Concept Formation Processes regarding Height and Base in Triangles

Felicitas Pielsticker

doi:10.29333/iejme/10891

Full Text (PDF)

Felicitas Pielsticker ¹ ^*

More Detail

¹ University of Siegen, GERMANY^* Corresponding Author

Abstract

The paper addresses concept formation processes of a student in the field of geometry. More precisely, the paper deals with the question of how to assist a student in a mathematical concept formation process – in the context of area calculations in triangles – with a specially designed learning environment based on the usage of 3D printed objects. Methodologically, we use the descriptive framework of the concept of Domains of Subjective Experience for the analysis of our case study. The objective is to describe the use of a learning environment to initiate targeted, theoretical concepts with a student. The student develops an empirical notion of bases and heights in triangles (with respect to the calculation of surface areas), using 2D drawings and 3D printed objects. In this article we argue, teaching mathematics consistently in an object-oriented and practical (visual) way, and additionally promoting the development of concept formation as a mathematical activity, brings a student in a situation where he develops hypotheses, tries them out, and transfers them to other fields of application; thus engages in concept formation processes.

Keywords

concept formation processes
empirical objects
3D printing

License

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, 2021, Volume 16, Issue 2, Article No: em0633

https://doi.org/10.29333/iejme/10891

Publication date: 07 May 2021

Article Views: 1480

Article Downloads: 1016

Open Access Disclosures References How to cite this article

Disclosure

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

Bauersfeld, H. (1983). Subjektive erfahrungsbereiche als grundlage einer interaktionstheorie des mathematiklernens und -lehrens [Domains of subjective experience as the basis of an interaction theory of learning and teaching mathematics]. In H. Bauersfeld, H. Bussmann, & G. Krummheuer (Eds.), Lernen und Lehren von Mathematik: Analysen zum Unterrichtshandeln II (pp. 1-56). Aulis.
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.
Böer, H. (2017). Mathe live 8: Mathematik für die Sekundarstufe I [Maths live 8: Mathematics for lower secondary level]. Klett.
Bruner, J. S. (1971). Über kognitive entwicklung [On cognitive development]. In J. S. Bruner, R. R. Oliver, & P. M. Greenfield (Eds.), Studien zur kognitiven Entwicklung (pp. 21-53). Klett.
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
Coles, A. (2015). On enactivism and language: Towards a methodology for studying talk in mathematics classrooms. ZDM, 47, 235-246. https://doi.org/10.1007/s11858-014-0630-y
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
Fetzer, M., & Tiedemann, K. (2017). Talking with objects. CERME 10, Feb 2017. Dublin, Ireland.
Gopnik, A. (2003). The theory theory as an alternative to the innateness hypothesis. https://doi.org/10.1002/9780470690024.ch10
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 (pp. 15-30). Springer. https://doi.org/10.1007/978-3-658-10261-6_2
Holland, G. (2007). Geometrie in der sekundarstufe: Entdecken – Konstruieren – Deduzieren: Didaktische und methodische fragen [Geometry in secondary school: Discovering – Constructing – Deducing: educational and methodological questions]. Franzbecker.
Kadunz, G., & Sträßer, R. (2008). Didaktik der geometrie in der sekundarstufe I [Geometry in secondary school]. Franzbecker.
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
Lawler, R. W. (1980). The progressive construction of mind. Cognitive Science, 5, 1-34. https://doi.org/10.1111/j.1551-6708.1981.tb00867.x
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. Routledge.
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 (pp. 49-82). Hildesheim: Franzbecker.
National Council of Teachers of Mathematics (2000). Executive Summary. Principles and Standards for School Mathematics. NCTM.
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
Polya, G. (1945). How to solve it. https://notendur.hi.is/hei2/teaching/Polya_HowToSolveIt.pdf
Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
Schoenfeld, A. H. (2010). How we think: A theory of goal-oriented decision making and its educational applications. Routledge. https://doi.org/10.4324/9780203843000
Stake, R. E. (1995). The art of case study research. Sage.
Steinbring, H. (2015). Mathematical interaction shaped by communication, epistemological constraints and enactivism. ZDM, 47, 281-293. https://doi.org/10.1007/s11858-014-0629-4
Struve, H. (1990). Grundlagen einer geometriedidaktik [Foundations of school geometry]. Lehrbücher und Monographien zur Didaktik der Mathematik, 17. BI-Wissenschaftsverlag.
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 (Vol. 3, pp. 1839-1893). https://doi.org/10.1007/978-94-007-7654-8_57
Tall, D. (2002). Using technology to support an embodied approach to learning concepts in mathematics. In L. M. Carvalho, & L. C. Guimaraes (Eds.), História e tecnologia no ensino da matemática, vol. 1 (pp. 1-28). Cincia Moderna.
Tall, D. (2013). How humans learn to think mathematically. Exploring the three worlds of mathematics. Cambridge University Press. https://doi.org/10.1017/CBO9781139565202
Tall, D. (2020). Building long-term meaning in mathematical thinking: Aha! and Uh-Huh!.
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
Tiedemann, K. (2016). “Ich habe mir einfach die Rechenmaschine in meinem Kopf gebaut!” Zur Entwicklung fachsprachlicher Fähigkeiten bei Grundschulkindern [“I just built the adding machine in my head!” For developing technical language skills in elementary school children]. Beiträge zum Mathematikunterricht 2016 (pp. 991-994). WTM-Verlag. http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2020b-aha-uhu.pdf
Vollrath, H.-J. (1984). Methodik des Begriffslehrens im Mathematikunterricht [Methodology of teaching concepts in mathematics lessons]. Klett.
Weigand, H.-G., Filler, A., Hölzl, R., Kuntze, S., Ludwig, M., Roth, J., Schmidt-Thieme, B., & Wittmann, G. (2014). Didaktik der geometrie für die sekundarstufe I [Geometry in secondary school]. Springer. https://doi.org/10.1007/978-3-642-37968-0
Wittenberg, A. I. (1957). Vom denken in begriffen [About thinking in concepts]. Birkhäuser.
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.

How to cite this article

APA

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

Vancouver

Pielsticker F. Concept Formation Processes regarding Height and Base in Triangles. INT ELECT J MATH ED. 2021;16(2):em0633. https://doi.org/10.29333/iejme/10891

AMA

Pielsticker F. Concept Formation Processes regarding Height and Base in Triangles. INT ELECT J MATH ED. 2021;16(2), em0633. https://doi.org/10.29333/iejme/10891

Chicago

Pielsticker, Felicitas. "Concept Formation Processes regarding Height and Base in Triangles". International Electronic Journal of Mathematics Education 2021 16 no. 2 (2021): em0633. https://doi.org/10.29333/iejme/10891

Harvard

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

MLA

Pielsticker, Felicitas "Concept Formation Processes regarding Height and Base in Triangles". International Electronic Journal of Mathematics Education, vol. 16, no. 2, 2021, em0633. https://doi.org/10.29333/iejme/10891