Investigating Pre-Service Teachers’ Instrumented Action Schemes Related to Inverse Functions
DOI:
https://doi.org/10.5281/zenodo.17948158Keywords:
Instrumental approach, pre-service teachers, technology-assisted teaching, representations, instrumented action schemesAbstract
This study aims to examine pre-service teachers’ processes of representing inverse functions and the instrumental genesis that emerges during these processes. Focusing on the instrumented action schemes developed for writing and graphing inverse functions in both paper-and-pencil and dynamic geometry environments, the study evaluates the contribution of technological tools to mathematics teaching. The research was conducted with six pre-service elementary mathematics teachers enrolled in the second year of a public university in Türkiye. Data were collected through worksheets, written and oral interviews, screen recordings, and audio recordings, and were analyzed using content analysis. The data were coded, categorized, and organized into themes within the framework of the components of the instrumental approach, then interpreted in line with the research purpose. The findings revealed that the instrumented action schemes developed by pre-service teachers in the paper-and-pencil environment remained at an operational level, whereas in the Cabri-Geometry environment they developed conceptual, interactive, and multi-representational schemes. The visual and interactive features of the dynamic environment facilitated pre-service teachers’ understanding of algebraic concepts and their transitions among representations. This indicates that technology integration should be adopted as an approach that supports conceptual learning in mathematics teaching.
References
Ainsworth, S., 1999. The functions of multiple representations. Computers ve Education, 33(2–3): 131–152.
Baccaglini-Frank, A., Antonini, S., 2016. From conjecture generation by maintaining dragging to proof. Arxiv Preprint Arxiv, 1605.02583.
Barabash, M., 2019. Dragging as a geometric construction tool: Continuity considerations inspired by students’ attempts. Digital Experiences in Mathematics Education, 5(2): 124–144.
Bokhove, C., Drijvers, P., 2012. Effects of a digital intervention on the development of algebraic expertise. Computers ve Education, 58(1): 197–208.
Cresswell, J., 2013. Qualitative inquiry ve research design: Choosing among five approaches. SAGE.
Dede, Y., Argün, Z., 2003. Cebir, öğrencilere niçin zor gelmektedir? Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 24: 180–185.
Dreyfus, T., Tsamir, P., 2004. Ben’s consolidation of knowledge structures about infinite sets. The Journal of Mathematical Behavior, 23(3): 271–300.
Dreyfus, T., Hershkowitz, R., Schwarz, B., 2015. The nested epistemic actions model for abstraction in context: Theory as methodological tool and methodological tool as theory. In A. Bikner-Ahsbahs, C. Knipping, ve N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods. Springer, pp. 185–217.
Drijvers, P., 2019. Embodied instrumentation: Combining different views on using digital technology in mathematics education. In Eleventh Congress of the European Society for Research in Mathematics Education (No. 1). Freudenthal Group; Freudenthal Institute; ERME.
Drijvers, P., Sinclair, N., 2024. The role of digital technologies in mathematics education: Purposes and perspectives. ZDM–Mathematics Education, 56(2): 239–248.
Drijvers, P., Trouche, L., 2008. From artifacts to instruments: A theoretical framework behind the orchestra metaphor. In G. W. Blume ve M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Cases and perspectives. Information Age Publishing, pp. 363–392.
Eisenberg, T. Dreyfus, T., 1991. On the reluctance to visualize in mathematics. In W. Zimmermann ve S. Cunningham (Eds.), Visualization in teaching and learning mathematics. Mathematical Association of America, pp. 25–37.
Engelhardt, P.V., Corpuz, E.G., Ozimek, D.J., Rebello, N.S., 2004. The teaching experiment—What it is and what it isn’t. AIP Conference Proceedings, American Institute of Physics, pp. 157–160.
Fitrianna, A.Y., Rosjanuardi, R., Prabawanto, S., 2024. Algebraic reasoning of high school students in solving inverse function problems: Viewed from mathematical resilience. Indonesian Journal of Science and Mathematics Education, 7(2): 322–336.
Frankel, R. M., Devers, K.J., 2000. Study design in qualitative research-1: Developing questions and assessing resource needs. Education for Health, 13(2): 251–261.
Haspekian, M., 2005. An “instrumental approach” to study the integration of a computer tool into mathematics teaching: The case of spreadsheets. International Journal of Computers for Mathematical Learning, 10(2): 109–141.
Janvier, C.E., 1987. Problems of representation in the teaching and learning of mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics, Lawrence Erlbaum Associates, pp. 1–9.
Kaput, J.J., 1998. Representations, inscriptions, descriptions and learning: A kaleidoscope of windows. Journal of Mathematical Behavior, 17(2): 265–281.
Kieran, C., 1992. The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning. Macmillan Publishing Company, pp. 390–419.
Kieran, C., Drijvers, P., 2006. The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11: 205–263.
Komatsu, K., Jones, K., 2020. Interplay between paper-and-pencil activity and dynamic-geometry-environment use during generalisation and proving. Digital Experiences in Mathematics Education, 6(2): 123–143.
Laborde, C., 2002. Integration of technology in the design of geometry tasks with Cabri-Geometry. International Journal of Computers for Mathematical Learning, 6: 283–317.
MacGregor, M., Stacey, K., 1993. Cognitive models underlying students’ formulation of simple linear equations. Journal for Research in Mathematics Education, 24(3): 217–232.
Mariotti, M.A., 2000. Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44(1): 25–53.
National Council of Teachers of Mathematics, 2000. Principles and standards for school mathematics.
Ndlovu, M., Wessels, D., De Villiers, M., 2011. An instrumental approach to modelling the derivative in Sketchpad. Pythagoras, 32(2): 1–15.
Nolasco, J., 2018. The struggle with inverse functions doing and undoing process. Master’s Thesis, California State University, San Bernardino.
Öcal, M.F., Kar, T., Güler, G., İpek, A.S., 2020. Comparison of prospective mathematics teachers’ problem posing abilities in paper-pencil test and on dynamic geometry environment in terms of creativity. Journal of Research in Mathematics Education, 9(3): 243–272.
Paoletti, T., Stevens, I.E., Hobson, N.L., Moore, KC., LaForest, K.R., 2018. Inverse function: Pre-service teachers’ techniques and meanings. Educational Studies in Mathematics, 97(1): 93–109.
Pittalis, M., Drijvers, P., 2023. Embodied instrumentation in a dynamic geometry environment: Eleven-year-old students’ dragging schemes. Educational Studies in Mathematics, 113(2): 181–205.
Segal, R., Oxman, V., Stupel, M., 2021. Using dynamic geometry software to enhance specialized content knowledge: Pre-service mathematics teachers’ perceptions. International Electronic Journal of Mathematics Education, 16(3): em0647.
Song, Y.,2021. A review of how class orchestration with technology has been conducted for pedagogical practices. Educational Technology Research and Development, 69(3): 1477–1503.
Steffe, L.P., Ulrich, C., 2014. Constructivist teaching experiment. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer, pp. 102–109.
Tall, D., Gray, E., Ali, M.B., Crowley, L., DeMarois, P., McGowen, M., ... ve Yusof, Y., 2001. Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Math, Science ve Technology Education, 1(1): 81–104.
Tapan, M.S., 2006. Différents types de savoirs mis en œuvre dans la formation initiale d’enseignants de mathématiques à l’intégration de technologies de géométrie dynamique. Yayımlanmamış Doktora Tezi, Université Joseph-Fourier – Grenoble I.
Taşlıbeyaz, E., 2010. Ortaöğretim öğrencilerinin bilgisayar destekli matematik öğretiminde matematik algılarına yönelik durum çalışması: Lise 3. sınıf uygulaması. Yayımlanmamış Yüksek Lisans Tezi, Atatürk Üniversitesi.
Trouche, L., 2004. Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3): 281–307.
Trouche, L., 2005. Instrumental genesis, individual and social aspects. In D. Guin, K. Ruthven, ve L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument. Springer, pp. 197–230.
Verillon, P., Rabardel, P., 1995. Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity. European Journal of Psychology of Education, 10(1): 77–101.
Xu, C., 2025. Improving mathematical skills of mathematics education students through integrated use of graphing software and programming techniques. International Journal of Education and Humanities, 5(1): 277.
Yuhasriati, Y., Johar, R., Khairunnisak, C., Rohaizati, U., Zubaidah, T., 2022. Students’ mathematical representation ability in learning algebraic expression using realistic mathematics education. Jurnal Didaktik Matematika, 9(1): 151–165.
Zhang, Y., Wang, P., Jia, W., Zhang, A., Chen, G., 2025. Dynamic visualization by GeoGebra for mathematics learning: A meta-analysis of 20 years of research. Journal of Research on Technology in Education, (2): 437–458.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Ejons International Journal on Mathematic, Engineering and Natural Sciences
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
