Supporting Variation in Task Design Through the Use of Technology

  • Christian BokhoveEmail author
Part of the Mathematics Education in the Digital Era book series (MEDE, volume 8)


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.


Task Design Sequence Crisis Feedback Fading Variation 


  1. Abels, M., Boon, P., & Tacoma, S. (2013). Designing in the digital mathematics environment. Retrieved from
  2. Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom Assessment. Phi Delta Kappan, 80(2), 139–149.Google Scholar
  3. 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
  4. 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
  5. Bokhove, C. (2011). Use of ICT for acquiring, practicing and assessing algebraic expertise. Utrecht: Freudenthal Institute, Utrecht University.Google Scholar
  6. Bokhove, C., & Drijvers, P. (2012a). Effects of a digital intervention on the development of algebraic expertise. Computers & Education, 58(1), 197–208. doi: 10.1016/j.compedu.2011.08.010.CrossRefGoogle Scholar
  7. Bokhove, C., & Drijvers, P. (2012b). Effects of feedback in an online algebra intervention. Technology, Knowledge and Learning., 7(1–2), 43–59. doi: 10.1007/s10758-012-9191-8.CrossRefGoogle Scholar
  8. Bokhove, C. (2014). Using crises, feedback and fading for online task design. PNA, 8(4), 127–138.Google Scholar
  9. Clifford, M. M. (1984). Thoughts on a theory of constructive failure. Educational Psychologist, 19(2), 108–120.CrossRefGoogle Scholar
  10. De Jong, T. (2010). Cognitive load theory, educational research, and instructional design: Some food for thought. Instructional Science, 38(2), 105–134. doi: 10.1007/s11251-009-9110-0.CrossRefGoogle Scholar
  11. 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.
  12. Fan, L., Wong, N. Y., Cai, J., & Li, S. (Eds.). (2004). How Chinese learn mathematics: Perspectives from insiders. Singapore: World Scientific.Google Scholar
  13. 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
  14. Gu, L. (1994). Theory of teaching experiment: The methodology and teaching principle of Qingpu [in Chinese]. Beijing, China: Educational Science Press.Google Scholar
  15. 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
  16. Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1), 81–112.CrossRefGoogle Scholar
  17. Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38(6), 523–550.CrossRefGoogle Scholar
  18. Kapur, M. (2011). A further study of productive failure in mathematical problem solving: Unpacking the design components. Instructional Science, 39(4), 561–579.CrossRefGoogle Scholar
  19. Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13, 135–157.CrossRefGoogle Scholar
  20. Leung, A., Baccaglini-Frank, A., & Mariotti, M. A. (2013). Discernment in dynamic geometry environments. Educational Studies in Mathematics, 84(3), 439–460. doi: 10.1007/s10649-013-9492-4.CrossRefGoogle Scholar
  21. Marton, F., & Booth, S. (1997). Learning and Awareness. Mahwah: Lawrence Erlbaum.Google Scholar
  22. Marton, F., & Pang, M. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193–220.CrossRefGoogle Scholar
  23. Marton, F., & Trigwell, K. (2000). Variatio est Mater Studiorum. Higher Education Research and Development, 19(3), 381–395.CrossRefGoogle Scholar
  24. 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
  25. Marton, F., & Tsui, A. (Eds.). (2004). Classroom discourse and the space for learning. Marwah: Erlbaum.Google Scholar
  26. Mason, J., & Johnston-Wilder, S. (2006). Designing and using mathematical tasks. London: QED Publishing.Google Scholar
  27. Ohlsson, S. (2011). Deep learning: How the mind overrides experience? Cambridge: Cambridge University Press.Google Scholar
  28. Pea, R. D. (2004). The social and technological dimensions of scaffolding and related theoretical concepts of learning, education, and human activity. Journal of the Learning Sciences, 13(3), 423–451.CrossRefGoogle Scholar
  29. 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
  30. Renkl, A., Atkinson, R. K., & Große, C. S. (2004). How fading worked solution steps works—a cognitive load perspective. Instructional Science, 32(1/2), 59–82.CrossRefGoogle Scholar
  31. Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27(4), 587–597.CrossRefGoogle Scholar
  32. Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253–286. doi: 10.1177/0895904803260042.CrossRefGoogle Scholar
  33. Schoenfeld, A. H. (2009). Bridging the cultures of educational research and design. Educational Designer, 1(2).Google Scholar
  34. Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.Google Scholar
  35. Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285. doi: 10.1016/0364-0213(88)90023-7.CrossRefGoogle Scholar
  36. 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
  37. 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.
  38. Van Hiele, P. M. V. (1985). Structure and Insight: A theory of mathematics education. Orlando: Academic Press.Google Scholar
  39. Watson, A., & Mason, J. (2002). Student-generated examples in the learning of mathematics. Canadian Journal of Science, Mathematics and Technology Education, 2(2), 237–249.CrossRefGoogle Scholar
  40. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah: Erlbaum.Google Scholar
  41. Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.University of SouthamptonSouthamptonUK

Personalised recommendations