Advertisement

Frameworks and Principles for Task Design

  • Carolyn Kieran
  • Michiel Doorman
  • Minoru Ohtani
Part of the New ICMI Study Series book series (NISS)

Abstract

This chapter gives an overview of the current state of the art related to frameworks and principles for task design so as to provide a better understanding of the design process and the various interfaces between teaching, researching, and designing. In so doing, it aims at developing new insights and identifying areas related to task design that are in need of further study. The chapter consists of three main sections. The first main section begins with a historical overview, followed by a conceptualization of current frameworks for task design in mathematics education and a description of the characteristics of the design principles offered by these frames. The second main section presents a set of cases that illustrate the relations between frameworks for task design and the nature of the tasks that are designed within a given framework. Because theoretical frameworks and principles do not account for all aspects of the process of task design, the third main section addresses additional factors that influence task design, as well as the diversity of design approaches across various professional communities in mathematics education. The chapter concludes with a discussion of the progress made in the area of task design within mathematics education over the past several decades and includes some overall recommendations with respect to frameworks and principles for task design and for future design-related research.

Keywords

Task design frameworks in mathematics Task design principles in mathematics History of task design in mathematics Task design and learning theory in mathematics Mathematics task design communities Approaches to task design in mathematics education 

References

  1. Ainley, J., & Pratt, D. (2005). The significance of task design in mathematics education: Examples from proportional reasoning. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 103–108). Melbourne: PME.Google Scholar
  2. Aizikovitsh-Udi, E., Clarke, D., & Kuntze, S. (2013). Hybrid tasks: Promoting statistical thinking and critical thinking through the same mathematical activities. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 451–460). Available from hal.archives-ouvertes.fr/hal-00834054
  3. Anderson, J. R. (1995/2000). Learning and memory. New York: Wiley.Google Scholar
  4. Arcavi, A., Kessel, C., Meira, L., & Smith, J. P. (1998). Teaching mathematical problem solving: An analysis of an emergent classroom community. Research in Collegiate Mathematics Education III, 7, 1–70.CrossRefGoogle Scholar
  5. Artigue, M. (1992). Didactical engineering. In R. Douady & A. Mercier (Eds.), Recherches en Didactique des Mathématiques, Selected papers (pp. 41–70). Grenoble: La Pensée Sauvage.Google Scholar
  6. Artigue, M. (2009). Didactical design in mathematics education. In C. Winslow (Ed.), Nordic research in mathematics education: Proceedings from NORMA08 in Copenhagen (pp. 7–16). Rotterdam: Sense Publishers.Google Scholar
  7. Artigue, M., Cerulli, M., Haspekian, M., & Maracci, M. (2009). Connecting and integrating theoretical frames: The TELMA contribution. International Journal of Computers for Mathematical Learning, 14, 217–240.CrossRefGoogle Scholar
  8. Artigue, M., & Mariotti, M. A. (2014). Networking theoretical frames: The ReMath enterprise. Educational Studies in Mathematics, 85, 329–355.CrossRefGoogle Scholar
  9. Arzarello, F., & Sabena, C. (2011). Semiotic and theoretic control in argumentation and proof activities. Educational Studies in Mathematics, 77, 189–206. doi: 10.1007/s10649-010-9280-3.CrossRefGoogle Scholar
  10. Barquero, B., & Bosch, M. (2015). Didactic engineering as a research methodology: From fundamental situations to study and research paths. In A. Watson & M. Ohtani (Eds.), Task design in mathematics education: An ICMI Study 22. New York: Springer.Google Scholar
  11. Bartolini Bussi, M. (1991). Social interaction and mathematical knowledge. In F. Furinghetti (Ed.), Proceedings of the 15th Conference of the International Group for the Psychology of Mathematics Education (Vol. I, pp. 1–16). Assisi: PME.Google Scholar
  12. Bell, A. W. (1979). Research on teaching methods in secondary mathematics. In D. Tall (Ed.), Proceedings of the Third Conference of the International Group for the Psychology of Mathematics Education (pp. 4–12). Warwick: PME.Google Scholar
  13. Bell, A. (1993a). Guest editorial. Educational Studies in Mathematics, 24, 1–4.CrossRefGoogle Scholar
  14. Bell, A. (1993b). Principles for the design of teaching. Educational Studies in Mathematics, 24, 5–34.CrossRefGoogle Scholar
  15. Bishop, A. J. (1979). Visual abilities and mathematics learning. In D. Tall (Ed.), Proceedings of the Third Conference of the International Group for the Psychology of Mathematics Education (pp. 17–28). Warwick: PME.Google Scholar
  16. Black, P., Harrison, C., Lee, C., Marshall, B., & Wiliam, D. (2003). Assessment for learning: Putting it into practice. Buckingham: Open University Press.Google Scholar
  17. Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom assessment. Phi Delta Kappan, 80(2), 139–148. Also available from: http://weaeducation.typepad.co.uk/files/blackbox-1.pdf
  18. Boer, W., et al. (Eds.). (2004). Moderne wiskunde (edite 8), vwo B1 deel 2. Groningen: Wolters-Noordhoff.Google Scholar
  19. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (1999). How people learn. Washington, DC: National Academy Press.Google Scholar
  20. Brousseau, G. (1997). Theory of didactical situations in mathematics (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. & Trans.). Dordrecht, The Netherlands: Kluwer.Google Scholar
  21. Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. Journal of the Learning Sciences, 2(2), 141–178.CrossRefGoogle Scholar
  22. Burkhardt, H. (2014). Curriculum design and systemic change. In Y. Li & G. Lappen (Eds.), Mathematics curriculum in school education (pp. 13–34). New York: Springer.CrossRefGoogle Scholar
  23. Burkhardt, H., & Swan, M. (2013). Task design for systemic improvement: principles and frameworks. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 431–440). Available from hal.archives-ouvertes.fr/hal-00834054
  24. Cheng, E. C., & Lo, M. L. (2013). Learning study: Its origins, operationalisation, and implications (OECD Education Working Papers, No. 94). Paris: OECD Publishing. Available from doi: 10.1787/5k3wjp0s959p-en
  25. Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19, 221–266.Google Scholar
  26. Chevallard, Y. (2012). Teaching mathematics in tomorrow’s society: A case for an oncoming counterparadigm. Plenary at ICME 12. Retrieved from http://www.icme12.org/upload/submission/1985_F.pdf
  27. Clark, K., & Holquist, M. (1986). Mikhail Bakhtin. Cambridge, MA: Harvard University Press.Google Scholar
  28. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York: Macmillan.Google Scholar
  29. Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 3–67). Charlotte, NC: Information Age.Google Scholar
  30. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.CrossRefGoogle Scholar
  31. Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education (pp. 68–95). London: Routledge.Google Scholar
  32. Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14, 83–95.CrossRefGoogle Scholar
  33. Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190.CrossRefGoogle Scholar
  34. Collins, A. (1992). Toward a design science of education. In E. Scanlon & T. O’Shea (Eds.), New directions in educational technology (pp. 15–22). Berlin: Springer.CrossRefGoogle Scholar
  35. Collins, A., Brown, J. S., & Newman, S. (1989). Cognitive apprenticeship: Teaching the craft of reading, writing, and mathematics. In L. B. Resnick (Ed.), Knowing, learning, and instruction: Essays in honor of Robert Glaser. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  36. Collins, A., Joseph, D., & Bielaczyc, K. (2004). Design research: Theoretical and methodological issues. Journal of the Learning Sciences, 13, 15–42.CrossRefGoogle Scholar
  37. Collopy, F. (2009). Lessons learned – Why the failure of systems thinking should inform the future of design thinking. Fast Company blog. http://www.fastcompany.com/1291598/lessons-learned-why-failure-systems-thinking-should-inform-future-design-thinking
  38. Corey, D. L., Peterson, B. E., Lewis, B. M., & Bukarau, J. (2010). Are there any places that students use their heads? Principles of high-quality Japanese mathematics instruction. Journal for Research in Mathematics Education, 41, 438–478.Google Scholar
  39. Cross, N. (2001). Designerly ways of knowing: Design discipline versus design science. Design Issues, 17(3), 49–55.CrossRefGoogle Scholar
  40. de Lange, J. (1979). Exponenten en logaritmen. Utrecht: I.O.W.O.Google Scholar
  41. de Lange, J. (1987). Mathematics, insight, and meaning: Teaching, learning and testing of mathematics for the life and social sciences. Utrecht: OW & OC.Google Scholar
  42. de Lange, J. (2012). Dichotomy in design: And other problems from the swamp. Plenary address at ISDDE. Utrecht: Freudenthal Institute.Google Scholar
  43. de Lange, J. (2013). There is, probably, no need for this presentation. Plenary presentation at the ICMI Study-22 Conference, The University of Oxford. http://www.mathunion.org/icmi/digital-library/icmi-study-conferences/icmi-study-22-conference/
  44. de Lange, J., & Kindt, M. (1984). Groei, (Een produktie ten behoove van de experiment in het kader van de Herverkaveling Eindexamenprogramma’s Wiskunde I en II VWO, 1e herziene versie). Utrecht: OW & OC.Google Scholar
  45. Ding, L., Jones, K., & Pepin, B. (2013). Task design in a school-based professional development programme. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 441–450). Available from hal.archives-ouvertes.fr/hal-00834054
  46. Ding, L., Jones, K., Pepin, B., & Sikko, S. A. (2014). How a primary mathematics teacher in Shanghai improved her lessons: A case study of ‘angle measurement’. In S. Pope (Ed.), Proceedings of the 8th British Congress of Mathematics Education (pp. 113–120). Nottingham: BCME.Google Scholar
  47. Doig, B., Groves, S., & Fujii, T. (2011). The critical role of task development in lesson study. In L. C. Hart, A. Alston, & A. Murata (Eds.), Lesson study research and practice in mathematics education. Learning together (pp. 181–199). New York: Springer.CrossRefGoogle Scholar
  48. Doorman, L. M., & Gravemeijer, K. P. E. (2009). Emergent modeling: Discrete graphs to support the understanding of change and velocity. ZDM: The International Journal on Mathematics Education, 41(1), 199–211.CrossRefGoogle Scholar
  49. Duval, R. (2006). Les conditions cognitives de l’apprentissage de la géométrie: développement de la visualisation, différenciation des raisonnements et coordination de leur fonctionnement [Cognitive conditions of learning geometry: development of visualization, differentiation of reasoning and coordination of its functioning]. Annales de Didactique et de Sciences Cognitives, 10, 5–53.Google Scholar
  50. Ernest, P. (1991). Philosophy of mathematics education. London: Falmer Press.Google Scholar
  51. Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  52. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.Google Scholar
  53. Fisk, A. D., & Gallini, J. K. (1989). Training consistent components of tasks: Developing an instructional system based on automatic-controlled processing principles. Human Factors, 31, 453–463.Google Scholar
  54. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.Google Scholar
  55. Freudenthal, H. (1978). Weeding and sowing. Dordrecht: Reidel.Google Scholar
  56. Freudenthal, H. (1979). How does reflective thinking develop? In D. Tall (Ed.), Proceedings of the Third Conference of the International Group for the Psychology of Mathematics Education (pp. 92–107). Warwick: PME.Google Scholar
  57. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.Google Scholar
  58. Freudenthal, H. (1991). Revisiting mathematics education. China lectures. Dordrecht: Kluwer.Google Scholar
  59. Fujii, T. (2013, July). The critical role of task design in lesson study. Plenary paper presented at the ICMI Study 22 Conference on Task Design in Mathematics Education, Oxford. http://www.mathunion.org/icmi/digital-library/icmi-study-conferences/icmi-study-22-conference/
  60. Gagné, R. M. (1965). The conditions of learning. New York: Holt, Rinehart & Winston.Google Scholar
  61. García, F. J., & Ruiz-Higueras, L. (2013). Task design within the Anthropological Theory of the Didactics: Study and research courses for pre-school. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 421–430). Available from hal.archives-ouvertes.fr/hal-00834054
  62. Glaser, R. (1976). Components of a psychology of instruction: Toward a science of design. Review of Educational Research, 46(1), 1–24.CrossRefGoogle Scholar
  63. Goddijn, A. (2008). Polygons, triangles and capes: Designing a one-day team task for senior high school. In ICME-11 – Topic Study Group 34: Research and development in task design and analysis. Available from http://tsg.icme11.org/tsg/show/35
  64. Goldenberg, E. P. (2008). Task Design: How? In ICME-11 – Topic Study Group 34: Research and development in task design and analysis. Available from http://tsg.icme11.org/tsg/show/35
  65. Goris, T. (2006). Math B day, Olympiad and a few words of Japanese. Nieuwe Wiskurant, 26(2), 4–5.Google Scholar
  66. Gravemeijer, K. (1994). Educational development and developmental research. Journal for Research in Mathematics Education, 25, 443–471.CrossRefGoogle Scholar
  67. Gravemeijer, K. (1998). Developmental research as a research method. In J. Kilpatrick & A. Sierpinska (Eds.), What is research in mathematics education and what are its results? (Vol. 2, pp. 277–295). Dordrecht: Kluwer.Google Scholar
  68. Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In J. van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.), Educational design research (pp. 45–85). Available from http://www.fisme.science.uu.nl/publicaties/literatuur/EducationalDesignResearch.pdf
  69. Gravemeijer, K., & Cobb, P. (2013). Design research from the learning design perspective. In T. Plomp & N. Nieveen (Eds.), Educational design research (pp. 72–113). London: Routledge.Google Scholar
  70. Gravemeijer, K., & Stephan, M. (2002). Emergent models as an instructional design heuristic. In K. P. E. Gravemeijer, R. Lehrer, B. V. Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 145–169). Dordrecht: Kluwer.CrossRefGoogle Scholar
  71. Gravemeijer, K., van Galen, F., & Keijzer, R. (2005). Designing instruction on proportional reasoning with average speed. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 103–108). Melbourne: PME.Google Scholar
  72. Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2001). The role of surprise and uncertainty in promoting the need to prove in computerized environment. Educational Studies in Mathematics, 44, 127–150.CrossRefGoogle Scholar
  73. Hart, L., Alston, A., & Murata, A. (Eds.). (2011). Lesson study research and practice in mathematics education: Learning together. New York: Springer.Google Scholar
  74. Hershkowitz, R. (1990). Psychological aspects of geometry learning – Research and practice. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition (pp. 70–95). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  75. Hilton, P. (1976). Education in mathematics and science today: The spread of false dichotomies. In H. Athen & H. Kunle (Eds.), Proceedings of the Third International Congress on Mathematical Education (pp. 75–97). Karlsruhe, FRG: University of Karlsruhe.Google Scholar
  76. Huang, R., & Bao, J. (2006). Towards a model for teacher professional development in China: Introducing Keli. Journal of Mathematics Teacher Education, 9, 279–298.Google Scholar
  77. Jacobs, J. K., & Morita, E. (2002). Japanese and American teachers’ evaluations of videotaped mathematics lessons. Journal for Research in Mathematics Education, 33, 154–175.CrossRefGoogle Scholar
  78. Janvier, C. (1979). The use of situations for the development of mathematical concepts. In D. Tall (Ed.), Proceedings of the Third Conference of the International Group for the Psychology of Mathematics Education (pp. 135–143). Warwick: PME.Google Scholar
  79. Johnson, D. C. (1980). The research process. In R. J. Shumway (Ed.), Research in mathematics education (pp. 29–46). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  80. Kali, Y. (2008). The design principles database as a means for promoting design-based research. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education (pp. 423–438). London: Routledge.Google Scholar
  81. Kalmykova, Z. I. (1966). Methods of scientific research in the psychology of instruction. Soviet Education, 8(6), 13–23.Google Scholar
  82. Kelly, A. E., Lesh, R. A., & Baek, J. Y. (Eds.). (2008). Handbook of design research methods in education. London: Routledge.Google Scholar
  83. Kieran, C., & Drijvers, P. (2006). The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra. International Journal of Computers for Mathematical Learning, 11, 205–263.CrossRefGoogle Scholar
  84. Kieran, C., Krainer, K., & Shaughnessy, J. M. (2013). Linking research to practice: Teachers as key stakeholders in mathematics education research. In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Third international handbook of mathematics education (pp. 361–392). New York: Springer.Google Scholar
  85. Kilpatrick, J. (1992). A history of research in mathematics education. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 3–38). New York: Macmillan.Google Scholar
  86. Koedinger, K. R. (2002). Toward evidence for instructional design principles: Examples from Cognitive Tutor Math 6. In D. S. Mewborn, et al. (Eds.), Proceedings of the 24th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 1–20). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.Google Scholar
  87. Koichu, B. (2013). Variation theory as a research tool for identifying learning in the design of tasks. Plenary panel at the ICMI Study-22 Conference, The University of Oxford. http://www.mathunion.org/icmi/digital-library/icmi-study-conferences/icmi-study-22-conference/
  88. Koichu, B., Zaslavsky, O., & Dolev, L. (2013). Effects of variations in task design using different representations of mathematical objects on learning: A case of a sorting task. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 461–470). Available from: hal.archives-ouvertes.fr/hal-00834054
  89. Komatsu, K., & Tsujiyama, Y. (2013). Principles of task design to foster proofs and refutations in mathematical learning: Proof problem with diagram. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 471–480). Available from hal.archives-ouvertes.fr/hal-00834054
  90. Komatsu, K., Tsujiyama, Y., Sakamaki, A., & Koike, N. (2014). Proof problems with diagrams: An opportunity for experiencing proofs and refutations. For the Learning of Mathematics, 34(1), 36–42.Google Scholar
  91. Krainer, K. (2011). Teachers as stakeholders in mathematics education research. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 47–62). Ankara: PME.Google Scholar
  92. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  93. Leikin, R. (2013). On the relationships between mathematical creativity, excellence and giftedness. In S. Oesterle & D. Allen (Eds.), Proceedings of 2013 Annual Meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d’Étude en Didactique des Mathématiques (pp. 3–17). Burnaby, BC: CMESG/GCEDM.Google Scholar
  94. Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education, 27, 133–150.CrossRefGoogle Scholar
  95. Lerman, S., Xu, G., & Tsatsaroni, A. (2002). Developing theories of mathematics education research: The ESM story. Educational Studies in Mathematics, 51, 23–40.CrossRefGoogle Scholar
  96. Lesh, R. A. (2002). Research design in mathematics education: Focusing on design experiments. In L. English (Ed.), Handbook of international research in mathematics education (pp. 27–50). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  97. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 591–645). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  98. Levav-Waynberg, A., & Leikin, R. (2009). Multiple solutions for a problem: A tool for evaluation of mathematical thinking in geometry. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 776–785). Lyon, FR: CERME6.Google Scholar
  99. Lewis, C. (2002). Lesson study: A handbook of teacher-led instructional change. Philadelphia, PA: Research for Better Schools.Google Scholar
  100. Limón, M. (2001). On the cognitive conflict as an instructional strategy for conceptual change. Learning & Instruction, 11, 357–380. doi: 10.1016/S0959-4752(00)00037-2.
  101. Lin, F.-L., Yang, K.-L., Lee, K.-H., Tabach, M., & Stylianides, G. (2012). Principles of task design for conjecturing and proving. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education (pp. 305–325). New York: Springer.Google Scholar
  102. Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. Newbury Park, CA: Sage.Google Scholar
  103. Margolinas, C. (Ed.). (2013). Task design in mathematics education (Proceedings of ICMI Study 22). Available from hal.archives-ouvertes.fr/hal-00834054
  104. Martinez, M. V., & Castro Superfine, A. (2012). Integrating algebra and proof in high school: Students’ work with multiple variables and a single parameter in a proof context. Mathematical Thinking and Learning, 14, 120–148.CrossRefGoogle Scholar
  105. Marton, F., Runesson, U., & Tsui, B. M. (2004). The space of learning. In F. Marton & A. B. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3–40). Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  106. Mathematics Assessment Resource Service (MARS). (2012). Estimating: Counting trees (p. T-2). Nottingham: Shell Centre. Available from http://map.mathshell.org
  107. McKenney, S., & Reeves, T. (2012). Conducting educational design research. London: Routledge.Google Scholar
  108. Menchinskaya, N. A. (1969). Fifty years of Soviet instructional psychology. In J. Kilpatrick & I. Wirszup (Eds.), Soviet studies in the psychology of learning and teaching mathematics (Vol. 1, pp. 5–27). Stanford, CA: School Mathematics Study Group.Google Scholar
  109. Morselli, F. (2013). The “Language and argumentation” project: researchers and teachers collaborating in task design. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 481–490). Available from hal.archives-ouvertes.fr/hal-00834054
  110. Movshovitz-Hadar, N., & Edri, Y. (2013). Enabling education for values with mathematics teaching. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 377–388). Available from hal.archives-ouvertes.fr/hal-00834054
  111. Newell, A., & Simon, H. A. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice Hall.Google Scholar
  112. Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52.CrossRefGoogle Scholar
  113. Ohtani, M. (2011). Teachers’ learning and lesson study: Content, community, and context. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 63–66). Ankara: PME.Google Scholar
  114. Okamoto, K., Koseki, K., Morisugi, K., Sasaki, T., et al. (2012). Mathematics for the future. Osaka: Keirinkan (in Japanese).Google Scholar
  115. Piaget, J. (1971). Genetic epistemology. New York: W.W. Norton.Google Scholar
  116. Pirie, S., & Kieren, T. E. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 61–86.CrossRefGoogle Scholar
  117. Pólya, G. (1945/1957). How to solve it: A new aspect of mathematical method. Princeton, NJ: Princeton University Press.Google Scholar
  118. Ponte, J. P., Mata-Pereira, J., Henriques, A. C., & Quaresma, M. (2013). Designing and using exploratory tasks. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 491–500). Available from hal.archives-ouvertes.fr/hal-00834054
  119. Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM: The International Journal on Mathematics Education, 40, 165–178.CrossRefGoogle Scholar
  120. Prusak, N., Hershkowitz, R., & Schwarz, B. B. (2013). Conceptual learning in a principled design problem solving environment. Research in Mathematics Education. doi: 10.1080/14794802.2013.836379
  121. Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.CrossRefGoogle Scholar
  122. Runesson, U. (2005). Beyond discourse and interaction. Variation: A critical aspect for teaching and learning mathematics. The Cambridge Journal of Education, 35(1), 69–87.CrossRefGoogle Scholar
  123. Ruthven, K., Laborde, C., Leach, J., & Tiberghien, A. (2009). Design tools in didactical research: Instrumenting the epistemological and the cognitive aspects of the design of teaching sequences. Educational Researcher, 38, 329–342.CrossRefGoogle Scholar
  124. Sawada, T., & Sakai, Y. (Eds.). (2013). Elementary mathematics 2 (Part 1). Tokyo: Kyoiku Shuppan (in Japanese).Google Scholar
  125. Schein, E. (1972). Professional education: Some new directions. New York: McGraw-Hill.Google Scholar
  126. Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic.Google Scholar
  127. Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 53–69). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  128. Schoenfeld, A. H. (1999). Looking toward the 21st century: Challenge of educational theory and practice. Educational Researcher, 28(7), 4–14.CrossRefGoogle Scholar
  129. Schoenfeld, A. H. (2009). Bridging the cultures of educational research and design. Educational Designer, 1(2). http://www.educationaldesigner.org/ed/volume1/issue2/article5/pdf/ed_1_2_schoenfeld_09.pdf
  130. Schön, D. (1983). The reflective practitioner: How professionals think in action. London: Basic Books.Google Scholar
  131. Schunn, C. (2008). Engineering educational design. Educational Designer, 1(1). http://www.educationaldesigner.org/ed/volume1/issue1/article2/index.htm
  132. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.CrossRefGoogle Scholar
  133. Sfard, A. (2008). Thinking as communicating. New York: Cambridge University Press.CrossRefGoogle Scholar
  134. Shimizu, S. (1981). Characteristics of “problem” in mathematics education (II). Epsilon: Bulletin of Department of Mathematics Education, Aichi University of Education, 23, 29–43 (in Japanese).Google Scholar
  135. Sierpinska, A. (2003). Research in mathematics education: Through a keyhole. In E. Simmt & B. Davis (Eds.), Proceedings of the 2003 Annual Meeting of the Canadian Mathematics Education Study Group/Groupe Canadien d’Étude en Didactique des Mathématiques (pp. 11–35). Edmonton, AB: CMESG/GCEDM.Google Scholar
  136. Simon, H. A. (1969). The sciences of the artificial. Cambridge, MA: MIT Press.Google Scholar
  137. Simon, M. (2013). Developing theory for design of mathematical task sequences: Conceptual learning as abstraction. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 501–508). Available from hal.archives-ouvertes.fr/hal-00834054
  138. Simon, M. A., Saldanha, L., McClintock, E., Karagoz Akar, G., Watanabe, T., & Ozgur Zembat, I. (2010). A developing approach to studying students’ learning through their mathematical activity. Cognition and Instruction, 28, 70–112.CrossRefGoogle Scholar
  139. Skemp, R. R. (1979). Goals of learning and qualities of understanding. In D. Tall (Ed.), Proceedings of the Third Conference of the International Group for the Psychology of Mathematics Education (pp. 250–261). Warwick: PME.Google Scholar
  140. Steffe, L. P., & Kieren, T. E. (1994). Radical constructivism and mathematics education. Journal for Research in Mathematics Education, 25, 711–733.CrossRefGoogle Scholar
  141. Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43, 428–464.CrossRefGoogle Scholar
  142. Stephan, M., & Akyuz, D. (2013). An instructional design collaborative in one middle school. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 509–518). Available from hal.archives-ouvertes.fr/hal-00834054
  143. Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: Free Press.Google Scholar
  144. Stokes, D. E. (1997). Pasteur’s quadrant: Basic science and technical innovation. Washington, DC: Brookings.Google Scholar
  145. Streefland, L. (1990). Fractions in realistic mathematics education, a paradigm of developmental research. Dordrecht: Kluwer.Google Scholar
  146. Streefland, L. (1993). The design of a mathematics course. A theoretical reflection. Educational Studies in Mathematics, 25(1–2), 109–135.CrossRefGoogle Scholar
  147. Swan, M. (2008). The design of multiple representation tasks to foster conceptual development. In ICME-11 – Topic Study Group 34: Research and development in task design and analysis. Available from http://tsg.icme11.org/tsg/show/35
  148. Swan, M., & Burkhardt, H. (2012). Designing assessment of performance in mathematics. Educational Designer, 2(5). Available from http://www.educationaldesigner.org/ed/volume2/issue5/article19/
  149. Sweller, J., van Merriënboer, J. J. G., & Paas, F. G. W. C. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10, 251–296.CrossRefGoogle Scholar
  150. Tejima, K. (1987). How many children in a line?: Task on ordinal numbers (video). Tokyo: Tosho Bunka Shya (in Japanese).Google Scholar
  151. Treffers, A. (1987). Three dimensions: A model of goal and theory description in mathematics instruction – The Wiskobas project. Dordrecht: Reidel.CrossRefGoogle Scholar
  152. Vamvakoussi, X., & Vosniadou, S. (2012). Bridging the gap between the dense and the discrete. The number line and the “rubber line” bridging analogy. Mathematical Thinking & Learning, 14(4), 265–284.CrossRefGoogle Scholar
  153. Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35.CrossRefGoogle Scholar
  154. Van den Heuvel-Panhuizen, M., & Drijvers, P. (2013). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521–525). New York: Springer.Google Scholar
  155. Van Dooren, W., Vamvakoussi, X., & Verschaffel, L. (2013). Mind the gap – Task design principles to achieve conceptual change in rational number understanding. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 519–527). Available from: hal.archives-ouvertes.fr/hal-00834054
  156. van Merriënboer, J. J. G., Clark, R. E., & de Croock, M. B. M. (2002). Blueprints for complex learning: The 4C/ID-model. Educational Technology Research and Development, 50(2), 39–64.CrossRefGoogle Scholar
  157. van Nes, F. T., & Doorman, L. M. (2011). Fostering young children’s spatial structuring ability. International Electronic Journal of Mathematics Education, 6(1), 27–39.Google Scholar
  158. von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 3–17). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  159. Vosniadou, S., Vamvakoussi, X., & Skopeliti, I. (2008). The framework theory approach to conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (pp. 3–34). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  160. 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
  161. Watson, A., et al. (2013). Introduction. In C. Margolinas (Ed.), Task design in mathematics education (Proceedings of ICMI Study 22, pp. 7–14). Available from: hal.archives-ouvertes.fr/hal-00834054
  162. Wittmann, E. (1984). Teaching units as the integrating core of mathematics education. Educational Studies in Mathematics, 15, 25–36.CrossRefGoogle Scholar
  163. Wittmann, E. C. (1995). Mathematics education as a ‘design science’. Educational Studies in Mathematics, 29, 355–374.CrossRefGoogle Scholar
  164. Yang, Y., & Ricks, T. E. (2013). Chinese lesson study: Developing classroom instruction through collaborations in school-based teaching research group activities. In Y. Li & R. Huang (Eds.), How Chinese teach mathematics and improve teaching (pp. 51–65). London: Routledge.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Carolyn Kieran
    • 1
  • Michiel Doorman
    • 2
  • Minoru Ohtani
    • 3
  1. 1.Département de mathématiquesUniversité du Québec à MontréalMontrealCanada
  2. 2.Freudenthal InstituteUtrecht UniversityUtrechtThe Netherlands
  3. 3.School of Education, College of Human and Social SciencesKanazawa UniversityKanazawa, IshikawaJapan

Personalised recommendations