Abstract
The emergence of interactive mathematics software in the early 1990shas invited mathematics education researchers to consider ways that dynamic geometry environments (DGEs) might influence K–16 learners’ growth of geometric understanding. This article examines the applicability of the Pirie–Kieren theory to trace this growth. Data for this article came from a series of task-based interviews that aimed at examining the relationship between The Geometer’s Sketchpad(GSP) usage and the development of geometric knowledge about compositions of geometric transformations. Data analysis showed that the Pirie–Kierentheory provides a useful tool to characterize, at the micro-level, a learner’s process of gaining a particular geometric understanding of a composition of geometric transformations in this DGE. By supporting the participants to transform geometric objects into their desired form, and to become acquainted with their properties or structures, tools in this DGE facilitated not only their transition between different levels of understanding, but also the to-and-fro movements within each specific level of understanding. The presence of a DGE challenges mathematics education researchers to reconsider the Primitive Knowing level of the Pirie–Kieren theory, and to attend to the difference between a visual image brought to mind by a GSP diagram and a mental image developed from manipulating the diagram.
Similar content being viewed by others
References
Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. ZDM: The International Journal on Mathematics Education, 34(3), 66–72.
Baccaglini-Frank, A. (2019). Dragging, instrumented abduction and evidence, in processes of conjecture generation in a dynamic geometry environment. ZDM: The International Journal on Mathematics Education, 51(5), 779–791.
Baccaglini-Frank, A., & Mariotti, M. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. International Journal of Computers for Mathematical Learning, 15(3), 225–253.
Bu, L., & Schoen, R. (Eds.). (2011). Model-centred learning: Pathways to mathematical understanding using GeoGebra. Rotterdam, The Netherlands: Sense Publishers.
Cavey, L., & Berenson, S. (2005). Learning to teach high school mathematics: Patterns of growth in understanding right-triangle trigonometry during lesson plan study. The Journal of Mathematical Behavior, 24(2), 171–190.
Christou, C., Mousoulides, N., Pittalis, M., & Pitta-Pantazi, D. (2004). Proofs through exploration in dynamic geometry environments. TheInternational Journal of Science and Mathematics Education, 2(3), 339–352.
Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Edwards, L. (2003). The nature of mathematics as viewed from cognitive science. Paper presented at the Third Congress of the European Society for Research in Mathematics. Italy: Bellaria https://www.mathematik.unidortmund.de/~erme/CERME3/Groups/TG1/TG1_edwards_cerme3.pdf.
Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.
Goldin, G. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. Kelly & R. Lesh (Eds.), Research design in mathematics and science education (pp. 517–545). Mahwah, NJ: Lawrence Erlbaum Associates.
González, G., & Herbst, P. (2009). Students’ conceptions of congruency through the use of dynamic geometry software. International Journal of Computers for Mathematical Learning, 14(2), 153–182.
Gülkilik, H. (2016). The role of virtual manipulatives in high school students’ understanding of geometric transformations. In P. Moyer-Packenham (Ed.), International perspectives on teaching and learning mathematics with virtual manipulatives (pp. 213–243). Cham, Switzerland: Springer.
Gülkilik, H., Ugurlu, H., & Yürük, N. (2015). Examining students’ mathematical understanding of geometric transformations using the Pirie–Kieren model. Educational Sciences: Theory and Practice, 15(6), 1531–1548.
Guven, B. (2012). Using dynamic geometry software to improve eighth grade students’ understanding of transformation geometry. Australasian Journal of Educational Technology, 28(2), 364–382.
Hollebrands, K. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. The Journal of Mathematical Behavior, 22(1), 55–72.
Hollebrands, K. (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.
Jackiw, N. (1991). The Geometer’s Sketchpad. Berkeley, CA: KeyCurriculum Press. (Computer software).
Johnson-Gentile, K., Clements, D., & Battista, M. (1994). Effects of computer and non-computer environments on students’ conceptualizations of geometric motions. Journal of Educational Computing Research, 11(2), 121–140.
Jung, I. (2002). Student representation and understanding of geometric transformations with technology experience. Unpublished doctoral dissertation. Athens, GA: The University of Georgia.
Laborde, C. (2002). Integration of technology in the design of geometry tasks with Cabri-geometry. International Journal of Computers for Mathematical Learning, 6(3), 283–317.
Laborde, C. (2005). Robust and soft constructions: Two sides of the use of dynamic geometry environments. In S. Chu, W. Yang, & H. Lew (Eds.), Proceedings of the tenth Asian technology conference in mathematics (pp. 22–35). Cheong-Ju, South Korea: Korea National University of Education http://epatcm.any2any.us/EP/EP2005/2005P279/fullpaper.pdf.
Leung, A., Baccaglini-Frank, A., & Mariotti, M. (2013). Discernment of invariants in dynamic geometry environments. Educational Studies in Mathematics, 84(3), 439–460.
Mariotti, M. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44(1–2), 25–53.
Ng, O.-L.& Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. ZDM: The International Journal on Mathematics Education, 51(3), 421–434.
Olivero, F., & Robutti, O. (2007). Measuring in dynamic geometry environments as a tool for conjecturing and proving. International Journal of Computers for Mathematical Learning, 12(2), 135–156.
Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26(2–3), 165–190.
Rabardel, P., & Beguin, P. (2005). Instrument-mediated activity: From subject development to anthropocentric design. Theoretical Issues in Ergonomics, 6(5), 429–461.
Sinclair, N., & Jackiw, N. (2010). Modeling practices with the Geometer’s sketchpad. In R. Lesh, P. Galbraith, C. Haines, & A. Hurford (Eds.), Modeling students’mathematical modeling competencies (pp. 541–554). Boston, MA: Springer.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.
Thom, J., & Pirie, S. (2006). Looking at the complexity of two young children’s understanding of number. The Journal of Mathematical Behavior, 25(3), 185–195.
van Hiele, P. (1986). Structure and insight: A theory of mathematics education. Orlando, FL: Academic Press.
von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the learning and teaching of mathematics (pp. 3–18). Hillsdale, NJ: Lawrence Erlbaum Associates.
Warner, L. (2008). How do students’ behaviors relate to the growth of their mathematical ideas? The Journal of Mathematical Behavior, 27(3), 206–227.
Yanik, H. (2011). Prospective middle school mathematics teachers’ preconceptions of geometric translations. Educational Studies in Mathematics, 78(2), 231–260.
Yao, X. &Manouchehri, A. (2019). Middle school students’ generalizations about properties of geometric transformations in a dynamic geometry environment. The Journal of Mathematical Behavior, 55,(#100703).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yao, X. Characterizing Learners’ Growth of Geometric Understanding in Dynamic Geometry Environments: a Perspective of the Pirie–Kieren Theory. Digit Exp Math Educ 6, 293–319 (2020). https://doi.org/10.1007/s40751-020-00069-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40751-020-00069-1