Revisiting Theory for the Design of Tasks: Special Considerations for Digital Environments

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


Teachers should and do design tasks for the mathematics classroom, with specific mathematical learning as the objective. Completing the tasks should require students to engage in dialectics of action, formulation and validation (Brousseau in Theory of didactical situations in mathematics : didactique des mathematiques, Dordrecht: Kluwer Academic Publishers, 1997) and to move between the pragmatic/empirical field and the mathematical/systematic field (Noss et al. in Educational Studies in Mathematics, 33(2), 203–233, 1997). In the classroom, students act within a milieu, and where computers are part of this milieu, particular considerations with respect to task design include questions about the mathematics the student does and the mathematics the computer does, and the role of feedback from the computer. Whilst taking into account the role of the computer, the design of tasks can also be guided by theoretical constructs related to obstacles of various kinds; ontogenic, didactical and epistemological (Brousseau in Theory of didactical situations in mathematics : didactique des mathematiques,Dordrecht: Kluwer Academic Publishers, 1997), and, whereas the first two should be avoided, the third should be encouraged. An example of a task taken from empirical research in an ordinary classroom is used to illustrate some of these ideas, also demonstrating how difficult and complex it is for many teachers to design tasks that use computer software in ways that provoke the sort of student activity that would be likely to lead to mathematical learning. Implications for teacher professional development are discussed.


Task design Feedback Modes of production Pragmatic/empirical field Mathematical/systematic field Digital tools Epistemological obstacles 


  1. Ainley, J., & Pratt, D. (2002). Purpose and utility in pedagogic task design. Proceedings of the 26th Annual Conference of the International Group for the Psychology of Mathematics Education, 2, 17–24.Google Scholar
  2. Balacheff, N. (1990). Towards a problematique for research on mathematics teaching. Journal for Research in Mathematics Education, 21(4), 258–272. Retrieved from<258:TAPFRO>2.0.CO;2-U.CrossRefGoogle Scholar
  3. Balacheff, N., & Kaput, J. (1996). Computer-based learning environments in mathematics. In A. J. Bishop (Ed.), International handbook of mathematics education (pp. 469–502). Dordrecht: Kluwer.Google Scholar
  4. Balacheff, N., & Sutherland, R. (1994). Epistemological domain of validity of microworlds: the case of Logo and Cabri-geometre. In R. Lewis & P. Mendelsohn (Eds.), Lessons from learning. Amsterdam: Elsevier Science BV. (N. Holland), IFIP.Google Scholar
  5. Becta. (2004). What the research says about the use of ICT in maths. Retrieved from
  6. Bokhove, C., & Drijvers, P. (2010). Digital tools for Algebra education: Criteria and evaluation. International Journal of Computers for Mathematical Learning, 15(1), 45–62. doi: 10.1007/s10758-010-9162-x.CrossRefGoogle Scholar
  7. Bokhove, C., & Drijvers, P. (2011). Effects of a digital intervention the development of algebraic expertise. Computers & Education, (3), 3770–3777. Retrieved from
  8. Bokhove, C., & Drijvers, P. (2012). Effects of feedback in an online algebra intervention. Technology, Knowledge and Learning, 17(1–2), 43–59. doi: 10.1007/s10758-012-9191-8.CrossRefGoogle Scholar
  9. Brousseau, G. (1997). Theory of didactical situations in mathematics: didactique des mathematiques, 1970–1990. Dordrecht: Kluwer Academic Publishers.Google Scholar
  10. Christiansen, B., & Walther, G. (1986). Task and activity. In B. Christiansen, A. G. Howson, & M. Otte (Eds.), Perspectives on mathematics education (pp. 243–307). Dordrecht: Kluwer.Google Scholar
  11. Condie, R., & Munro, B. (2007). The impact of ICT in schools—a landscape review. Report: Becta.Google Scholar
  12. Dörfler, W. (2000). Means for meaning. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms; Perspectives on discourse, tools, and instructional design (pp. 99–131). Mahwah, New Jersey: Lawrence Erlbaum.Google Scholar
  13. Goldenberg, E. P. (1988). mathematics, metaphors, and human factors: mathematical, technical, and pedagogical challenges in the educational use of graphical representation of functions. Journal of Mathematical Behavior, 7(2), 135–173.Google Scholar
  14. Granberg, C., & Olsson, J. (2015). ICT-supported problem solving and collaborative creative reasoning: Exploring linear functions using dynamic mathematics software. Journal of Mathematical Behavior, 37, 48–62. doi: 10.1016/j.jmathb.2014.11.001.CrossRefGoogle Scholar
  15. Harel, G., & Sowder, L. (2005). Advanced mathematical—thinking at any age: its nature and its development. Mathematical Thinking and Learning, 7(1), 27–50.CrossRefGoogle Scholar
  16. Hillel, J. (1992). The computer as a problem-solving tool; it gets a job done, but is it always appropriate? In J. P. Ponte, J. F. Matos, J. M. Matos, & D. Fernandes (Eds.), Mathematical problem solving and new information technologies. New York: Springer.Google Scholar
  17. Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? In A. J. Bishop (Ed.), Second international handbook of mathematics education. Dordrecht: Kluwer.Google Scholar
  18. Hoyles, C., Noss, R., Vahey, P., & Roschelle, J. (2013). Cornerstone mathematics: Designing digital technology for teacher adaptation and scaling. ZDM—International Journal on Mathematics Education, 45(7), 1057–1070. doi: 10.1007/s11858-013-0540-4.CrossRefGoogle Scholar
  19. Joubert, M. (2007). Classroom mathematical learning with computers: The mediational effects of the computer, the teacher and the task. University of Bristol.Google Scholar
  20. Laborde, C. (1998). Relationship between the spatial and theoretical in geometry—the role of computer dynamic representations in problem solving (pp. 183–195). London: Chapman and Hall.Google Scholar
  21. Laborde, C. (2002). Integration of technology in the design of geometry tasks with Cabri-Geometry. International Journal of Computers for Mathematical Learning, 6(3), 283–317.CrossRefGoogle Scholar
  22. Laborde, C., Kilpatrick, J., Hoyles, C., & Skovsmose, O. (2004). The hidden role of diagrams in students’ construction of meaning in geometry (pp. 1–21). Netherlands: Kluwer.Google Scholar
  23. Laborde, C., & Sträßer, R. (2010). Place and use of new technology in the teaching of mathematics: ICMI activities in the past 25 years. ZDM, 42(1), 121–133. doi: 10.1007/s11858-009-0219-z.CrossRefGoogle Scholar
  24. Love, E., Burton, L., & Jaworski, B. (1995). Software for mathematics education. Lund: Chartwell-Bratt.Google Scholar
  25. Love, E., & Mason, J. (1992). Teaching mathematics: Action and awareness. Milton Keynes: Open University.Google Scholar
  26. Margolinas, C., & Drijvers, P. (2015). Didactical engineering in France; an insider’s and an outsider’s view on its foundations, its practice and its impact. ZDM—Mathematics Education, 47(6), 893–903. doi: 10.1007/s11858-015-0698-z.Google Scholar
  27. Mariotti, M. A. (2000). Introduction to proof: The mediation of a dynamic software environment. Educational Studies in Mathematics, 44(1), 25–53.CrossRefGoogle Scholar
  28. Martin, L., & Pirie, S. (2003). Making images and noticing properties: The role of graphing software in mathematical generalisation. Mathematics Education Research Journal, 15(2), 171–186. doi: 10.1007/BF03217377.CrossRefGoogle Scholar
  29. Mason, J. (1989). Mathematical abstraction as the result of a delicate shift of attention. For the Learning of Mathematics, 9(2), 2–8.Google Scholar
  30. Mavrikis, M., Noss, R., Hoyles, C., & Geraniou, E. (2013). Sowing the seeds of algebraic generalization: Designing epistemic affordances for an intelligent microworld. Journal of Computer Assisted learning, 29(1), 68–84. doi: 10.1111/j.1365-2729.2011.00469.x.CrossRefGoogle Scholar
  31. McNeil, N., & Jarvin, L. (2007). When theories don’t add up: Disentangling he manipulatives debate. Theory into Practice, 46(4), 309–316. doi: 10.1080/00405840701593899.CrossRefGoogle Scholar
  32. Monaghan, J., & Trouche, L. (2016). Tasks and digital tools. In J. Monaghan, L. Trouche, & J. M. Borwein (Eds.), Tools and mathematics (pp. 391–415). Switzerland: Springer.CrossRefGoogle Scholar
  33. Nevile, L., Burton, L., & Jaworski, B. (1995). Looking at, through, back at: Useful ways of viewing mathematical software. Lund: Chartwell-Bratt.Google Scholar
  34. Noss, R., Healy, L., & Hoyles, C. (1997). The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, 33(2), 203–233.CrossRefGoogle Scholar
  35. Olivero, F. (2003). The proving process within a dynamic geometry environment. Advanced mathematical thinking (Vol. PhD). PhD thesis. Graduate School of Education: University of Bristol.Google Scholar
  36. Orton, A., & Frobisher, L. J. (1996). Insights into teaching mathematics. London: Cassell.Google Scholar
  37. Pimm, D. (1995). Symbols and meanings in school mathematics. London: Routledge.CrossRefGoogle Scholar
  38. Resnick, L. B. (1984). A developmental theory of number understanding. Pittsburg: Learning Research and Development Center, University of Pittsburg.Google Scholar
  39. Romberg, T., & Kaput, J. (1999). Mathematics worth teaching, mathematics worth understanding. In E. Fennema & T. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 3–19). Mahwah, New Jersey: Lawrence Erlbaum Associates.Google Scholar
  40. Roschelle, J. (1997). Learning in interactive environments prior knowledge and new experience. Exploratorium Institute for Inquiry: University of Massachusetts, Dartmouth.Google Scholar
  41. Ruthven, K., Hennessy, S., & Brindley, S. (2004). Teacher representations of the successful use of computer-based tools and resources in secondary-school English, Mathematics and Science. Teaching and Teacher Education, 20(3), 259–275.CrossRefGoogle Scholar
  42. Säljö, R. (1999). Learning as the use of tools: A sociocultural perspective on the human-technology link. In Learning with Computers: Analysing Productive Interaction Table of Contents, pp. 144–161.Google Scholar
  43. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145–166.CrossRefGoogle Scholar
  44. Schwarz, B. B., & Hershkowitz, R. (2001). Production and transformation of computer artifacts toward construction of meaning in Mathematics. Mind Culture and Activity, 8(3), 250–267.CrossRefGoogle Scholar
  45. Sfard, A. (2000). Symbolizing mathematical reality into being—or how mathematical discourse and mathematical objects create each other. In Symbolizing and communicating: Perspectives on Mathematical discourse, tools, and instructional design, pp. 37–98.Google Scholar
  46. Sierpinska, A. (2000). The “theory of didactic situations”: Lecture notes for a graduate course with samples of students’ work. master in the teaching of mathematics. Concordia University. Retrieved 2006 from
  47. Sierpinska, A. (2004). Research in mathematics education through a keyhole: Task problematization. For the Learning of Mathematics, 24(2), 7–15.Google Scholar
  48. Sutherland, R. (2007). Teaching for learning mathematics. Maidenhead: Open University Press.Google Scholar
  49. Sutherland, R., & Balacheff, N. (1999). Didactical Complexity of Computational Environments for the Learning of Mathematics. International Journal of Computers for Mathematical Learning, 4(1), 1–26.Google Scholar
  50. Swan, M. (2006). Collaborative learning in mathematics: A challenge to our beliefs and practices. In S. M. Wilson & J. Berne (Vol. 24). London: NRDC.Google Scholar
  51. Vygotsky, L. S. (1980). Mind in society. Cambridge, Mass: Harvard University Press.Google Scholar
  52. Watson, A., et al. (2013). Task design in mathematics education proceedings of ICMI study 22. In C. Margolinas (Ed.), Task Design in Mathematics Education Proceedings of ICMI Study 22. Oxford.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.African Institute for Mathematical SciencesCape TownSouth Africa

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