Approaching Euclidean proofs through explorations with manipulative and digital artifacts

References

• Arıcı, S., & Aslan-Tutak, F. (2015). The effect of origami-based instruction on spatial visualization, geometry achievement, and geometric reasoning. International Journal of Science and Mathematics Education, 13(1), 179–200. https://doi.org/10.1007/s10763-013-9487-8
   Web of Science ®Google Scholar
• Arzarello, F., Bartolini Bussi, M. G., Leung, A. Y. L., Mariotti, M. A., & Stevenson, J. (2012). Experimental approaches to theoretical thinking: Artefacts and proofs. In G. Hanna & M. De Villiers (Eds.), Proof and proving in mathematics education (pp. 97–143). Springer Netherlands. https://doi.org/10.1007/978-94-007-2129-6_5
   Google Scholar
• Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. Zentralblatt für Didaktik der Mathematik, 34(3), 66–72. https://doi.org/10.1007/BF02655708
   Google Scholar
• 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. https://doi.org/10.1007/s10758-010-9169-3
   Google Scholar
• Battista, M. T. (2009). Highlights of research on learning school geometry. In T. V. Craine, & R. Rubenstein (Eds.), Understanding geometry for a changing world (pp. 91–108). NCTM.
   Google Scholar
• Boakes, N. J. (2006). The effects of Origami lessons on students’ spatial visualization skills and achievement levels in a seventh grade classroom [Doctoral dissertation]. Temple University.
   Google Scholar
• Boakes, N. J. (2009). Origami instruction in the Middle School Mathematics classroom: Its impact on spatial visualization and geometry knowledge of students. RMLE Online, 32(7), 1–12. https://doi.org/10.1080/19404476.2009.11462060
   Google Scholar
• Budinski, N., Lavicza, Z., & Fenyvesi, K. (2018). Ideas for using GeoGebra and origami in teaching regular polyhedrons lessons. K-12 STEM Education, 4(1), 297–303. https://doi.org/10.29333/iejme/6266
   Google Scholar
• De Villiers, M. (2003). The value of experimentation in mathematics. In S. Jaffer, & L. Burgess (Eds.), Proceedings of the ninth National Congress of the Association for Mathematics Education of South Africa: Teaching and learning mathematics in and for changing times (pp. 174–185). Association for Mathematics Education of South Africa.
   Google Scholar
• De Villiers, M. (2011). Simply symmetric. Simply symmetric. Learning and Teaching Mathematics, 11, 22–26.
   Google Scholar
• Dimmel, J. K., & Herbst, P. G. (2015). The semiotic structure of geometry diagrams: How textbook diagrams convey meaning. Journal for Research in Mathematics Education, 46(2), 147–195. https://doi.org/10.5951/jresematheduc.46.2.0147
   Web of Science ®Google Scholar
• Duval, R. (1994). Les différentes fonctinnemments d’une figure dans une démarche géométrique. Repères-IREM, 17, 121–138.
   Google Scholar
• Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland, & J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (pp. 142–157). Springer-Verlag.
   Google Scholar
• Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt (Ed.), 21st proceedings of the annual meeting of the north American chapter of the international group for the psychology of mathematics education (pp. 3–26). ERIC Clearinghouse.
   Google Scholar
• Fenyvesi, K., Budinski, N., & Lavicza, Z. (2014). Two solutions to an unsolvable problem: Connecting origami and geoGebra in a serbian high school. In G. Greenfield, G. W. Hart, & R. Sarhangi (Eds.), Proceedings of bridges 2014: Mathematics, music, art, architecture, culture (pp. 95–102). Tessellations Publishing.
   Google Scholar
• Font, V., Godino, J. D., & Contreras, Á. (2008). From representations to onto-semiotic configurations in analysing Mathematics teaching and learning processes. In L. Radfrod, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 157–173). Sense Publishers. https://doi.org/10.1163/9789087905972_010
   Google Scholar
• Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82(1), 97–124. https://doi.org/10.1007/s10649-012-9411-0
   Web of Science ®Google Scholar
• Font, V., Planas, N., & Godino, J. D. (2010). Modelo para el análisis didáctico en educación matemática 1. Infancia y Aprendizaje, 33(2), 1–18. https://doi.org/10.1174/021037010790317243
   Web of Science ®Google Scholar
• Friedman, M. (2018). A History of folding in mathematics. Springer International Publishing. https://doi.org/10.1007/978-3-319-72487-4
   Google Scholar
• Frigerio, E. (2002). In praise of the papercup: Mathematics and origami at the university. In T. C. Hull (Ed.), 3rd international meeting of origami science, math, and education (pp. 291–298). A K Peters.
   Google Scholar
• Fujita, T., & Jones, K. (2003). The place of experimental tasks in geometry teaching: learning from the textbook designs of the early 20th century. Research in Mathematics Education, 5(1), 47–62. https://doi.org/10.1080/14794800008520114
   Google Scholar
• Geretschlager, R. (1995). Euclidean constructions and the geometry of origami. Mathematics Magazine, 68(5), 357–371. https://doi.org/10.1080/0025570X.1995.11996356
   Google Scholar
• Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM, 39(1–2), 127–135. https://doi.org/10.1007/s11858-006-0004-1
   Google Scholar
• Godino, J. D., Gonzato, M., Cajaraville, J. A., & Fernández, T. (2012). Una aproximación ontosemiótica a la visualización en Educación Matemática. Enseñanza de Las Ciencias, 30(2), 109–130.
   Google Scholar
• Golan, M., & Jackson, P. (2009). Origametria: A program to teach geometry and to develop learning skills using the art of origami Origametria in the classroom lesson structure. In R. J. Lang (Ed.), Fourth international meeting of origami science, mathematics, and education (pp. 1–13). Taylor & Francis.
   Google Scholar
• Gürbüz, MÇ, Ağsu, M., & Güler, H. K. (2018). Investigating geometric habits of mind by using paper folding. Acta Didactica Napocensia, 11(3–4), 157–174. https://doi.org/10.24193/adn.11.3-4.12
   Google Scholar
• Haga, K. (2008). Origamics. Mathematical explorations through paper folding. World Scientific.
   Google Scholar
• Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education. III (pp. 234–283). American Mathematical Society. https://doi.org/10.1090/cbmath/007/07
   Google Scholar
• Healy, L. (2000). Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. In T. Nakahara, & M. Koyama (Eds.), Proceedings of the 24th conference of the international group for the psychology of mathematics education (Vol. 1, pp. 103–117). Hiroshima University.
   Google Scholar
• Herbst, P. G., Fujita, T., Halverscheid, S., & Weiss, M. (2017). The learning and teaching of geometry in secondary schools - a modeling perspective (1st ed.). Routledge.
   Google Scholar
• Hoffer, A. (1981). Geometry is more than proof. The Mathematics Teacher, 74(1), 11–18. https://doi.org/10.5951/MT.74.1.0011
   Google Scholar
• Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry contexts. In C. Mammana, & V. Villani (Eds.), Perspective on the teaching of geometry for the 21st century (pp. 121–128). Kluwer.
   Google Scholar
• Hsieh, F. J., Horng, W. S., & Shy, H. Y. (2012). From exploration to proof production. In G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 279–303). Springer. https://doi.org/10.1007/978-94-007-2129-6_12
   Google Scholar
• Hull, T. C. (2013). Project origami: Activities for exploring mathematics. A K Peters/CRC Press.
   Google Scholar
• Hull, T. C. (2020). Origametry: Mathematical methods in paper folding. Cambridge University Press.
   Google Scholar
• Huzita, H. (1989). Axiomatic development of origami geometry. In H. Huzita (Ed.), First International meeting of origami science and technology (pp. 143–158). Università di Padova.
   Google Scholar
• Komatsu, K., & Jones, K. (2020). Interplay between paper-and-pencil activity and dynamic-geometry-environment use during generalisation and proving. Digital Experiences in Mathematics Education, 6(2), 123–143. https://doi.org/10.1007/s40751-020-00067-3
   Google Scholar
• Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1/2), 151–161. https://doi.org/10.1023/A:1012793121648
   Google Scholar
• Laborde, C. (2005). Robust and soft constructions: Two sides of the use of dynamic geometry environments. In S. C. Chu, H. C. Lew, & W. C. Yang (Eds.), Proceedings of the tenth Asian technology conference in mathematics (pp. 22–35). Blacksburg.
   Google Scholar
• Lam, T. K., & Pope, S. (2016). Learning mathematics with origami – a mathematics education resource. Association of Teachers of Mathematics.
   Google Scholar
• Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry environments. International Journal of Mathematical Education in Science and Technology, 37(6), 665–679. https://doi.org/10.1080/00207390600712539
   Google Scholar
• Mariotti, M. A. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44(1–3), 25–53. https://doi.org/10.1023/A:1012733122556
   Google Scholar
• Mariotti, M. A. (2006). Proof and Proving in Mathematics education. In A. Gutiérrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 173–204). Brill | Sense. https://doi.org/10.1163/9789087901127_008
   Google Scholar
• Marrades, R., & Gutiérrez, Á. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1/2), 87–125. https://doi.org/10.1023/A:1012785106627
   Google Scholar
• 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. https://doi.org/10.1007/s10758-007-9115-1
   Google Scholar
• Pedemonte, B. (2002). Relation between argumentation and proof in mathematics: Cognitive unity or break? In J. Novotná (Ed.), Proceedings of the 2nd conference of the European Society for research in mathematics education (pp. 70–80). Charles University. https://doi.org/10.1207/S15327884MCA0803
   Google Scholar
• Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23–41. https://doi.org/10.1007/s10649-006-9057-x
   Google Scholar
• Polya, G. (1954). Mathematics and plausible reasoning. Princeton University Press.
   Google Scholar
• Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
   Google Scholar
• Senk, S. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448–456. https://doi.org/10.5951/MT.78.6.0448
   Google Scholar
• Sinclair, N., Pimm, D., & Skelin, M. (2012). Developing essential understanding of geometry in grades 9-12. National Council of Teachers of Mathematics.
   Google Scholar
• Soldano, C., Luz, Y., Arzarello, F., & Yerushalmy, M. (2019). Technology-based inquiry in geometry: Semantic games through the lens of variation. Educational Studies in Mathematics, 100(1), 7–23. https://doi.org/10.1007/s10649-018-9841-4
   Web of Science ®Google Scholar
• Stylianides, G. (2008). An analytic framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 9–16.
   Google Scholar
• Tomaschko, M., Arkün Selay, K., & Hohenwarter, M. (2018). Opportunities for participation, productivity, and personalization through GeoGebra Mathematics apps. In A. A. Khan, & S. Umair (Eds.), Handbook of research on mobile devices and smart gadgets in K-12 education (pp. 45–56). IGI Global. https://doi.org/10.4018/978-1-5225-2706-0.ch004
   Google Scholar
• Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. CDASSG project. University of Chicago.
   Google Scholar
• Wares, A. (2011). Using origami boxes to explore concepts of geometry and calculus. International Journal of Mathematical Education in Science and Technology, 42(2), 264–272. https://doi.org/10.1080/0020739X.2010.519797
   Google Scholar
• Wiles, P. (2013). Folding corners of the habits of mind. Mathematics Teaching in the Middle School, 19(4), 208–213. https://doi.org/10.5951/mathteacmiddscho.19.4.0208
   Google Scholar
• Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I., & Landman, G. W. (2012). The need for Proof and Proving: Mathematical and pedagogical perspectives. In G. Hanna, & M. De Villiers (Eds.), Proof and Proving in Mathematics education (pp. 215–229). Springer. https://doi.org/10.1007/978-94-007-2129-6
   Google Scholar