Didactical usefulness of interactive mathematical maps – designing activities supporting prospective teachers’ learning
Abstract
Prospective teachers often have difficulty in linking school mathematics and university course content, which manifests itself as a lack of their understanding and the significance of university course content. This double discontinuity is experienced by future teachers in their transition from high school pupil to university student and then from university student to their school teaching career. Thus, it is necessary to improve university teaching and teacher education to try to bridge these ”gaps”. Using the educational context of a geometry course in the teacher education of upper secondary teachers, we explore the technical implementation and usefulness of the components of interactive mathematical maps. Such maps comprise a supplementing didactical tool that shows the interrelations between mathematical discoveries and the development of particular mathematical content – starting from an initial historical problem situated in time. The research findings showed the map in its current format to be perceived as useful and mostly easy to use. Further, the map seemed to promote both a process-oriented and an application-oriented approach as well as favourable beliefs, such as mathematics being an emerging science promoting a view of doing mathematics, in which an open error culture can be established.
References
Ableitinger, C., Hefendehl-Hebeker, L. & Herrmann, A. (2013). Aufgaben zur Vernetzung von Schul- und Hochschulmathematik. In H. Allmendinger, K. Lengnink, A. Vohns & G. Wickel (Eds.), Mathematik verständlich unterrichten (pp. 217-233). Springer Spektrum. https://doi.org/10.1007/978-3-658-00992-2_14
Ausubel, P. D. (1963). The psychology of meaningful verbal learning. Grune & Stratton.
Bauer, T. & Partheil, U. (2009). Schnittstellenmodule in der Lehramts-ausbildung im Fach Mathematik. Math Semesterber, 56, 85-103. https://doi.org/10.1007/s00591-008-0048-0
Brandl, M. (2009). The vibrating string - an initial problem for modern mathematics; historical and didactical aspects. In I. Witzke (Ed.), Mathematical practice and development throughout history: proceedings of the 18th novembertagung on the history, philosophy and didactics of mathematics (pp. 95-114). Logos.
Brandl, M. (2016). Narrative Mathematik-Didaktik mittels Elementen bildender Kunst. In Institut für Mathematik und Informatik der Pädagogischen Hochschule Heidelberg (Eds.), Beiträge zum Mathematikunterricht 2016 (pp. 1415-1418). WTM-Verlag.
Brandl, M. (2017). Narrative Didaktik als Vernetzungsinstrument: die Schule von Athen. In T. Borys, M. Brandl & A. Brinkmann (Eds.), Mathe vernetzt - Anregungen und Materialien für einen vernetzenden Mathematikunterricht (pp. 7-20). MUED.
Bruner, J. (1986). Actual minds, possible worlds. Harvard University Press. https://doi.org/10.4159/9780674029019
Davis, F. D. (1985). A technology acceptance model for empirically testing new end-user information systems: theory and results [Ph D thesis]. Massachusetts Institute of Technology]. http://hdl.handle.net/1721.1/15192
Egan, K. (1989a). The shape of the science text: a function of stories. In S. de Castell, A. Luke & C. Luke, (Eds.), Language, authority and criticism: readings on the school textbook (pp. 96-108). Falmer Press.
Egan, K. (1989b). Memory, imagination, and learning: connected by the story. Phi Delta Kappan, 70 (6), 455-473.
Ernest, P. (2014). What is mathematics, and why learn it? In P. Andrews & T. Rowland (Eds.), Masterclass in mathematics education: international perspectives on teaching and learning (pp. 3-14). Bloomsbury. https://doi.org/10.5040/9781350284807.ch-001
Felbrich, A., Kaiser, G. & Schmotz, C. (2012). The cultural dimension of beliefs: an investigation of future primary teachers' epistemological beliefs concerning the nature of mathematics in 15 countries. ZDM, 44, 355-366. https://doi.org/10.1007/s11858-012-0418-x
Felbrich, A., Müller, C. & Blömeke, S. (2008). Epistemological beliefs concerning the nature of mathematics among teacher educators and teacher education students in mathematics. ZDM, 40, 763-776. https://doi.org/10.1007/s11858-008-0153-5
Hefendehl-Hebeker, L. (2013). Doppelte Diskontinuität oder die Chance der Brückenschläge. In C. Ableitinger, J. Kramer & S. Prediger (Eds.), Zur doppelten Diskontinuität in der Gymnasiallehrerbildung. Konzepte und Studien zur Hochschuldidaktik und Lehrerbildung Mathematik (pp. 1-15). Springer. https://doi.org/10.1007/978-3-658-01360-8_1
Isaev, V. & Eichler, A. (2017). Measuring beliefs concerning the double discontinuity in secondary teacher education. In T. Dooley & G. Gueudet (Eds.), Proceedings of CERME 10 (pp. 2916-2923). DCU Institute of Education and ERME. https://hal.archives-ouvertes.fr/hal-01949039
Klassen, S. (2006). A theoretical framework for contextual science teaching. Interchange, 37 (1-2), 31-61. https://doi.org/10.1007/s10780-006-8399-8
Klein, F. (1932). Elementary mathematics from an advanced standpoint.
Macmillan. [Original work published 1908] https://archive.org/details/elementarymathem032765mbp
Klein, F. (2016). Elementary mathematics from higher standpoint. Volume I: arithmetic algebra analysis. Springer. [Original work published 1924] https://doi.org/10.1007/978-3-662-49442-4
Koch, M., Confrey, J., Clark-Wilson, A., Jameson, E. & Suurtamm, C. (2021). Digital maps of the connections in school mathematics: three projects to enhance teaching and learning. In A. Clark-Wilson, A. Donevska-Todorova, E. Faggiano, J. Trgalová & H.-G. Weigand (Eds.), Mathematics education in the digital age: learning practice and theory (pp. 121-137). Routledge. https://doi.org/10.4324/9781003137580-8
Kubli, F. (2002). Plädoyer für Erzählungen im Physikunterricht: Geschichte und Geschichten als Verstehenshilfen - Ergebnisse einer Untersuchung (2nd ed.). Aulis Deubner.
Kubli, F. (1999). Historical aspects in physics teaching: using Galileo's work in a new Swiss project. Science & Education, 8 (2), 137-150. https://doi.org/10.1023/A:1008613706212
Norris, S., Guilbert, M., Smith, M., Shahram, H. & Phillips, L. (2005). A theoretical framework for narrative explanation in science. Science Education, 89, 4, 535-554. https://doi.org/10.1002/sce.20063
Pinto, A. & Cooper, J. (2022). The road not taken - investigating affordances of infinitesimal calculus for enriching the repertoire of secondary mathematics teachers. International Journal of Research in Undergraduate Mathematics Education, 8 (2), 318-338. https://doi.org/10.1007/s40753-021-00161-w
Przybilla, J., Brandl, M., Vinerean, M. & Liljekvist, Y. (2021). Interactive mathematical maps - a contextualized way of meaningful learning. In G. A. Nortvedt, N. F. Buchholtz, J. Fauskanger, F. Hreinsdóttir, M. Hähkiöniemi et al. (Eds.), Bringing Nordic mathematics education into the future. Proceedings of NORMA 20 (pp. 209-216). SMDF. https://www.uv.uio.no/ils/english/about/events/2021/norma/proceedings
Przybilla, J., Brandl, M., Vinerean, M. & Liljekvist, Y. (2022). Digital mathematical maps - results from iterative research cycles. In J. Hodgen, E. Geraniou, G. Bolondi & F. Ferretti. (Eds.), Proceedings of CERME12 (pp. 1-8). Free University of Bozen-Bolzano and ERME. https://hal.archives-ouvertes.fr/hal-03754749/
Resnick, L. B. & Resnick, D. P. (1992). Assessing the thinking curriculum: new tools for educational reform. In B. R. Gifford & M. C. O'Connor (Eds.), Changing assessments: alternative views of aptitude, achievement and instruction (pp. 37-75). Kluwer. https://doi.org/10.1007/978-94-011-2968-8_3
Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula, T. Buttery & E. Guyton (Eds.), Handbook of research on teacher education (2nd ed., pp. 102-119). Macmillan. https://doi.org/10.12691/education-2-12-14
Scherer, R., Siddiq, F. & Tondeur, J. (2019). The technology acceptance model (TAM): a meta-analytic structural equation modeling approach to explaining teachers' adoption of digital technology in education. Computers & Education, 128, 13-35. https://doi.org/10.1016/j.compedu.2018.09.009
Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20 (2), 5-24. https://doi.org/10.1007/BF03217474
Winsløw, C. & Grønbæk, N. (2014). Klein's double discontinuity revisited: contemporary challenges for universities preparing teachers to teach calculus. Recherches en Didactique des Mathématiques, 34 (1), 59-86. https://www.researchgate.net/publication/243963587
Wood, D., Bruner, J. & Ross, G. (1976). The role of tutoring in problem-solving. Journal of Child Psychology and Psychiatry, 17, 89-100. https://doi.org/10.1111/j.1469-7610.1976.tb00381.x
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.