Task Design In Mathematics Education pp 19-81 | Cite as

# Frameworks and Principles for Task Design

## 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

- 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 - 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 - Anderson, J. R. (1995/2000).
*Learning and memory*. New York: Wiley.Google Scholar - 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 - 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 - 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 - 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 - Artigue, M., & Mariotti, M. A. (2014). Networking theoretical frames: The ReMath enterprise.
*Educational Studies in Mathematics, 85*, 329–355.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - Bell, A. (1993a). Guest editorial.
*Educational Studies in Mathematics, 24*, 1–4.CrossRefGoogle Scholar - Bell, A. (1993b). Principles for the design of teaching.
*Educational Studies in Mathematics, 24*, 5–34.CrossRefGoogle Scholar - 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 - Black, P., Harrison, C., Lee, C., Marshall, B., & Wiliam, D. (2003).
*Assessment for learning: Putting it into practice*. Buckingham: Open University Press.Google Scholar - 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 - Boer, W., et al. (Eds.). (2004).
*Moderne wiskunde (edite 8), vwo B1 deel 2*. Groningen: Wolters-Noordhoff.Google Scholar - Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). (1999).
*How people learn*. Washington, DC: National Academy Press.Google Scholar - 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 - 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 - 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 - 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 - 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 - Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique.
*Recherches en Didactique des Mathématiques, 19*, 221–266.Google Scholar - 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 - Clark, K., & Holquist, M. (1986).
*Mikhail Bakhtin*. Cambridge, MA: Harvard University Press.Google Scholar - 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 - 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 - Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.
*Educational Researcher, 32*(1), 9–13.CrossRefGoogle Scholar - 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 - 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 - Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research.
*Educational Psychologist, 31*, 175–190.CrossRefGoogle Scholar - 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 - 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 - Collins, A., Joseph, D., & Bielaczyc, K. (2004). Design research: Theoretical and methodological issues.
*Journal of the Learning Sciences, 13*, 15–42.CrossRefGoogle Scholar - 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 - 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 - Cross, N. (2001). Designerly ways of knowing: Design discipline versus design science.
*Design Issues, 17*(3), 49–55.CrossRefGoogle Scholar - de Lange, J. (1979).
*Exponenten en logaritmen*. Utrecht: I.O.W.O.Google Scholar - 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 - de Lange, J. (2012).
*Dichotomy in design: And other problems from the swamp*. Plenary address at ISDDE. Utrecht: Freudenthal Institute.Google Scholar - 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/ - 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
- 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 - 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 - 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 - 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 - 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 - Ernest, P. (1991).
*Philosophy of mathematics education*. London: Falmer Press.Google Scholar - Fernandez, C., & Yoshida, M. (2004).
*Lesson study: A Japanese approach to improving mathematics teaching and learning*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra.
*For the Learning of Mathematics, 9*(2), 19–25.Google Scholar - 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 - Freudenthal, H. (1973).
*Mathematics as an educational task*. Dordrecht: Reidel.Google Scholar - Freudenthal, H. (1978).
*Weeding and sowing*. Dordrecht: Reidel.Google Scholar - 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 - Freudenthal, H. (1983).
*Didactical phenomenology of mathematical structures*. Dordrecht: Reidel.Google Scholar - Freudenthal, H. (1991).
*Revisiting mathematics education. China lectures*. Dordrecht: Kluwer.Google Scholar - 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/ - Gagné, R. M. (1965).
*The conditions of learning*. New York: Holt, Rinehart & Winston.Google Scholar - 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 - Glaser, R. (1976). Components of a psychology of instruction: Toward a science of design.
*Review of Educational Research, 46*(1), 1–24.CrossRefGoogle Scholar - 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 - 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 - Goris, T. (2006). Math B day, Olympiad and a few words of Japanese.
*Nieuwe Wiskurant, 26*(2), 4–5.Google Scholar - Gravemeijer, K. (1994). Educational development and developmental research.
*Journal for Research in Mathematics Education, 25*, 443–471.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Hart, L., Alston, A., & Murata, A. (Eds.). (2011).
*Lesson study research and practice in mathematics education: Learning together*. New York: Springer.Google Scholar - 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 - 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 - 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 - 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 - Janvier, C. (1979). The use of situations for the development of mathematical concepts. In D. Tall (Ed.),
- 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 - 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 - Kalmykova, Z. I. (1966). Methods of scientific research in the psychology of instruction.
*Soviet Education, 8*(6), 13–23.Google Scholar - Kelly, A. E., Lesh, R. A., & Baek, J. Y. (Eds.). (2008).
*Handbook of design research methods in education*. London: Routledge.Google Scholar - 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 - 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 - 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 - 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 - 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/ - 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 - 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 - 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 - 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 - Lakatos, I. (1976).
*Proofs and refutations: The logic of mathematical discovery*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Lewis, C. (2002).
*Lesson study: A handbook of teacher-led instructional change*. Philadelphia, PA: Research for Better Schools.Google Scholar - 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. - 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 - Lincoln, Y. S., & Guba, E. G. (1985).
*Naturalistic inquiry*. Newbury Park, CA: Sage.Google Scholar - Margolinas, C. (Ed.). (2013).
*Task design in mathematics education*(Proceedings of ICMI Study 22). Available from hal.archives-ouvertes.fr/hal-00834054 - 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 - 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 - Mathematics Assessment Resource Service (MARS). (2012).
*Estimating: Counting trees*(p. T-2). Nottingham: Shell Centre. Available from http://map.mathshell.org - McKenney, S., & Reeves, T. (2012).
*Conducting educational design research*. London: Routledge.Google Scholar - 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 - 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 - 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 - Newell, A., & Simon, H. A. (1972).
*Human problem solving*. Englewood Cliffs, NJ: Prentice Hall.Google Scholar - 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 - 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 - Okamoto, K., Koseki, K., Morisugi, K., Sasaki, T., et al. (2012).
*Mathematics for the future*. Osaka: Keirinkan (in Japanese).Google Scholar - Piaget, J. (1971).
*Genetic epistemology*. New York: W.W. Norton.Google Scholar - 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 - Pólya, G. (1945/1957).
*How to solve it: A new aspect of mathematical method*. Princeton, NJ: Princeton University Press.Google Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Sawada, T., & Sakai, Y. (Eds.). (2013).
*Elementary mathematics 2 (Part 1)*. Tokyo: Kyoiku Shuppan (in Japanese).Google Scholar - Schein, E. (1972).
*Professional education: Some new directions*. New York: McGraw-Hill.Google Scholar - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. New York: Academic.Google Scholar - 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 - Schoenfeld, A. H. (1999). Looking toward the 21st century: Challenge of educational theory and practice.
*Educational Researcher, 28*(7), 4–14.CrossRefGoogle Scholar - 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 - Schön, D. (1983).
*The reflective practitioner: How professionals think in action*. London: Basic Books.Google Scholar - Schunn, C. (2008). Engineering educational design.
*Educational Designer, 1*(1)*.*http://www.educationaldesigner.org/ed/volume1/issue1/article2/index.htm - 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 - Sfard, A. (2008).
*Thinking as communicating*. New York: Cambridge University Press.CrossRefGoogle Scholar - 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 - 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 - Simon, H. A. (1969).
*The sciences of the artificial*. Cambridge, MA: MIT Press.Google Scholar - 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 - 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 - Skemp, R. R. (1979). Goals of learning and qualities of understanding. In D. Tall (Ed.),
- Steffe, L. P., & Kieren, T. E. (1994). Radical constructivism and mathematics education.
*Journal for Research in Mathematics Education, 25*, 711–733.CrossRefGoogle Scholar - 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 - 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 - Stigler, J. W., & Hiebert, J. (1999).
*The teaching gap*. New York: Free Press.Google Scholar - Stokes, D. E. (1997).
*Pasteur’s quadrant: Basic science and technical innovation*. Washington, DC: Brookings.Google Scholar - Streefland, L. (1990).
*Fractions in realistic mathematics education, a paradigm of developmental research*. Dordrecht: Kluwer.Google Scholar - Streefland, L. (1993). The design of a mathematics course. A theoretical reflection.
*Educational Studies in Mathematics, 25*(1–2), 109–135.CrossRefGoogle Scholar - 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 - 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/ - 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 - Tejima, K. (1987).
*How many children in a line?: Task on ordinal numbers (video)*. Tokyo: Tosho Bunka Shya (in Japanese).Google Scholar - Treffers, A. (1987).
*Three dimensions: A model of goal and theory description in mathematics instruction – The Wiskobas project*. Dordrecht: Reidel.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Wittmann, E. (1984). Teaching units as the integrating core of mathematics education.
*Educational Studies in Mathematics, 15*, 25–36.CrossRefGoogle Scholar - Wittmann, E. C. (1995). Mathematics education as a ‘design science’.
*Educational Studies in Mathematics, 29*, 355–374.CrossRefGoogle Scholar - 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