Skip to main content
SpringerLink
Log in

Transforming Images in a DGS: The Semiotic Potential of the Dragging Tool for Introducing the Notion of Conditional Statement

  • Chapter
  • First Online:
Transformation - A Fundamental Idea of Mathematics Education

Abstract

Research has shown that the tools provided by dynamic geometry systems (DGSs) impact on students’ approach to Euclidean Geometry and specifically on investigating open problems asking for producing conjectures. Building on the work of Arzarello, Olivero, and other researchers, the study addresses the use of specific dragging modalities in the solution of conjecture problems. Within the frame of the theory of semiotic mediation (TSM), the investigation aims at describing the semiotic potential of the dragging tool: how personal meanings emerging from students’ activities in a DGS can potentially be transformed into mathematical meanings. A theoretical discussion is presented, concerning the possible meanings, emerging in respect to the different dragging modalities, their relationship with mathematical meanings concerning conjectures, and conditional statements. Further, it is described how meanings emerge during different exploratory processes and how they may be related to the basic components of a conditional statement: premise, conclusion, and conditional link between them. Some examples discussed are drawn from a teaching experiment where participants were introduced to certain ways of dragging and then interviewed while working on open problem activities.

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
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
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.

    Someone who mediates, i.e., a mediator; something that is mediated, i.e., a content/force/energy released by mediation; someone/something subjected to mediation, i.e., the “mediatee” to whom/which mediation makes some difference; the circumstances for mediation, viz, (a) the means of mediation, i.e., modality; (b) the location, i.e., site in which mediation might occur. For a full discussion, see Hasan (2002).

  2. 2.

    Referring to the intentionality of the action, Baccaglini-Frank (2010) calls this kind of invariant the intentionally induced invariant. For the objective of this contribution, it is not necessary to introduce the terminology elaborated by Baccaglini-Frank.

References

  • Arsac, G., & Mante, M. (1983). Des “problème ouverts” dans nos classes du premier cycle. Petit x, 2, 5–33.

    Google Scholar 

  • Arzarello, F. (2000). Inside and outside: Spaces, times and language in proof production. Proceedings of PME 24: Psychology of Mathematics Education 24th International Conference, (1, pp. 23–38). Hiroshima, Japan.

    Google Scholar 

  • Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practices in dynamic geometry environments. ZDM The International Journal on Mathematical Education, 34(3), 66–72.

    Google Scholar 

  • Baccaglini-Frank, A. (2010). Conjecturing in dynamic geometry: A model for conjecture-generation through maintaining dragging. Doctoral dissertation, University of New Hampshire, Durham, NH.

    Google Scholar 

  • Baccaglini-Frank, A. (in press). The maintaining dragging scheme and the notion 2 instrumented abduction. Proceedings of the 10th Conference of the PME-NA, Columbus, OH.

    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., & Antonini, S. (2009). Different perceptions of invariants and generality of proof in dynamic geometry. In M. Tzekaki & H. Sakonidis (Eds.), Proceedings of PME 33: Psychology of mathematics education 33rd international conference (2, pp. 89–96). Thessaloniki: PME.

    Google Scholar 

  • Bartolini Bussi, M. G. 1998. Verbal interaction in mathematics classroom: a Vygotskian analysis. In H. Steinbring, M. G. Bartolini Bussi & A. Sierpinska (Eds.), Language and communication in mathematics classroom (pp. 65–84). Reston: NCTM.

    Google Scholar 

  • Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones & R. Lesh, D. Tirosh (Eds.), Handbook of international research in mathematics education, second revised edition (pp. 746–805). Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Battista, M. T. (2007). The development of geometric and spatial thinking. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843–908). Reston, VA: NCTM.

    Google Scholar 

  • Boero, P., Garuti, R., & Lemut, E. (2007). Approaching theorems in grade VIII: Some mental processes underlying producing and proving conjectures, and conditions suitable to enhance them. In P. Boero (Ed.), Theorems in schools: From history, epistemology and cognition to classroom practice (pp. 247–262). Rotterdam: Sense Publishers.

    Google Scholar 

  • Borba, M. C., & Villarreal, M. E. (2005). Humans-with-media and the reorganization of mathematical thinking: Information and communication technologies, modelling, visualization and experimentation. New York: Springer.

    Google Scholar 

  • Goldenberg, E. P. (1995). Rumination about dynamic imagery. In R. Sutherland & J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (pp. 203–223). New York: Springer.

    Google Scholar 

  • Hadas, N., Hershkowitz, R., Schwarz, B. B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44(1), 127–150.

    Article  Google Scholar 

  • Hasan, R. (2002). Semiotic mediation, language and society: three exotropic theories – Vygotsky, Halliday Bernstein. In J. Webster (Ed.) Language, society and consciousness: the collected works of Ruqaya Hasan (Vol. 1). Equinox, London. http://lchc.ucsd.edu/MCA/Paper/JuneJuly05/HasanVygHallBernst.pdf.Accessed 20 March 2008.

  • Healy, L. (2000). Identifying and explaining geometrical relationship: interactions with robust and soft Cabri constructions. Proceedings of PME 24: Psychology of Mathematics Education 24th International Conference (1, pp. 103–117), Hiroshima: Hiroshima University.

    Google Scholar 

  • Healy, L., Hoyles, C. (2001). Software tools for geometrical problem solving: Potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6(3), 235–256.

    Article  Google Scholar 

  • Hölzl, R. (1996). How does ‘dragging’ affect the learning of geometry. International Journal of Computers for Mathematical Learning, 1, 169–187.

    Article  Google Scholar 

  • Hölzl, R. (2001). Using DGS to add contrast to geometric situations: A case study. International Journal of Computers for Mathematical Learning, 6(1), 63–86.

    Article  Google Scholar 

  • Hoyles, C. (1993). Microworlds/schoolworlds: The transformation of an innovation. In C. Keitel & K. Ruthven (Eds.) Learning from computers: Mathematics education and technology. NATO ASI Series. New York: Springer-Verlag.

    Google Scholar 

  • Jackiw, N. (2009). The Geometer’s sketchpad (Version 5)(computer software). Berkeley, CA: Key Curriculum Press.

    Google Scholar 

  • Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software. Educational Studies in Mathematics, 44(1 & 2), 55–85.

    Article  Google Scholar 

  • Kozulin, A. (2003). Psychological tools and mediated learning. In A. Kozulin, B. Gindis, V. S. Ageyev & S. M. Miller (Eds.) Vygotsky’s educational theory in cultural context (pp. 15–38). Cambridge: Cambridge University Press.

    Chapter  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), 151–156.

    Article  Google Scholar 

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

    Google Scholar 

  • Laborde, J.-M., Bellemain, F. (1995). Cabri-géomètre II and Cabri-géomètre II plus (computer software). Dallas, USA: Texas Instruments and Grenoble. France: Cabrilog.

    Google Scholar 

  • Laborde, C., Laborde, J. M. (1991). Problem solving in geometry: From microworlds to intelligent computer environments. In Ponte et al. (Eds.), Mathematical problem solving and new information technologies (pp. 177–192). NATO AS1 Series F, 89.

    Google Scholar 

  • Laborde, J. M., & Sträßer, R. (1990). Cabri Géomètre, a microworld of geometry for guided discovery learning. Zentralblatt fuÌr Didaktik der Mathematik, 22(5), 171–177.

    Google Scholar 

  • Laborde, C., Kynigos, C., Hollebrands, K., Sträßer, R. (2006). Teaching and learning geometry with technology. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education. Past, present and future (pp. 275–304). Rotterdam: Sense.

    Google Scholar 

  • Leung, A., Lopez-Real, F. (2000). An analysis of students’ explorations and constructions using Cabri geometry. In M. A. Clements, H. Tairab & W. K. Yoong (Eds.) Science, mathematics and technical education in the 20th and 21st centuries (pp. 144–154). Brunei: Universiti Brunei Darussalam.

    Google Scholar 

  • Leung, A., 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, 145–165.

    Article  Google Scholar 

  • Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry. International Journal of Mathematical Education in Science and Technology, 37(6), 665–679.

    Article  Google Scholar 

  • Mariotti, M. A., Maracci, M. (2010). Un artefact comme outils de médiation sémiotique: une ressource pour l’enseignant. In G. Gueudet & L. Trouche. Ressources vives. Le travail documentaire des professeurs en mathématiques (pp. 91–107). Rennes: Presses Universitaires de Rennes et INRP.

    Google Scholar 

  • Mariotti, M. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44(1 and 2), 25–53.

    Article  Google Scholar 

  • Mariotti, M. A. (2001). Justifying and proving in the cabri environment. International Journal of Computers for Mathematical Learning, 6(3), 257–281.

    Article  Google Scholar 

  • Mariotti, M. A. (2007). Geometrical proof: the mediation of a microworld. In P. Boero (Ed.) Theorems in school: From history epistemology and cognition to classroom practice (pp. 285–304). Rotterdam: Sense Publishers.

    Google Scholar 

  • Mariotti, M. A. (2009). Artifacts and signs after a Vygotskian perspective: the role of the teacher. ZDM The International Journal on Mathematics Education, 41, 427–440.

    Article  Google Scholar 

  • Meira, L. (1998). Making sense of instructional devices: the emergence of transparency in mathematical activity. Journal for Research in Mathematics Education, 29(2), 121–142.

    Google Scholar 

  • Noss, R., Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers. Dordrecht: Kluwer Academic.

    Book  Google Scholar 

  • Olivero, F. (2001). Conjecturing in open geometric situations in a dynamic geometry environment: an exploratory classroom experiment. In C. Morgan & K. Jones (Eds.), Research in Mathematics Education (Vol. 3, pp. 229–246). London.

    Google Scholar 

  • Olivero, F. (2002). Proving within dynamic geometry environments, Doctoral Dissertation, Graduate School of Education, Bristol.

    Google Scholar 

  • Pedemonte, B. (2008). Argumentation and algebraic proof. ZDM The International Journal on Mathematical Education, 40, 385–400.

    Article  Google Scholar 

  • Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.

    Google Scholar 

  • Restrepo, A. M. (2008). Génèse instrumentale de deplacement en géométrie dinamyque shez des élèves de 6eme, Ph.D Thesis, Ecole doctorale des Mathématiques, Sciences et Technologies de l’Information, Informatique, Université Joseph Fourier, Grenoble, France.

    Google Scholar 

  • Silver, E. A. (1995). The nature and use of open problems in mathematics education: Mathematical and pedagogical perspectives. ZDM The International Journal on Mathematical Education, 27(2), 67–72.

    Google Scholar 

  • Sträßer, R. (2001). Cabri-geometre: Does dynamic geometry software (DGS) change geometry and its teaching and learning? International Journal of Computers for Mathematical Learning, 6(3), 319–333.

    Article  Google Scholar 

  • Talmon, V., Yerushalmy, M. (2004). Understanding dynamic behavior: Parent-child relations in dynamic geometry environments. Educational Studies in Mathematics, 57, 91–119.

    Article  Google Scholar 

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

    Google Scholar 

  • Vygotsky, L. S. (1981). The genesis of higher mental functions. In J. V. Wertsch (Ed.), The concept of activity in soviet psychology. Armonk, NY: Sharpe.

    Google Scholar 

  • Wertsch, J. V., Addison Stone, C. (1985). The concept of internalization in Vygotsky’s account of the genesis of higher mental functions. In J. V. Wertsch (Ed.), Culture, communication and cognition: Vygotskian perspectives (pp. 162–166). New York: Cambridge University Press.

    Google Scholar 

Download references

Acknowledgments

Special thanks to Anna Baccaglini-Frank for the thoughtful discussions that we had during the preparation of her dissertation and for sharing with me the rich set of data collected for her investigation.

Author information

Authors and Affiliations

  1. Department of Information Ingeneering and Mathematics, University of Siena, Via Roma, 56, 53100, Siena, Italy

    Prof. Maria Alessandra Mariotti

Authors
  1. Prof. Maria Alessandra Mariotti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Alessandra Mariotti .

Editor information

Editors and Affiliations

  1. IEIM - Institut für Mathematik, Universität Paderborn, Paderborn, Germany

    Sebastian Rezat

  2. Fakultät für Mathematik - IDM, Universität Bielefeld, Bielefeld, Germany

    Mathias Hattermann

  3. Fakultät für Mathematik - IDM, Universität Bielefeld, Bielefeld, Germany

    Andrea Peter-Koop

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Mariotti, M. (2014). Transforming Images in a DGS: The Semiotic Potential of the Dragging Tool for Introducing the Notion of Conditional Statement. In: Rezat, S., Hattermann, M., Peter-Koop, A. (eds) Transformation - A Fundamental Idea of Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3489-4_8

Download citation

Publish with us

Policies and ethics

Search

Navigation