way we teach mathematical concepts but also the nature of knowledge, and thus what is possible to learn. While geometric transformations are recognized to be foundational to the formation of students’ geometric conceptions, little research has focused on how these notions can be introduced in elementary schooling. Problem: This project addressed the need for development of students’ reasoning about and with geometric transformations in elementary school. We investigated the nature of students’ understandings of translations, rotations, scaling, and stretching in two dimensions in the context of use of the software application Graphs ’n Glyphs. More specifically, we explored the question “What is the nature of elementary students’ reasoning of geometric transformations when instruction exploits the technological tool Graphs ’n Glyphs?” Method: Using a design research perspective, we present the conduct of a teaching experiment with one pair of fourth-graders, studying translation and rotation. The project investigated how and to what extent activity using Graphs ’n Glyphs can elicit students’ reasoning about geometric transformations, and explored the constraints and affordances of Graphs ’n Glyphs for thinking-in-change about geometric transformations. Results: The students proved adept using the software with carefully designed tasks to explore the behavior of two-dimensional shapes, distinguish among transformations, and develop predictions. In relation to varied conditions of transformations, they formed generalizations about the way a shape behaves, including the use of referent points in predicting outcomes of translations, and recognizing the role of the center of rotation. Implications: The generalizations that the students developed are foundational for developing an understanding of the properties of transformations in the later years of schooling. Dynamic technological approaches to geometry may prove as valuable to elementary students’ understanding of geometry as it is for older students. Constructivist content: This study contributes to ongoing constructivism/constructionism conversations by concentrating on the transformation of ideas when engaging learners in activity through particular contexts and tools. Key Words: Geometry, transformations, constructionist technologies.


Panorkou N. & Maloney A. (2015) Elementary students’ construction of geometric transformation reasoning in a dynamic animation environment. Constructivist Foundations 10(3): 338–347. http://constructivist.info/10/3/338

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Boulter D. R. & Kirby J. R. (1994) Identification of strategies used in solving transformational geometry problems. Journal of Educational Research 87(5): 298–303. ▸︎ Google︎ Scholar
Cobb P., Confrey J., diSessa A. A., Lehrer R. & Schauble L. (2003) Design experiments in educational research. Educational Researcher 32 (1): 9–13. ▸︎ Google︎ Scholar
Confrey J. & Maloney A. P. (2015) A design research study of a curriculum and diagnostic assessment system for a learning trajectory on equipartitioning. ZDM – The International Journal on Mathematics Education 47(6) In press. ▸︎ Google︎ Scholar
Confrey J., Hoyles C., Jones D., Kahn K., Maloney A. P., Nguyen K. H., Noss R. & Pratt D. (2010) Designing software for mathematical engagement through modeling. In: Hoyles C. &Lagrange J.-B. (eds.) Mathematics education and technology – Rethinking the terrain. Springer, New York: 19–45. ▸︎ Google︎ Scholar
Confrey J., Maloney A., Ford L. & Nguyen K. (2006) Graphs ’n Glyphs as a means to teach animation and graphics to motivate proficiency in mathematics by middle grade urban students. In: Hoyles C., Lagrange J.-B., Son L. H. & Sinclair N. (eds.) Procedings of the 17th ICMI study conference “Technology Revisited” Volume 2: 112–119. Hanoi University of Technology, Hanoi. ▸︎ Google︎ Scholar
Coxford Jr. A. F. (1991) Geometry from multiple perspectives. Curriculum and evaluation standards for school mathematics addenda series, grades 9–12. NCTM, Reston VA. ▸︎ Google︎ Scholar
diSessa A. (1988) Knowledge in pieces. In: Forman G. & Pufall P. B. (eds.) Constructivism in the computer age. Lawrence Erlbaum Associates, New Jersey: 49–70. ▸︎ Google︎ Scholar
Edwards L. D. (1991) Children’s learning in a computer microworld for transformation geometry. Journal for Research in Mathematics Education 22(2): 122–137. ▸︎ Google︎ Scholar
Edwards L. D. (1997) Exploring the territory before proof: Students’ generalizations in a computer microworld for transformational geometry. International Journal of Computers for Mathematical Learning 2: 187–215. ▸︎ Google︎ Scholar
Edwards L. D. (2003) The nature of mathematics as viewed from cognitive science. In: European Research in Mathematics Education III. Proceedings of the Third Conference of the European Society for Research in Mathematics Education, 28 February–3 March 2003, Bellaria. Italy. http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG1/TG1_edwards_cerme3.pdf
Healy L. & Hoyles C. (2001) Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning 6: 235–256. ▸︎ Google︎ Scholar
Hollebrands K. F. (2003) High school students’ understandings of geometric transformations in the context of a technological environment. Journal of Mathematical Behavior 22: 55–72. ▸︎ Google︎ Scholar
Hollebrands K. F. (2004) High school students’ intuitive understandings of geometric transformations. Mathematics Teacher 97(3): 207–214. ▸︎ Google︎ Scholar
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 38(2): 164–192. ▸︎ Google︎ Scholar
Hoyles C. & Noss R. (1992) A pedagogy for mathematical microworld. Educational Studies in Mathematics 23(1): 31–57. ▸︎ Google︎ Scholar
Kidder F. R. (1976) Elementary and middle school children’s comprehension of Euclidean transformations. Journal for Research in Mathematics Education 7(1): 40–52. ▸︎ Google︎ Scholar
Kirby J. R. & Boulter D. R. (1999) Spatial ability and transformational geometry. European Journal of Psychology of Education 14(2): 283–294. ▸︎ Google︎ Scholar
Kynigos C. (2015) Constructionism: Theory of learning or theory of design? In: Cho S. J. (ed.) Selected regular lectures from the 12th International Congress on Mathematical Education. Springer, Heidelberg: 417–438. ▸︎ Google︎ Scholar
Laborde C. (1995) Designing tasks for learning geometry in a computer-based environment. Technology in mathematics teaching: 35–67. ▸︎ Google︎ Scholar
Laborde C. (2001) Integration of technology in the design of geometry tasks with Carbi-Geometry. International Journal of Computers for Mathematical Leaning 6: 283–317. ▸︎ Google︎ Scholar
Lehrer R., Jenkins M. & Osana H. (1998) Longitudinal study of children’s reasoning about space and geometry. In: Lehrer R. & Chazan D. (eds.) Designing learning environments for developing understanding of geometry and space. Lawrence Erlbaum Associates, Mahwah NJ: 137–168. ▸︎ Google︎ Scholar
Maloney A. P., Nguyen K. H. & Confrey J. (2008) Graphs ’n Glyphs as professional transitional software to assist middle school students in rational number reasoning topics. In: Proceedings of the XI International Conference of Mathematics Education, Topic Study Group 22. Monterrey, Mexico, 22 July 2008. http://tsg.icme11.org/document/get/468
Moyer J. C. (1978) The relationship between the mathematical structure of Eucliedean transformations and the spontaneously developed cognitive structures of young children. Journal for Research in Mathematics Education 9: 83–92. ▸︎ Google︎ Scholar
National Council of Teachers of Mathematics (1989) Curriculum and evaluation standards for school mathematics. NCTM, Reston VA. ▸︎ Google︎ Scholar
National Council of Teachers of Mathematics (2000) Principles and standards for school mathematics. NCTM, Reston VA. ▸︎ Google︎ Scholar
Noss R. & Hoyles C. (1996) Windows on mathematical meanings: Learning cultures and computers. Springer, Berlin. ▸︎ Google︎ Scholar
Olive J., Makar K., Hoyos V., Kor L. K., Kosheleva O. & Straber R. (2010) Mathematical knowledge and practices resulting from access to digital technologies. In: Hoyles C. & Lagrange J.-B. (eds.) Mathematics education and technology: Rethinking the terrain. Springer, New York: 133–177. ▸︎ Google︎ Scholar
Osta I. (1998) CAD tools and the teaching of geometry. In: Mammana C. & Villani V. (eds.) Perspectives on the teaching of geometry for the 21st century: An ICMI study. Springer, New York: 128–144. ▸︎ Google︎ Scholar
Papert S. (1980) Mindstorms: Children, computers, and powerful ideas. Basic Books, New York. ▸︎ Google︎ Scholar
Papert S. (1992) The children’s machine: Rethinking school in the age of the computer. Basic Books, New York NY. ▸︎ Google︎ Scholar
Perkins D., Crismond D., Simmons R. & Unger C. (1995) Inside understanding. In: Perkins D., Schwartz J., West M. & Wiske M. (eds.) Software goes to school: Teaching for understanding with new technologies. Oxford University Press, New York NY: 70–89. ▸︎ Google︎ Scholar
Pratt D. & Noss R. (2010) Designing for mathematical abstraction. International Journal of Computers for Mathematical Learning 15(2): 81–97. ▸︎ Google︎ Scholar
Riegler P. & Steffe L. (eds.) (2014) Forty years of radical constructivism in educational research. Special issue of Constructivist Foundations 9(3) http://www.univie.ac.at/constructivism/journal/9/3
Schoenfeld A. (2000) Purposes and methods of research in mathematics education. Notices of the AMS 47(6): 641–649. ▸︎ Google︎ Scholar
Schultz K. A. & Austin J. D. (1983) Directional effects in transformation tasks. Journal for Research in Mathematics Education 14(2): 95–101. ▸︎ Google︎ Scholar
Schwartz J. L. & Yerushalmy M. (1993) Getting students to function in algebra. In: Harel G. & Dubinsky E. (eds.) The concept of function: Aspects of epistemology and pedagogy. Mathematical Association of America, Washington DC: 261–289. Retrieved from http://www.edu.haifa.ac.il/personal/michalyr/pdf/getting-students-to-fn.pdf on 18 February 2015. ▸︎ Google︎ Scholar
Sinclair N. (2008) The history of the geometry curriculum in the United States. Information Age Publishing, Charlotte NC. ▸︎ Google︎ Scholar
Underwood J. S., Hoadley C., Lee H. S., K. F., DiGiano C. & Renninger K. A. (2005) IDEA: Identifying design principles in educational applets. Educational Technology Research and Development 53(2): 99–112. ▸︎ Google︎ Scholar
Willford H. J. (1972) A study of transformational geometry instruction in the primary grades. Journal for Research in Mathematics Education3(4): 260–271. ▸︎ Google︎ Scholar
Xistouri X. & Pitta-Pantazi D. (2011) Elementary students’ transformational geometry ability and cognitive style. Congress of the European society for research in mathematics education. ERME, Rzeszów, Poland: 1–10. ▸︎ Google︎ Scholar
Xistouri X. & Pitta-Pantazi D. (2012) The influence of cognitive style on children’s ability to solve reflection and rotation tasks. In: Proceedings of the 17th annual conference of the Education, Learning, Styles and Individual Differences network: 404–416. Tribun EU, Brno. ▸︎ Google︎ Scholar
Yaglom I. M. (1962) Geometric transformations I. Mathematical Association of America, Washington DC. ▸︎ Google︎ Scholar
Yanik H. B. (2014) Middle-school students’ concept images of geometric translations. The Journal of Mathematical Behavior 36: 33–50. ▸︎ Google︎ Scholar

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