Learner-generated Examples Within a System for Computer-aided Assessment as a Tool for Engaging In-service Teachers with Linear Functions
Keywords:
linear functions, learner-generated examples, example spaces, computer-aided assessment, in-service teachersAbstract
This paper examines how learner-generated examples (LGE) tasks within a system for computer-aided assessment (CAA) facilitate in-service teachers' (ISTs') engagement with and explorations of the features of linear functions. Employing the concept of example spaces, the study explores how ISTs approach and engage with linear functions through these tasks. Seven ISTs participated in a series of LGE tasks in a CAA system, and their digital responses were collected. Semi-structured interviews were conducted with four participants following task completion. The results reveal common procedures adopted by the ISTs, such as using the "one-unit-right-a-up/down" procedure, as well as varying approaches to generation points and plotting them into a coordinate system. ISTs, however, faced challenges in effectively communicating their mathematical explanations. This study highlights the potential of LGE tasks in a CAA system to enhance ISTs' example spaces and improve their understanding of mathematical concepts.
References
Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111–129. https://doi.org/10.1007/s10649-008-9149-x
Antonini, S. (2006). Graduate students’ processes in generating examples of mathematical objects. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková, Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 57–64). PME.
Ball, D. L. (2003). What mathematical knowledge is needed for teaching mathematics? Secretary’s Summit on Mathematics, U.S. Department of Education.
Birgin, O. (2012). Investigation of eighth-grade students' understanding of the slope of the linear function. Bolema: Boletim de Educação Matemática, 26, 139–162. https://doi.org/10.1590/S0103-636X2012000100008
Birgin, O., Gurbuz, R., & Catlıoglu, H. (2012). Determining eighth grade students’ understandings and difficulties of linear functions. Energy Education Science and Technology Part B: Social and Educational Studies, 4(3), 1345–1354.
Breen, S., & O’Shea, A. (2019). Designing mathematical thinking tasks. PRIMUS, 29(1), 9–20. https://doi.org/10.1080/10511970.2017.1396567
Breen, S., O'Shea, A., & Pfeiffer, K. (2016). Students’ views of example generation tasks. Teaching Mathematics and Its Applications: International Journal of the IMA, 35(1), 27–40. https://doi.org/10.1093/teamat/hrv017
Brunström, M., Fahlgren, M., Vinerean, M., & Wondmagegne, Y. (2022). Designing for a combined use of a dynamic mathematics software environment and a computer-aided assessment system. In J. Hodgen, E. Geraniou, G. Bolondi, & F. E. Ferretti, Proceedings of the Twelfth Congress of the European Research Society in Mathematics Education (CERME12).
Dinkelman, M. O., & Cavey, L. O. (2015). Learning about functions through learner-generated examples. The Mathematics Teacher, 109(2), 104–110.
Fahlgren, M., & Brunström, M. (2023). Designing example-generating tasks for a technology-rich mathematical environment. International Journal of Mathematical Education in Science and Technology, 1–17. https://doi.org/10.1080/0020739X.2023.2255188
Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69, 183–194. https://doi.org/10.1007/s10649-008-9143-3
Iannone, P., Inglis, M., Mejía-Ramos, J. P., Simpson, A., & Weber, K. (2011). Does generating examples aid proof production? Educational Studies in Mathematics, 77, 1–14. https://doi.org/10.1007/s10649-011-9299-0
Jukic Matic, L., Kehler-Poljak, G., & Rukavina, S. (2022). The influence of curriculum on the concept of function: An empirical study of pre-service teachers. European Journal of Science and Mathematics Education, 10(3), 380–395. https://doi.org/10.30935/scimath/12042
Kinnear, G. (2024). Comparing example generation with classification in the learning of new mathematics concepts. Research in Mathematics Education, 26(1), 109–132. https://doi.org/10.1080/14794802.2022.2152086
Kinnear, G., Jones, I., Sangwin, C., Alarfaj, M., Davies, B., Fearn, S., Foster, C., Heck, A., Henderson, K., Hunt, T., Iannone, P., Kontorovich, I., Larson, N., Lowe, T., Meyer, J. C., O'Shea, A., Rowlett, P., Sikurajapathi, I., & Wong, T. (2024). A collaboratively-derived research agenda for e-assessment in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 10(1), 203–231. https://doi.org/10.1007/s40753-022-00189-6
Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. In V. Richardson (Ed.), Handbook of research on teaching (4th ed., pp. 333–357).
Lloyd, G. M., & Wilson, M. (1998). Supporting innovation: The impact of a teacher's conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education, 29(3), 248–274. https://doi.org/10.5951/jresematheduc.29.3.0248
Moschkovich, J. N. (1996). Moving up and getting steeper: Negotiating shared descriptions of linear graphs. The Journal of the Learning Sciences, 5(3), 239–277.
Nilsen, H. K. (2013). Learning and teaching functions and the transition from lower secondary to upper secondary school. [Doctoral dissertation, University of Agder]. http://hdl.handle.net/11250/139758
Ovedal-Hakestad, S., & Larson, N. (2024). Pre-service teachers’ experiences of working with learner-generated example tasks through a computer-aided assessment system. Presented at NORMA24: The Tenth Nordic Conference on Mathematics Education.
Pierce, R., Stacey, K., & Bardini, C. (2010). Linear functions: Teaching strategies and students' conceptions associated with y = mx + c. Pedagogies: An International Journal, 5(3), 202–215. https://doi.org/10.1080/1554480X.2010.486151
Rønning, F. (2017). Influence of computer-aided assessment on ways of working with mathematics. Teaching Mathematics and Its Applications: International Journal of the IMA, 36(2), 94–107. https://doi.org/10.1093/teamat/hrx001.
Sangwin, C. (2013). Computer aided assessment of mathematics. OUP Oxford.
Sangwin, C. (2019). Developing and evaluating an online linear algebra examination for university mathematics. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. Utrecht University and ERME.
Sangwin, C. J., & Köcher, N. (2016). Automation of mathematics examinations. Computers & Education, 94, 215–227. https://doi.org/10.1016/j.compedu.2015.11.014
Schwarz, B., & Dreyfus, T. (1995). New actions upon old objects: A new ontological perspective on functions. Educational Studies in Mathematics, 29(3), 259–291. https://doi.org/10.1007/BF01274094
Sinclair, N., Watson, A., Zazkis, R., & Mason, J. (2011). The structuring of personal example spaces. The Journal of Mathematical Behavior, 30(4), 291–303. https://doi.org/10.1016/j.jmathb.2011.04.001
Watson, A., & Mason, J. (2006). Mathematics as a constructive activity: Learners generating examples. Routledge.
Yerushalmy, M., Nagari-Haddif, G., & Olsher, S. (2017). Design of tasks for online assessment that supports understanding of students’ conceptions. ZDM Mathematics Education, 49(5), 701–716. https://doi.org/10.1007/s11858-017-0871-7
Zaslavsky, O. (2019). There is more to examples than meets the eye: Thinking with and through mathematical examples in different settings. The Journal of Mathematical Behavior, 53, 245–255. https://doi.org/10.1016/j.jmathb.2017.10.001
Zazkis, R., & Leikin, R. (2007). Generating examples: From pedagogical tool to a research tool. For the Learning of Mathematics, 27(2), 15–21.
Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. Educational Studies in Mathematics, 69, 131–148. https://doi.org/10.1007/s10649-008-9131-7
Zodik, I., & Zaslavsky, O. (2008). Characteristics of teachers’ choice of examples in and for the mathematics classroom. Educational Studies in Mathematics, 69(2), 165–182. https://doi.org/10.1007/s10649-008-9140-6
