Skip to main content
SpringerLink
Log in

Developing the Mathematical Eye Through Problem-Solving in a Dynamic Geometry Environment

  • Chapter
  • First Online:
Book cover Broadening the Scope of Research on Mathematical Problem Solving

Part of the book series: Research in Mathematics Education ((RME))

Abstract

The aim of this chapter is that of discussing some aspects characterizing the processes of problem solving in Euclidean geometry. After defining specific visualization skills actively involved in solving geometrical problems, we illustrate how those are differently involved in problem-solving processes; the comparison between solving processes that take place with paper and pen or in a dynamic geometry environment shows that a dynamic geometry environment can provide effective support for mobilizing but also for developing visualization skills. Therefore, we claim that problem-solving activities can be designed with the aim of fostering the student’s development of specific visualization skills and contributing to the development of a mathematical eye.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This construct has also been used in a recent study by Miragliotta, Baccaglini-Frank and Tomasi (2017) and is being used in the doctoral work of Miaragliotta (Miragliotta and Baccaglini-Frank 2017, 2018).

  2. 2.

    The process is described in further detail by Baccaglini-Frank and Mariotti (Baccaglini-Frank, 2010a, 2010b; Baccaglini-Frank & Mariotti, 2011).

  3. 3.

    Adapted from Baccaglini-Frank and Mariotti (2010, pp. 238, 240, 241) with permission from International Journal of Computers for Mathematical Learning, copyright 2010, by Springer

References

  • Arcavi, A., & Hadas, N. (2002). Computer mediated learning: An example of an approach. In F. Hitt (Ed.), Representations and mathematical visualization. Mexico D.F.: Cinvestav – IPN.

    Google Scholar 

  • Baccaglini-Frank, A. (2010a). Conjecturing in Dynamic Geometry: A Model for Conjecture-generation through Maintaining Dragging. Doctoral dissertation, University of New Hampshire, Durham, NH, USA.

    Google Scholar 

  • Baccaglini-Frank, A. (2010b). The maintaining dragging scheme and the notion of instrumented abduction. In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd annual meeting of the PMENA (Vol. VI, pp. 607–615). Columbus, OH: The Ohio State University.

    Google Scholar 

  • Baccaglini-Frank, A., & Antonini, S. (2016). From conjecture generation by maintaining dragging to proof. In C. Csíkos, A. Rausch, & J. Szitányi (Eds.), Proceedings of the 40th conference of the International Group for the Psychology of mathematics education (Vol. 2, pp. 43–50). Szeged, Hungary: PME.

    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.

    Article  Google Scholar 

  • Baccaglini-Frank, A., & Mariotti, M. A. (2011). Conjecture-generation through dragging and abduction in dynamic geometry. In A. Méndez-Vilas (Ed.), Education in a technological world: Communicating current and emerging research and technological efforts (pp. 100–107). Badajoz, Spain: Formatex.

    Google Scholar 

  • Boero, P., Garuti, R., & Lemut, E. (1999). About the generation of conditionality. In O. Zaslavsky (Ed.), Proceedings of the 23rd conference of the International Group for the Psychology of mathematics education (Vol. 2, pp. 137–144). Haifa, Israel: Technion.

    Google Scholar 

  • Boero, P., Garuti, R., & Mariotti, M. A. (1996). Some dynamic mental process underlying producing and proving conjectures. In L. Puig (Ed.), Proceedings of 20th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 121–128). Valencia, Spain: Universitat de València.

    Google Scholar 

  • Bishop, A. J. (1980). Spatial abilities and mathematics education: A review. Educational Studies in Mathematics, 11, 257–269.

    Article  Google Scholar 

  • Bishop, A. J. (1983). Space and geometry. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 175–203). New York, USA: Academic Press.

    Google Scholar 

  • Bishop, A. (1989). Review of research in visualization in mathematics education. Focus on Learning Problems in Mathematics, 11(1), 7–16.

    Google Scholar 

  • Cornoldi, C., & Vecchi, T. (2004). Visuo-spatial working memory and individual differences. Hove, UK: Psychology Press.

    Book  Google Scholar 

  • Duval, R. (1994). Les différents fonctionnements d'une figure dans une démarche géométrique. Repères-IREM, 17, 121–138.

    Google Scholar 

  • Duval, R. (1995a). Geometrical pictures: Kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (Vol. 138 of the series NATO ASI Series, pp. 142–157). Berlin, Heidelberg: Springer.

    Chapter  Google Scholar 

  • Duval, R. (1995b). Sémiosis et noésis. Berne, Switzerland: Peter Lang.

    Google Scholar 

  • Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 37–52). Dordrecht, The Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139–162.

    Article  Google Scholar 

  • Hadamard, J. (1944). The psychology of mathematical invention. New York: Dover Publications.

    Google Scholar 

  • Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago, USA: University of Chicago Press.

    Google Scholar 

  • Laborde, C., & Laborde, J. M. (1992). Problem solving in geometry: From microworlds to intelligent computer environments. In J. P. Ponte, J. F. Matos, J. M. Matos, & D. Fernandes (Eds.), Mathematical problem solving and new information technologies (pp. 177–192). Berlin, Germany: Springer.

    Chapter  Google Scholar 

  • Laborde, J. M., & Straesser, R. (1990). Cabri-Géomètre: A microworld of geometry for guided discovery learning. Zentralblatt für Didaktik der Mathematik, 22(5), 171–177.

    Google Scholar 

  • Laborde, C. (2005). Proceedings of the 10th Asian technology conference in mathematics. In Robust and soft constructions (pp. 22–35). Korea: National University of Education.

    Google Scholar 

  • Leung, A., Baccaglini-Frank, A., & Mariotti, M. A. (2013). Discernment in dynamic geometry environments. Educational Studies in Mathematics, 84(3), 439–460.

    Article  Google Scholar 

  • Mariotti, M. A. (2014). Transforming images in a DGS: The semiotic potential of the dragging tool for introducing the notion of conditional statement. In S. Rezat et al. (Eds.), Transformation – a fundamental idea of mathematics education (pp. 155–172). New York: Springer. https://doi.org/10.1007/978-1-4614-3489-4_8

    Chapter  Google Scholar 

  • Miragliotta, E., Baccaglini-Frank, A., & Tomasi, L. (2017). Apprendimento della geometria e abilità visuo-spaziali: un possibile quadro teorico e un’esperienza didattica. L’Insegnamento della Matematica e delle Scienze Integrate, 40B(3), 339–360.

    Google Scholar 

  • Mariotti, M. A. (1995). In R. Sutherland & J. Mason Eds (Eds.), Images and concepts in geometrical reasoning. Exploiting mental imagery with computers in mathematics education (pp. 97–116).

    Chapter  Google Scholar 

  • Miragliotta, E., & Baccaglini-Frank, A. (2017). Visuo-spatial abilities and geometry: a first proposal of a theoretical framework for interpreting processes of visualization. In T. Dooley & G. Gueudet (Eds.), Proceedings of the tenth congress of the European society for research in mathematics education (CERME10, February 1–5, 2017) (pp. 3952–3959). Dublin, Ireland: DCU Institute of Education and ERME.

    Google Scholar 

  • Miragliotta, E., & Baccaglini-Frank, A. (2018). You see (only) what you predict: the power of geometric prediction. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd conference of the international group for the psychology of mathematics education (Vol. 3, pp. 387–394). Umeå, Sweden: PME.

    Google Scholar 

  • Neisser, U. (2014). Cognitive psychology. Psychology press classical editions. New York: Taylor and Francis.

    Book  Google Scholar 

  • Piaget, J., & Inhelder, B. (1966). L’image mentale chez l’enfant. Presses Universitaires de France.

    Google Scholar 

  • Pólya, G. (1957). How to Solve It. Garden City, NY: Doubleday.

    Google Scholar 

  • Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205–235). Rotterdam: Sense Publishers.

    Google Scholar 

  • Presmeg, N. C. (1986). Visualization in high school mathematics. For the Learning of Mathematics, 6(3), 42–46.

    Google Scholar 

  • Presmeg, N. C. (1997). Reasoning with metaphors and metonymies in mathematics learning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp. 267–279). Mahwah, NJ, USA: Lawrence Erlbaum.

    Google Scholar 

  • Vygotski, L. S. (1978). Mind in society. The development of higher psychological processes. Cambridge, MA: Harvard University Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Università di Siena, Siena, Italy

    Maria Alessandra Mariotti

  2. Università di Pisa, Pisa, Italy

    Anna Baccaglini-Frank

Authors
  1. Maria Alessandra Mariotti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Baccaglini-Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Baccaglini-Frank .

Editor information

Editors and Affiliations

  1. Universidade do Algarve and UIDEF Instituto de Educação, Universidade de Lisboa, Lisbon, Portugal

    Nélia Amado

  2. Universidade do Algarve and UIDEF Instituto de Educação, Universidade de Lisboa, Lisbon, Portugal

    Susana Carreira

  3. School of Education, University of Southampton, Southampton, UK

    Keith Jones

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mariotti, M.A., Baccaglini-Frank, A. (2018). Developing the Mathematical Eye Through Problem-Solving in a Dynamic Geometry Environment. In: Amado, N., Carreira, S., Jones, K. (eds) Broadening the Scope of Research on Mathematical Problem Solving. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-99861-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-99861-9_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-99860-2

  • Online ISBN: 978-3-319-99861-9

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation