Digital-Concrete Materials: Revisiting Fröbel in Sketchpad Tasks
Keywords:
Digital-concrete materials, DGS environment, Fröbel’s Gifts
Abstract
The present paper sets out to revisit Fröbel’s play theory, through the open-ended instructional materials, designed for pupils’ learning which he bequeathed to us. Many researchers have highlighted the advantages of digital or computer concrete materials including DGS manipulatives for teaching and learning. In terms of the present study, it is interesting to mention the introduction of Fröbel’s first Gift that I adapted in the DGS environment, designed to provide a play-based way of presenting/inquiring about geometric objects. The proposed DGS materials can be displayed, inquired about, and managed through properly set-up tasks, using linking visual active representations. The dynamic notions (e.g., dynamic point, segment, instrumental decoding, hybrid-dynamic objects, etc.), are taken as given and form the specific theoretical basis for the required processes. Dynamic interdependencies of tools in various sequential steps will be considered for the idea of building DGS Gifts, linked to the pupils’ level of conceptualization.
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References
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215-241.
Balacheff N. (2010) Bridging knowing and proving in Mathematics. An essay from a didactical perspective. In: Hanna G., Jahnke H. N., Pulte H. (eds.) Explanation and Proof in Mathematics (pp. 115-135). Heidelberg: Springer.
Baroody, A.J. (1989). Manipulatives don’t come with guarantees. Arithmetic Teacher 37(2), 4–5.
Barwise, J., &Etchemendy, J. (1991). Visual information and valid reasoning. In W. Zimmerman & S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 9-24). Washington, DC: Math. Assoc. of America.
Bartolini Bussi, Maria G., DainaTaimina, & Masami Isoda. 2010. Concrete models and dynamic instruments as early technology tools in classrooms at the dawn of ICMI: From Felix Klein to present applications in mathematics classrooms in different parts of the world. ZDM: The International Journal on Mathematics Education 42(1): 19–31.
Chen, C. & Herbst, P. (2005). The descriptive mode of interaction with diagrams in proving triangles congruent. Paper presented at the ICMI Regional Conference: The Third East Asia Regional Conference on Mathematics Education, Shanghai, China.
Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2(5), 270-279.
Clements, D. H., &Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38(2), 136-163.
Comenius, J.A. (1657/1896). Didactica Magna (tr. By M.W. Keatinge). London: Adam &Charles Black.
Deliyianni, E., Elia, I., Gagatsis, A., Monoyiou, A., &Panaoura, A. (2009). A theoretical model of students’ geometrical figure understanding. Proceedings of CERME 6, Lyon, France.
Dienes, Zoltan P. (1960). Building up Mathematics, Hutchinson Educational: London.
Diezmann, C. M. (2005) Primary students’ knowledge of the properties of spatially-oriented diagrams In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the InternationalGroup for the Psychology of Mathematics Education, Vol. 2, pp. 281-288. Melbourne: PME
Duval, Raymond. (1988). Pour uneapproche cognitive des problemes de geometrieentermes de congruence. Annales de Didactique et de Sciences Cognitives, I, 57-74.
Duval, R. (1995). Geometrical Pictures: kinds of representation and specific processings. In R. Sutherland and J. Mason (Eds), Exploiting Mental Imagery with Computers in Mathematics Education. Berlin: Springer. pp. 142-157
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.
Fischbein, E. (1980, August). Intuition and proof. Paper presented at the annual meeting of the International Group for the Psychology of Mathematics Education, Berkeley, CA.
Forsythe, K.S. (2014)The kite family and other animals: Does a dragging utilisation scheme generate only shapes or can it also generate mathematical meanings? PhD thesis . School of Education University of Leicester
Fröbel, F. (1885). The education of man (W. N. Hailmann, Trans.). New York: Appleton.
Glasersfeld, E. v. (1995). Radical constructivism: A way of knowing and learning. London: Falmer Press
Henton, M. (1996). Adventure in the Classroom: Using Adventure to Strengthen Learning and Build a Community of Life-long Learners. Dubuque, Iowa: Kendall/Hunt.
Jackiw, Nicholas (2006) "Mechanism and Magic in the Psychology of Dynamic Geometry," Mathematics and the Aesthetic: New Approaches to an Ancient Affinity. Heidelberg: Springer Verlag.
Jackiw, N. (1991). The Geometer's Sketchpad [Computer Software]. Berkeley, CA: Key Curriculum Press
Johansson, Jan-Erik. (2022). Didactic and Curriculum in ECEC from a Froebelian standpoint. Global Education Review, 9 (2), 67-82.
Jones, K. (1998), Theoretical Frameworks for the Learning of Geometrical Reasoning. Proceeding of the British Society for Research into Learning Mathematics, 18(1&2), 29-34.
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.
Karmiloff-Smith, A. (1990). Constraints on representational change: Evidence from children’s drawing. Cognition, 34, 57-83
Kolb, D. A. (1984). Experiential learning: Experience as the source of learning and development. New Jersey: Prentice
Kolb, A. Y. & Kolb, D. A. (2009). Experiential learning theory: A dynamic, holistic approach to management learning, education and development. Chapter 3 in Armstrong, S. J. &Fukami, C. (Eds.) Handbook of Management Learning, Education and Development. London: Sage Publications
Kolb, A.Y., & Kolb, D. A. (2013) The Kolb learning style invntory: Version 4.0 2011. A comprehensive guide to the theory, psycometrics, research on validity and educational applications. Experience Based Learning Systems. https://learningfromexperience.com/research-library/the-kolb-learning-style-inventory-4-0/
Lindsay, R. K. (1995). Images and Inferences. In J. Glasgow, N. H. Narayanan, & B. C.Karan. Diagrammatic reasoning (pp. 111- 135). Menlo Park, CA: AAI Press.
Mainali, B. (2021). Representation in teaching and learning mathematics. International Journal of Education in Mathematics, Science, and Technology (IJEMST), 9(1), 1-21. https://doi.org/10.46328/ijemst.1111
Maria –Kraus Boelte and John Kraus (1877). The Kindergarten Guide E. Steiger&Company,New York
Laborde, C. (2005). The hidden role of diagrams in students’ construction of meaning in geometry. In J. Kilpatrick, C. Hoyles, O. Shovsmose& P. Valero (Eds.), Meaning in mathematics education (pp. 159–179). New York: Springer
Mesquita, A. (1998). On conceptual obstacles linked with external representation in geometry. Journal of Mathematical Behavior, 17(2), 183-195.
Moritz, J. B. (1998). Observing students learning from students. In S. Groves, B. Jane, I. Robottom, & R. Tytler (Eds.), Contemporary Approaches to Research in Mathematics, Science, Health and Environmental Education 1997 (pp. 160-165). Geelong, VIC: Deakin University
Nesher, P. (1987). Towards an instructional theory: the role of student’s misconceptions. For the Learning of Mathematics, 7(3), 33-40.
Olive, J., & Makar, K., with V. Hoyos, L. K. Kor, O. Kosheleva, & R. Straesser (2010). Mathematical knowledge and practices resulting from access to digital technologies. In C. Hoyles& J. Lagrange (Eds.), Mathematics education and technology – Rethinking the terrain. The 17th ICMI Study (pp. 133–177). New York: Springer.
Papert, S. (1980). Mindstorms. New York: Basic Books
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.Interuniversity Postgraduate Program.
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. Book Chapter. Vol.5 pp.353-393.http://iisit.org/IssuesVol5.htm
Patsiomitou, S. (2008b) Linking Visual Active Representations and the van Hiele model of geometrical thinking. InYang, W-C, Majewski, M., Alwis T. and Klairiree, K. (Eds.) Proceedings of the13thAsian Conference in Technology in Mathematics. pp 163-178.ISBN 978-0-9821164-1-8. Bangkok, Thailand: http://atcm.mathandtech.org/EP2008/papers_full/2412008_14999.
Patsiomitou, S. (2010) Building LVAR (Linking Visual Active Representations) modes in a DGS environment. 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, ISBN 987 -975-429-297-8. Vol. 3, pp. 361-368. Ankara, Turkey: PME.
Patsiomitou, S. (2012) 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. (2018). 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).A Trajectory for the Teaching and Learning of the Didactics of Mathematics [using ICT]: Linking Visual Active Representations. Monograph. Published by Global Journal Incorporated. (September 5, 2019). ISBN: 978-1-7340132-0-7. http://doi.org/10.34257/SPatTrajICT
Patsiomitou, S. (2019b). Hybrid-dynamic objects: DGS environments and conceptual transformations. International Journal for Educational and Vocational Studies. Vol. 1, No. 1, May 2019, pp. 31-46. https://doi.org/10.29103/ijevs.v1i1.1416.
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. Publication date: March 31st 2021.
Patsiomitou, S. (2021b).A Research Synthesis Using Instrumental Learning Trajectories: Knowing How and Knowing Why.Information and Knowledge Management. ISSN 2224-5758 (Paper) ISSN 2224-896X (Online). 11(3). DOI:10.7176/IKM/11-3-02.
Patsiomitou, S. (2022a). DGS Cui-Rods: Reinventing Mathematical Concepts. GPH-International Journal of Educational Research, 5(09), 01-11. https://doi.org/10.5281/zenodo.7036045
Patsiomitou, S. (2022b).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. 5 (8). August 2022, 143-163.http://doi.org/10.37502/IJSMR.2022.5813
Pea, R.D. (1987). Cognitive technologies for mathematics education. In A.H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 89-122). Hillsdale, NJ:Lawrence Erlbaum Associates.
Piaget, J. (1937/1971). The construction of reality in the child (M. Cook, Trans.). New York: Basic Books.
Piaget, J., &Inhelder, B. (1967). The child’s conception of space. New York: W. W. Norton
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
Sedig, K., Sumner, M. (2006) Characterizing interaction with visual mathematical representations. International Journal of Computers for Mathematical Learning,11:1-55 . New York: Springer
Sedig, K. and Liang, H. (2008). Learner-information interaction: A macro-level framework characterizing visual cognitive tools. Journal of Interactive Learning Research, 19(1), 147-173
Sinclair N., & Jackiw, N. (2007). Modeling practices with The Geometer‘s Sketchpad. Proceedings of the ICTMA-13. Bloomington, IL: Indiana University
Stahl, G. (2013). "Translating Euclid: Designing a Human-Centered Mathematics." Morgan & Claypool Publishers.
Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction: An epistemological perspective. New York: Springer.
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.
Swedosh, P., & Clark, J. (1998). Mathematical misconceptions – we have an effective method for reducing their incidence but will the improvement persist? In C. Kanes, M. Goos, & E. Warren (Eds.), Teaching mathematics in new time (Proceedings of the 21st Annual Conference of the Mathematics Education Research Group of Australasia, Vol. 2, pp. 588-595). Brisbane: MERGA.
Taner Derman, M., Sahin Zeteroglu, E., & Ergisi Birgόl Arzu. (2020). The effect of play-basedmath activities on different areas of development in children 48 to 60 months of
age. Sage Open, 10(2).https://doi.org/10.1177/2158244020919531
Van de Walle, J., & L.H. Lovin. (2005). Teaching Student-Centered Mathematics: Grades K-3. Allyn& Bacon.
van Essen, G., &Hamaker, C. (1990). Using self-generated drawings to solve arithmetic word problems. Journal of Educational Research, 83(6), 301-312
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. In T. Nunes (Ed.), Giving meaning to mathematical signs: Psychological, pedagogical and cultural processes. Human Development [Special Issue], 52, 83–94.
Watson, J.M. & Moritz, J.B. (2001). The role of cognitive conflict in developing student’s understanding of chance measurement. In J.Bobis, B.Perry, &M.Mitchelmore (Eds), Numeracy and beyond (Proceedings of the 24th Annual Conferenc of the Mathematics Education Research Group of Australasia, (Vol. 2, pp.523-530). Sydney: MERGA
Zull, J. (2002). The art of changing the brain. Sterling VA: Stylus.
Zull, J. (2011). From brain to mind: Using neuroscience to guide change in education. Sterling VA: Stylus.
Balacheff N. (2010) Bridging knowing and proving in Mathematics. An essay from a didactical perspective. In: Hanna G., Jahnke H. N., Pulte H. (eds.) Explanation and Proof in Mathematics (pp. 115-135). Heidelberg: Springer.
Baroody, A.J. (1989). Manipulatives don’t come with guarantees. Arithmetic Teacher 37(2), 4–5.
Barwise, J., &Etchemendy, J. (1991). Visual information and valid reasoning. In W. Zimmerman & S. Cunningham (Eds.), Visualization in teaching and learning mathematics (pp. 9-24). Washington, DC: Math. Assoc. of America.
Bartolini Bussi, Maria G., DainaTaimina, & Masami Isoda. 2010. Concrete models and dynamic instruments as early technology tools in classrooms at the dawn of ICMI: From Felix Klein to present applications in mathematics classrooms in different parts of the world. ZDM: The International Journal on Mathematics Education 42(1): 19–31.
Chen, C. & Herbst, P. (2005). The descriptive mode of interaction with diagrams in proving triangles congruent. Paper presented at the ICMI Regional Conference: The Third East Asia Regional Conference on Mathematics Education, Shanghai, China.
Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2(5), 270-279.
Clements, D. H., &Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38(2), 136-163.
Comenius, J.A. (1657/1896). Didactica Magna (tr. By M.W. Keatinge). London: Adam &Charles Black.
Deliyianni, E., Elia, I., Gagatsis, A., Monoyiou, A., &Panaoura, A. (2009). A theoretical model of students’ geometrical figure understanding. Proceedings of CERME 6, Lyon, France.
Dienes, Zoltan P. (1960). Building up Mathematics, Hutchinson Educational: London.
Diezmann, C. M. (2005) Primary students’ knowledge of the properties of spatially-oriented diagrams In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the InternationalGroup for the Psychology of Mathematics Education, Vol. 2, pp. 281-288. Melbourne: PME
Duval, Raymond. (1988). Pour uneapproche cognitive des problemes de geometrieentermes de congruence. Annales de Didactique et de Sciences Cognitives, I, 57-74.
Duval, R. (1995). Geometrical Pictures: kinds of representation and specific processings. In R. Sutherland and J. Mason (Eds), Exploiting Mental Imagery with Computers in Mathematics Education. Berlin: Springer. pp. 142-157
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.
Fischbein, E. (1980, August). Intuition and proof. Paper presented at the annual meeting of the International Group for the Psychology of Mathematics Education, Berkeley, CA.
Forsythe, K.S. (2014)The kite family and other animals: Does a dragging utilisation scheme generate only shapes or can it also generate mathematical meanings? PhD thesis . School of Education University of Leicester
Fröbel, F. (1885). The education of man (W. N. Hailmann, Trans.). New York: Appleton.
Glasersfeld, E. v. (1995). Radical constructivism: A way of knowing and learning. London: Falmer Press
Henton, M. (1996). Adventure in the Classroom: Using Adventure to Strengthen Learning and Build a Community of Life-long Learners. Dubuque, Iowa: Kendall/Hunt.
Jackiw, Nicholas (2006) "Mechanism and Magic in the Psychology of Dynamic Geometry," Mathematics and the Aesthetic: New Approaches to an Ancient Affinity. Heidelberg: Springer Verlag.
Jackiw, N. (1991). The Geometer's Sketchpad [Computer Software]. Berkeley, CA: Key Curriculum Press
Johansson, Jan-Erik. (2022). Didactic and Curriculum in ECEC from a Froebelian standpoint. Global Education Review, 9 (2), 67-82.
Jones, K. (1998), Theoretical Frameworks for the Learning of Geometrical Reasoning. Proceeding of the British Society for Research into Learning Mathematics, 18(1&2), 29-34.
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.
Karmiloff-Smith, A. (1990). Constraints on representational change: Evidence from children’s drawing. Cognition, 34, 57-83
Kolb, D. A. (1984). Experiential learning: Experience as the source of learning and development. New Jersey: Prentice
Kolb, A. Y. & Kolb, D. A. (2009). Experiential learning theory: A dynamic, holistic approach to management learning, education and development. Chapter 3 in Armstrong, S. J. &Fukami, C. (Eds.) Handbook of Management Learning, Education and Development. London: Sage Publications
Kolb, A.Y., & Kolb, D. A. (2013) The Kolb learning style invntory: Version 4.0 2011. A comprehensive guide to the theory, psycometrics, research on validity and educational applications. Experience Based Learning Systems. https://learningfromexperience.com/research-library/the-kolb-learning-style-inventory-4-0/
Lindsay, R. K. (1995). Images and Inferences. In J. Glasgow, N. H. Narayanan, & B. C.Karan. Diagrammatic reasoning (pp. 111- 135). Menlo Park, CA: AAI Press.
Mainali, B. (2021). Representation in teaching and learning mathematics. International Journal of Education in Mathematics, Science, and Technology (IJEMST), 9(1), 1-21. https://doi.org/10.46328/ijemst.1111
Maria –Kraus Boelte and John Kraus (1877). The Kindergarten Guide E. Steiger&Company,New York
Laborde, C. (2005). The hidden role of diagrams in students’ construction of meaning in geometry. In J. Kilpatrick, C. Hoyles, O. Shovsmose& P. Valero (Eds.), Meaning in mathematics education (pp. 159–179). New York: Springer
Mesquita, A. (1998). On conceptual obstacles linked with external representation in geometry. Journal of Mathematical Behavior, 17(2), 183-195.
Moritz, J. B. (1998). Observing students learning from students. In S. Groves, B. Jane, I. Robottom, & R. Tytler (Eds.), Contemporary Approaches to Research in Mathematics, Science, Health and Environmental Education 1997 (pp. 160-165). Geelong, VIC: Deakin University
Nesher, P. (1987). Towards an instructional theory: the role of student’s misconceptions. For the Learning of Mathematics, 7(3), 33-40.
Olive, J., & Makar, K., with V. Hoyos, L. K. Kor, O. Kosheleva, & R. Straesser (2010). Mathematical knowledge and practices resulting from access to digital technologies. In C. Hoyles& J. Lagrange (Eds.), Mathematics education and technology – Rethinking the terrain. The 17th ICMI Study (pp. 133–177). New York: Springer.
Papert, S. (1980). Mindstorms. New York: Basic Books
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.Interuniversity Postgraduate Program.
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. Book Chapter. Vol.5 pp.353-393.http://iisit.org/IssuesVol5.htm
Patsiomitou, S. (2008b) Linking Visual Active Representations and the van Hiele model of geometrical thinking. InYang, W-C, Majewski, M., Alwis T. and Klairiree, K. (Eds.) Proceedings of the13thAsian Conference in Technology in Mathematics. pp 163-178.ISBN 978-0-9821164-1-8. Bangkok, Thailand: http://atcm.mathandtech.org/EP2008/papers_full/2412008_14999.
Patsiomitou, S. (2010) Building LVAR (Linking Visual Active Representations) modes in a DGS environment. 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, ISBN 987 -975-429-297-8. Vol. 3, pp. 361-368. Ankara, Turkey: PME.
Patsiomitou, S. (2012) 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. (2018). 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).A Trajectory for the Teaching and Learning of the Didactics of Mathematics [using ICT]: Linking Visual Active Representations. Monograph. Published by Global Journal Incorporated. (September 5, 2019). ISBN: 978-1-7340132-0-7. http://doi.org/10.34257/SPatTrajICT
Patsiomitou, S. (2019b). Hybrid-dynamic objects: DGS environments and conceptual transformations. International Journal for Educational and Vocational Studies. Vol. 1, No. 1, May 2019, pp. 31-46. https://doi.org/10.29103/ijevs.v1i1.1416.
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. Publication date: March 31st 2021.
Patsiomitou, S. (2021b).A Research Synthesis Using Instrumental Learning Trajectories: Knowing How and Knowing Why.Information and Knowledge Management. ISSN 2224-5758 (Paper) ISSN 2224-896X (Online). 11(3). DOI:10.7176/IKM/11-3-02.
Patsiomitou, S. (2022a). DGS Cui-Rods: Reinventing Mathematical Concepts. GPH-International Journal of Educational Research, 5(09), 01-11. https://doi.org/10.5281/zenodo.7036045
Patsiomitou, S. (2022b).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. 5 (8). August 2022, 143-163.http://doi.org/10.37502/IJSMR.2022.5813
Pea, R.D. (1987). Cognitive technologies for mathematics education. In A.H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 89-122). Hillsdale, NJ:Lawrence Erlbaum Associates.
Piaget, J. (1937/1971). The construction of reality in the child (M. Cook, Trans.). New York: Basic Books.
Piaget, J., &Inhelder, B. (1967). The child’s conception of space. New York: W. W. Norton
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
Sedig, K., Sumner, M. (2006) Characterizing interaction with visual mathematical representations. International Journal of Computers for Mathematical Learning,11:1-55 . New York: Springer
Sedig, K. and Liang, H. (2008). Learner-information interaction: A macro-level framework characterizing visual cognitive tools. Journal of Interactive Learning Research, 19(1), 147-173
Sinclair N., & Jackiw, N. (2007). Modeling practices with The Geometer‘s Sketchpad. Proceedings of the ICTMA-13. Bloomington, IL: Indiana University
Stahl, G. (2013). "Translating Euclid: Designing a Human-Centered Mathematics." Morgan & Claypool Publishers.
Steinbring, H. (2005). The construction of new mathematical knowledge in classroom interaction: An epistemological perspective. New York: Springer.
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.
Swedosh, P., & Clark, J. (1998). Mathematical misconceptions – we have an effective method for reducing their incidence but will the improvement persist? In C. Kanes, M. Goos, & E. Warren (Eds.), Teaching mathematics in new time (Proceedings of the 21st Annual Conference of the Mathematics Education Research Group of Australasia, Vol. 2, pp. 588-595). Brisbane: MERGA.
Taner Derman, M., Sahin Zeteroglu, E., & Ergisi Birgόl Arzu. (2020). The effect of play-basedmath activities on different areas of development in children 48 to 60 months of
age. Sage Open, 10(2).https://doi.org/10.1177/2158244020919531
Van de Walle, J., & L.H. Lovin. (2005). Teaching Student-Centered Mathematics: Grades K-3. Allyn& Bacon.
van Essen, G., &Hamaker, C. (1990). Using self-generated drawings to solve arithmetic word problems. Journal of Educational Research, 83(6), 301-312
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. In T. Nunes (Ed.), Giving meaning to mathematical signs: Psychological, pedagogical and cultural processes. Human Development [Special Issue], 52, 83–94.
Watson, J.M. & Moritz, J.B. (2001). The role of cognitive conflict in developing student’s understanding of chance measurement. In J.Bobis, B.Perry, &M.Mitchelmore (Eds), Numeracy and beyond (Proceedings of the 24th Annual Conferenc of the Mathematics Education Research Group of Australasia, (Vol. 2, pp.523-530). Sydney: MERGA
Zull, J. (2002). The art of changing the brain. Sterling VA: Stylus.
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Published
2022-10-11
How to Cite
Patsiomitou, S. (2022). 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
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Articles
Copyright (c) 2022 Stavroula Patsiomitou
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