Reconstruction of the Interpretation of Geometric Diagrams of Primary School Children Based on Actions on Various Materials -
A Semiotic Perspective on Actions

Lara Kristina Billion

doi:10.29333/iejme/11068

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Lara Kristina Billion ¹ ^*

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¹ Goethe University Frankfurt, Frankfurt am Main, GERMANY^* Corresponding Author

Abstract

This paper adopts a semiotic perspective on mathematical learning according to Peirce, in which the actions on material arrangements are considered the bases for diagrammatic work. The focus is on the learner’s actions using digital and analogue material arrangements which are the starting point for the reconstruction of the learner’s mathematical interpretations. In this paper a qualitative interpretative paradigm is adopted for the reconstruction. Specifically, a semiotic specification of the context analysis according to Mayring and an adoption of Vogel, is carried out. The reconstruction of the mathematical interpretation is presented with a focus on a geometrical problem that third graders are working on. Finally, the results of several cases are compared to identify possible differences between the analysed actions when using digital and analogue material arrangements.

Keywords

semiotics
mathematics education
actions on material
mathematical interpretation of diagrams
primary school

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 16, Issue 3, October 2021, Article No: em0650

https://doi.org/10.29333/iejme/11068

Publication date: 13 Jul 2021

Article Views: 2170

Article Downloads: 1246

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Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Aebli, H. (1980). Denken, das ordnen des tuns. Band 1 Kognitive aspekte der handlungstheorie [Thinking, ordering what is done. Volume 1 Cognitive Aspects of Action Theory]. Klett.
Bannert, M. (2009). Prompting self-regulated learning through prompts. Zeitschrift für Psychologie, 23(2), 139-145. https://doi.org/10.1024/1010-0652.23.2.139
Billion, L., & Vogel, R. (2020a). Grundschulkinder arbeiten digital an einem geometrischen problem – Rekonstruktion mathematischer deutungen [Elementary school children work digitally on a geometric problem - reconstruction of mathematical interpretations]. In S. Ladel, C. Schreiber, R. Rink, & D. Walter (Eds.), Aktuelle forschungsprojekte zu digitalen medien in der primarstufe (pp. 135-150). WTM-Verlag. https://doi.org/10.37626/GA9783959871747.0
Billion, L., & Vogel, R. (2020b). Material as an impulse for mathematical activities in primary school – A semiotic perspective on a geometric example. In M. Inprasitha, N. Changsri, & N. Boonsena (Eds), Interim Proceedings of the 44^th Conference of the International Group for the Psychology of Mathematics Education (pp. 28-36). PME.
Billion, L. (2018) Mathematical learning processes with varying types of material conditioning. 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. 5, pp. 207). PME.
Carey, S. (1988). Conceptual differences between children and adults. Mind and Language, 3(3), 167-181. https://doi.org/10.1111/j.1468-0017.1988.tb00141.x
De Ruiter, J. P. (2000). The production of gesture and speech. In D. McNeill (Ed.), Language and Gesture (pp. 284-311). University Press. https://doi.org/10.1017/CBO9780511620850
Dörfler, W. (1988). Die genese mathematischer objekte und operationen aus handlungen als kognitive konstruktion [The genesis of mathematical objects and operations from actions as a cognitive construction]. In W. Dörfler (Ed.), Schriftreihe Didaktik der Mathematik Band 16. Kognitive Aspekte mathematischer Begriffsentwicklung (pp. 55-126). Verlag Hölder-Pichler-Tempsky.
Dörfler, W. (1991). Der computer als kognitives werkzeug und kognitives medium [The computer as a cognitive tool and medium]. In W. Dörfler, W. Peschek, E. Schneider, K. Wegenkittl (Eds.), Computer – Mensch – Mathematik (pp. 51-75). Verlag Hölder-Pichler-Tempsky.
Dörfler, W. (2006a). Diagramme und mathematikunterricht [Diagrams and math lessons]. Journal der Mathematikdidaktik, 3/4, 200-219. https://doi.org/10.1007/BF03339039
Dörfler, W. (2006b). Inscriptions as objects of mathematical activities. In J. Maaz & W. Schlögelmann (Eds.), New mathematics education research and practice (pp. 97-111). Sense Publisher. https://doi.org/10.1163/9789087903510_011
Dörfler, W. (2015). Abstrakte objekte in der mathematik [Abstract objects in mathematics]. In G. Kadunz (Ed.), Semiotische perspektiven auf das lernen von mathematik (pp. 33-49). Springer-Verlag. https://doi.org/10.1007/978-3-642-55177-2
Floer, J. (1993). Lernmaterialien als stützen der anschauung im arithmetischen anfangsunterricht [Learning materials as support for the intuition in arithmetic beginners lessons]. In J.-H. Lorenz (Ed.), Anschauung und mathematik (Band 18) (pp. 106-121). Aulis Verlag Deubner & Co KG.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155-177. https://doi.org/10.1207/s15327833mtl0102_4
Gravemeijer, K. (2002). Preamble: from models to modelling. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 7-22). Kluwer Academic Publisher. https://doi.org/10.1007/978-94-017-3194-2
Hoffmann, M. (2005). Erkenntnisentwicklung. Ein semiotisch-pragmatischer ansatz [Knowledge development. A semiotic-pragmatic approach]. Vittorio Klostermann.
Hohenwarter, M. (2001). GeoGebra – Dynamic mathematics for everyone. Austria & USA.
Huber, H. (1972). OrbiMath. Mathematik konstruktiv [OrbiMath. Math constructive]. Herder Verlag.
Huth, M. (2018). Die bedeutung von gestik bei der konstruktion von fachlichkeit in mathematischen gesprächen junger lernender [The importance of gesture bi the construction of technicality in mathematical conversations of young learners]. In M. Martens, K. Rabenstein, K. Bräu, M. Fetzer, H. Gersch, I. Hardy, & C. Schelle (Eds.), Konstruktion von fachlichkeit. Ansätze, erträge und diskussionen in der empirischen unterrichtsforschung (pp. 219-231). Verlag Julius Klinkhardt.
Huth, M. (2020). Gestische darstellungen in mathematischen interaktionen [Gestural representations in mathematical interactions]. In H.-S. Siller, W. Weigel, & J. F. Wörler (Eds.), Beiträge zum mathematikunterricht (pp. 1381-1384). WTM-Verlag. https://doi.org/10.37626/GA9783959871402.0
Kadunz, G. (2015). Zum verhältnis von geometrischen zeichen und argumentationen [On the relationship between geometric signs and arguments]. In G. Kadunz (Ed.), Semiotische perspektiven auf das lernen von mathematik (pp. 71-88). Springer-Verlag. https://doi.org/10.1007/978-3-642-55177-2
Kadunz, G. (2016). Geometry, A means of argumentation. In A. Sáenz-Ludloy & G. Kadunz (Eds.), Semiotics as a tool for learning mathematics. How to describe the construction, visualisation, and communication of mathematical concepts (pp. 25-42). Sense Publishers. https://doi.org/10.1007/978-94-6300-337-7
Krämer, S. (2007). Immanenz und transzendenz der spur: Über das epistemologische doppelleben der spur [Immanence and transcendence of the trace: About the epistemological double life of the trace]. In S. Krämer, W. Kogge & G. Grube (Eds.), Spur. Spurenlesen als orientierungstechnik und wissenskunst (pp. 155-181). Suhrkamp.
Latour, B. (2012). Visualisation and cognition: Drawing things together. http://worrydream.com/refs/Latour%20-%20Visualisation%20and%20Cognition.pdf
Lorenz, J.-H. (1993). Veranschaulichungsmittel in arithmetischen anfangsunterricht [Illustrative means in beginning arithmetic lessons]. In J.-H. Lorenz (Ed.), Anschauung und mathematik (Band 18) (pp. 122-146). Aulis Verlag Deubner & Co KG.
Mayring, Ph. (2014). Qualitative content analysis: theoretical foundation, basic procedures and software solutions. Klagenfurt. https://nbn-resolving.org/urn:nbn:de:0168-ssoar-395173
Meyer, M., & Prediger, S. (2012). Sprachenvielfalt im mathematikunterricht – Herausforderung, chancen und förderansätze [Language diversity in mathematics lessons - challenges, opportunities and support approaches]. Praxis der mathematik in der schule, PM 54(45), 2-9.
Pape, H. (2007). Fußabdrücke und eigennamen: Peirces theorie des relationalen kerns der bedeutung indexikalischer zeichen [Footprints and proper names: Peirce's relational core theory of the meaning of indexical signs]. In S. Krämer, W. Kogge, & G. Grube (Eds.) Spur. Spurenlesen als orientierungstechnik und wissenskunst (pp. 37-54). Suhrkamp.
Peirce, C. S. (1865-1909). The logic notebook. MS [R] 339.
Peirce, C. S. (1901). Index (in exact logic). In J. M. Baldwin (Ed.), Dictionary of philosophy and psychology (Vol. I, pp. 531-532). Macmillan and Co.
Piaget, J. (1998). Der aufbau der wirklichkeit beim kinde [The construction of reality in the child] (2nd Ed.). Klett.
Posner, G. J., Strike, K. A., Hewson, P. W., & Gertzog, W. A. (1982). Accommodation of a scientific conception: Toward a theory of conceptual change. Science Education, 66(2), 211-227. https://doi.org/10.1002/sce.3730660207
Radatz, H. (1993). MARC bearbeitet aufgaben wie 72 - 59. Anmerkungen zu anschauung und verständnis im arithmetikunterricht [MARC handles tasks like 72 - 59. Notes on visualization and understanding in arithmetic lessons]. In J.-H. Lorenz (Ed.), Anschauung und mathematik (Band 18) (pp. 14-24). Aulis Verlag Deubner & Co KG.
Rosch, E. (1978). Principles of categorization. In E. Rosch & B. Lloyd (Eds.), Cognition and categorization (pp. 27-48). Erlbaum.
Schreiber, C. (2013). Semiotic processes in chat-based problem-solving situations. Educational Studies Mathematics, 82(1), 51-73. https://doi.org/10.1007/s10649-012-9417-7
Seiler, T. (2001). Begreifen und verstehen. Ein buch über begriffe und bedeutungen [Comprehend and understand. A book about terms and meanings] (pp. 129-154). Allgemeine Wissenschaft.
Shapiro, S. (1997). Philosophy of mathematics. Structure and ontology. Oxford University Press.
Sinclair, N., & de Freitas, E. (2014). The haptic nature of gestures. Rethinking gesture with new multitouch digital technologies. Gesture, 14(3), 351-374. https://doi.org/10.1075/gest.14.3.04sin
Thomas, R. (2009). Mathematics is not a game but… The Mathematical Intelligencer, 31(1), 4-8. https://doi.org/10.1007/s00283-008-9015-9
Villamor, G., Willis, D., & Wroblewski, L. (2010). Touch gesture reference guide. https://www.lukew.com/ff/entry.asp?1071
Vogel, R. (2017). "wenn man da von oben guckt sieht das aus als ob …“ – Die 'dimensionslücke‘ zwischen zweidimensionaler darstellung dreidimensionaler objekte im multimodalen austausch ["When you look from above, it looks like ..." - The 'dimension gap' between two-dimensional representation of three-dimensional objects in multimodal exchange]. In M. Beck & R. Vogel (Eds.), Geometrische aktivitäten und gespräche von kindern im blick qualitativen forschens. Mehrperspektivische ergebnisse aus den projekten erStMaL und MaKreKi (pp. 61-75). Waxmann-Verlag.
Vogel, R., & Huth, M. (2020). Modusschnittstellen in mathematischen lernprozessen. Handlungen am material und gesten als diagrammatische tätigkeit [Mode interfaces in mathematical learning processes. Actions on material and gestures as diagrammatic activity]. In G. Kadunz (Ed.), Zeichen und sprache im mathematikunterricht – Semiotik in theorie und praxis (pp. 215-255). Springer Spektrum. https://doi.org/10.1007/978-3-662-61194-4
Wessel, L., Büchter, A., & Prediger, S. (2018). Weil sprache zählt – Sprachsensibel mathematikunterricht planen, durchführen und auswerten [Because language counts - language-sensitive planning, implementation and evaluation of math lessons]. Mathematik Lehren, 206, 2-7.
Wittmann, E. Ch. (1981). Grundfragen des mathematikunterrichts [Basic questions of math class] (6th Ed.). Vieweg. https://doi.org/10.1007/978-3-322-91539-9

How to cite this article

APA

Billion, L. K. (2021). Reconstruction of the Interpretation of Geometric Diagrams of Primary School Children Based on Actions on Various Materials -
A Semiotic Perspective on Actions. International Electronic Journal of Mathematics Education, 16(3), em0650. https://doi.org/10.29333/iejme/11068

Vancouver

Billion LK. Reconstruction of the Interpretation of Geometric Diagrams of Primary School Children Based on Actions on Various Materials -
A Semiotic Perspective on Actions. INT ELECT J MATH ED. 2021;16(3):em0650. https://doi.org/10.29333/iejme/11068

AMA

Billion LK. Reconstruction of the Interpretation of Geometric Diagrams of Primary School Children Based on Actions on Various Materials -
A Semiotic Perspective on Actions. INT ELECT J MATH ED. 2021;16(3), em0650. https://doi.org/10.29333/iejme/11068

Chicago

Billion, Lara Kristina. "Reconstruction of the Interpretation of Geometric Diagrams of Primary School Children Based on Actions on Various Materials -
A Semiotic Perspective on Actions". International Electronic Journal of Mathematics Education 2021 16 no. 3 (2021): em0650. https://doi.org/10.29333/iejme/11068

Harvard

Billion, L. K. (2021). Reconstruction of the Interpretation of Geometric Diagrams of Primary School Children Based on Actions on Various Materials -
A Semiotic Perspective on Actions. International Electronic Journal of Mathematics Education, 16(3), em0650. https://doi.org/10.29333/iejme/11068

MLA

Billion, Lara Kristina "Reconstruction of the Interpretation of Geometric Diagrams of Primary School Children Based on Actions on Various Materials -
A Semiotic Perspective on Actions". International Electronic Journal of Mathematics Education, vol. 16, no. 3, 2021, em0650. https://doi.org/10.29333/iejme/11068