Characterizing Interaction with Visual Mathematical Representations

  • Kamran Sedig
  • Mark Sumner
Article

Abstract

This paper presents a characterization of computer-based interactions by which learners can explore and investigate visual mathematical representations (VMRs). VMRs (e.g., geometric structures, graphs, and diagrams) refer to graphical representations that visually encode properties and relationships of mathematical structures and concepts. Currently, most mathematical tools provide methods by which a learner can interact with these representations. Interaction, in such cases, mediates between the VMR and the thinking, reasoning, and intentions of the learner, and is often intended to support the cognitive tasks that the learner may want to perform on or with the representation. This paper brings together a diverse set of interaction techniques and categorizes and describes them according to their common characteristics, goals, intended benefits, and features. In this way, this paper aims to provide a preliminary framework to help designers of mathematical cognitive tools in their selection and analysis of different interaction techniques as well as to foster the design of more innovative interactive mathematical tools. An effort is made to demonstrate how the different interaction techniques developed in the context of other disciplines (e.g., information visualization) can support a diverse set of mathematical tasks and activities involving VMRs.

Keywords

computer-supported mathematical reasoning human–computer interfaces interaction techniques and design interactive visual representations mathematical cognitive tools visual mathematical representations 

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References

  1. Ainsworth S.E., Bibby P.A. and Wood D.J. (1997). Information technology and multiple representations: New opportunities – new problems. Journal of Information Technology for Teacher Education 6(1): 93–105Google Scholar
  2. Arcavi A. and Hadas N. (2000). Computer mediated learning: An example of an approach. International Journal of Computers for Mathematical Learning 5: 25–45CrossRefGoogle Scholar
  3. Banks D. (1992). Interactive manipulation and display of two-dimensional surfaces in four-dimensional space. Computer Graphics 25(2): 197–207Google Scholar
  4. Barwise J. and Etchemendy J. (1998). Computers, visualization and the nature of reasoning. In: Bynum, T.W. and Moor, J.H. (eds) The Digital Phoenix: How Computers are Changing Philosophy, pp 93–116. Blackwell, LondonGoogle Scholar
  5. Bates M.J. (1986). An exploratory paradigm for online information retrieval. In: Brookes, B.C. (eds) Intelligent Information Systems for the Information Society, pp. North-Holland, AmsterdamGoogle Scholar
  6. (2001). Cognitive Technology: Instruments of Mind – Proceedings of the International Cognitive Technology Conference. Springer-Verlag, Warwick, UKGoogle Scholar
  7. Botsmanova M.E. (1972). On the role of graphic analysis in solving arithmetic problems. In: Kilpatrick, J. and Wirzup, I. (eds) Soviet Studies in the Psychology of Learning and Teaching Mathematics, pp 119–123. University of Chicago Press, ChicagoGoogle Scholar
  8. Bridger M. and Bridger M. (2001). Mapping diagrams: Another view of functions. In: Cuocoand, A.A. and Curcio, F.R. (eds) The Roles of Representation in School Mathematics: 2001 Yearbook, pp. National Council of Teachers of Mathematics, Reston, VAGoogle Scholar
  9. Card S., MacKinlay J. and Shneiderman B. (1999). Readings in Information Visualization: Using Vision to Think. Morgan Kaufmann Publishers, San FranciscoGoogle Scholar
  10. Chen C. (1999). Information Visualization and Virtual Environments. Springer-Verlag, London, UKGoogle Scholar
  11. Cheng P. (2002). Electrifying diagrams for learning: Principles for complex representational systems. Cognitive Science 26: 685–736CrossRefGoogle Scholar
  12. (2001). The Roles of Representation in School Mathematics: 2001 Yearbook. National Council of Teachers of Mathematics, Reston, VAGoogle Scholar
  13. (2003). The Psychology of Problem Solving. Cambridge University Press, Cambridge, UKGoogle Scholar
  14. de Souza, C.S. and Sedig, K. (2001). Semiotic considerations on direct concept manipulation as a distinct interface style for learnware. In Proceedings of the Brazilian Human–Computer Interaction Conference (IHC2001) (pp. 229–241), 15–17 October. Florianopolis, Santa Catarina, BrazilGoogle Scholar
  15. Diezmann C.M. and English L.D. (2001). Promoting the use of diagrams as tools for thinking. In: Cuocoand, A.A. and Curcio, F.R. (eds) The Roles of Representation in School Mathematics: 2001 Yearbook, pp. National Council of Teachers of Mathematics, Reston, VAGoogle Scholar
  16. Dix, A.J. and Ellis, G. (1998). Starting simple – adding value to static visualization through simple interaction. In Proceedings of the 4th International Working Conference on Advanced Visual Interfaces (AVI’ 98) (pp. 124–134). L’Aquilla, Italy. New York: ACM PressGoogle Scholar
  17. (1997). Mathematical Reasoning: Analogies, Metaphors and Images. Lawrence Erlbaum Associate, Mahwah, NJGoogle Scholar
  18. Fisher R. (1990). Teaching Children to Think. Basil Blackwell, OxfordGoogle Scholar
  19. Frederickson G. (2002). Hinged Dissections: Swinging and Twisting. Cambridge University Press, Cambridge, MAGoogle Scholar
  20. Frederickson G. (2003). Dissection: Plane and Fancy. Cambridge University Press, Cambridge, MAGoogle Scholar
  21. Gadanidis G, Sedig K and Liang H (2004). Designing online mathematical investigation. Journal of Computers in Mathematics and Science Teaching 23(3): 275–298Google Scholar
  22. (1995). Diagrammatic Reasoning: Cognitive and Computational Perspectives. The MIT Press, Cambridge, MAGoogle Scholar
  23. Gobet F., Lane P., Croker S., Cheng P., Jones G., Oliver I. and Pine J. (2001). Chunking mechanisms in human learning. Trends in Cognitive Sciences 5(6): 236–243CrossRefGoogle Scholar
  24. Golightly, D. (1996). Harnessing the interface for domain learning. In M.J. Tauber (Ed), Proceedings of the CHI ’96 Conference Companion on Human Factors in Computing Systems: Common Ground (CHI ’96) (pp. 37–38), 13–18 April. Vancouver, British Columbia, Canada. New York: ACM PressGoogle Scholar
  25. Golledge R.G. (1999). Human wayfinding and cognitive maps. In: Golledge, R.G. (eds) Wayfinding Behavior: Cognitive Mapping and Other Spatial Processes, pp 5–45. The Johns Hopkins University Press, Baltimore, LondonGoogle Scholar
  26. Gonzalez-Lopez M. (2001). Using dynamic geometry software to simulate physical motion. International Journal of Computers for Mathematical Learning 6: 127–142CrossRefGoogle Scholar
  27. Greeno J.G. (1978). Natures of problem-solving abilities. In: Estes, W.K. (eds) Handbook of Learning and Cognitive Processes, pp 239–270. Lawrence Erlbaum Associates, Hillsdale, NJGoogle Scholar
  28. Halpern D.F. (2003). Knowledge and Thought. Lawrence Erlbaum Associates, Hillsdale, NJGoogle Scholar
  29. Hanna G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics 44: 5–23CrossRefGoogle Scholar
  30. Hansen Y.M. (1999). Graphic tools for thinking, planning and problem solving. In: Jacobson, R. (eds) Information Design, pp 193–221. The MIT Press, Cambridge, MAGoogle Scholar
  31. Hanson, A. and Ma, H. (1995). Space walking. In Proceedings of IEEE Visualization ’95 (pp. 126–133), Atlanta, GA, USAGoogle Scholar
  32. Hanson A., Munzner T. and Francis G. (1994). Interactive methods for visualizable geometry. IEEE Computer 27(7): 73–83Google Scholar
  33. Hazzan O. and Goldenberg P. (1997). Students’ understanding of the notion of function in dynamic geometry environments. International Journal of Computers for Mathematical Learning 1: 263–291CrossRefGoogle Scholar
  34. Healy L. and Hoyles C. (2001). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning 6: 235–256CrossRefGoogle Scholar
  35. Hitt, F. (Ed.) (2002). Representations and mathematics visualization. In Proceedings of the Twenty-Fourth North American Chapter of the International Group for the Psychology of Mathematics Education, Georgia, USAGoogle Scholar
  36. Holst, S.J. (1996). Directing learner attention with manipulation styles. In Tauber M.J. (Ed.), Proceedings of the CHI ’96 Conference Companion on Human Factors in Computing Systems: Common Ground (CHI ’96) (pp. 43–44), 13–18 April. Vancouver, British Columbia, Canada. New York: ACM PressGoogle Scholar
  37. Holzl R. (1996). How does dragging affect the learning of geometry?. International Journal of Computers for Mathematical Learning 1: 169–187CrossRefGoogle Scholar
  38. Hutchins E.L., Hollan J.D. and Norman D.A. (1986). Direct manipulation interfaces. In: Norman, D.A. and Draper, S.W. (eds) User Centered System Design: New Perspectives in Human–Computer Interaction, pp. Lawrence Erlbaum Associates, Hillsdale, NJGoogle Scholar
  39. Jackiw N. (1995). The Geometer’s Sketchpad, v. 3.0. Key Curriculum Press, Berkeley, CAGoogle Scholar
  40. (1993). Structural Knowledge: Techniques for Representing, Conveying and Acquiring Structural Knowledge. Lawrence Erlbaum Associates, Hillsdale, NJGoogle Scholar
  41. Jones K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics 44: 55–85CrossRefGoogle Scholar
  42. Jones S. and Scaife M. (2000). Animated diagrams: An investigation into the cognitive effects of using animation to illustrate dynamic processes. In: Anderson, M. and Cheng, P. (eds) Theory and Applications of Diagrams – Lecture Notes in Artificial Intelligence no. 1889, pp 231–244. Springer-Verlag, BerlinGoogle Scholar
  43. Kaput J. (1993). Urgent need for proleptic research. In: Romberg, T.A., Fennema, E. and Carpenter, T.P. (eds) Integrating Research on the Graphical Representation of Functions, pp. Lawrence Erlbaum Associates, Hillsdale, NJGoogle Scholar
  44. Keller P.R. and Keller M.M. (1993). Visual Cues: Practical Data Visualization. IEEE Computer Society Press, Los Alamitos, CAGoogle Scholar
  45. Kordaki M. and Potari D. (2002). The effects of area measurement tools on student strategies: The role of a computer microworld. International Journal of Computers for Mathematical Learning 7: 65–100CrossRefGoogle Scholar
  46. Labeke, N.V. (2001). Multiple external representations in dynamic geometry: A domain-informed design. In S. Ainsworth and R. Cox (Eds.), AI-ED 2001, Workshop Papers, External Representations of AIED: Multiple Forms and Multiple Roles (pp. 24–31), 20 May, Texas, USAGoogle Scholar
  47. (2000). Computers as Cognitive Tools. Lawrence Erlbaum Associates, New JerseyGoogle Scholar
  48. Larkin J. and Simon H. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science 11: 65–99CrossRefGoogle Scholar
  49. Laurillard D. (1993). Rethinking University Teaching: A Framework for the Effective Use of Educational Technology. Routledge Falmer, LondonGoogle Scholar
  50. Leung Y.K. and Apperley M.D. (1994). A review and taxonomy of distortion-oriented presentation techniques. ACM Transactions on Computer–Human Interaction 1(2): 126–160CrossRefGoogle Scholar
  51. Leung A. and 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: 145–165CrossRefGoogle Scholar
  52. Marrades R. and Gutierrez A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics 44: 87–125CrossRefGoogle Scholar
  53. Matsuda, N. and Toshio, O. (1998). Diagrammatic reasoning for geometry ITS to teach auxiliary line construction problems. In Proceedings of the International Conference on Intelligent Tutoring Systems (pp. 244–253), CanadaGoogle Scholar
  54. Morey J. and Sedig K. (2003). Archimedean kaleidoscope: A cognitive tool to support thinking and reasoning about geometric solids. In: Sarfraz, M. (eds) Geometric Modeling: Techniques, Applications, Systems and Tools, pp. Kluwer Academic Publisher, BerlinGoogle Scholar
  55. Morey, J. and Sedig, K. (2004). Using indexed-sequential geometric glyphs to explore visual patterns. In Proceedings of Interactive Visualisation and Interaction Technologies, ICCS 2004 (pp. 996–1003), June, Krakow, PolandGoogle Scholar
  56. Morey, J. and Sedig, K. (2004). Adjusting degree of visual complexity: An interactive approach for exploring four-dimensional polytopes. The Visual Computer: International Journal of Computer Graphics 20, 1–21. Berlin: Springer-VerlagGoogle Scholar
  57. Morey, J., Sedig, K. and Mercer R. (2001). Interactive metamorphic visuals: Exploring polyhedra relationships. In Proceedings of the 5th International Conference on Information Visualization (pp. 483–488). England: IEEE Computer Society PressGoogle Scholar
  58. Morrison, J., Tversky, B. and Betrancourt, M. (2000). Animation: Does it facilitate learning? In Proceedings of the AAAI 2000 Spring Symposium Smart Graphics (pp. 53–60), Stanford, CA, USAGoogle Scholar
  59. Moyer, P., Bolyard, J. and Spikell, M. (2001). Virtual manipulatives in the K-12 classroom. In A. Rogerson (Ed.), Proceedings of the International Conference on New Ideas in Mathematics Education (pp. 184–187), AustraliaGoogle Scholar
  60. (2000). Principles and Standards for School Mathematics. The National Council of Teachers of Mathematics, Reston, VAGoogle Scholar
  61. Norman D.A. (1993). Things that Make Us Smart: Defining Human Attributes in the Age of the Machine. Addison-Wesley, New YorkGoogle Scholar
  62. (1986). User Centered System Design: New Perspectives in Human–Computer Interaction. Lawrence Erlbaum Associates, Mahwah, NJGoogle Scholar
  63. Olive J. (2000). Computer tools for interactive mathematical activity in the elementary school. International Journal of Computers for Mathematical Learning 5: 241–262CrossRefGoogle Scholar
  64. Ormrod J.E. (1995). Human Learning. Prentice-Hall Inc, Englewood Cliffs, NJGoogle Scholar
  65. Otero, N., Rogers, Y. and du Boulay, B. (2001). Is interactivity a good thing? Assessing its benefits for learning. In Proceedings of the 9th International Conference on HCI (pp. 790–794), New Orleans, USA. New Jersey: Lawrence Erlbaum AssociatesGoogle Scholar
  66. Paivio A. (1983). The empirical case for dual coding. In: Yuille, J.C. (eds) Imagery, Memory and Cognition, pp. Lawrence Erlbaum Associates, Hillsdale, NJGoogle Scholar
  67. Papert S. (1980). Mindstorms: Children, Computers and Powerful Ideas. Basic Books, New YorkGoogle Scholar
  68. Park O. (1998). Visual displays and contextual presentations in computer-based instruction. Educational Technology, Research and Development 46(3): 37–51CrossRefGoogle Scholar
  69. Parker G., Franck G. and Ware C. (1997). Visualization of large nested graphs in 3D: Navigation and interaction. Journal of Visual Languages and Computing 9(5): 299–317Google Scholar
  70. Peper R.J. and Mayer R.E. (1986). Generative effects of note-taking during science lectures. Journal of Educational Psychology 78: 34–38CrossRefGoogle Scholar
  71. (1996). Forms of Representation. Intellect Books, Exeter, UKGoogle Scholar
  72. Pettersson R. (1989). Visuals for Information Research and Practice. Educational Technology Publications, Englewood Cliffs, NJGoogle Scholar
  73. Pimm D. (1995). Symbols and Meanings in School Mathematics. Routledge Falmer, New YorkGoogle Scholar
  74. Preece J., Rogers Y. and Sharp H. (2002). Interaction Design: Beyond Human–Computer Interaction. John Wiley and Sons, New YorkGoogle Scholar
  75. Raskin J. (2000). The Humane Interface: New Directions for Designing Interactive Systems. Addison-Wesley, Reading, MAGoogle Scholar
  76. Roth S., Chuah M., Kerpedjiev S., Kolojejchick J. and Lucas P. (1997). Towards an information visualization workspace: Combining multiple means of expression. Human–Computer Interaction Journal 12: 131–185CrossRefGoogle Scholar
  77. Salomon G. (1979). Interaction of Media, Cognition and Learning: An Exploration of How Symbolic Forms Cultivate Mental Skills and Affect Knowledge Acquisition. Jossey-Bass Publishers, San Francisco, CAGoogle Scholar
  78. Scaife M. and Rogers Y. (1996). External cognition: How do graphical representations work?. International Journal of Human–Computer Studies 45: 185–213CrossRefGoogle Scholar
  79. Scher D. and Goldenberg E.P. (2001). A multirepresentational journey through the law of cosines. In: Cuocoand, A.A. and Curcio, F.R. (eds) The Roles of Representation in School Mathematics: 2001 Yearbook, pp 1–1. National Council of Teachers of Mathematics, Reston, VAGoogle Scholar
  80. Sedig, K. (2004). Need for a prescriptive taxonomy of interaction for mathematical cognitive tools. In Proceedings of Interactive Visualisation and Interaction Technologies, ICCS 2004 (pp. 1030–1037), June, Krakow, PolandGoogle Scholar
  81. Sedig K., Klawe M. and Westrom M. (2001). Role of interface manipulation style and scaffolding on cognition and concept learning in learnware. ACM Transactions on Computer–Human Interaction 1(8): 34–59CrossRefGoogle Scholar
  82. Sedig, K. and Morey, J. (2005). A descriptive framework for designing interaction for visual abstractions. In G. Malcolm (Ed.), Multidisciplinary Approaches to Visual Representations and Interpretations 239–254. Elsevier ScienceGoogle Scholar
  83. Sedig, K., Morey, J. and Chu, B. (2002). TileLand: A microworld for creating mathematical art. In Proceedings of ED-MEDIA 2002: World Conference on Educational Multimedia and Hypermedia, July, Denver, USAGoogle Scholar
  84. Sedig, K., Morey, J., Mercer, R. and Wilson, W. (2005). Visualizing, interacting and experimenting with lattices using a diagrammatic representation. In G. Malcolm (Ed.), Multidisciplinary Approaches to Visual Representations and Interpretations 255–268. Elsevier ScienceGoogle Scholar
  85. Sedig K., Rowhani S., Morey J. and Liang H. (2003). Application of information visualization techniques to the design of a mathematical mindtool: A usability study. Information Visualization 2(3): 142–160CrossRefGoogle Scholar
  86. Shedroff N. (1999). Information interaction design: A unified field theory of design. In: Jacobson, R. (eds) Information Design, pp 267–292. The MIT Press, Cambridge, MAGoogle Scholar
  87. Sherman W.R. and Craig A.B. (2003). Understanding Virtual Reality: Interface, Application and Design. Morgan Kaufmann Publishers, San Francisco, CAGoogle Scholar
  88. Shneiderman B. (1988). We can design better user interfaces: A review of human–computer interaction styles. Ergonomics 3(15): 699–710CrossRefGoogle Scholar
  89. Shneiderman, B. and Kang, H. (2000). Direct annotation: A drag-and-drop strategy for labeling photos. In Proceedings of the International Conference on Information Visualization (pp. 88–95), July. Los Alamitos, CA: IEEE Computer Society PressGoogle Scholar
  90. Sims R. (1999). Interactivity on stage: Strategies for learner–designer communication. Australian Journal of Educational Technology 15(3): 257–272Google Scholar
  91. Sims R. (2000). An interactive conundrum: Constructs of interactivity and learning theory. Australian Journal of Educational Technology 16(1): 45–57Google Scholar
  92. Spence R. (1999). A framework for navigation. International Journal of Human–Computer Studies 51: 919–945CrossRefGoogle Scholar
  93. Spence R. (2001). Information Visualization. Addison-Wesley, New YorkGoogle Scholar
  94. Stojanov G. and Stojanoski K. (2001). Computer interfaces: From communication to mind-prosthesis metaphor. In: Beynon, M., Nehaniv, C.L. and Dautenhahn, K. (eds) Cognitive Technology: Instruments of Mind – Proceedings of the International Cognitive Technology Conference, pp 301–310. Springer-Verlag, Warwick, UKCrossRefGoogle Scholar
  95. Straesser R. (2001). Cabri-géomètre: Does dynamic geometry software (DGS) change geometry and its teaching and learning?. International Journal of Computers for Mathematical Learning 6: 319–333CrossRefGoogle Scholar
  96. Strothotte T. (1998). Computational Visualization: Graphics, Abstraction and Interactivity. Springer-Verlag, BerlinGoogle Scholar
  97. Stylianou D. (2002). On the interaction of visualization and analysis: The negotiation of a visual representation in expert problem solving. Journal of Mathematical Behavior 21: 303–317CrossRefGoogle Scholar
  98. Tall D. and West B. (1992). Graphic insight into mathematical concepts. In: Cornu, B. and Ralston, A. (eds) The Influence of Computers and Informatics on Mathematics and its Teaching, pp 117–123. UNESCO, ParisGoogle Scholar
  99. Thompson S. and Riding R. (1990). The effect of animated diagrams on the understanding of a mathematical demonstration in 11- to 14-year-old pupils. British Journal of Educational Psychology 60: 93–98Google Scholar
  100. Travaglini, S. (2003). An exploratory study of interaction design issues using a tiling microworld. Unpublished Master’s thesis. Department of Computer Science, The University of Western Ontario, London, Ontario, CanadaGoogle Scholar
  101. Tufte E.R. (1997). Visual Explanations. Graphics Press, Cheshire, CTGoogle Scholar
  102. Tweedie L., Spence R., Dawkes H. and Su H. (1996). Externalizing abstract mathematical models. Proceedings of CHI’96 (pp. 406–412). New York: ACM PressGoogle Scholar
  103. Ware C. (2000). Information Visualization: Perception for Design. Morgan Kaufmann Publishers, San Francisco, CAGoogle Scholar
  104. Webb R. (2003). Stella: Polyhedron Navigator. Symmetry: Culture and Science 11(1–4): 231–268Google Scholar
  105. Weisstein E.W. (2003). CRC Concise Encyclopedia of Mathematics. Chapman and Hall/CRC, Boca Raton, FLGoogle Scholar
  106. West T.G. (1995). Forward into the past: A revival of old visual talents with computer visualization. ACM SIGGRAPH Computer Graphics 29(4): 14–19CrossRefGoogle Scholar
  107. Winn W. (1993). Perception principles. In: Fleming, M. and Levie, W.H. (eds) Instructional Message Design: Principles from the Behavioral and Cognitive Sciences, pp. Educational Technology Publications, Englewood Cliffs, NJGoogle Scholar
  108. Yacci M. (2000). Interactivity demystified: A structural definition for distance education and intelligent computer-based instruction. Educational Technology 40(4): 5–16Google Scholar
  109. Yerushalmy M. and Shternberg B. (2001). Charting a visual course to the concept of function. In: Cuocoand, A.A. and Curcio, F.R. (eds) The Roles of Representation in School Mathematics: 2001 Yearbook, pp. National Council of Teachers of Mathematics, Reston, VAGoogle Scholar
  110. Zhang J. (1997). The nature of external representations in problem solving. Cognitive Science 21(2): 179–217CrossRefGoogle Scholar

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© Springer 2006

Authors and Affiliations

  • Kamran Sedig
    • 1
  • Mark Sumner
    • 1
  1. 1.Cognitive Engineering LaboratoryThe University of Western OntarioOntarioCanada

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