Description of the activated mathematical knowledge of the triangle concept in three empirical contexts

Felicitas Pielsticker

doi:10.29333/iejme/12170

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Felicitas Pielsticker ¹ ^*

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¹ University of Siegen, Siegen, GERMANY^* Corresponding Author

Abstract

The paper addresses concept formation processes of students in the field of geometry. More precisely, the paper deals with the questions of what knowledge do students activate about triangles in different contexts with different (digital) tools and furthermore what content-related meaning do they give to the concept of triangles? Methodologically, we use the descriptive framework of empirical theories for the analysis of our case study. The students develop different notion of the concept of triangles, using three empirical contexts like 3D pen, dynamic geometry software and pencil paper in an activity that allow students to construct the characteristics of triangles. In this article, we focus on how the students in our case study (further) develop and give meaning to the concept of triangle, which is central in geometry, in three empirical contexts; thus, engage in concept formation processes.

Keywords

concept formation processes
empirical contexts
3D pen
triangle
geometry

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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, 2022, Volume 17, Issue 4, Article No: em0697

https://doi.org/10.29333/iejme/12170

Publication date: 16 Jun 2022

Article Views: 743

Article Downloads: 701

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Funding

No funding source is reported for this study.

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Declaration of interest: No conflict of interest is declared by the author.

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

References

Balzer, W. (1982). Empirische Theorien: Modelle–Strukturen–Beispiele. Die Grundzüge der modernen Wissenschaftstheorie [Empirical theories: Models–structures–examples. The main features of modern philosophy of science]. Friedr. Vieweg & Sohn. https://doi.org/10.1007/978-3-663-00169-0_6
Bauersfeld, H. (1985). Ergebnisse und Probleme von Mikroanalysen mathematischen Unterrichts. In W. Dörfler, & R. Fischer (Eds.), Empirische Untersuchungen zum Lehren und Lernen von Mathematik [Empirical investigations into the teaching and learning of mathematics] (pp. 7-25). Hölder-Pichler-Tempsky.
Bauersfeld, H. (1988). Interaction, construction, and knowledge: Alternative perspectives for mathematics education. In D. A. Grouws, & T. J. Cooney (Eds.), Perspectives on research on effective mathematics teaching (pp. 27-46). Lawrence Erlbaum.
Burscheid, H. J., & Struve, H. (2020). Mathematikdidaktik in Rekonstruktionen: Grundlegung von Unterrichtsinhalten [Mathematics education in reconstruction: A contribution to its foundation]. Springer. https://doi.org/10.1007/978-3-658-29452-6
Clements, D. H., Sarama, J., & Joswick, C. (2018). Learning and teaching geometry in early childhood. Quadrante, XXVII(2), 7-31.
Clements, D.H., & Sarama, J. (2011). Early childhood teacher education: The case of geometry. Journal of Mathematics Teacher Education, 14(2), 133-148. https://doi.org/10.1007/s10857-011-9173-0
Dilling, F., & Witzke, I. (2020). The use of 3D-printing technology in calculus education–Concept formation processes of the concept of derivative with printed graphs of functions. Digital Experiences in Mathematics Education, 6(3), 320-339. https://doi.org/10.1007/s40751-020-00062-8
Elschenbroich, H.-J. (2018). Leibniz calculus. GeoGebra book. https://www.geogebra.org/m/hymsqdyg
Elschenbroich, H.-J., & Seebach, G. (2018). Funktionen erkunden. Ideenreiche Arbeitsblätter mit GeoGebra [Explore functions. Worksheets with GeoGebra]. Friedrich Verlag.
Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24, 139-162. https://doi.org/10.1007/BF01273689
Fischer, R., & Malle, G. (1985). Mensch und Mathematik. Eine Einführung in didaktisches Denken und Handeln [Man and mathematics. An introduction to didactic thinking and acting]. Bibliographisches Institut.
Hefendehl-Hebeker, L. (2016). Mathematische Wissensbildung in Schule und Hochschule [Mathematical knowledge development in school and university]. In A. Hoppenbrock, R. Biehler, R. Hochmuth, & H.-G. Rück (Eds.), Lehren und Lernen von Mathematik in der Studieneingangsphase [Teaching and learning mathematics in the introductory phase] (pp. 15-30). Springer. https://doi.org/10.1007/978-3-658-10261-6_2
Hempel, C. G. (1945). Geometry and empirical science. The American Mathematical Monthly, 52(1), 7-17. https://doi.org/10.1080/00029890.1945.11991492
Kaenders, R., & Schmidt, R. (2014). Mit GeoGebra mehr Mathematik verstehen. Beispiele für die Förderung eines tieferen Mathematikverständnisses aus dem GeoGebra Institut Köln/Bonn [Understanding more mathematics with GeoGebra. Examples for the promotion of a deeper understanding of mathematics from the GeoGebra Institute Cologne/Bonn]. Springer. https://doi.org/10.1007/978-3-658-04222-6
KMK. (2005). Bildungsstandards im Fach Mathematik für den Primarbereich [Educational standards in mathematics for primary education]. https://www.kmk.org/fileadmin/veroeffentlichungen_beschluesse/2004/2004_10_15-Bildungsstandards-Mathe-Primar.pdf
Krummheuer, G. (1984). Zur unterrichtsmethodischen Dimension von Rahmungsprozessen [On the methodological dimension of framing processes]. Journal für Mathematik-Didaktik, 84(4), 285-306. https://doi.org/10.1007/BF03339250
MacDonald, M., Hill, C., & Sinclair, N. (2020). The problem and potential of representation: Being and becoming. In S. Smythe, K. Toohey, D. Dagenais, & M. Forte (Eds.), Rethinking language and literacy. Pedagogies with new materialities (pp. 153-174). Routledge. https://doi.org/10.4324/9780429491702-9
Maier, A. S., & Benz, C. (2014). Children’s constructions in the domain of geometric competencies in two different instructional settings. In U. Kortenkamp, B. Brandt, C. Benz, G. Krummheuer, S. Ladel, & R. Vogel (Eds.), Early mathematics learning: Selected papers of the POEM conference 2012 (pp. 173-187). Springer. https://doi.org/10.1007/978-1-4614-4678-1
Meyer, M. (2010). Wörter und ihr Gebrauch–Analyse von Begriffsbildungsprozessen im Mathematikunterricht [Words and their use–Analysis of concept formation processes in mathematics lessons]. In G. Kadunz (Ed.). Sprache und Zeichen [Language and characters] (pp. 49-82). Franzbecker.
Panorkou, N., & Pratt, D. (2016). Using Google SketchUp to develop students’ experiences of dimension in geometry. Digital Experiences in Mathematics Education, 2(3), 199-227. https://doi.org/10.1007/s40751-016-0021-9
Pielsticker, F. (2020). Mathematische Wissensentwicklungsprozesse von Schülerinnen und Schülern. Fallstudien zu empirisch-orientiertem Mathematikunterricht mit 3D-Druck [Students’ processes of knowledge development in relation to empirically-oriented mathematics classes using the example of the 3D printing technology]. Springer. https://doi.org/10.1007/978-3-658-29949-1
Pielsticker, F. (2021). Concept formation processes regarding height and base in triangles. International Electronic Journal of Mathematics Education, 16(2), em0633. https://doi.org/10.29333/iejme/10891
Polya, G. (1945). How to solve it. https://doi.org/10.1515/9781400828678
Poon, K.-K., & Leung, C.-K. (2016). A study of geometric understanding via logical reasoning in Hong Kong. International Journal for Mathematics Teaching and Learning, 17(3), 92-123.
Satlow, E., & Newcombe, N. (1998). When is a triangle not a triangle? Young children’s developing concepts of geometric shape. Cognitive Development, 13(4), 547-559. https://doi.org/10.1016/S0885-2014(98)90006-5
Schiffer, K. (2019). Probleme beim Übergang von Arithmetik zu Algebra [Problems in the transition from arithmetic to algebra]. Springer. https://doi.org/10.1007/978-3-658-27777-2
Schlicht, S. (2016). Zur Entwicklung des Mengen und Zahlbegriffs [On the development of the concept of quantity and number]. Springer. https://doi.org/10.1007/978-3-658-15397-7
Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
Sneed, J.D. (1971). The logical structure of mathematical physics. Reidel. https://doi.org/10.1007/978-94-010-3066-3
Stake, R. E. (1995). The art of case study research. SAGE.
Stegmüller, W. (1986). Theorie und Erfahrung: Probleme und Resultate der Wissenschaftstheorie und Analytischen Philosophie, Band II, 3. Teilband: Die Entwicklung des neuen Strukturalismus seit 1973 [Theory and experience: Problems and results in the philosophy of science and analytic philosophy, volume II, part 3: The development of the new structuralism since 1973]. Springer.
Stoffels, G. (2020). (Re-)konstruktion von Erfahrungsbe-reichen bei Übergängen von empirisch-gegenständlichen zu formal-abstrakten Auffassungen theoretisch grundlegen, historisch reflektieren und beim Übergang Schule-Hochschule anwenden [(Re)constructing domains of experience in transitions from empirical-objective to formal-abstract understandings theoretically, reflecting historically and applying in the transition from school to higher education]. Universitätsverlag Siegen.
Struve, H. (1990). Grundlagen einer Geometriedidaktik [Foundations of school geometry]. Vide Descriptio.
Taber, K. S. (2014). Methodological issues in science education research: a perspective from the philosophy of science. In M. R. Matthews (Ed.), International handbook of research in history, philosophy and science teaching (pp. 1839-1893). Springer. https://doi.org/10.1007/978-94-007-7654-8_57
Tall, D. (2013). How humans learn to think mathematically. Exploring the three worlds of mathematics. Cambridge University Press. https://doi.org/10.1017/CBO9781139565202
Thompson, P. W. (1994). Concrete materials and teaching for mathematical understanding. Arithmetic Teacher, 41(9), 556-558. https://doi.org/10.5951/AT.41.9.0556
Winter, H. (1995). Mathematikunterricht und Allgemeinbildung [Mathematics education and general education]. Mitteilungen der Gesellschaft für Didaktik der Mathematik [Communications of the Society for Didactics of Mathematics], 61, 37-46.
Witzke, I. (2009). Die Entwicklung des Leibnizschen Calculus: Eine Fallstudie zur Theorieentwicklung in der Mathematik [The development of the Leibniz calculus: A case study on the development of theory in mathematics]. Franzbecker.
Youkap, P., T. (2021). Student comprehension of the concept of a geometrical figure: The case of straight lines and parallel line. Journal of Mathematics Education, 6(2) 149-158. https://doi.org/10.31327/jme.v6i2.1408

How to cite this article

APA

Pielsticker, F. (2022). Description of the activated mathematical knowledge of the triangle concept in three empirical contexts. International Electronic Journal of Mathematics Education, 17(4), em0697. https://doi.org/10.29333/iejme/12170

Vancouver

Pielsticker F. Description of the activated mathematical knowledge of the triangle concept in three empirical contexts. INT ELECT J MATH ED. 2022;17(4):em0697. https://doi.org/10.29333/iejme/12170

AMA

Pielsticker F. Description of the activated mathematical knowledge of the triangle concept in three empirical contexts. INT ELECT J MATH ED. 2022;17(4), em0697. https://doi.org/10.29333/iejme/12170

Chicago

Pielsticker, Felicitas. "Description of the activated mathematical knowledge of the triangle concept in three empirical contexts". International Electronic Journal of Mathematics Education 2022 17 no. 4 (2022): em0697. https://doi.org/10.29333/iejme/12170

Harvard

Pielsticker, F. (2022). Description of the activated mathematical knowledge of the triangle concept in three empirical contexts. International Electronic Journal of Mathematics Education, 17(4), em0697. https://doi.org/10.29333/iejme/12170

MLA

Pielsticker, Felicitas "Description of the activated mathematical knowledge of the triangle concept in three empirical contexts". International Electronic Journal of Mathematics Education, vol. 17, no. 4, 2022, em0697. https://doi.org/10.29333/iejme/12170