An ‘Alive’ DGS Tool for Students’ Cognitive Development

Stavroula Patsiomitou


Abstract— A basic goal of the current study, which is an excerpt from a larger study, is to analyse students’ interactions in the context of their working on transformations of tools, and specifically of custom tools in a microworld, the Geometer’s Sketchpad. Custom tools are encapsulated objects created in a DGS environment. The construction of a custom tool and its subsequent implementation in a pair of students are the focus of this study. Custom tools can serve as structural units of knowledge, as conceptual objects and hence as ‘schemes’. Moreover, they can become an ‘alive’ active tool for students’ cognitive development. The paper will include the following parts: (a) how students learn in a constructivist framework; (b) a description of the van Hiele model, and especially the meanings of ‘symbol and signal character’; (c) how a DGS environment functions as an ‘alive’ microworld; (d) the role of artifacts-[custom] tools as instruments-[custom] tools; (e) the research methodology of the current study (f) a detailed description of the experimental process (g) discussion and conclusion.

Full Text:




Artigue, M. (2000). Instrumentation issues and the integration of computer technologies into secondary mathematics teaching. Proceedings of the Annual Meeting of the GDM. Potsdam, 2000: available on line at

Balacheff, N. Kaput, J. (1997) Computer-based learning environment in mathematics. Alan Bischop. International Handbook of Mathematics Education, Kluwer Academic publisher, pp.469501, 1997.

Bayram, S. (2004). The effect of instruction with concrete models on eighth grade students’ geometry achievement and attitudes toward geometry. Master of Science in Secondary Science and Mathematics Education. Middle East Technical University.

Battista, M. T. (2007).The development of geometric and spatial thinking.In Lester, F. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 843-908). NCTM. Reston, VA: National Council of Teachers of Mathematics.

Beguin, P. & Rabardel, P. (2000). “Designing for Instrument Mediated Activity”, Scandinavian Journal of Information Systems 12(1-2), 173–90. Special Issue: Information Technology in Human Activity.

Bodgan, R. C., & Biklen, S. K. (1998). Qualitative Research in Education. Boston London: Allyn and Bacon, Inc.

Bransford, J., Brown, A., & Cocking, R. (2000). How People Learn. Washington, DC: National Research Council.

Bruner, J.S. (1961). The act of discovery. Harvard Educational Review, 31(1), 2132.

Bruner, J. S. (1966). Towards a Theory of Instruction, New York: Norton.

Bruner, J.S., Olver, R.R., & Greenfield, P.M. (1966). Growth. New York: John Wiley & Sons.

Burger, W. F., & Shaughnessy, J. M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17, 31-48.

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.

Choi-Koh Sang Sook (2001). A student’s learning of geometry using the computer, Journal of Educational Research, pp. 301–311

Christou, C., Mousoulides, N., Pittalis, M., & Pitta-Pantazi, D. (2005). Problem solving and problem posing in a dynamic geometry environment. The Montana Mathematics Enthousiast, 2(2), 125-143.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464). NewYork: Macmillan.

Confrey, J. (1995). How compatible are radical constructivism, sociocultural approaches, and social constructivism? In L. Steffe & J. Gale (Eds.), Constructivism in education (pp. 185-225). Hillsdale, NJ: Lawrence Erlbaum Associates.

Coxford, A. F., Usiskin Z. P.(1975). Geometry: A Transformation Approach, Laidlaw Brothers, Publishers.

Crowley, M. (1987). The van Hiele model of development of geometric thought. In M. M. Lindquist, (Ed.), Learning and teaching geometry, K-12 (pp.1-16). Reston, VA: NCTM.

Davis, G. E., & Tall, D. O. (2002). What is a scheme? In D. Tall & M. Thomas (Eds.), Intelligence, learning and understanding: A tribute to Richard Skemp (pp. 141-160). Flaxton, Queensland: Post Pressed

Dewey, J. (1938/1988). Experience and education. In J. A. Boydston (Ed.), John Dewey: The later works, 1925-1953 (Vol. 13, pp. 1-62). Carbondale, IL: Southern Illinois University Press. (Reprinted from: 1997).

Dörfler, W. (1991). Der Computer als kognitives Werkzeug und kognitives Medium. Computer - Mensch - Mathematik.Dörfler W. et al. (Eds.) Wien, Stuttgart, Hölder-Pichler-Tempsky; B.G.Teubner: 51-76.

Dubinsky, E. (1988). On Helping Students Construct the Concept of Quantification. 12th International Conference Psychology of Mathematics Education. Veszprem. vol. I:255-262.

Duval, R. (1995). Sémiosis et pensée humaine. Berne: Peter Lang

Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Volume 1 (pp. 3-26). Cuernavaca, Morelos, Mexico.

Drijvers, P., & Trouche, L. (2008). From artefacts to instruments: A theoretical framework behind the orchestra metaphor. In G. W. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics (Cases and perspectives, Vol. 2, pp. 363–392). Charlotte: Information Age.

Edelman, G. M. (1989, 1992). Neural Darwinism: The Theory of Neuronal Group Selection. New York: Basic Books

Eisenhardt, M. K. (2002). Building theories from case study research. In A. Huberman & M. Miles (Eds.), The qualitative researcher's companion (pp. 5-36). Thousand Oaks: Sage Publications

Ernest, P. Ed. (1994) Constructing Mathematical Knowledge: Epistemology and Mathematics Education, London: Routledge.

Edwards, L. (1998). Embodying mathematics and science: Microworls as representations. Journal of Mathematics Behavior, 17(1), 53-78.

El-Demerdash, M.E-S. A. (2010). The effectiveness of an enrichment program using dynamic geometry software in developing mathematically gifted students’ geometric creativity in high schools. PhD thesis. University of Education Schwabisch Gmund

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

Fosnot, C. T. (2003) Teaching and Learning in the 21st Century Plenary Address, AMESA Conference, Capetown, South Africa, June, 2003

Frick, R. W. (1989). Explanations of grouping in immediate ordered recall. Memory and Cognition, 17: 551-562.

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).

Fuys, D., Geddes, D., & Tischler, R. (1988). The Van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education: Monograph Number 3.

Gawlick, T. (2005). Connecting arguments to actions –Dynamic geometry as means for the attainment of higher van Hiele levels. Zentralblatt für Didaktik der Mathematik, 37(5), 361-370.

Von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London: Falmer Press.

Glaser, BG., Strauss, Al. (1967) . The discovery of grounded theory: Strategies for qualitative research New York: Aldine de Gruyter

Gobet, F., Lane, P., Croker, S., Cheng, P., Jones, G., Oliver, I. and Pine, J. (2001). Chunking mechanisms in human learning. Trends in Cognitive Sciences 5(6): 236–243.

Goldenberg, P., & Cuoco, A. (1996). What is dynamic geometry? In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 351– 368). Mahwah, NJ: Lawrence Erlbaum Associates

Goldin, G., & Janvier, C. (1998). Representations and the psychology of mathematics education. Journal of Mathematical Behavior, 17(1), 1-4.

Govender, R., & de Villiers, M. (2002). Constructive evaluation of definitions in a Sketchpad context. Paper presented at the meeting of the Association for Mathematics Education of South Africa, Durban, South Africa.

Govender, R. & De Villiers, M. (2004) “A dynamic approach to quadrilateral definitions.” Pythagoras 58, pp.34-45.

Haj-Yahya, A., & Hershkowitz, R. (2013). When visual and verbal representations meet the case of geometrical figures. Proceedings of PME 37, 2, pp.409–416.

Hayes, R., & Oppenheim, R. (1997). Constructivism: Reality is what you make it. In T. Sexton & B. Griffin (Eds.), Constructivist thinking in counseling practice, research and training (pp. 19-41). New York: Teachers College Press.

Hershkovitz, R. (1990). ‘Psychological aspects of learning geometry’, in P. Nesher and J. Kilpatrick (eds), Mathematics and Cognition, Cambridge University Press, Cambridge

Hohenwarter, M. (2001). GeoGebra. Online at:

Hollebrands, K. F. (2003). High school students’ understanding of geometric transformations in the context of a technological environment. Journal of Mathematical Behavior 22, 55-72.

Jackiw, N. (1991/2001). The Geometer's Sketchpad [Computer Software].Berkeley, CA: Key Curriculum Press

Jackiw, N. and Sinclair, N. (2009). Sounds and pictures: dynamism and dualism in dynamic geometry. ZDM, 41, 413–426.

Janvier, C. (1987). Problems of Representations in the Learning and Teaching of Mathematics. New Jersey: Lawrence Erlbaum Associates.

Kadunz, G. (2001) Modularity and Geometry In: G. Törner, R. Bruder, N. Neill, A. Peter-Koop, B. Wollring (Hrsg.): Developments in Mathematics Education in German speaking Countries. Selected Papers from the Annual Conference on Didactics of Mathematics. Hildesheim: Franzbecker 2004, pp. 63-72.

Kadunz, G. (2002). Macros and modules in geometry. ZDM, 34(3), 73-77.

Kant, I. (1965). The Critique of Pure Reason, translated by N. Kemp Smith, Macmillan, London.

Karmiloff-Smith, A., & Inhelder, B. (1974). If you want to get ahead, get a theory. Cognition, 3(3), 195-212.

Felix Klein (1948). Elementary Mathematics from an Advanced Standpoint: Geometry; Dover reprint of a translation.

Kress, G. (1994). Learning to Write (2nd edition), Routledge, London.

Kortenkamp, U. (2004). Experimental mathematics and proofs in the classroom. Zentralblatt fur Didaktik der Mathematik, 36(2), 61–66.

Laborde, C. (1993). The Computer as Part of the Learning Environment: The Case of Geometry, in C. Keitel & K. Ruthven (Eds.) Learning from Computers: Mathematics Education and Technology (pp. 48-67). Berlin: Springer-Verla

Laborde, J. M. (2004). Cabri 3D. Online at:

Laborde, J-M., Baulac, Y., & Bellemain, F. (1988) Cabri Géomètre [Software]. Grenoble, France: IMAG-CNRS, Universite Joseph Fourier

Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education, 27, 133-150.

Lin Fou-Lai, Yang Kai-Lin (2002), Defining a Rectangle under a Social and Practical Setting by Two Seventh Graders. Zentralblatt für Didaktik der Mathematik, 34 (1). pp. 17-28.

Lionni, L. (1970) Fish Is Fish. New York: Scholastic Press.

Littlefield Cook, Joan and Cook, Greg (2005) Child development: principles and perspectives, Boston, Mass: Pearson

Mason, M. M. (1998). The van Hiele levels of geometric understanding. Retrieved from

Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63(2), 81-97.

Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A new framework for teacher knowledge. Teachers College Record, 108(6), 1017–1054.

Monaghan, F. (2000) What difference does it make? Children views of the difference between some quadrilaterals, Educational Studies in Mathematics, 42 (2), 179–196.

Oldknow, A. (2003). Geometric and algebraic modeling with dynamic geometry soft

ware. Micromath, 19(2), 16–19.

O’Toole, T., Plummer, C. (2004) Building Mathematical Understandings in the Classroom: A Constructivist Teaching Approach. A project funded under the Australian Government’s Numeracy Research Initiative and conducted by Catholic Education South Australia.

Pape, S., & Tchoshanov, M. (2001). The role of representation(s) in developing mathematical understanding. Theory into Practice, 40(2), 118-125.

Patsiomitou, S. (2005). Fractals as a context of comprehension of the meanings of the sequence and the limit in a Dynamic Computer Software environment. Master Thesis. Department of Mathematics. National and Kapodistrian University of Athens. (in Greek)

Patsiomitou, S. (2006a): DGS ‘custom tools/scripts’ as building blocks for the formulation of theorems-in-action, leading to the proving process. Proceedings of the 5th Pan-Hellenic Conference with International Participation "ICT in Education" (HCICTE 2006), pp. 271-278, Thessaloniki, 5-8 October. ISBN 960-88359-3-3 (in Greek). Http://

Patsiomitou, S. (2006b): 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), pp. 55-78 (in Greek)

Patsiomitou, S. (2007). Fractals as a context of comprehension of the meanings of the sequence and the limit in a Dynamic Computer Software environment. Electronic Proceedings of the 8th International Conference on Technology in Mathematics Teaching (ICTMT8) in Hradec Králové (E. Milková, Pavel Prazák, eds.), University of Hradec Králové, 2007. ISBN 978-80-7041-285-5 (cd-rom).

Patsiomitou, S. (2008a). The construction of the number φ and the Fibonacci sequence using “The Geometer's Sketchpad v4” Dynamic Geometry software. Proceedings of the 1st Pan-Hellenic ICT Educational Conference, "Digital Material to support Primary and Secondary-level teachers' pedagogical work", p.307-315 Naoussa, 9-11 May 2008.(in Greek)

Patsiomitou, S., (2008b). The development of students’ geometrical thinking through transformational processes and interaction techniques in a dynamic geometry environment. Issues in Informing Science and Information Technology journal. Vol.5 pp.353-393. Available on line

Patsiomitou, S. (2008c) Linking Visual Active Representations and the van Hiele model of geometrical thinking. In Yang, W-C, Majewski, M., Alwis T. and Klairiree, K. (Eds.) “Enhancing Understanding and Constructing Knowledge in Mathematics with Technology”.Proceedings of the 13th Asian Conference in Technology in Mathematics. pp 163-178. Published by Mathematics and Technology,LLC. ISBN 978-0-9821164-1-8. Bangkok, Thailand: Suan Shunanda Rajabhat University. Available on line

Patsiomitou, S. (2008d) Custom tools and the iteration process as the referent point for the construction of meanings in a DGS environment. In Yang, W-C, Majewski, M., Alwis T. and Klairiree, K. (Eds.) Proceedings of the 13th Asian Conference in Technology in Mathematics. pp. 179-192.Available on line

Patsiomitou, S. (2009) Learning Mathematics with The Geometer’s Sketchpad v4. Klidarithmos Publications. Volume B. ISBN: 978-960-461-309-0.

Patsiomitou, S. (2010). Building LVAR (Linking Visual Active Representations) modes in a DGS environment at the Electronic Journal of Mathematics and Technology (eJMT), pp. 1-25, Issue 1, Vol. 4, February, 2010, ISSN1933-2823.

Patsiomitou, S. (2011) Theoretical dragging: A non-linguistic warrant leading to dynamic propositions. In Ubuz, B (Ed.). Proceedings of the 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

Patsiomitou, S. (2012a). The development of students’ geometrical thinking through transformational processes and interaction techniques in a dynamic geometry environment. PhD thesis. University of Ioannina (December 2012).

Patsiomitou, S. (2012b) A Linking Visual Active Representation DHLP for student’s cognitive development. Global Journal of Computer Science and Technology, Vol. 12 Issue 6, March 2012. pp. 53-81. ISSN 9754350. Available at:

Patsiomitou, S. (2013) Students learning paths as ‘dynamic encephalographs’ of their cognitive development". Ιnternational journal of computers & technology [Online], 4(3) pp.802-806 (18 April 2013) ISSN 2277-3061

Patsiomitou, S. (2014). Student’s Learning Progression Through Instrumental Decoding of Mathematical Ideas. Global Journal of Computer Science and Technology, Vol. 14 Issue 1,pp. 1-42. Online ISSN: 0975-4172.

Patsiomitou, S. (2018). A dynamic active learning trajectory for the construction of number pi (π): transforming mathematics education. International Journal of Education and Research. 6 (8) pp. 225-248.

Patsiomitou, S. & Emvalotis A. (2009a). 'Economy' and 'Catachrèse' in the use of custom tools in a Dynamic geometry problem-solving process Electronic Proceedings of the 9th International Conference on Technology in Mathematics Teaching (ICTMT8) in Metz.

Patsiomitou, S., and Emvalotis A. (2009b) Composing and testing a DG research–based curriculum designed to develop students’ geometrical thinking. Paper at the annual European Conference on Educational Research (ECER) Vienna, Sept. 25. - 28., 2009 .

Patsiomitou, S & Emvalotis, A. (2010).The development of students’ geometrical thinking through a DGS reinvention process. In M. Pinto, & Kawasaki, T. (Eds), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education, Vol.4 (pp. 33-40), Belo Horizonte. Brazil: PME

Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed ? Educational Studies in Mathematics, 66, 23-41.

Piaget, J. (1970). Genetic Epistemology, W. W. Norton, New York.

Piaget, J. (1937/1971). The construction of reality in the child (M. Cook, Trans.) The Elaboration of the Universe (Conclusion). New York:Basic Books.

Piaget, J. (1952/1977). The origins of intelligence in children (M. Cook, Trans.). New York: International University Press.

Pieron, H. (1957). Vocabulaire de la Psychologic, PUE Paris.

Rabardel, P.(1995) Les hommes et les technologies - Approche cognitive des instruments contemporains. A. Colin, Paris.

Richter-Gebert, J. and Kortenkamp, U. (1999). User manual of the Interactive Geometry Software Cinderella. Springer-Verlag, Heidelberg.

Rittle-Johnson, B., & Schneider, M. (2014). Developing conceptual and procedural knowledge of mathematics. Oxford handbook of numerical cognition. Oxford, UK: Oxford University Press

Scher, D. (2002). Students’ Conceptions of Geometry in a Dynamic Geometry Software Environment. PhD thesis. New York University

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.

Schulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1–22.

Sedig, K., Sumner, M. (2006) Characterizing interaction with visual mathematical representations. International Journal of Computers for Mathematical Learning,11:1-55 New York: Springer

Sfard, A. ( 2001). Learning mathematics as developing a discourse. In R. Speiser, C. Maher, C. Walter (Eds), Proceedings of 21st Conference of PME-NA (pp. 23-44). Columbus, Ohio: Clearing House for Science, mathematics, and Environmental Education.

Simon, H. (1980). Problem solving and education. In D. Tuma & F. Reif (Eds.), Pro blem solving and education: Issues in teaching and research (pp. 81-96). Hillsdale, NJ: Erlbaum.

Simon, M.A. (1996) ‘Beyond inductive and deductive reasoning: the search for a sense of knowing’, Educational Studies in Mathematics, v.30, 197-210.

Sinclair N., & Jackiw, N. (2007). Modeling practices with The Geometer‘s Sketchpad. Proceedings of the ICTMA-13. Bloomington, IL: Indiana University

Skemp, R. (1987). The psychology of learning mathematics. Hillsdale, N J: Erlbaum.

Straesser, R. (2001). Cabri-Geometre: Does Dynamic Geometry software (DGS) change geometry and it’s teaching and learning? International Journal of Computers for Mathematical Learning ,6, pp. 319–333

Straesser, R. (2002). Research on dynamic geometry software (DGS) - An introduction. ZDM, Vol. 34,p. 65

Sträßer, R. (2003) Macros and Proofs: Dynamical Geometry Software as an Instrument to Learn Mathematics Proceedings of 11th International Conference on Artificial Intelligence in Education, Sydney Australia.

Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage Publications, Inc.

Steffe, L. P., & Tzur, R. (1994). Interaction and children’s mathematics. In P. Ernest (Ed.), Constructing mathematical knowledge: Epistemology and mathematical education (pp. 8-32). London, England: Routledge.

Teppo, A. (1991). Van Hiele levels of geometric thought revisited. Mathematics Teacher, 84, 210-221.

Toulmin, S.E. (1958). The uses of argument. Cambridge: Cambridge University Press.

Trouche, L. (2003) From Artifact to Instrument : Mathematics Teaching Mediated by Symbolic Calculators, in P. Rabardel and Y. Waern (eds), special issue of Interacting with Computers in, vol.15 (6).

Trouche, L. (2004). Managing the complexity of the human/machine interaction in computerized learning environments: guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning 9, pp. 281-307, Kluwer academic publishers

Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. Chicago, IL: University of Chicago.

van Hiele, P. M. (1986). Structure and insight: A theory of mathematics education. Orlando, Florida: Academic Press.

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. Brooklyn: Brooklyn College. 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. (ERIC Document Reproduction Service, No. ED 287 697).

Vergnaud, G. (1998) A Comprehensive Theory of Representation for Mathematics Education. Journal of Mathematical Behavior, 17 (2), 167-181 ISSN 0364-0213

Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83–94.

Vico, G. (1710) De antiquissima Italorum sapientia, NICOLINI, F. (Ed) BARI, LATZERA (1914).

Vygotsky, L. (1978). Mind in Society: The development of higher psychological processes. Cambridge, MA: Harvard University Press

Vygotsky, L.S. (1934/1962). Thought and language. MIT Press

Weibell, Christian J. (2011) "Principles of Learning: A Conceptual Framework for Domain-Specific Theories of Learning". All Theses and Dissertations.Paper 2759. Brigham Young University - Provo

Whiteley Walter (1999). The Decline and Rise of Geometry in 20th Century North America. Proceedings of the 1999 CMESG Conference.

Winn, W. (1993). Perception principles. In M. Fleming and W.H. Levie (Eds), Instructional Message Design: Principles from the Behavioral and Cognitive Sciences, 2nd edition, Englewood Cliffs, NJ: Educational Technology Publications.


Archimedes Geo3D. Online at: http://www.raumgeometr


  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.