Functions of Interactive Visual Representations in Interactive Mathematical Textbooks

  • Michal Yerushalmy


The paper explores changes in technology that have implications for the teaching and learning of school mathematics. To this end, it examines aspects of interactive mathematical textbooks; specifically it analyzes functions authors may intend to be carried out by embedded interactive diagrams. The paper analyzes theoretical as well as practical lessons that I learned while designing such a book. It is the purpose of this article to provide a rough, preliminary collection of categories of diagram function that would allow an orderly discussion of the subject. While this is not an empirical study, the hypotheses about student practices with interactive diagrams are based on a long series of studies related to the learning of algebra in a technologically-rich environment.


algebra interactive diagrams semiotic functions text design visual design 


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  1. Arzarello, F., Olivero, F., Paola, D., Robutti, O. 2002A cognitive analysis of dragging practises in cabri environmentsZentralblatt Fur Didaktik Der Mathematik346672Google Scholar
  2. Ball, L.D., Cohen, K.D. 1996Reform by the book: What is- or might be- the role of curriculum materials in teacher learning and instructional reform?Educational Researcher2568Google Scholar
  3. Borba, M.C., Confrey, J. 1996A student’s construction of transformations of functions in a multiple representational environmentEducational Studies in Mathematics31319337CrossRefGoogle Scholar
  4. Chazan, D. 2000Beyond Formulas in Mathematics and Teaching: Dynamics of the High School Algebra ClassroomTeachers College PressNYGoogle Scholar
  5. Chazan, D., Yerushalmy, M. 2003On appreciating the cognitive complexity of school algebra: Research on algebra learning and directions of curricular changeKilpatrick, J.Schifter, D.Martin, G. eds. A Research Companion to the Principles and Standards for School MathematicsNCTMReston123135Google Scholar
  6. diSessa, A. (2000). Changing Minds: computers, Learning and Literacy. MIT Press.Google Scholar
  7. Edwards, L.D. 1998Embodying mathematics and science: Microworlds as representationsJournal of Mathematical Behavior175378CrossRefGoogle Scholar
  8. Freudenthal, H. 1973Mathematics as an Educational TaskReidelDordrechtGoogle Scholar
  9. Gilead, S. and Yerushalmy, M. (2001). Deep structures of algebra word problems: Is it approach (in)dependent? Proceedings of the 25th Annual meeting of the International Group of Psychology of Mathematics Education, UtrechtGoogle Scholar
  10. Goldenberg, P.E. 1999Principles, arts, and craft in curriculum design: The case of connected geometryInternational Journal of Computers for Mathematical Learning4191224CrossRefGoogle Scholar
  11. Hershkovitz, R. (1987). The acquisition of concepts and misconceptions in basic geometry – or when a little learning is a dangerous thing. In Novak, J.D. (Eds), Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics 3 (pp. 238–251). Ithaca NY.Google Scholar
  12. Hershkovitz, R., Schwarz, B.B. 1999Reflective processes in a technology-based mathematics classroomCognition and Instruction1766591Google Scholar
  13. Hershkovitz, R., Dreyfus, T., Ben-Zvi, D., Freidlander, A., Hadas, N., Resnick, T., Tabach, M., Schwarz, B. 2002Mathematics curriculum development for computerized environments: A designer-researcher-teacher-learner-activityEnglish, L. eds. Handbook of International Research in Mathematics EducationLawrence Erlbaum AssociatesMahwah, New Jersey & LondonGoogle Scholar
  14. Holzl, R. 1996How does “Dragging” affect the learning of geometryInternational Journal of Computers for Mathematical Learning1169187Google Scholar
  15. Hoyles, C. (1993). Microworlds/Schoolworlds: The transformation of an innovation. In C. Keitel and K. Ruthven (Eds), Learning from Computers: Mathematics Education and Technology. pp 1–16Google Scholar
  16. Hoyles, C. and Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education. innovation . In A.J. Bishop, M.A. Clements, C. Keitel, J. Kilpatrick and F.K.S. Leung (Eds), Second International Handbook of Mathematics Education Dordrecht, Vol. 1. (pp. 323–350) KluwerGoogle Scholar
  17. Jewitt, C., Oyama, R. 2001Visual meaning: A social semiotic approachLeeuwen, T.Jewitt, C. eds. Handbook of Visual AnalysisSAGE PublicationsThousand Oaks, CAGoogle Scholar
  18. Kieran, C. and Yerushalmy, M. (2004). Research on the role of technological environments in algebra learning and teaching. In K. Stacey and H. Shick (Eds), ICMI Studies in Algebra (Chapter 5)Google Scholar
  19. Kress, G., Leeuwen, T. 1996Reading Images The Grammar of Visual DesignRoutledgeLondonGoogle Scholar
  20. Kress, G. 2003Literacy in the New Media AgeRoutledgeLondonGoogle Scholar
  21. Laborde, C. 2001Integration of technology in the design of geometry tasks with cabri-geometryInternational Journal of Computers for Mathematical Learning.6283317CrossRefGoogle Scholar
  22. Lagrange, J.-B., Artigue, M.,  et al. 2003Technology and Math Education: A Multidimensional Study of the Evolution of Research and Innovation. The 2nd International Handbook of Mathematics EducationKluwer Academic PublisherNetherlandsGoogle Scholar
  23. Lemke J.L. (1998). Multiplying meaning: visual and verbal semiotics in scientific text. London: Routledge (pp. 87–113) In J.R. martin and R. Veel (Eds), Reading Science. RoutledgeGoogle Scholar
  24. Leikin, R. and Dinur, S. (2003). Patterns of flexibility: Teachers’ behavior in mathematical discussion. In Electronic Proceedings of the Third Conference of the European Society for Research in Mathematics Education. (10 p.)
  25. Levin, J.R. (1989). A transfer- appropriate-processing perspective of pictures in prose. In H. Mandl and J.R. Levin (Eds), Knowledge Acquisition from Text and Pictures. Elsevier Science Publishers B.V. 58.Google Scholar
  26. Love, E., Pimm, D. 1996This is so: A text on textsBishop, A.J. eds. International Handbook of Mathematics EducationKluwer Academic PublisherNetherlands371409Google Scholar
  27. Mandl, H., Levin, J.R. 1989Knowledge Acquisition from Text and PicturesElsevier Science PublishersAmsterdamGoogle Scholar
  28. Molitor, S. and Ballstaedt, S.-P. (1989). Problems in knowledge acquisition from text and pictures. In H. Mandl and J.R. Levin (Eds), Knowledge Acquisition from text and Pictures. vol. 58. (pp. 3–35) Elsevier Science Publishers B.V.Google Scholar
  29. Morgan, C. eds. 1998Writing mathematically, The Discourse of InvestigationFalmer PressLondonGoogle Scholar
  30. Nunokawa, K. 1994Improving diagrams gradually: One approach to using diagrams in problem solvingFor the Learning of Mathematics143438Google Scholar
  31. Netz, R. 1999The Shaping of Deduction in Greek MathematicsCambridge University PressUKGoogle Scholar
  32. Romberg, T.A., Fennema, E., Carpenter, T. 1993Integrating Research on Graphical Representations of FunctionsErlbaum IncNJGoogle Scholar
  33. Shternberg, B., Yerushalmy, M. 2003Models of functions and models of situations: On design of a modeling based learning environmentDoerr, H.M.Lesh, R. eds. Beyond Constructivism: A Model and Modeling Perspective on Teaching, Learning, and Problem Solving in Mathematics EducationLawrence AlbumMahwah, NJ479500Google Scholar
  34. Vinner, S. 1983Concept definition, concept image and the notion of functionInternational Journal of Mathematical Education in Science and Technology14293305Google Scholar
  35. Vinner, S., Dreyfus, T. 1989Images and definitions for the concept of functionJournal for Research in Mathematics Education20356366Google Scholar
  36. Visual-Mathematics (1995) Algebra and functions, (Curriculum In Hebrew) Center for Educational Technology, Tel-Aviv.
  37. Yerushalmy, M. 2000Problem Solving Strategies and Mathematical Resources: A Longitudinal View on Problem Solving in a Functional Based Approach to AlgebraEducational Studies in Mathematics43125147CrossRefGoogle Scholar
  38. Yerushalmy, M. 1999Making exploration visible: On software design and school algebra curriculumInternational Journal of Computers for Mathematical Learning4169184Google Scholar
  39. Yerushalmy, M. 1997aReaching the unreachable: Technology and the semantics of asymptotesInternational Jounral of Computers for Mathematical Learning.2125Google Scholar
  40. Yerushalmy, M. 1997bMathematizing qualitative verbal descriptions of situations: A language to support modelingCognition and Instruction15207264Google Scholar
  41. Yerushalmy, M. 1991Students perceptions of aspects of algebraic function using multiple representation softwareJournal of Computer Assisted Learning. Blackwell Scientific Publications74257Google Scholar
  42. Yerushalmy, M., Chazan, D. 2002Flux in school algebra: Curricular change, graphing technology, and research on student learning and teacher knowledgeEnglish, L.D. eds. Handbook of International Research in Mathematics Education.Lawrence Erlbaum AssociatesMahwah, New Jersey and London725755Google Scholar
  43. Yerushalmy, M., Chazan, D. 1990Overcoming visual obstacles with the aid of the supposerEducational Studies in Mathematics21199219CrossRefGoogle Scholar
  44. Yerushalmy, M., Katriel, H. and Shternberg, B. (2002). The Function Web-book.
  45. Yerushalmy, M and Gafni, R. (1992). Syntactic Manipulations and Semantic Interpretations in Algebra: The Effect of Graphic Representation. Learning and Instruction. Vol. 2. (pp. 303–319) Pergamon Press Ltd.Google Scholar
  46. Yerushalmy, M., Schwartz, J.L. 1993Seizing the opportunity to make algebra mathematically and pedagogically interestingRomberg, T.A.Fennema, E.Carpenter, T. eds. Integrating Research on Graphical Representations of FunctionsErlbaum IncNJGoogle Scholar
  47. Yerushalmy, M. and Shternberg, B. (2001). A visual course to functions. In Cuoco, A.A. and Curcio, F. (Eds), The Roles of Representations in School Mathematics (pp. 125–147). NTCM Yearbook.Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of EducationUniversity of HaifaHaifaIsrael

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