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Abstract
phenomenological standpoint, in which interpretative activities are relevant. This approach requires a careful analysis of the semiotic resources’ evolution, including those related to the used digital tools. The paper aims to introduce an analytical tool, the Timeline. This tool is an elaboration on previous analysis tools, like the interaction flowchart and the semiotic bundle. Such a tool allows the analysis of relationship among interactions, semiotic resources, and meaning-making. In this paper, the Timeline is used to analyze two episodes from two different learning experiments where GeoGebra and augmented reality are used. High school students from Italy and Israel participated in this study. Video recording has been used to document the entire learning experiments. The analysis provides evidence that the Timeline enables investigating the relationship between students-teacher-artifacts interactions and meaning-making. Moreover, results may give teachers ideas for using digital tools to foster students’ meaning-making.
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References
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- Arzarello, F. (2008). Mathematical landscapes and their inhabitants: Perceptions, languages, theories. In: Niss, M. (Editor), Proceedings of 10th International Congress on Mathematical Education, Plenary lecture, 158–181.
- Arzarello, F., Ascari, M., Baldovino, C., & Sabena, C. (2011). The teacher’s activity under a phenomenological lens. In: U. Behiye, Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education. Vol. 2, pp. 49–56.
- Arzarello, F., & Robutti, O. (2010). Framing the embodied mind approach within a multimodal paradigm. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp. 730-759).
- Arzarello, F., Bazzini, L., Politano, L., & Sabena, C. (2010). Multimodal processes in teaching and learning mathematics: A case study in primary school. In G. Pérez-Bustamante, K. Phusavat, F. Ferreira (Eds.) Proceedings of the IASK International Conference (pp. 286–292).
- Arzarello, F., & Edwards, L. (2005). Gesture and the construction of mathematical meaning. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 123-154). Retrieved from https://www.emis.de/proceedings/PME29/PME29ResearchForums/PME29RFArzarelloEdwards.pdf
- Bagossi, S. (2022). Second-order covariation: an analysis of students’ reasonings and teacher’s interventions when modelling real phenomena. Unpublished Ph.D. Thesis. University of Modena and Reggio Emilia in agreement with the University of Ferrara and the University of Parma, Italy.
- Cañigueral, R., & Hamilton, A. F. de C. (2019). The role of eye gaze during natural social interactions in typical and autistic people, Frontiers in Psychology, 10, 560. https://doi.org/10.3389/fpsyg.2019.00560
- Drijvers, P., Ball, L., Barzel, B., Heid, M.K., Cao, Y., & Maschietto, M. (2016). Uses of Technology in Lower Secondary Mathematics Education: A Concise Topical Survey. Springer Open. https://doi.org/10.1007/978-3-319-33666-4
- Faggiano, E., Montone, A., & Mariotti, M. A. (2018). Synergy between manipulative and digital artefacts: a teaching experiment on axial symmetry at primary school. International Journal of Mathematical Education in Science and Technology, 49(8), 1165–1180. https://doi.org/10.1080/0020739X.2018.1449908
- Gallese, V., & Lakoff, G. (2005). The brain's concepts: The role of the sensory-motor system in conceptual knowledge. Cognitive Neuropsychology, 22(3-4), 455–479. https://doi.org/10.1080/02643290442000310
- Javorski, B., & Potari, D. (2009). Bridging the macro- and micro-divide: Using an activity Theory model to capture sociocultural complexity in mathematics teaching and its development. Educational Studies in Mathematics, 72(2), 219–236. https://doi.org/10.1007/s10649-009-9190-4
- Liljedahl, P., & Andrà, C. (2014). Students’ gazes: new insights into student interactions. In: Views and beliefs in mathematics education - contributions of the 19th MAVI conference, 213–226. Ed: Springer.
- Loncke, F. T., Campbell, J., England, A. M., & Haley, T. (2006). Multimodality: A basis for augmentative and alternative communication–psycholinguistic, cognitive, and clinical/educational aspects. Disability and Rehabilitation, 28(3), 169–174. https://doi.org/10.1080/09638280500384168
- McNeill, D. (1992). Hand and mind: What gestures reveal about thought. University of Chicago press.
- Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2–7. https://doi.org/10.2307/20749442
- Radford, L. (2011). Book Review: Classroom Interaction: Why is it Good, Really? Baruch Schwarz, Tommy Dreyfus and Rina Hershkowitz (Eds.) (2009) Transformation of knowledge through classroom interaction. Educational Studies in Mathematics, 76(1), 101–115. https://doi.org/10.1007/s10649-010-9271-4
- Radford, L., Schubring, G., & Seeger, F. (2011). Signifying and meaning-making in mathematics thinking, teaching and learning: Semiotic perspectives. Educational Studies in Mathematics, 77(2-3). 149–156. https://doi.org/10.1007/s10649-011-9322-5
- Rota, G. C. (1991). The End of Objectivity: The Legacy of Phenomenology. Lectures at MIT.
- Roth, W.-M. (2001). Gestures: Their role in teaching and learning. Review of Educational Research, 71, 365–392. https://doi.org/10.3102%2F00346543071003365
- Sabena, C., Robutti, O., Ferrara, F., & Arzarello, F. (2012). The development of a semiotic framework to analyze teaching and learning processes: Examples in pre- and post-algebraic contexts, Recherches en Didactique des Mathématiques, Enseignement de l'algèbre élémentaire, Numéro spécial, 237–251.
- Saxe, G.B., Gearhart, M., Shaughnessy, M., Earnest, D., Cremer, S., Sitabkhan, Y., Platas, L., & Young, A. (2009). A methodological framework and empirical techniques for studying the travel of ideas in classroom communities. In B. Schwarz, T. Dreyfus, & R. Hershkowish (Eds.) Transformation of Knowledge Through Classroom Interaction. Routledge: London, 2009; 211–230.
- Schwarz, B., Dreyfus, T., & Hershkowitz, R. (Eds.) (2009). Transformation of knowledge through classroom interaction. London: Routledge.
- Seeger, F. (2011). On meaning making in mathematics education: Social, emotional, semiotic. Educational Studies in Mathematics, 77(2-3), 207–226. https://doi.org/10.1007/s10649-010- 9279-9
- Sfard, A., & Kieran, C. (2001). Cognition as communication: Rethinking learning-by-talking through multi-faceted analysis of students' mathematical interactions. Mind, Culture and Activity, 8(1), 42–76. https://doi.org/10.1207/S15327884MCA0801_04
- Smedlund, J., Hemmi, K., & Röj-Lindberg, A. S. (2018). Structuring students’ mathematical conversations with flowcharts and intention analysis–affordances and constraints. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.). Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 203-210). Umeå, Sweden: PME.
- Swidan, O. (2022). Meaning making through collective argumentation: The role of students’ argumentative discourse in their exploration of the graphic relationship between a function and its anti-derivative, Teaching Mathematics and its Applications: An International Journal of the IMA. https://doi.org/10.1093/teamat/hrab034
- Swidan, O., Sabena, C., & Arzarello, F. (2020). Disclosure of mathematical relationships with a digital tool: A three layer-model of meaning. Educational Studies in Mathematics, 103(1), 83–101. https://doi.org/10.1007/s10649-019-09926-2
- Swidan, O., Bagossi, S., Beltramino, S., & Arzarello, F. (2022). Adaptive instruction strategies to foster covariational reasoning in a digitally rich environment. The Journal of Mathematical Behavior, 66, 100961. https://doi.org/10.1016/j.jmathb.2022.100961
- Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for Research in Mathematics Education, Reston, VA: National Council of Teachers of Mathematics, 421–456.
- Wilson, M. (2002). Six views of embodied cognition. Psychonomic Bulletin & Review, 9(4), 625–636. https://doi.org/10.3758/BF03196322
References
Arzarello, F. (2006). Semiosis as a multimodal process. Revista Latinoamericana de Investigación en Matemática Educativa RELIME, 9(Extraordinario 1), 267–299. Retrieved from https://revistas.ucr.ac.cr/index.php/cifem/article/view/39888/40429
Arzarello, F. (2008). Mathematical landscapes and their inhabitants: Perceptions, languages, theories. In: Niss, M. (Editor), Proceedings of 10th International Congress on Mathematical Education, Plenary lecture, 158–181.
Arzarello, F., Ascari, M., Baldovino, C., & Sabena, C. (2011). The teacher’s activity under a phenomenological lens. In: U. Behiye, Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education. Vol. 2, pp. 49–56.
Arzarello, F., & Robutti, O. (2010). Framing the embodied mind approach within a multimodal paradigm. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (pp. 730-759).
Arzarello, F., Bazzini, L., Politano, L., & Sabena, C. (2010). Multimodal processes in teaching and learning mathematics: A case study in primary school. In G. Pérez-Bustamante, K. Phusavat, F. Ferreira (Eds.) Proceedings of the IASK International Conference (pp. 286–292).
Arzarello, F., & Edwards, L. (2005). Gesture and the construction of mathematical meaning. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 123-154). Retrieved from https://www.emis.de/proceedings/PME29/PME29ResearchForums/PME29RFArzarelloEdwards.pdf
Bagossi, S. (2022). Second-order covariation: an analysis of students’ reasonings and teacher’s interventions when modelling real phenomena. Unpublished Ph.D. Thesis. University of Modena and Reggio Emilia in agreement with the University of Ferrara and the University of Parma, Italy.
Cañigueral, R., & Hamilton, A. F. de C. (2019). The role of eye gaze during natural social interactions in typical and autistic people, Frontiers in Psychology, 10, 560. https://doi.org/10.3389/fpsyg.2019.00560
Drijvers, P., Ball, L., Barzel, B., Heid, M.K., Cao, Y., & Maschietto, M. (2016). Uses of Technology in Lower Secondary Mathematics Education: A Concise Topical Survey. Springer Open. https://doi.org/10.1007/978-3-319-33666-4
Faggiano, E., Montone, A., & Mariotti, M. A. (2018). Synergy between manipulative and digital artefacts: a teaching experiment on axial symmetry at primary school. International Journal of Mathematical Education in Science and Technology, 49(8), 1165–1180. https://doi.org/10.1080/0020739X.2018.1449908
Gallese, V., & Lakoff, G. (2005). The brain's concepts: The role of the sensory-motor system in conceptual knowledge. Cognitive Neuropsychology, 22(3-4), 455–479. https://doi.org/10.1080/02643290442000310
Javorski, B., & Potari, D. (2009). Bridging the macro- and micro-divide: Using an activity Theory model to capture sociocultural complexity in mathematics teaching and its development. Educational Studies in Mathematics, 72(2), 219–236. https://doi.org/10.1007/s10649-009-9190-4
Liljedahl, P., & Andrà, C. (2014). Students’ gazes: new insights into student interactions. In: Views and beliefs in mathematics education - contributions of the 19th MAVI conference, 213–226. Ed: Springer.
Loncke, F. T., Campbell, J., England, A. M., & Haley, T. (2006). Multimodality: A basis for augmentative and alternative communication–psycholinguistic, cognitive, and clinical/educational aspects. Disability and Rehabilitation, 28(3), 169–174. https://doi.org/10.1080/09638280500384168
McNeill, D. (1992). Hand and mind: What gestures reveal about thought. University of Chicago press.
Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2–7. https://doi.org/10.2307/20749442
Radford, L. (2011). Book Review: Classroom Interaction: Why is it Good, Really? Baruch Schwarz, Tommy Dreyfus and Rina Hershkowitz (Eds.) (2009) Transformation of knowledge through classroom interaction. Educational Studies in Mathematics, 76(1), 101–115. https://doi.org/10.1007/s10649-010-9271-4
Radford, L., Schubring, G., & Seeger, F. (2011). Signifying and meaning-making in mathematics thinking, teaching and learning: Semiotic perspectives. Educational Studies in Mathematics, 77(2-3). 149–156. https://doi.org/10.1007/s10649-011-9322-5
Rota, G. C. (1991). The End of Objectivity: The Legacy of Phenomenology. Lectures at MIT.
Roth, W.-M. (2001). Gestures: Their role in teaching and learning. Review of Educational Research, 71, 365–392. https://doi.org/10.3102%2F00346543071003365
Sabena, C., Robutti, O., Ferrara, F., & Arzarello, F. (2012). The development of a semiotic framework to analyze teaching and learning processes: Examples in pre- and post-algebraic contexts, Recherches en Didactique des Mathématiques, Enseignement de l'algèbre élémentaire, Numéro spécial, 237–251.
Saxe, G.B., Gearhart, M., Shaughnessy, M., Earnest, D., Cremer, S., Sitabkhan, Y., Platas, L., & Young, A. (2009). A methodological framework and empirical techniques for studying the travel of ideas in classroom communities. In B. Schwarz, T. Dreyfus, & R. Hershkowish (Eds.) Transformation of Knowledge Through Classroom Interaction. Routledge: London, 2009; 211–230.
Schwarz, B., Dreyfus, T., & Hershkowitz, R. (Eds.) (2009). Transformation of knowledge through classroom interaction. London: Routledge.
Seeger, F. (2011). On meaning making in mathematics education: Social, emotional, semiotic. Educational Studies in Mathematics, 77(2-3), 207–226. https://doi.org/10.1007/s10649-010- 9279-9
Sfard, A., & Kieran, C. (2001). Cognition as communication: Rethinking learning-by-talking through multi-faceted analysis of students' mathematical interactions. Mind, Culture and Activity, 8(1), 42–76. https://doi.org/10.1207/S15327884MCA0801_04
Smedlund, J., Hemmi, K., & Röj-Lindberg, A. S. (2018). Structuring students’ mathematical conversations with flowcharts and intention analysis–affordances and constraints. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.). Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 203-210). Umeå, Sweden: PME.
Swidan, O. (2022). Meaning making through collective argumentation: The role of students’ argumentative discourse in their exploration of the graphic relationship between a function and its anti-derivative, Teaching Mathematics and its Applications: An International Journal of the IMA. https://doi.org/10.1093/teamat/hrab034
Swidan, O., Sabena, C., & Arzarello, F. (2020). Disclosure of mathematical relationships with a digital tool: A three layer-model of meaning. Educational Studies in Mathematics, 103(1), 83–101. https://doi.org/10.1007/s10649-019-09926-2
Swidan, O., Bagossi, S., Beltramino, S., & Arzarello, F. (2022). Adaptive instruction strategies to foster covariational reasoning in a digitally rich environment. The Journal of Mathematical Behavior, 66, 100961. https://doi.org/10.1016/j.jmathb.2022.100961
Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for Research in Mathematics Education, Reston, VA: National Council of Teachers of Mathematics, 421–456.
Wilson, M. (2002). Six views of embodied cognition. Psychonomic Bulletin & Review, 9(4), 625–636. https://doi.org/10.3758/BF03196322