# Transforming Mathematics: Using Dynamic Geometry Software to Strengthen Understanding of Enlargement and Similarity

### Abstract

This article discusses the potential to use Dynamic Geometry Software (DGS) to form conceptual links across enlargement and similarity by developing learners’ understanding of scale factor and ratio. From the theoretical perspective of situated abstraction, a combination of both situated cognition and mathematical abstraction, it analyses existing literature on the teaching, learning and assessment of enlargement and similarity as well as literature on DGS and how it acts as a microworld, where an array of situations in a mathematically consistent environment can be created. Particular focus is given to how the dragging and measurement facilities in DGS support abstractions through both amplification and reorganisation of traditional pencil and paper methods. The empirical element of this article describes a small scale classroom based project on the use of DGS as a microworld for transformation geometry. Through analysing learners’ dialogue and written responses to tasks, it proposes that a combination of minimally pre-constructed tasks, peer-discussion and utilising the dragging and measurement facilities, can enhance the observation of patterns in transformation geometry and concludes that these conditions can support learners to move from the particular to the general, allowing abstractions to be conceived and strengthening learners’ understanding of enlargement and similarity.

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Ainley, J., Pratt, D., and Hansen, A. (2006), Connecting engagement and focus in pedagogic task design, British Educational Research Journal, 32(1), 23-38

Baccaglini-Frank, A. and Mariotti, M. (2010), Generating Conjectures in Dynamic Geometry:

The Maintaining Dragging Model, International Journal of Computers for Mathematical Learning 15, 225–253

Banks, T. and Alcorn, A. (2003), Mathematics for AQA GCSE – Higher Tier, Lancashire: Causeway Press Limited

Baston, C., Bolter, J., Cole, G., Dyer, G., Flowers, M., Hughes, K., Jolly, P., Knott, J., Linsky, J., Newman, G., Pepper, R., Petran, J., Pledger, K., Summerson, R., Tanner, K. and Western, B. (2010), Mathematics A - Linear Foundation, Harlow: Edexcel Limited

Becta (2009), Secondary Mathematics with ICT: A pupil’s entitlement to ICT in secondary mathematics, http://www.nationalstemcentre.org.uk/dl/2a99cf80d5efac70875df99f983cfac1de333a72/13810-entitlement_sec_maths.pdf, accessed 13 April 2013

BERA (2011), Ethical Guidelines for Educational Research, London: British Educational Research Association

Boaler, J. (1999), Participation, knowledge and beliefs: A community perspective on mathematics learning, Educational Studies in Mathematics, 40(3), 259-281

Brown, J. S., Collins, A. and Duguid, P. (1989), Situated Cognition and the Culture of Learning, Educational Researcher, 18(1), 32-42

Clements, D.H. (2002), Computers in Early Childhood Mathematics, Contemporary Issues in Early Childhood, 3(2), 160-181

Cox, D. (2013), Similarity in Middle School Mathematics: At the Crossroads of

Geometry and Number, Mathematical Thinking and Learning, 15(1), 3-23

Denscombe, M. (2007), The Good Research Guide for small-scale social research projects (third ed.), Maidenhead: Oxford University Press (first published 1998)

deVilliers, M. (2006), Some Pitfalls of Dynamic Geometry Software, Learning and Teaching Mathematics, 4, 46-52

diSessa, A. (1993), Toward an Epistemology of Physics, Cognition and

Instruction, 10(2-3), 105-225

Dubinsky, E. and McDonald, M.A. (2002), APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research, in Hoton, D. (ed.), The Teaching and Learning of Mathematics at University Level, New ICMI Study Series, 2002, Vol. 7, Section 3, pp.275-282.

Edexcel (2012), Principal Examiner Feedback Summer 2012 GCSE Mathematics (Linear) 1MA0, Pearson Education Ltd.

Edexcel (2011), Examiner Report June 2011 GCSE Mathematics (1380), Edexcel Ltd, Publications Code UG028357

Edexcel (2010), Principal Examiner Feedback Summer 2010 GCSE Mathematics (1380), Edexcel Ltd

French (2012), Similar Triangle Match Up, http://www.tes.co.uk/teaching-resource/Similar-Triangle-Match-Up-6185944/, accessed 24 March 2013

Glover, D., Miller, D., Averis, D. and Door, V. (2007), The evolution of an effective pedagogy for teachers using the interactive whiteboard in mathematics and modern languages: an empirical analysis from the secondary sector, Learning, Media and

Technology, 32(1), 5-20

Hershkowitz, R., Schwarz, B. and Dreyfus, T. (2001), Abstraction in Context: Epistemic Actions, Journal for Research in Mathematics Education, 32(2), 195-222

Hollebrands, K. (2003), High school students’ understandings of geometric transformations in the context of a technological environment, Journal of Mathematical Behavior 22, 55–72

Hölzl, R. (1996), How does ‘dragging’ affect the learning of geometry, International Journal of Computers for Mathematical Learning , 1(2), 169-187

JMC (2011), Digital technologies and mathematics education, Clark-Wilson, A., Oldknow, A. and Sutherland, R. (eds.), Joint Mathematical Council of the United Kingdom (JMC)

http://cme.open.ac.uk/cme/JMC/Digital%20Technologies%20files/JMC_Digital_Technologies_Report_2011.pdf, accessed 26 March 2013

Jones, K. (2012), Using Dynamic Geometry Software in Mathematics Teaching, Mathematics Teaching 229, ATM

Jones, K. (2011), The value of learning geometry with ICT: lessons from innovative

educational research, in Oldknow, A. and Knights, C. (eds.), Mathematics

Education with Digital Technology, London: Continuum, pp.39-45

Jones, K. (2002), Research on the use of dynamic geometry software: implications for the classroom, MicroMath, 18(3), 18-20

Lack (2011), Exploring combined transformations group activity, http://www.tes.co.uk/teaching-resource/GCSE-Maths-Game-Transformations-Lesson-6140178/, accessed 9 March 2013

Lamon, S. (1995), Ratio and Proportion: Elementary Didactical Phenomenology, in Sowder, J. and Schappelle, B. (eds.), Providing a foundation for teaching mathematics in the middle grades, Albany: SUNY Press

Lesh, R., Post, T. and Behr, M. (1988), Proportional reasoning, Number concepts and operations in the middle grades, 2, 93-118

Maymon-Erez, M. and Yerushalmy, M. (2006), “If You Can Turn a Rectangle into a Square, You Can Turn a Square into a Rectangle...” Young Students Experience the Dragging Tool, International Journal of Computers for Mathematical Learning, 11(3), 271-299

Nardi, E. and Steward, S. (2003), Is Mathematics T.I.R.E.D.? A Profile of Quiet Disaffection in the Secondary Mathematics Classroom, British Educational Research Journal, 29(3), 345–367

Noss, R. and Hoyles, C. (1996), Windows on Mathematical Meanings: Learning Cultures and Computers, Volume 17, London: Kluwer Academic Publishers

OCR (2012), General Certificate of Secondary Education Mathematics A (J562) OCR Report to Centres, Cambridge: Oxford Cambridge and RSA Publications

OCR (2011), General Certificate of Secondary Education Mathematics A (J562) Examiners’ Reports, Nottingham: Oxford Cambridge and RSA Publications

Olive, J. (2000), Implications of Using Dynamic Geometry Technology for Teaching and Learning, Teaching and Learning Problems in Geometry conference, Fundão, Portugal , 6-9 May 2000, http://math.coe.uga.edu/Olive/Portugal/Portugal_paper.html, accessed 9 March 2013

Olivero, F. and 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

Ozmantar, M. F. and Roper, T. (2004), Mathematical abstraction through scaffolding, Proceedings of the 28th Conference of the International, 3, 481-488

Papert, S. (1991), Situating Constructionism, in Papert, S. and Harel, I (eds), Constructionism, Norwood, NJ: Ablex Publishing Corporation

Papert, S. (1987), Microworlds: transforming education, Artificial intelligence and education, 1, 79-94

Papert, S. (1972), Teaching children to be mathematicians versus teaching about mathematic’, International Journal of Mathematical Education in Science and Technology, 3(3), 249-262

Payne, G., Holt, R., Rayment, M. and Robinson, I. (2006), AQA GCSE Mathematics – Higher, Oxford: Heinemann

Pea, R. D. (1985), Beyond amplification: Using the computer to reorganize mental functioning, Educational Psychologist, 20(4), 167-182

Pratt, D. and Noss, R. (2002), The Microevolution of Mathematical Knowledge: The Case of Randomness, The Journal of the Learning Sciences, 11(4), 455–488

Pratt, D. and Noss, R. (2010), Designing for Mathematical Abstraction, International Journal of Computers for Mathematical Learning, 15(2), 81-97

Rayner, D. (2003), Modular Mathematics Higher GCSE for AQA, Oxford: Oxford University Press

Ruthven, K. (2003), Linking algebraic and geometric reasoning with dynamic geometry software - Final report to the Qualifications and Curriculum Authority https://www.educ.cam.ac.uk/people/staff/ruthven/RuthvenQCA03.pdf, accessed 26 March 2013

Ruthven, K., Hennessy, S. and Deaney, R. (2008), Constructions of dynamic geometry: A study of the interpretative flexibility of educational software in classroom practice, Computers & Education, 51, 297–317

Ruthven, K. (2012), The didactical tetrahedron as a heuristic for analysing the incorporation of digital technologies into classroom practice in support of investigative approaches to teaching mathematics, ZDM , 44(5), 627-640

Stewart, W. (2009), Pupils take hold of the teaching, TES2 January 2009, http://www.tes.co.uk/article.aspx?storycode=6006663, accessed 14 April 2013

Tall, D. (2004), Building Theories: The Three Worlds of Mathematics, For the Learning of Mathematics, 24(1), 29-32.

Taylor, R. P. (1980), The computer in school: Tutor, tool, tutee, New York: Teachers College Press

White, P. and Mitchelmore, M. (2007), Conceptual Knowledge in Introductory Calculus, Mathematics Education Research Journal, 19(2), 1-9

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