Skip to main content
SpringerLink
Log in

Generating Conjectures in Dynamic Geometry: The Maintaining Dragging Model

  • Published:
International Journal of Computers for Mathematical Learning Aims and scope Submit manuscript

Abstract

Research has shown that the tools provided by dynamic geometry systems (DGSs) impact students’ approach to investigating open problems in Euclidean geometry. We particularly focus on cognitive processes that might be induced by certain ways of dragging in Cabri. Building on the work of Arzarello, Olivero and other researchers, we have conceived a model describing some cognitive processes that can occur during the production of conjectures in dynamic geometry and that seem to be related to the use of specific dragging modalities. While describing such cognitive processes, our model introduces key elements and describes how these are developed during the exploratory phase and how they evolve into the basic components of the statement of the conjecture (premise, conclusion, and conditional link between them). In this paper we present our model and use it to analyze students’ explorations of open problems. The description of the model and the data presented are part of a more general qualitative study aimed at investigating cognitive processes during conjecture-generation in a DGS, in relation to specific dragging modalities. During the study the 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 article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Notes

  1. We note that a base point is a free point (or a semi-free point if it is linked to an object) that therefore can be dragged anywhere on the screen (or along the object it is linked to).

  2. Although during the introductory lessons we did not explicitly specify particular points to activate the trace on, we only proposed to activate it on the base point selected to be dragged.

  3. It can be useful to make a distinction between construction-invariants that are described in the steps of the step-by-step construction, as opposed to ones that are consequences of such steps.

  4. Notice how this choice of the radius makes the constructed circle a D-invariant, (Baccaglini-Frank et al. 2009).

  5. Notice the transition from “D is dragged along the circle” to this crystallized form.

  6. Before the problem analyzed in the excerpts, Fra and Gia had (during the introductory lessons and the preceding part of their interview) explored constructions in which maintaining dragging had led to lines (or segments), circles (or arcs of circles), and a parabola.

  7. It is beyond the scope of this paper (and of the general study) to describe the processes of instrumental genesis of the maintaining dragging. However the analysis of solvers’ behaviors during successive activities, highlighted some elements that may characterize a process of instrumental genesis. This will be the object of future studies we plan to carry out.

  8. The extent to which the path is associated to a curve or a locus seems to depend on the mathematical expertise of the solver. For a mathematician it will unconsciously be considered a locus (or a subset of a locus), but this will not necessarily be the case for a student.

Abbreviations

CL:

Conditional link

DGS:

Dynamic geometry system

GDP:

Geometric description of the path

III:

Intentionally induced invariant

IOD:

Invariant observed during dragging

MD:

Maintaining dragging

References

  • Arsac, G. (1999). Variations et variables de la démostration géométriques. Recherches en Didactique de Mathématiques, 19(3), 357–390.

    Google Scholar 

  • Arzarello, F. (2000). Inside and outside: Spaces, times and language in proof production. In Proceedings of PME XXIV, (Vol. 1, pp. 23–38), Hiroshima, Japan.

  • Arzarello, F. (2001). Dragging, perceiving and measuring: Physical practices and theoretical exactness in Cabri-environments. In Proceedings of CabriWorld 2, Montreal.

  • Arzarello, F., Micheletti, C., Olivero, F., Paola, D., & Robutti, O. (1998). A model for analysing the transition to formal proofs in geometry. In A. Olivier & K. Newstead (Eds.), In Proceedings of the 22nd PME, (Vol. 2, pp. 24–31), Stellenbosch, South Africa.

  • Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practices in cabri environments. ZDM, 34(3), 66–72.

    Article  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.

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

  • Baccaglini-Frank, A., Mariotti, M. A., & Antonini, S. (2009). Different perceptions of invariants and generality of proof in dynamic geometry. In Tzekaki, M., & Sakonidis, H. (Eds.), In Proceedings of the 33rd conference of the IGPME, (Vol. 2, pp. 89–96), Thessaloniki, Greece.

  • Bartolini Bussi, M. G. (1998). Joint activity in mathematics classrooms: a Vygotskian analysis. In F. Seeger, J. Voigt, & U. Waschescio (Eds.), The culture of the mathematics classroom (pp. 13–49). Cambridge: Cambridge University Press.

    Google Scholar 

  • Boero, P., Garuti, R., & Lemut, E. (1999). About the generation of conditionality. In Proceedings of the 23rd meeting of the IGPME, (Vol. 2, pp. 137–144). Haifa.

  • Boero, P., Garuti, R., & Lemut, E. (2007). Approaching theorems in grade VIII. In P. Boero (Ed.), Theorems in school: From history epistemology and cognition to classroom practice (pp. 249–264). Rotterdam: Sense Publishers.

    Google Scholar 

  • Boero, P., Garuti, R., Lemut, E., & Mariotti, M. A. (1996b). Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems. In P. Luis, & A. Gutierrez (Eds.), Proceedings of 20th conference of the IGPME, (Vol. 1, pp. 113–120), Valencia, Spain.

  • Boero. P., Garuti, R., & Mariotti, M. A. (1996a). Some dynamic mental processes underlying producing and proving conjectures. In P. Luis, & A Gutierrez (Eds.), Proceedings of 20th conference of the IGPME, (Vol. 2, pp. 121–128), Valencia, Spain.

  • diSessa, A. (2007). An interactional analysis of clinical interviewing. Cognition and Instruction, 25(4), 523–565

    Article  Google Scholar 

  • Ginsburg, H. (1981). The clinical interview in psychological research on mathematical thinking: Aims, rationales, techniques. For the Learning of Mathematics, 1(3), 4–11.

    Google Scholar 

  • Gousseau-Coutat, S. (2006). Intégration de la géométrie dinamique dans l’enseignement de la géométrie pour favoriser la liaison école primaire collège : une ingégnierie didactique au college sur la notion de propriété. Ph.D Thesis, Ecole doctorale des Mathématiques, Sciences et Technologies de l’Information, Informatique, Université Joseph Fourier, Grenoble, France.

  • Healy, L. (2000). Identifying and explaining geometric relationship: interactions with robust and soft Cabri constructions. In Proceedings of the 24th conference of the IGPME, (Vol. 1, pp. 103–117) Hiroshima, Japan.

  • Hunting, R. P. (1997). Clinical interview methods in mathematics education research and practice. Journal of Mathematical Behavior, 16(2), 145–165.

    Article  Google Scholar 

  • Laborde, C. (2001). Integration of technology in the design of geometry tasks with cabri-geometry. International Journal of Computers for Mathematical Learning, 6, 283–317.

    Article  Google Scholar 

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

  • 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.

  • Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13(2), 135–157.

    Article  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 

  • Leung, A., Chan, Y. C., & Lopez-Real, F. (2006). Instrumental genesis in dynamic geometry environments. In L. H. Son, N. Sinclair, J. B. Lagrange, & C. Hoyles (Eds.), Proceedings of the ICMI 17 study conference: part 2 (pp. 346–353). Vietnam: Hanoi University of Technology.

  • 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. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics. Special issue, 44, 25–53.

    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). Rotterdam, The Netherlands: Sense Publishers. ISBN 9077874194.

    Google Scholar 

  • Mariotti, M. A., Bartolini Bussi, M. G., Boero, P., Ferri, F., & Garuti, R. (1997). Approaching geometry theorems in contexts: from history and epistemology to cognition. In Pehkonen, E. (Ed.), Proceedings of 21st conference of the IGPME, (Vol. 1, pp. 180–195). Lahti, Finland.

  • Mogetta, C., Olivero, F., & Jones, K. (1999). Providing the motivation to prove in a dynamic geometry environment. In Proceedings of the British society for research into learning mathematics, St Martin’s University College, Lancaster, pp. 91–96.

  • 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.

  • Olivero, F. (2002). Proving within dynamic geometry environments, Ph. D. Thesis, Graduate School of Education, Bristol.

  • 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, 135–156.

    Article  Google Scholar 

  • Rabardel, P. (2002). People and technology: A cognitive approach to contemporary instruments. Retrieved online June, 16th 2009 from: http://ergoserv.psy.univ-paris8.fr.

  • Rabardel, P., & Samurçay, R., (2001). Artifact mediation in learning. New challenges to research on learning. In International symposium organized by the center for activity theory and developmental work research, University of Helsinki, March 21–23.

  • Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof? Educational Studies in Mathematics, 52, 319–325.

    Article  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.

  • Schoenfeld, A. (2000). Purposes and methods of research in mathematics education. Notices of the AMS, 641–649.

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

    Google Scholar 

  • Strässer, R. (2009). Instruments for learning and teaching mathematics an attempt to theorize about the role of textbooks, computers and other artefacts to teach and learn mathematics. In Tzekaki, M., & Sakonidis, H. (Eds.), Proceedings of the 33rd PME (Vol. 1, pp. 67–81). Thessaloniki, Greece.

  • 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 

  • Vérillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1), 77–101.

    Article  Google Scholar 

Download references

Acknowledgments

The study this article reports on was carried out within the research project PRIN 2007B2M4EK (Instruments and representations in the teaching and learning of mathematics: theory and practice)—Università di Siena & Università di Modena e Reggio Emilia (Italy), with additional support from a dissertation fellowship from the University of New Hampshire (USA). We wish to thank the anonymous reviewers for their very helpful comments, questions, and suggestions on a previously submitted draft.

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Informatiche e Matematiche, Università di Siena, Piano dei Mantellini, 44, 53100, Siena, Italy

    Anna Baccaglini-Frank & Maria Alessandra Mariotti

Authors
  1. Anna Baccaglini-Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maria Alessandra Mariotti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Alessandra Mariotti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baccaglini-Frank, A., Mariotti, M.A. Generating Conjectures in Dynamic Geometry: The Maintaining Dragging Model. Int J Comput Math Learning 15, 225–253 (2010). https://doi.org/10.1007/s10758-010-9169-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10758-010-9169-3

Keywords

Search

Navigation