Skip to main content
SpringerLink
Log in

Supporting Variation in Task Design Through the Use of Technology

  • Chapter
  • First Online:
Digital Technologies in Designing Mathematics Education Tasks

Part of the book series: Mathematics Education in the Digital Era ((MEDE,volume 8))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    An English translation of part of the module can be found at http://www.fi.uu.nl/dwo/soton. Log in as guest, and choose ‘Demo for 22nd ICMI study’. Java is needed.

  2. 2.

    It is attributed to Keats but he probably used a different wording.

References

  • 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., & 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Bokhove, C. (2014). Using crises, feedback and fading for online task design. PNA, 8(4), 127–138.

    Google Scholar 

  • Clifford, M. M. (1984). Thoughts on a theory of constructive failure. Educational Psychologist, 19(2), 108–120.

    Article  Google Scholar 

  • 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.

    Article  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 

  • Hattie, J., & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1), 81–112.

    Article  Google Scholar 

  • Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38(6), 523–550.

    Article  Google Scholar 

  • Kapur, M. (2011). A further study of productive failure in mathematical problem solving: Unpacking the design components. Instructional Science, 39(4), 561–579.

    Article  Google Scholar 

  • Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13, 135–157.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Marton, F., & Booth, S. (1997). Learning and Awareness. Mahwah: Lawrence Erlbaum.

    Google Scholar 

  • Marton, F., & Pang, M. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193–220.

    Article  Google Scholar 

  • Marton, F., & Trigwell, K. (2000). Variatio est Mater Studiorum. Higher Education Research and Development, 19(3), 381–395.

    Article  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 

  • 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.

    Article  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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253–286. doi:10.1177/0895904803260042.

    Article  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 

  • 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.

    Article  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. (2002). Student-generated examples in the learning of mathematics. Canadian Journal of Science, Mathematics and Technology Education, 2(2), 237–249.

    Article  Google Scholar 

  • Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah: Erlbaum.

    Google Scholar 

  • 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.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Southampton, Southampton, UK

    Christian Bokhove

Authors
  1. Christian Bokhove
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Bokhove .

Editor information

Editors and Affiliations

  1. Department of Education Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

    Allen Leung

  2. Department of Mathematics “G. Castelnuovo”, “Sapienza” University of Rome, Rome, Italy

    Anna Baccaglini-Frank

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bokhove, C. (2017). Supporting Variation in Task Design Through the Use of Technology. In: Leung, A., Baccaglini-Frank, A. (eds) Digital Technologies in Designing Mathematics Education Tasks. Mathematics Education in the Digital Era, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-43423-0_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-43423-0_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43421-6

  • Online ISBN: 978-3-319-43423-0

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation