Details some small-scale teaching experiments that show some of the benefits, and some of the drawbacks, of using. 3-D g
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USinG dYnaMic GeoMetRY SoFtWaRe in MatHeMaticS teacHinG: Keith Jones offers a revised research bibliography updating that written for Micromath in 2002
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ynamic geometry software (dGS) provides an environment in which to create and then manipulate geometric constructions thereby providing learners with opportunities to interact with geometrical theorems and see the results. Well-known examples of dGS include Cabri, GeoGebra, Sketchpad, plus many others; a useful comparative review is provided by Mackrell (2011). in an earlier survey of research on the use of dGS in mathematics teaching (Jones, 2002), i identified three main threads in research up to that time. the first was the ways in which learners interact with the software, especially how they learn to use the capability of the software to drag objects, such as points, when constructing geometrical figures. an example of such research is Hölzl (1996) who provide insights into how students use the drag-mode in dGS. a second thread concerned the design of teaching activities so that learners come to understand geometrical ideas and concepts; laborde (2001) provides an example, which illustrates the time it can take to utilise the full potential of dGS, in the classroom. the third thread was the impact of using dGS on learning proof, especially whether the opportunities offered by dGS environments to ‘see’ mathematical properties on-screen might reduce, or even replace, any motivation for proof, or, on the contrary, whether dGS might open up new meaningful approaches to learning to prove. an example of research on these issues is provided by Jones (2000), showing how, over time, students’ mathematical explanations can evolve to ones that should help provide a foundation on which to build further notions of deductive reasoning. Since that earlier survey of research, dynamic geometry software has evolved; so much so that such software might now be called dynamic mathematics software, or perhaps mathematics visualisation software, as increasingly algebraic and other capabilities are being added. at the same time, research has both continued to delve into the issues identified above, while also expanding to cover newly surfacing matters. the aim of this revised research bibliography is to update the main themes and findings of research into the use of dGS in mathematics teaching and to illustrate newly emerging themes. other useful reviews are those by laborde et al (2006) and Hollebrands et al (2007).
the issue of how learners interact with the software, and learn to use the tools that are provided, continues to be the focus of research. examples of such studies include Hollebrands (2007), Maymon-erez and Yerushalmy (2007) and olivero and Robutti (2007). likewise, the impact of using dGS on learning to prove continues, an example being Baccaglini-Frank and Mariotti (2010). a newly emerging theme in the research is the use with learners of pre-constructed dGS files; see, for example, leung (2011), Sinclair (2003) and trgalová et al (2011). another theme, given the challenges faced by teachers when using dGS for the first time (erfjord, 2011), is how collaboration between teachers and education researchers can help teachers learn about using dGS from other teachers (lavicza et al, 2010). a third newlyemerging theme concerns new developments with the software, such as the facility to import digital pictures, and the development of software for 3-d geometry; examples include accascina and Rogora (2006), Jackiw and Sinclair (2009) and pierce and Stacey (2011). it continues to be the case that research on using dGS to teach transformation geometry is relatively limited, as is the use of dGS to teach ideas of loci. Research into the impact of linking the geometry capabilities of dGS with the increasingly incorporated algebraic capabilities is only just beginning. research Bibliography - publications in alphabetic order by first author; accascina, G. & Rogora, e. (2006). Using 3d diagrams for teaching geometry. International Journal for Technology in Mathematics Education, 13(1), 11–22. Details some small-scale teaching experiments that show some of the benefits, and some of the drawbacks, of using 3-D geometry software. Baccaglini-Frank, a. & Mariotti, M. a. (2010) Generating conjectures in dynamic geometry: the maintaining dragging model. International Journal of Computers for Mathematical Learning, 15(3), 225–253. Demonstrates how a form of object dragging employed by learners to maintain a particular property not only supports the generation of suitable conjectures but also provides elements that can help deductively in the generation of a proof.
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© ATM 2012 • No reproduction (including Internet) except for legitimate academic purposes •
[email protected] for permissions.
USinG dYnaMic GeoMetRY SoFtWaRe in MatHeMaticS teacHinG
erfjord, i. (2011). teachers’ initial orchestration of students’ dynamic geometry software use: consequences for students’ opportunities to learn mathematics. Technology, Knowledge and Learning, 16(1), 35-54. Outlines the challenges faced by teachers when using a new computer software tool for the first time when it can be difficult to see potential benefits, and disadvantages, of the new tool. Jackiw, n. & Sinclair, n. (2009). Sounds and pictures: dynamism and dualism in dynamic geometry. ZdM: International Journal on Mathematics Education, 41(4), 413-426. Explores the potential benefits of being able to use sounds and pictures in dynamic geometry software, and ponders whether such developments decrease learners’ sense of immediacy of interacting with geometrical theory. Jones, K. (2000). providing a foundation for deductive reasoning: students’ interpretations when using dynamic geometry software. Educational Studies in Mathematics, 44(1-3), 55-85. Shows how students’ mathematical explanations can evolve from imprecise ‘everyday’ expressions through to ones that should help provide a foundation on which to build further notions of deductive reasoning. Jones, K. (2002). Research on the use of dynamic geometry software: implications for the classroom, MicroMath, 18(3), 18-20 & 44-45. An overview of research published up to 2002. 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. Shows how the strategies that students employ are influenced by their understandings of geometric properties and relations, and by their perceived affordances of the DGS environment. Hollebrands, K., laborde, c. & Strasser, R. (2008). technology and the learning of geometry at the secondary level. in M. K. Heid & G. H. Blume (eds.), Research on technology and the teaching and learning of Mathematics: Vol. 1. Research syntheses (pp. 155–206). charlotte, nc: information age. A useful review of research into the use of technology in geometry education. Hölzl, R. (1996): How does ‘dragging’ affect the learning of geometry? international Journal of computers for Mathematical learning, 1(2), 169-187. A study that provides insight into how students use the dragmode in DGS. laborde, c. (2001), integration of technology in the design of geometry tasks with cabri-Geometry. international Journal of computers for Mathematical learning, 6(3), 283-317. Illustrates how long it can take to utilise fully the potential of DGS in the classroom. laborde, c., Kynigos, c., Hollebrands, K. & Strasser, R. (2006), teaching and learning geometry with technology. in a. Gutiérrez & p. Boero (eds.), Handbook of Research on the psychology of Mathematics education (pp. 275–304). Rotterdam: Sense publishers. A useful review of research into the use of technology in geometry education, especially the research reported at annual PME conferences. lavicza, Z., Hohenwarter, M., Jones, K., lu, a. & dawes, M. (2010) establishing a professional development network around dynamic mathematics software in england. international Journal for technology in Mathematics education, 17(4), 177-182. Shows how collaboration between teachers and education
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researchers can help teachers learn about using DGS from other teachers. leung, a. (2011). an epistemic model of task design in dynamic geometry environment. ZdM: international Journal on Mathematics education, 43(3), 325 -336. Outlines a way of designing DGS tasks that aim to motivate learners to make mathematical conjectures and, with careful attention to the role of the teacher, arrive at mathematical proofs. Mackrell, K. (2011) design decisions in interactive geometry software. ZdM: international Journal on Mathematics education, 43(3), 373–387. Helpful comparison of the capabilities of some of the main DGS packages. Maymon-erez, M. & Yerushalmy, M. (2007) “if you can turn a rectangle to a square then you can turn a square to a rectangle…”: on the complexity and importance of psychologizing the dragging tool by young students. international Journal of computers for Mathematical learning. 11(3), 271-299. Illustrates the complexities that 11-12 year olds experience when using dragging in DGS and how it can help to provide pre-constructed geometric figures. 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. Shows how measuring is a powerful DGS tool, but also how it is a complex tool that requires appropriate management and interpretation. pierce, R. & Stacey, K. (2011). Using dynamic geometry to bring the real world into the classroom. in l. Bu & R. Schoen (eds.), Model-centered learning: pathways to mathematical understanding using GeoGebra (pp. 41-55). Rotterdam: Sense publishers. Provides example of how, with DGS, the “real world” may be brought into the mathematics classroom in two ways: through the use of digital images and through the use of simulations, in each case using pre-prepared DGS files. Sinclair, M. (2003). Some implications of the results of a case study for the design of pre-constructed, dynamic geometry sketches and accompanying materials. educational Studies in Mathematics. 52(3), 289-317. Reports on how decisions about the design of pre-constructed DGS files support, or impede, learners’ development of exploration strategies and geometric thinking skills trgalová, J., Soury-lavergne, S. & Jahn, a. p. (2011), Quality assessment process for dynamic geometry resources in intergeo project: rationale and experiments. ZdM: international Journal on Mathematics education, 43(3), 337-351. Using experience from a project that aimed to provide teachers with ‘”good quality’’ DGS resources, the involvement of teachers into the quality assessment of such resource is shown to be a promising way of stimulating the use of DGS (provided that teachers can make the quality process their own). Keith Jones is deputy director of the Mathematics and Science education Research centre at the University of Southampton, UK. His research focuses on the teaching and learning of geometrical and spatial reasoning, and on the use of digital technologies; see: www.soton.ac.uk/education/research/centres.page the original article in Micromath can be found at www.atm.org.uk/mt229
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Mathematics Teaching 229 Journal of the Association of Teachers of Mathematics
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