Supporting Variation in Task Design Through the Use of Technology
This chapter describes a digital intervention for algebraic expertise that was built on three principles, crises, feedback and fading, as described by Bokhove and Drijvers (Technology, Knowledge and Learning. 7(1–2), 43–59, 2012b). The principles are retrospectively scrutinized through Marton’s Theory of Variation, concluding that the principles share several elements with the patterns of variation: contrast, generalisation, separation and fusion. The integration of these principles in a digital intervention suggests that technology has affordances and might be beneficial for task design with variation. The affordances in the presented technology comprise (i) authoring features, which enable teacher-authors to design their own contrasting task sequences, (ii) randomisation, which automates the creation of a vast amount of tasks with similar patterns and generalisations, (iii) feedback, which aids students in improving students’ learning outcomes, and (iv) visualisations, which allow fusion through presenting multiple representations. The principles are demonstrated by discussing a sequence of tasks involving quadratic formulas. Advantages and limitations are discussed.
KeywordsTask Design Sequence Crisis Feedback Fading Variation
- Abels, M., Boon, P., & Tacoma, S. (2013). Designing in the digital mathematics environment. Retrieved from http://www.fisme.uu.nl/wisweb/dwo/DWO_handleidingen/2013-10-11manual_DMEauthoringtool.pdf.
- Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom Assessment. Phi Delta Kappan, 80(2), 139–149.Google Scholar
- Bokhove, C. (2008, June). Use of ICT in formative scenarios for algebraic skills. Paper presented at the 4th Conference of the International Society for Design and Development in Education, Egmond aan Zee, The Netherlands.Google Scholar
- Bokhove, C. (2010). Implementing feedback in a digital tool for symbol sense. International Journal for Technology in Mathematics Education, 17(3), 121–126.Google Scholar
- Bokhove, C. (2011). Use of ICT for acquiring, practicing and assessing algebraic expertise. Utrecht: Freudenthal Institute, Utrecht University.Google Scholar
- Bokhove, C. (2014). Using crises, feedback and fading for online task design. PNA, 8(4), 127–138.Google Scholar
- Fan, L., & Bokhove, C. (2014). Rethinking the role of algorithms in school mathematics: a conceptual model with focus on cognitive development. ZDM-International Journal on Mathematics Education, 46(3). doi: 10.1007/s11858-014-0590-2.
- Fan, L., Wong, N. Y., Cai, J., & Li, S. (Eds.). (2004). How Chinese learn mathematics: Perspectives from insiders. Singapore: World Scientific.Google Scholar
- Gu, L. (1981). The visual effect and psychological implication of transformation of figures in geometry. Paper presented at Annual Conference of Shanghai Mathematics Association, Shanghai, China.Google Scholar
- Gu, L. (1994). Theory of teaching experiment: The methodology and teaching principle of Qingpu [in Chinese]. Beijing, China: Educational Science Press.Google Scholar
- Gu, L., Huang, R., & Marton, F (2004) Teaching with variation: A Chinese way of promoting effective Mathematics learning. In L. Fan, N. Y. Wong, J. Cai & S. Li (Eds.), How Chinese learn mathematics: Perspectives from insiders (2nd ed.). Singapore. World Scientific Publishing.Google Scholar
- Marton, F., & Booth, S. (1997). Learning and Awareness. Mahwah: Lawrence Erlbaum.Google Scholar
- Marton, F., Runesson, U., & Tsui, A. B. M. (2004). The space of learning. In F. Marton & A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3–40). Mahwah: Lawrence Erlbaum Associates, Inc. Publishers.Google Scholar
- Marton, F., & Tsui, A. (Eds.). (2004). Classroom discourse and the space for learning. Marwah: Erlbaum.Google Scholar
- Mason, J., & Johnston-Wilder, S. (2006). Designing and using mathematical tasks. London: QED Publishing.Google Scholar
- Ohlsson, S. (2011). Deep learning: How the mind overrides experience? Cambridge: Cambridge University Press.Google Scholar
- Piaget, J. (1964). Development and learning. In R. E. Ripple & V. N. Rockcastle (Eds.), Piaget Rediscovered (pp. 7–20). New York: Cornell University Press.Google Scholar
- Schoenfeld, A. H. (2009). Bridging the cultures of educational research and design. Educational Designer, 1(2).Google Scholar
- Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.Google Scholar
- Tall, D. (1977). Cognitive conflict and the learning of mathematics. In Proceedings of the First Conference of The International Group for the Psychology of Mathematics Education. Utrecht: PME. Retrieved from http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1977a-cog-confl-pme.pdf.
- Van der Kleij, F. M., Feskens, R. C. W., & Eggen, T. J. H. M. (2015). Effects of feedback in a computer-based learning environment on students’ learning outcomes: A meta-analysis. Review of Educational Research. Advance online publication. doi: 10.3102/0034654314564881.
- Van Hiele, P. M. V. (1985). Structure and Insight: A theory of mathematics education. Orlando: Academic Press.Google Scholar
- Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah: Erlbaum.Google Scholar