REIMAGINING SUSTAINABLE SCHOOL EDUCATION THROUGH A FRACTAL-BASED DYNAMIC PROGRAM (FDP): AN EXPERIENTIAL AND INTERACTIVE APPROACH

Author: Stavroula Patsiomitou

ABSTRACT

The paper presents the design, development, and implementation of a Fractal-based Dynamic Program (FDP), which synthesizes findings and design principles derived from my prior empirical research with newly generated evidence. The FDP is organized around a learning trajectory comprising five instructional units, each briefly outlined in terms of its learning goals, activities, and conceptual progression. In the methodology section, I describe the essential elements of the FDP that I designed and implemented, and I also propose strategies for its effective classroom implementation. The study is situated within a design-based research methodology, adopting a qualitative case study perspective. The FDP is proposed as a flexible informal curriculum framework grounded in principles of transformation geometry and fractals, and designed to support interdisciplinary, sustainability-oriented school education. The implementation of the FDP demonstrates how fractals can function as a powerful interdisciplinary teaching context that connects mathematics and art while enhancing students’ engagement and enjoyment of learning. Moreover, the study identifies pedagogical principles emerging from the implementation process, particularly concerning dynamic visualization, transformation-based reasoning, and students’ gradual transition from empirical exploration to formal mathematical thinking. Finally, the study addresses the need to empower educators to design and mediate learning experiences that foster sustainability citizenship in school education.

Keywords: Fractal-Based Learning, Transformation geometry, Dynamic Geometry, Curriculum Design

REFERENCES

  1. Apostel, L. (1972). Interdisciplinarity: Problems of teaching and research in universities. OECD Publications Center.
  2. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215-241.
  3. Battista, M. T. (2011). Conceptualizations and issues  related to learning progressions, learning trajectories, and levels  of sophistication. The Mathematics Enthusiast, 8, 507–570. http://www.math.umt.edu/TMME/vol8no3/Battista_TME_2011_article4_pp.507_570.pdf
  4. Bloom, B. S., Engelhart,M. D., Furst,E. J.,Hill,W. H.,& Krathwohl,D. R. (Eds.).(1956). Taxonomy of educational objectives: The classification of educational goals. Handbook I: Cognitive domain. New York: David McKay
  5. Brown, A. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning Sciences, 2(2), 141–178.
  6. Burkhardt, H. (1988). Teaching problem solving. In H. Burkhardt, S. Groves, A. Schoenfeld, & K. Stacey (Eds.), Problem solving – A world view (Proceedings of the Problem-Solving Theme Group, ICME5, pp. 17–42). Nottingham: Shell Centre.
  7. Chevallard, Y. (1989). On didactic transposition theory: Some introductory notes. In H. G. Steiner & M. Hejny (Eds.), Proceedings of the International Symposium on Selected Domains of Research and Development in Mathematics Education (pp. 51–62). University of Bielefeld, and University of Bratislava. http://yves.chevallard.free.fr/spip/spip/IMG/pdf/On_Didactic_Transposition_Theory.pdf
  8. Choppin, J. M. (1994). Spiral through recursion. Mathematics Teacher, 87, 504-508.
  9. Claudia, A. (2009). Gestalt configurations in geometry learning. Proceedings of CERME 6 (pp. 706-715), January 28-February 1, Lyon France.
  10. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13. https://www.researchgate.net/publication/244457271
  11. Driscoll, A. & Nagel, N. (2002). Early childhood education. Boston: Allyn and Bacon.
  12. Freudenthal, H. (1973) Mathematics as an Educational Task, Reidel, Dordrecht
  13. Fischbein, E. (1993) The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139-162.
  14. Fuys, D., Geddes, D., & Tischler, R. (Eds). (1984). English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele. Brooklyn: Brooklyn College. (ERIC Document Reproduction Service No. ED 287 697).
  15. Gravemeijer, K. and Terwel, J. (2000) Hans Freudenthal: a mathematician on didactics and curriculum theory. Journal of Curriculum Studies, 32(6), 777–796.
  16. Richter-Gebert, J. and Kortenkamp, U. (1999). User manual of the Interactive Geometry Software Cinderella. Springer-Verlag, Heidelberg
  17. Ghosh, Jonaki (2016) Fractal constructions leading to algebraic thinking. At Right Angles, 5 (3). pp. 59-66. ISSN 2582-1873.
  18. Golomb, S. W. (1964) “Replicating Figures in the Plane.” Mathematical Gazette. 48, 403-12
  19. Hohenwarter, M. (2001). GeoGebra [Computer software]. FCR-STEM, Learning Institute, Florida State University. http://www.geogebra.org/cms/.
  20. Hohenwarter, M. (2002) GeoGebra – Ein Software system für dynamische Geometrie und Algebra der Ebene. Master thesis, University of Salzburg
  21. Hohenwarter, M., Hohenwarter, J., Kreis, Y.,  and Lavicza, Z.  (2008) Teaching and Learning Calculus with Free Dynamic Mathematics Software GeoGebra. Research and development in the teaching and learning of calculus ICME 11, Monterrey, Mexico 2008
  22. Hasrul, H., & Irawan, B. (2023). Linking citizenship to education for sustainability: A theory of change conceptual framework. BIO Web of Conferences, ICOME 2023. https://www.bionferences.org/articles/bioconf/pdf/2023/24/bioconf_icome2023_02005.pdf
  23. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum Associates.
  24. Jackiw, N. (1991).The Geometer’s Sketchpad [Computer Software].Berkeley, CA: Key Curriculum Press.
  25. Jaworski, B. (2003). Inquiry as a pervasive pedagogic process in mathematics educa-tion development, Proceedings of the Third Conference of the European Society for Research in Mathematics Education. Bellaria, Italy. http://www.dm.unipi.it/~didattica/CERME3
  26. Benoit Mandelbrot (1975). Les objets fractals, forme, hasard et dimension. Paris:Flammarion.
  27. Kinach, Barbara M. (2014). “Generalizing: the core of algebraic thinking”. Mathematics Teacher, 107(6), 432-439. https://www.academia.edu/15631454/
  28. Klein, F. (1893). A comparative review of recent researches in geometry (M. W. Haskell, Trans.). Bulletin of the New York Mathematical Society, 2(10), 215–249. (Original work published 1872)
  29. KoehlerJ., Mishra P., Akcaoglu M., Rosenberg J.M. (2013). The technological pedagogical content knowledge framework for teachers and teacher educators. Commonwealth Educational Media Center for Asia, p.1-8.
  30. Kolb, D. A. (1984). Experiential learning: Experience as the source of learning and development. New Jersey: Prentice-Hall.
  31. Kolb , A.& Kolb, D. A. (2005) The Kolb Learning Style Inventory –version 3.1 2005 Technical http://learningfromexperience.com/media/2010/08/tech_spec_ls
  32. Laborde, J-, Baulac, Y., & Bellemain, F. (1988) Cabri Géomètre [Software]. Grenoble, France: IMAG-CNRS, Universite Joseph Fourier
  33. Laborde, J. M. (2004). Cabri 3D. Online at: http://www.cabri.com/
  34. Lieberman, N. (1977). Playfulness. It’s Relationship to Imagination and Creativity. New York: Academic Press Inc.
  35. Löfstedt, T. (2008). Fractal geometry, graph and tree constructions (Master’s thesis). Umeå University, Sweden.
  36. Mishra, P., & Koehler, M.J. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. Teachers College Record, 108(6), 1017–1054. doi:10.1111/j.1467-9620.2006.00684.x
  37. Novak D. & Cañas A. J. (2006). The Theory Underlying Concept Maps and How to Construct Them (Technical Report No. IHMC CmapTools 2006-01). Pensacola, FL: Institute for Human and Machine Cognition. Available at http://cmap.ihmc.us/Publications/ResearchPapers/TheoryUnderlyingConceptMaps.pdf
  38. Patsiomitou, S. (2005a). Fractals as a context of comprehension of the meanings of the sequence and the limit in a Dynamic Computer Software environment (Master’s thesis). National and Kapodistrian University of Athens, Department of Mathematics, Interuniversity Postgraduate Program. Available at http://me.math.uoa.gr/dipl/dipl_patsiomitou.pdf
  39. Patsiomitou, S. (2005b). Fractals as a context of comprehension of the meanings of the sequence and the limit in a Dynamic Software environment. In Proceedings of the 22nd Panhellenic Conference of the Hellenic Mathematical Society (pp. 311–323). Lamia, 18–20 November 2005. (in Greek)
  40. Patsiomitou, S. (2006a). Transformations on mathematical objects through animation and trace of their dynamic parameters. In Proceedings of the 5th Pan-Hellenic Conference with International Participation: Informatics and Education-ETPE (pp. 1070–1073). Thessaloniki, 5–8 October 2006. http://www.etpe.gr/custom/pdf/etpe1213.pdf (in Greek)
  41. Patsiomitou, S. (2006b). An approximation process generating number π (pi) through inscribed/circumscribed parametric polygons in a circle or through Riemann integrals’ approximation process: Experimentation and research in a Dynamic Geometry Software Environment. In Proceedings of the 23rd Panhellenic Conference of the Hellenic Mathematical Society (pp. 502–514). Patras, 24–26 November 2006. (in Greek)
  42. Patsiomitou, S. (2006c). Dynamic geometry software as a means of investigating, verifying, and discovering new relationships of mathematical objects. EUCLID C: Scientific journal of Hellenic Mathematical Society, 65, 55–78. (in Greek)
  43. Patsiomitou, S. (2007a). Fractals as a context of comprehension of the meanings of the sequence and the limit in a Dynamic Computer Software environment. In E. Milková & P. Prazák (Eds.), Electronic Proceedings of the 8th International Conference on Technology in Mathematics Teaching (ICTMT8) (pp. cd-rom). University of Hradec Králové. ISBN 978-80-7041-285-5
  44. Patsiomitou, S. (2007b). Sierpinski triangle, Baravelle spiral, Pythagorean tree: The Geometer’s Sketchpad v4 as a means for the construction of meanings. In Proceedings of the 4th Pan-Hellenic ICT Conference: Exploiting Information and Communication Technologies in Educational Practices (pp. 28–37). Greek Ministry of Education, Syros, 4–6 May 2007. (in Greek)
  45. Patsiomitou, S. (2008a). The development of students’ geometrical thinking through transformational processes and interaction techniques in a dynamic geometry environment. Issues in Informing Science and Information Technology, 5, 353–393. https://doi.org/10.28945/1015
  46. Patsiomitou, S. (2008b). Linking Visual Active Representations and the van Hiele model of geometrical thinking. In W.-C. Yang, M. Majewski, T. Alwis, & K. Klairiree (Eds.), Proceedings of the 13th Asian Conference in Technology in Mathematics (pp. 163–178). Bangkok, Thailand: Suan Shunanda Rajabhat University. http://atcm.mathandtech.org/EP2008/papers_full/2412008_14999.pdf
  47. Patsiomitou, S. (2008c). The construction of the number φ and the Fibonacci sequence in The Geometer’s Sketchpad v4 Dynamic Geometry software. In Proceedings of the 1st Pan-Hellenic ICT Educational Conference: Digital Material to support Primary and Secondary-level teachers’ pedagogical work (pp. 307–315). Naoussa, 9–11 May 2008. (in Greek)
  48. Patsiomitou, S. (2008d). The construction of a Baravelle spiral as a means for students’ intuitive understanding of ascending and descending sequences. In Proceedings of the 1st Panhellenic ICT Educational Conference: Digital Material to support Primary and Secondary-level teachers’ pedagogical work (pp. 316–324). Naoussa, 9–11 May 2008. (in Greek)
  49. Patsiomitou, S. (2009a). Learning Mathematics with The Geometer’s Sketchpad v4: Volume A (Monograph). Klidarithmos Publications. (in Greek)
  50. Patsiomitou, S. (2009b). Learning Mathematics with The Geometer’s Sketchpad v4: Volume B (Monograph). Klidarithmos Publications. (in Greek)
  51. Patsiomitou, S. (2009c). Tessellations, Pentominos, Structural Algebraic Units, Rep-Tiles, Tangram: A proposal for a qualitative upgrading of math curricula. In Proceedings of the 5th Pan-Hellenic ICT Conference: Exploiting Information and Communication Technologies in Didactic Practice (pp. 601–609). Syros, 8–10 May 2009. (in Greek)
  52. Patsiomitou, S. (2009d). Tessellations constructed using Geometer’s Sketchpad v4 as an intuitive means for the development of students’ deductive reasoning. In Proceedings of the 1st Educational Conference: Integration and Use of ICT in the Educational Process, 1, 154–160. Volos, 24–26 April. (in Greek)
    1. https://eproceedings.epublishing.ekt.gr/index.php/cetpe/article/view/6426
  53. Patsiomitou, S. (2010). Building LVAR (Linking Visual Active Representations) modes in a DGS environment. Electronic Journal of Mathematics and Technology (eJMT), 4(1), 1–25. https://ejmt.mathandtech.org/Contents/eJMT_v4n1p1.pdf
  54. Patsiomitou, S. (2011). Theoretical dragging: A non-linguistic warrant leading to dynamic propositions. 35th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 361-368. Ankara, Turkey: PME. ISBN 978-975-429-297-8. Available at https://www.researchgate.net/publication/230648462
  55. Patsiomitou, S. (2012a). The development of students’ geometrical thinking through transformational processes and interaction techniques in a dynamic geometry environment: Linking Visual Active Representations (PhD thesis). University of Ioannina. https://www.didaktorika.gr/eadd/handle/10442/35816 (in Greek)
  56. Patsiomitou, S. (2012b). A Linking Visual Active Representation DHLP for student’s cognitive development. Global Journal of Computer Science and Technology, 12(6), 53–81. http://computerresearch.org/index.php/computer/article/view/479/479
  57. Patsiomitou, S. (2012c). Didactic approaches to teaching Mathematics to students with different learning styles: Mathematics in the Real World. Self–publishing. ISBN 978-960-93-4456. https://www.academia.edu/2054056 (in Greek)
  58. Patsiomitou, S. (2013). Instrumental decoding of students’ conceptual knowledge through the modeling of real problems in a dynamic geometry environment. EUCLID C: Scientific journal of Hellenic Mathematical Society, 79, 107–136. (in Greek)
  59. Patsiomitou, S. (2014). Student’s learning progression through instrumental decoding of mathematical ideas. Global Journal of Computer Science and Technology, 14(1), 1–42. http://computerresearch.org/index.php/computer/article/view/41/41
  60. Patsiomitou, S. (2016a). Synthesis, application and evaluation of a “dynamic” curriculum: Transformations of fractals objects, parametric regular polygons and number π. Linking Visual Active Representations. A keynote speech. In 3rd Panhellenic Conference of “The New Educator (Neos Paidagogos)” (pp. 3563–3602). Eugenides Foundation, 16–17 April. (in Greek)
  61. Patsiomitou, S. (2016b). Linking Visual Active Representations: Synthesis, implementation and evaluation of a “dynamic” curriculum based on dynamic transformations of mathematical objects with the utilization of interaction techniques. The New Educator (Neos Paidagogos), 7, 315–347. (in Greek)
  62. Patsiomitou, S. (2016c). Environment & computer environments: The role of games in the development of students’ competencies and their sense for a substantial school environment. In Proceedings of the 13th Panhellenic Conference: The Education in the era of ICT and innovation (pp. 967–994). (in Greek)
  63. Patsiomitou, S. (2018a). A dynamic active learning trajectory for the construction of number π: Transforming mathematics education. International Journal of Education and Research, 6(8), 225–248. http://www.ijern.com/journal/2018/August-2018/18.pdf
  64. Patsiomitou, S. (2018b). An ‘alive’ DGS tool for students’ cognitive development. International Journal of Progressive Sciences and Technologies (IJPSAT), 11(1), 35–54. http://ijpsat.ijsht-journals.org/index.php/ijpsat/article/view/636
  65. Patsiomitou, S. (2019). A trajectory for the teaching and learning of the didactics of mathematics [using ICT]: Linking Visual Active Representations. Monograph. United States: Global Journal Incorporated. ISBN 978-1-7340132-0-7. http://doi.org/10.34257/SPatTrajICT
  66. Patsiomitou, S. (2021a). Instrumental learning trajectories: The case of GeoGebra. Athens: Angelakis Publications. ISBN 978-960-616-193. https://www.academia.edu/79248541/ (in Greek)
  67. Patsiomitou, S. (2021b). Creativity and skills in mathematics. ISBN 978-618-00-3221-5. https://www.academia.edu/51047627/ (in Greek)
  68. Patsiomitou, S. (2022). Conceptual and instrumental trajectories using linking visual active representations created with the Geometer’s Sketchpad. Athens: Klidarithmos Publications. ISBN 978-960-645-302-1. (in Greek)
  69. Patsiomitou, S. (2023a). Developing and managing knowledge through the eyes of the young learner: ‘Alive’ manipulatives before abstract notions. International Journal of Scientific and Management Research, 6(3), 18–40. http://doi.org/10.37502/IJSMR.2023.6302
  70. Patsiomitou, S. (2023b). A brief review on my studies: Managing the complexity of using Linking Visual Active Representations (LVAR). International Journal of Scientific and Management Research, 6(5), 1–33. http://doi.org/10.37502/IJSMR.2023.6501
  71. Patsiomitou, S. (2024a). The influence of artificial intelligence and digital media on the evaluation of school units and the enhancement of educational quality. In Proceedings of the 3rd International Conference of Educational Assessment, II(8), 351–373. https://eletea.gr/el/τεύχος-8-τόμος-ιι-2024/ (in Greek)
  72. Patsiomitou, S. (2024b). Investigating artificial intelligence technologies in generating LVAR. International Journal of Research in Education Humanities and Commerce, 5(5), 194–217. https://doi.org/10.37602/IJREHC.2024.5515
  73. Patsiomitou, S. (2025). A proposal for a fractal-based “Dynamic” Program: The Pythagorean Tree structure generated through instrumental schemata. International Journal of Research in Education Humanities and Commerce, 6(2), 342–388
  74. Paulus, P. B., & Brown, V. R. (2007). Toward more creative and innovative group idea generation: A cognitive-social-motivational perspective of brainstorming. Social and Personality Psychology Compass, 1(1), 248–265.
  75. Peirce, C. S. (1992, c. 1878a). Deduction, induction, and hypothesis. In N. Houser & C. Kloesel, (Eds.), The Essential Peirce: Selected philosophical writings, 1, 186-199. Bloomington: Indiana University Press.
  76. Piaget, (1951). Play, Dreams and Imitation in Childhood. London: William Heinmann Ltd.
  77. Piaget, J. (1985). The equilibration of cognitive structures: The central problem of intellectual development (Original work published 1975). Chicago, IL: University of Chicago Press.
  78. Rotem, H., & Ayalon, M. (2022). Building a model for characterizing critical events: Noticing classroom situations using multiple dimensions. The Journal of Mathematical Behavior, 66, 100947
  79. Schmidt, H. G. (1983). Problem-based learning: Rationale and description. Medical Education, 17, 11–16.
  80. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26 (2), 114-145
  81. Simon, M.A. (1996) ‘Beyond inductive and deductive reasoning: the search for a sense of knowing’, Educational Studies in Mathematics, v.30, 197-210.
  82. Tall, D., Vinner, S. (1981), Concept image and concept definition in mathematics with particular reference to limits and continuity, Educational Studies in Mathematics, 12, 151-169
  83. Tilbury, D. (1995). Environmental education for sustainability: Defining the new focus of environmental education in the 1990s. Environmental Education Research, 1(2), 195–212.
  84. Tirosh, D., Tsamir, P., Levenson, E., Barkai, R. (2019) Using theories and research to analyze a case: Learning about example use. Journal of Mathematics Teacher Education 22, 205–225.
  85. Treisman, A. M., & Gelade, G. (1980). A feature-integration theory of attention. Cognitive Psychology, 12(1), 97–136. https://doi.org/10.1016/0010-0285(80)90005-5
  86. Van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Orlando, Florida: Academic Press.
  87. Van Hiele-Geldof, D. (1957/1984). The didactics of geometry in the lowest class of secondary school. In D. Fuys, D. Geddes, & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele (ERIC Document Reproduction Service No. ED 287 697). Original document in Dutch: De didakteik van de meetkunde in de eerste klas van het V.H.M.O. Unpublished thesis, University of Utrecht, 1957.
  88. Vygotsky, L.S. (1934/1962). Thought and language. MIT Press
  89. UNESCO (2017). Education for Sustainable Development Goals. Learning Objectives. Au-thor. https://unesdoc.unesco.org/ark:/48223/pf0000247444
  90. Wang, H. (2012). A new era of science education: science teachers’ perceptions and classroom practices of science, Technology Engineering, and Mathematics (STEM) Integration. Unpublished doctoral dissertation, University of Minnesota. https://hdl.handle.net/11299/120980