Developing and Managing Knowledge through the Eyes of the Young Learner: ‘Alive’ Manipulatives before Abstract Notions

Stavroula Patsiomitou
Education Advisor of Mathematics, Researcher, Greek Ministry of Education and Religious Affairs,
PhD, University of Ioannina, MEd, National and Kapodistrian University of Athens, Greece
DOI – http://doi.org/10.37502/IJSMR.2023.6302
Email – spatsiom@gmail.com

Abstract

The current study has been influenced by that aspect of the philosophy of Realistic Mathematics Education which holds that mathematics should be learned as an activity of progressive mathematization. In this study I have used the Geometer’s Sketchpad to design tasks for young learners, anticipating their reactions. My aim is to propose and introduce types of tasks /problems designed for primary-level learners, concentrating on two aspects: (a) using play-based “alive” tasks to demonstrate mathematical concepts (b) linking the sequential visual representations with instrumentally decoded math notions, thus making the whole figural diagram “alive”, while giving the students the potential to focus their attention on simultaneous modifications (and transformations) of objects on the screen. In the current study, I paraphrase Pestalozzi’s words “things before words, concrete before abstract” as “active or alive manipulatives before words, ‘dynamic’ concrete before abstract”.

Keywords: Alive manipulatives, DGS environments, math concepts, RME, progressive mathematization

References

  • Arcavi, A. & Hadas, N. (1996). Promoting changes in the curriculum by means of dynamic technologies-an example. http://www.fi.ruu.nl/en/Icme-8/WG13_13.html
  • Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. A. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29, 503-532.
  • Baroody, A.J. (1989). Manipulatives don’t come with guarantees. Arithmetic Teacher 37(2), 4–5.
  • Bishop, A. J. (1980). Spatial abilities and mathematics education—A review. Educational studies in mathematics, 11(3), 257-269.
  • Christou, C., Jones, K., Mousoulides, N. and Pittalis, M. (2006), Developing the 3DMath Dynamic Geometry Software: theoretical perspectives on design, International Journal for Technology in Mathematics Education, 13(4), 168-174
  • 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). New York: Macmillan.
  • Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23(1) 2-33.
  • De Corte, E., Verschaffel, L., & Greer, B. (2000). Connecting mathematics problem solving to the real world. In: Proceedings of the International Conference on Mathematics Education into the 21st Century: Mathematics for living (pp. 66-73). Amman, Jordan: The National Center for Human Resource Developmen
  • De Corte, E. (2000). Marrying theory building and the improvement of school practice: A permanent challenge for instructional psychology. Learning and Instruction, 10, 249-266.
  • Dienes, Z. (1960). Building Up Mathematics (4th edition). London: Hutchinson Educational Ltd.
  • Drijvers, P.H.M. (2003). Learning algebra in a computer algebra environment. Design research on the understanding of the concept of parameter, Dissertation. Utrecht: CD-press. Retrieved from www.fi.uu.nl/~pauld/dissertation
  • De Villiers, M. (1996). The future of secondary school mathematics. Paper presented at the SOSI Geometry Imperfect Conference, UNISA, Pretoria.
  • Edwards, L. (1998). Embodying mathematics and science: Microworls as representations. Journal of Mathematics Behavior, 17(1), 53-78.
  • Figueiredo, N., van Galen, F., Gravemeijer, K. (2009) The actor’s and observer’s point of view: A geometry applet as an example. Educational Designer, 1(3). https://www.educationaldesigner.org/ed/volume1/issue3/article10/
  • Furinghetti, F. & Paola, D.(2003). Revisiting guided reinvention: icons, indexes, and symbols in the construction of mathematical objects. http://math.uncc.edu//~sae/dg7/fuinghetti.pdf
  • Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: D. Reidel.
  • Freudenthal, H. (1971) Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3 (3/4), 413- 435.
  • Freudenthal, H.: 1978, Weeding and Sowing. Preface of a Science of Mathematics Reidel. Publishing Company, Dordrecht.
  • Gillis, J.M. (2005). An investigation of student conjectures in static and dynamic geometry environments. (Unpublished doctoral thesis). Auburn University https://etd.auburn.edu/bitstream/handle/10415/854/GILLIS_JOHN_2.pdf?sequence=1
  • Gorev, Gurevich I., Barabash M. (2004) How is the efficiency of the computer usage in geometry related to the levels of students’ learning abilities? The 10th International Congress on Mathematical Education (ICMI), Copenhagen, Danemark,.http://www.icme10.dk.
  • Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137–165.
  • Goldin, G., & Janvier, C. (1998). Representations and the psychology of mathematics education. Journal of Mathematical Behavior, 17(1), 1-4.
  • Goldin, G., & Shteingold, N. (2001). Systems of representations and development of mathematical concepts. In A. Cucoco, & Curcio, F (Ed.). The roles of representation in school mathematics (pp. 1-23). Reston: National Council of Teachers Mathematics.
  • Goldin, G. (2000). Affective Pathways and Representation in Mathematical Problem Solving. Mathematical thinking and learning, 2(3),pp. 209-219.
  • Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht, The Netherlands: CD-B Press.
  • Gravemeijer, K. and Terwel, J. (2000) Hans Freudenthal: a mathematician on didactics and curriculum theory. Journal of Curriculum Studies, 32(6), 777–796.
  • Guttierez, A. (1996) Vizualization in 3-dimensional geometry: In search of a framework. In L. Puig and A. Guttierez (Eds.) Proceedings of the 20 th conference of the international group for the psychology of mathematics education, vol. 1, pp. 3-19. Valencia: Universidad de Valencia
  • Hadas, N., Hershkowitz, R., Schwarz, B. (2000). The Role of Contradiction and Uncertainty in Promoting the Need to Prove in Dynamic Geometry Environment. Educational Studies in Mathematics. 44, 127-150.
  • Hohenwarter, M. (2001). GeoGebra [Computer software]. FCR-STEM, Learning Institute, Florida State University. http://www.geogebra.org/cms/.
  • Jackiw, N. (1991). The Geometer’s Sketchpad [Computer Software].Berkeley, CA: Key Curriculum Press
  • Kaput, J. J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner & C. Kieran, Research issues in the learning and teaching of algebra (pp. 167-194). Hillsdale, NJ: Erlbaum
  • Laborde, J. M. (2004). Cabri 3D. Online at: http://www.cabri.com/
  • Laborde, J-M., Baulac, Y., & Bellemain, F. (1988) Cabri Géomètre [Software]. Grenoble, France: IMAG-CNRS, Universite Joseph Fourier
  • Moyer, P. S., Bolyard, J.J., & Spikell, M.A. (2002). What are virtual manipulatives? Teaching Children Mathematics, 8(6), 372-377.
  • Papert, S. (1980). MindStorms – Children, Computers and Powerful Ideas. The Harvester Press Limited, London
  • Patsiomitou, S. (2005a). Fractals as a context of comprehension of the meanings of the sequence and the limit in a Dynamic Computer Software environment. Master Thesis. National and Kapodistrian University of Athens. Department of Mathematics. Interuniversity Postgraduate Program.
  • Patsiomitou, S. (2005b): Fractals as a context of comprehension of the meanings of the sequence and the limit in a Dynamic Software environment. Proceedings of the 22nd Panhellenic Conference of the Hellenic Mathematical Society (p.311-323), Lamia, 18-20 November 2005. ISSN: 1105-7955
  • Patsiomitou, S. (2006a): Transformations on mathematical objects through animation and trace of their dynamic parameters. Proceedings of the 5th Pan-Hellenic Conference with International Participation. Informatics and Education-ETPE, pp. 1070-1073, Thessaloniki, 5-8 October 2006. ISBN 960-88359-3-3.http://www.etpe.gr/custom/pdf/etpe1213.pdf (in Greek)
  • Patsiomitou, S. (2006b): Conclusions from the Experimental Teaching with NCTM interactive math applets (illuminations.nctm.org): Student’s perception of meanings. Proceedings of the 5th Pan-Hellenic Conference with International Participation. Informatics and Education-ETPE,  1053-1057, Thessaloniki, 5-8 October 2006. ISBN 960-88359-3-3 (in Greek)
  • Patsiomitou, S. (2006c): The Dynamic Geometry Environment as a means for the instructional design of activities. The results of a research process in a high school. 1st ICT Conference. University of Nicosia, Cyprus (in Greek)
  • Patsiomitou, S. (2006d): 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.http://www.etpe.gr/custom/pdf/etpe1102.pdf (in Greek)
  • Patsiomitou, S. (2006e): Aesthetics and technology, as means for students’ motivation to the proving process in a dynamic geometry environment. Proceedings of the 23rd Panhellenic Conference of the Hellenic Mathematical Society,515-526, Patras, 24-26 November 2006. ISSN 1105-7985  (in Greek)
  • Patsiomitou, S. (2006f): An approximation process generating number π (pi) through inscribed/circumscribed parametric polygons in a circle or through Riemann integrals’ approximation process. Experimentation and research in a Dynamic Geometry Software Environment. Proceedings of the 23rd Panhellenic Conference of the Hellenic Mathematical Society,502-514, Patras, 24-26 November 2006. ISSN 1105-7985 (in Greek)
  • Patsiomitou, S. (2006g): 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. (2007a) 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. (2007b) Sierpinski triangle, Baravelle spiral, Pythagorean tree: The Geometer’s Sketchpad v4 as a means for the construction of meanings. Proceedings of the 4th Pan-Hellenic ICT Conference titled: “Exploiting Information and Communication Technologies in Educational Practices.”, Greek Ministry of Education, pp. 28-37, Syros, 4-6 May 2007(in Greek)
  • Patsiomitou, S. (2007c) Circle’s Area and number pi: An interdisciplinary teaching approach using the Geometer’s Sketchpad v4 and the History of Mathematics. Proceedings of the 4th Pan-Hellenic ICT Conference titled: “Exploiting Information and Communication Technologies in Educational Practices.”, Greek Ministry of Education, pp. 59-68, Syros, 4-6 May 2007(in Greek)
  • Patsiomitou, S. (2007d) The Conic Sections: A teaching approach using internet -Quick movies, Geometer’s Sketchpad v4, Cabri 3D and Function Probe. Proceedings of the 4th Pan-Hellenic ICT Conference titled: “Exploiting Information and Communication Technologies in Educational Practices.”, Greek Ministry of Education, pp. 38-47, Syros, 4-6 May 2007(in Greek)
  • Patsiomitou, S. (2007e): Modeling Euclid’s Elements in The Geometer’s Sketchpad v4 dynamic geometry software. “Astrolavos”: Scientific journal of New Technologies of the Hellenic Mathematical Society, (8), pp. 61-89 (in Greek)
  • Patsiomitou, S., (2008a). 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. 5 pp.353-393. ISSN 1547-5840 , https://doi.org/10.28945/1015
  • Patsiomitou, S. (2008b) Linking Visual Active Representations and the van Hiele model of geometrical thinking. In Yang, W-C, Majewski, M., Alwis T. and Klairiree, K. (Eds.) Proceedings of the 13th Asian Conference in Technology in Mathematics. pp 163-178. ISBN 978-0-9821164-1-8. Bangkok, Thailand: Suan Shunanda Rajabhat University. Available on line http://atcm.mathandtech.org/EP2008/papers_full/2412008_14999.pdf
  • Patsiomitou, S. (2008c). Do geometrical constructions affect students algebraic expressions? In Yang, W., Majewski, M., Alwis T. and Klairiree, K. (Eds.) Proceedings of the 13th Asian Conference in Technology in Mathematics. pp 193-202. ISBN 978-0-9821164-1-8. http://atcm.mathandtech.org/EP2008/pages/regular.html
  • 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. 179-192 . ISBN 978-0-9821164-1-8. http://atcm.mathandtech.org/EP2008/pages/regular.html
  • Patsiomitou, S. (2008e) The Geometer’s Sketchpad v4 Dynamic Geometry software as a means for interpreting and designing Euclid’s Elements. Proceedings of the 1st Pan-Hellenic ICT Educational Conference, “Digital Material to support Primary and Secondary-level teachers’ pedagogical work“, pp.325-333, Naoussa, 9-11 May 2008. (in Greek)
  • Patsiomitou, S. (2008f). The construction of the number φ and the Fibonacci sequence in “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. (2008g). The construction of a Baravelle spiral as a means for students’ intuitive understanding of ascending and descending sequences. Proceedings of the 1st Panhellenic ICT Educational Conference, “Digital Material to support Primary and Secondary-level teachers’ pedagogical work“, pp.316-324, Naoussa, 9-11 May 2008. (in Greek)
  • Patsiomitou, S. (2008h) Problem solving through Linking Visual Active Representations in a Dynamic Geometry Software package, leading to rigorous proof. Proceedings of the 6th Pan-Hellenic Conference with International Participation “Information and Communication Technologies in Education” (HCICTE 2008), University of Cyprus, pp. 81-88, Cyprus 25-28 September 2008. Http://www.etpe.gr/custom/pdf/etpe1232.pdf (in Greek)
  • Patsiomitou, S. (2009a) The Impact of Structural Algebraic Units on Students’ Algebraic Thinking in a DGS Environment at the Electronic Journal of Mathematics and Technology (eJMT), 3(3), 243-260. ISSN 1933-2823
  • Patsiomitou, S. (2009b) Learning Mathematics with The Geometer’s Sketchpad v4. Monograph. Klidarithmos Publications. Volume A . ISBN: 978-960-461-308-3) (in Greek)
  • Patsiomitou, S. (2009c) Learning Mathematics with The Geometer’s Sketchpad v4. Monograph. Klidarithmos Publications. Volume B. ISBN: 978-960-461-309-0) (in Greek)
  • Patsiomitou, S. (2009d). Cognitive and theoretical (gnostikotheoretices) links using the Geometer’s Sketchpad software’s interaction techniques. Proceedings of the 5th Pan-Hellenic ICT Conference, entitled “Exploiting Information and Communication Technologies in Didactic Practice“, Greek Ministry of Education,pp. 583-591. Syros 8, 9, 10 May 2009(in Greek)
  • Patsiomitou, S. (2009e) Demonstration of visual proofs through decomposition and rearrangement of equivalent figures, created in a dynamic geometry software. Proceedings of the 5th Pan-Hellenic ICT Conference, entitled “Exploiting Information and Communication Technologies in Didactic Practice“, Greek Ministry of Education, pp. 592-600. 8, 9, 10 May 2009 Syros. (in Greek)
  • Patsiomitou, S. (2009f) Tessellations, Pentominos, Structural Algebraic Units, Rep-Tiles, Tangram. A proposal for a qualitative upgrading of math curricula. Proceedings of the 5th Pan-Hellenic ICT Conference, entitled “Exploiting Information and Communication Technologies in Didactic Practice“, Greek Ministry of Education, pp. 601-609. Syros 8, 9, 10 May 2009 (in Greek)
  • Patsiomitou, S. (2009g) Students’ cognitive interactions through constructions created with the Geometer’s Sketchpad v4 DG environment. Proceedings of the 1st Educational Conference entitled “Integration and Use of ICT in the Educational Process”, 129-134. Volos, 24-26 April. Http://www.etpe.gr/custom/pdf/etpe1440.pdf (in Greek)
  • Patsiomitou, S. (2009h) Tessellations constructed using Geometer’s Sketchpad v4 as an intuitive means for the development of students’ deductive reasoning. Proceedings of the 1st Educational Conference entitled “Integration and Use of ICT in the Educational Process”, pp. 154-160. Volos, 24-26 April(in Greek)
  • Patsiomitou, S. (2010) Building LVAR (Linking Visual Active Representations) modes in a DGS environment at the Electronic Journal of Mathematics and Technology (eJMT), 1-25, Issue 1, Vol. 4, February, 2010, ISSN1933-2823
  • Patsiomitou, S. (2011a) 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. Available at https://www.academia.edu/1764209
  • Patsiomitou, S. (2011b). Theoretical dragging: a non-linguistic warrant leading students to develop ‘dynamic’ propositions. 28th Panhellenic Conference of Hellenic Mathematical Society,562-574, Department of Mathematics of the University of Athens.https://www.academia.edu/3544047
  • Stavroula Patsiomitou (2012a). The development of students’ geometrical thinking through transformational processes and interaction techniques in a dynamic geometry environment: Linking Visual Active Representations. PhD thesis. University of Ioannina (December 2012). https://www.didaktorika.gr/eadd/handle/10442/35816) (in Greek)
  • 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: http://computerresearch.org/index.php/computer/article/view/479/479
  • Patsiomitou, S. (2012c) Didactic approaches to teaching Mathematics to students with different learning styles: Mathematics in the Real World. Self–publishing. ISBN 978-960-93-4456. (in Greek) https://www.academia.edu/2054056/(in Greek)
  • Patsiomitou, S. (2012d). Building and Transforming Linking Visual Active Representations – Implementation of LVARs for the teaching of mathematics in class. Proceedings of 8th Pan-Hellenic Conference with International Participation “ICT in Education” (HCICTE, 2012), University of Thessaly http://www.etpe.gr/custom/pdf/etpe1895.pdf (in Greek)
  • Patsiomitou, S. (2013a) 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, https://doi.org/10.24297/ijct.v4i3.4207, http://cirworld.com/index.php/ijct/ article/view/3038/pdf
  • Patsiomitou, S. (2013b) Instrumental decoding of students’ conceptual knowledge through the modeling of real problems in a dynamic geometry environment. “EUCLID C”: Scientific journal of Hellenic Mathematical Society (79), pp.107-136. (in Greek)
  • Patsiomitou, S. (2014). Student’s Learning Progression Through Instrumental Decoding of Mathematical Ideas. Global Journal of Computer Science and Technology, 14 Issue 1,pp. 1-42. Online ISSN: 0975-4172 & Print ISSN: 0975-4350.http://computerresearch.org/index.php/computer/ article/view/41/41
  • Patsiomitou, S. (2015a). A Dynamic Teaching Cycle of Mathematics through Linking Visual Active Representations”. Scientific Journal “ERKYNA”, PanHellenic Pedagogical Society for Secondary-level Education. Vol.7, pp. 70-86. https://erkyna.gr/e_docs/periodiko/dimosieyseis/thet_ epistimes/t07-05.pdf(in Greek)
  • Patsiomitou, S. (2015b) Gender equality in [STEM] education: Why Do Women Teach and Men Manage? Proceedings of the Panhellenic conference “The New Educator (Neos Paidagogos)”, 23-24 May, Eugenides Foundation, pp. 1267-1290(in Greek)
  • Patsiomitou, S. (2015c). The development of students’ competence on instrumental decoding as a non-linguistic warrant for the development of their geometric thinking. Scientific journal “The New Educator (Neos Paidagogos)”. 5th issue, pp. 29-60. (in Greek)
  • Patsiomitou, S. (2015d). The Open eClass platform as a means for instructional design and curriculum management. Scientific journal “The New Educator (Neos Paidagogos)”. 6th issue, pp.211-244. (The paper has been presented at the conference of “EDUCATION IN THE ERA OF ICT”, 7 November 2015, Eugenides Foundation, conference proceedings: pp. 700-738) (in Greek)
  • Patsiomitou, S. (2016a). Synthesis, application and evaluation of a “dynamic” curriculum: Transformations of fractals objects, parametric regular polygons and number π. Linking Visual Active Representations. Invited Speech. 3rd Panhellenic Conference of “The New Educator (Neos Paidagogos)”April 16-17. Eugenides Foundation, pp.3563-3602. (in Greek)
  • Patsiomitou, S. (2016b). Linking Visual Active Representations: Synthesis, implementation and evaluation of a “dynamic” curriculum based to dynamic transformations of mathematical objects with the utilization of interaction techniques. Scientific journal “The New Educator (Neos Paidagogos)”. 7th issue, pp. 315-347(in Greek)
  • Patsiomitou, S. (2016c) Environment & computer environments: The role of games to the development of students’ competencies and their sense for a substantial school environment. Proceedings of the 13rd Panhellenic conference “The Education in the era of ICT and innovation” pp. 967-994. ISBN: 978-618-82301-1-8(in Greek)
  • Patsiomitou, S. (2018a). 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.http://www.ijern.com/journal/2018/August-2018/18.pdf
  • Patsiomitou, S. (2018b). An ‘alive’ DGS tool for students’ cognitive development. International Journal of Progressive Sciences and Technologies (IJPSAT) ISSN: 2509-0119. Vol. 11 No. 1 October 2018, pp. 35-54. http://ijpsat.ijsht-journals.org/index.php/ijpsat/article/view/636
  • Patsiomitou, S. (2019a). From Vecten’s Theorem to Gamow’s Problem: Building an Empirical Classification Model for Sequential Instructional Problems in Geometry. Journal of Education and Practice.10, No.5, pp.1-23. DOI: 10.7176/JEP/10-5-01. https://iiste.org/Journals/index.php/JEP/article/view/46479
  • Patsiomitou, S. (2019b). Hybrid-dynamic objects: DGS environments and conceptual transformations. International Journal for Educational and Vocational Studies. 1, No. 1, May 2019, pp. 31-46. DOI: https://doi.org/10.29103/ijevs.v1i1.1416. Available online at http://ojs.unimal.ac.id/index.php/ijevs
  • Patsiomitou, S. (2019c). A Trajectory for the Teaching and Learning of the Didactics of Mathematics [using ICT]: Linking Visual Active Representations. Monograph. Published by Global Journal Incorporated. United States. (September 5, 2019) . ISBN: 978-1-7340132-0-7. http://doi.org/10.34257/SPatTrajICT
  • Patsiomitou, S. (2020a). Didactics of Mathematics I: Linking Visual Active Representations. Anatolikos. Athens, ISBΝ: 978-618-5136-46-8. https://www.academia.edu/42019703/ (in Greek)
  • Patsiomitou, S. (2020b). Didactics Instruction and Assessment of Mathematics: Learning Trajectories and Curriculum. Anatolikos. Athens ISBΝ: 978-618-5136-49-9. https://www.academia.edu/43702210/(in Greek)
  • Patsiomitou, S. (2020c). Didactics and Instruction of Mathematics: From theory to action using microworlds. Monograph. Angelakis Publications. Athens. ISBN: 978-960-616-155-1. https://www.academia.edu/43795275/(in Greek)
  • Patsiomitou, S. (2020d). Formulation of a gender “theory” in education: established trends. Angelakis Publications. Athens. ISBN: 978-960-616-171-1 , https://www.academia.edu/44760742/(in Greek)
  • Patsiomitou, S. (2021a). Dynamic Euclidean Geometry: pseudo-Toulmin modeling transformations and instrumental learning trajectories. International Institute for Science, Technology and Education (IISTE): E-Journals. Journal of Education and Practice. ISSN 2222-1735 (Paper) ISSN 2222-288X (Online). 12 (9). pp. 80-96. DOI: 10.7176/JEP/12-9-09.
  • Patsiomitou, S. (2021b). A Research Synthesis Using Instrumental Learning Trajectories: Knowing How and Knowing Why. International Institute for Science, Technology and Education (IISTE): E-Journals. Information and Knowledge Management. ISSN 2224-5758 (Paper) ISSN 2224-896X (Online). 11(3). DOI: 10.7176/IKM/11-3-02.
  • Patsiomitou, S. (2021c). Instrumental learning trajectories: the case of Geogebra. Angelakis publications. Athens. ISBN:978-960-616-193. https://www.academia.edu/79248541/ (in Greek)
  • Patsiomitou, S. (2022a). Conceptual and instrumental trajectories using linking visual active representations created with the Geometer’s Sketchpad. Klidarithmos publications.Athens.ISBN:978-960-645-302-1. https://service.eudoxus.gr/search/#a/id:112691124/0(in Greek)
  • Patsiomitou, S. (2022b). DGS Cui-Rods: Reinventing Mathematical Concepts. GPH-International Journal of Educational Research, 5(09), 01-11. https://doi.org/10.5281/zenodo.7036045, http://www.gphjournal.org/index.php/er/article/view/693
  • Patsiomitou, S. (2022c). Inquiring and learning with DGS Cui-Rods: a proposal for managing the complexity of how primary-school pupils’ mathematically structure odd-even numbers”. International Journal of Scientific and Management Research. Volume 5 Issue 8 August 2022, 143-163. http://doi.org/10.37502/IJSMR.2022.58.
  • Patsiomitou, S. (2022d). Digital-Concrete Materials: Revisiting Fröbel in Sketchpad Tasks. GPH-International Journal of Educational Research, 5(10), 01-15. https://doi.org/10.5281/zenodo.7185215
  • Pestalozzi, H. (1803). ABC der Anschauung, oder Anschauungs-Lehre der Massverhaltnisse. Tubingen, Germany: J.G. Cotta.
  • Resnick, M. (1998) Technologies for lifelong kindergarten. Educational Technology Research and Development, 46, 43–55 https://web.media.mit.edu/~mres/papers/lifelongk/lifelongk.pdf
  • Richter-Gebert, J. and Kortenkamp, U. (1999). User manual of the Interactive Geometry Software Cinderella. Springer-Verlag, Heidelberg
  • Ronge, J., & Ronge, B. (1858). A practical guide to the English kinder-garten (children’s garden): For the use of mothers, nursery governesses, and infant teachers. (2 ed.). London: Hodson. https://archive.org/details/practicalguideto00rong
  • Ross, C. J. (2004) The effect of mathematical manipulative materials on third grade students’ participation, engagement, and academic performance. Master of Education in K-8 Mathematics and Science. University of Central Florida
  • Sriraman, Bharath, & English, Lyn D. (2005) On the teaching and learning of Dienes’ principles. International Reviews in Mathematics Education (ZDM) 37(3):258-262.
  • Treffers A (1987) Three dimensions. A model of goal and theory description in mathematics instruction – The Wiskobas Project. D. Reidel Publishing, Dordrecht.
  • Van de Walle, J., & L.H. Lovin. (2005). Teaching Student-Centered Mathematics: Grades K-3. Allyn & Bacon.
  • Van den Heuvel-Panhuizen, M. (2000). Mathematics education in the Netherlands: A guided tour. Freudenthal Institute Cd-rom for ICME9. Utrecht: Utrecht University.
  • Van den Heuvel-Panhuizen, M.: 1996, Assessment and Realistic Mathematics Education, CD-ß Press Utrecht University, Utrecht, The Netherlands.
  • Wubbels, T., Korthagen, F., & Broekman, H. (1997). Preparing teachers of realistic mathematics education. Educational Studies in Mathematics, 32, 1–28.