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Structured affordances in the use of open-ended tasks to facilitate collaborative problem solving

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Abstract

Contemporary curriculum demands the development of both problem solving skills and negotiative skills required for collaborative group work. Tasks are required that create the best possible conditions for students to develop both of these skill sets. Open-ended tasks have been shown to provide these conditions, but they also require the teacher to relinquish some level of control over student activity, since the open-endedness of the tasks provides students with the opportunity to realize their intentions rather than just the teacher’s. It is necessary to investigate the nature of the affordances created by the use of open-ended tasks in order to anticipate how they might best be used for both instruction and learning. What is needed is a structured way to identify the affordances offered by open-ended tasks. This paper reports a study conducted in a laboratory classroom equipped to record classroom social interactions in great detail using advanced video technology. The reported analysis addresses the following question: What are the foci of the students’ social interactions during collaborative problem solving while attempting open-ended mathematical tasks? Analysis was carried out in relation to each type of data generated: written response, transcripts and videos. Our research has identified structures to represent these affordances so that teachers can both anticipate them and draw on them in monitoring and facilitating student collaborative problem solving. The structured identification of affordances in relation to (a) written product and (b) negotiative focus provides teachers with two frameworks by which to scaffold student collaborative work while attempting open-ended tasks.

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References

  • Australian Curriculum Assessment and Reporting Authority (2015). The Australian curriculum v7.5: Mathematics. http://v7-5.australiancurriculum.edu.au/. Accessed 9 June 2017.

  • Binkley, M., Erstad, O., Herman, J., Raizen, S., Ripley, M., Miller-Ricci, M., et al. (2012). Defining twenty-first century skills. In P. Griffin, B. McGaw & E. Care (Eds.), Assessment and teaching of 21st century skills (pp. 17–66). Dordrecht: Springer.

    Chapter  Google Scholar 

  • Brousseau, G. (1986). Fondements et méthodes de la didactique des mathématiques. Recherches en didactique des mathématiques, 7(2), 33–115.

    Google Scholar 

  • Brousseau, G., Sarrazy, B., & Novotná, J. (2014). Didactic contract in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 153–159). Dordrecht: Springer.

    Google Scholar 

  • Capar, G., & Tarim, K. (2015). Efficacy of the cooperative learning method on mathematics achievement and attitude: A meta-analysis research. Educational Sciences: Theory and Practice, 15(2), 553–559.

    Google Scholar 

  • Chambers, D. L. (1978). Report on the Conference on Competency Based Mathematics Education, Madison, Wisconsin, January 13, 1978. Madison, WI: Wisconsin Department of Public Instruction.

    Google Scholar 

  • Choy, B. (2016). Snapshots of mathematics teacher noticing during task design. Mathematics Education Research Journal, 28(3), 421–440. doi:10.1007/s13394-016-0173-3.

    Article  Google Scholar 

  • Clarke, D. J. (1996). Assessment. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education (pp. 327–370). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Clarke, D. J. (Ed.). (2001a). Perspectives on practice and meaning in mathematics and science classrooms. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Clarke, D. J. (2001b). Untangling uncertainty, negotiation and intersubjectivity. In D. Clarke (Ed.), Perspectives on practice and meaning in mathematics and science classrooms (pp. 33–52). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Clarke, D. J. (2011). Open-ended tasks and assessment: The nettle or the rose. In B. Kaur & K. Y. Wong (Eds.), Assessment in the mathematics classroom (pp. 131–163). Singapore: World Scientific.

    Chapter  Google Scholar 

  • Clarke, D. J., Goos, M., & Morony, W. (2007). Problem solving and working mathematically: An Australian perspective. ZDM-The International Journal on Mathematics Education, 39(5–6), 475–490. doi:10.1007/s11858-007-0045-0.

    Article  Google Scholar 

  • Clarke, D. J., Keitel, C., & Shimizu, Y. (Eds.). (2006). Mathematics classrooms in twelve countries: The insider’s perspective. Rotterdam: Sense Publishers.

    Google Scholar 

  • Clarke, D. J., Strømskag, H., Johnson, H. L., Bikner-Ahsbahs, A., & Gardner, K. (2014). Mathematical tasks and the student. In P. Liljedahl, C. Nicol, S. Oesterle & D. Allan (Eds.), Proceedings of the 38th conference of the International Group for Psychology of Mathematics Education and the 36th conference of the North American chapter of the Psychology of Mathematics Education (Vol. 1, pp. 117–143). Vancouver: The International Group for the Psychology of Mathematics Education.

    Google Scholar 

  • Clarke, D. J., & Sullivan, P. (1990). Is a question the best answer? Australian Mathematics Teacher, 46(3), 30–33.

    Google Scholar 

  • Clarke, D. J., & Sullivan, P. (1992). Responses to open-ended tasks in mathematics: Characteristics and implications. In W. Geeslin & K. Graham (Eds.), Proceedings of the 16th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 137–144). Durham, NH.

  • Clarke, D. J., Xu, L. H., Arnold, J., Seah, L. H., Hart, C., Tytler, R., et al. (2012). Multi-theoretic approaches to understanding the science classroom. In C. Bruguière, A. Tiberghien & P. Clément (Eds.), E-Book proceedings of the ESERA 2011 biennial conference: Part 3 (pp. 26–40). Lyon: European Science Education Research Association.

    Google Scholar 

  • Collet, C., & Hine, D. (2010), Preparing graduates for both academia and industry. Australian Society for Medical Research (ASMR) Newsletter, p. 2.

  • CREA [Centre of Research in Theories and Practices that Overcome Inequalities] (2012). INCLUD-ED Strategies for inclusion and social cohesion in Europe from education: Final report. Spain: CREA, University of Barcelona.

    Google Scholar 

  • Deming, D. J. (2015). The growing importance of social skills in the labor market. Cambridge, MA: National Bureau of Economic Research.

    Book  Google Scholar 

  • Goos, M., & Galbraith, P. (1996). Do it this way! Metacognitive strategies in collaborative mathematical problem solving. Educational Studies in Mathematics, 30(3), 229–260.

    Article  Google Scholar 

  • Griffin, P. E., McGaw, B., & Care, E. (Eds.). (2012). Assessment and teaching of 21st century skills. Dordrecht: Springer.

    Google Scholar 

  • Hmelo-Silver, C. E., Chinn, C. A., Chan, C. K. K., & O’Donnell, A. (Eds.). (2013). The international handbook of collaborative learning. New York: Routledge.

    Google Scholar 

  • Hummers-Pradier, E., Beyer, M., Chevallier, P., Eilat-Tsanani, S., Lionis, C., Peremans, L., et al. (2010). Series: The research agenda for general practice/family medicine and primary health care in Europe. Part 4. Results: specific problem solving skills. The European Journal Of General Practice, 16(3), 174–181. doi:10.3109/13814788.2010.504982.

    Article  Google Scholar 

  • Inoue, N., & Buczynski, S. (2011). You asked open-ended questions, now what? Understanding the nature of stumbling blocks in teaching inquiry lessons. Mathematics Educator, 20(2), 10–23.

    Google Scholar 

  • Kosyvas, G. (2016). Levels of arithmetic reasoning in solving an open-ended problem. International Journal of Mathematical Education in Science and Technology, 47(3), 356–372. doi:10.1080/0020739x.2015.1072880.

    Article  Google Scholar 

  • Langley, P., & Trivedi, N. (2013). Elaborations on a theory of human problem solving. Advances in Cognitive Systems, 3, 1–12.

    Google Scholar 

  • Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Marriott, P. (1976). Two hundred ways to subtract - most of them wrong. Paper submitted as part of the course requirements of the Bachelor of Special Education. Clayton, Victoria: Monash University.

    Google Scholar 

  • National Research Council (2011). Assessing 21st Century skills: Summary of a workshop. Washington, DC: National Academy Press.

    Google Scholar 

  • Newell, A., & Simon, H. A. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Nohda, N. (2000). Teaching by open-approach method in Japanese mathematics classroom. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 1, pp. 39–53). Hiroshima, Japan.

  • Office of the Chief Scientist (2014). Science, technology, engineering and mathematics: Australia’s future. Canberra, ACT: Australian Government.

    Google Scholar 

  • OECD [Organisation for Economic Co-operation and Development] (2013a). PISA 2012 results: Creative problem solving: Students’ skills in tackling real-life problems (Vol. V). Paris: OECD [Organisation for Economic Co-operation and Development].

    Google Scholar 

  • OECD [Organisation for Economic Co-operation and Development] (2013b). PISA 2015 draft collaborative problem solving framework. Paris: OECD [Organisation for Economic Co-operation and Development].

    Google Scholar 

  • Olive, J., & Steffe, L. P. (1990). Constructing fractions in computer microworlds. In G. Booker, P. Cobb & T. N. de Mendicuti (Eds.), Proceedings of the 14th annual conference of the International Group for the Psychology of Mathematics Education, with the North American Chap. (12th PME-NA Conference) (Vol. 3, pp. 59–66). Mexico: International Group for the Psychology of Mathematics Education.

    Google Scholar 

  • Pazos, P., Micari, M., & Light, G. (2010). Developing an instrument to characterise peer-led groups in collaborative learning environments: Assessing problem-solving approach and group interaction. Assessment and Evaluation in Higher Education, 35(2), 191–208. doi:10.1080/02602930802691572.

    Article  Google Scholar 

  • Pólya, G. (1971). How to solve it: A new aspect of mathematical method (2nd edn.). Princeton, NJ: Princeton University Press.

    Google Scholar 

  • Prediger, S., Clarkson, P., & Boses, A. (2016). Purposefully relating multilingual registers: Building theory and teaching strategies for bilingual learners based on an integration of three traditions. In R. Barwell, P. Clarkson, A. Halai, M. Kazima, J. Moschkovich, N. Planas, et al. (Eds.), Mathematics education and language diversity: The 21st ICMI Study (pp. 193–215). Cham: Springer International Publishing.

    Chapter  Google Scholar 

  • Remedios, L., Clarke, D. J., & Hawthorne, L. (2008). The silent participant in small group collaborative learning contexts. Active Learning in Higher Education, 9(3), 201–216. doi:10.1177/1469787408095846.

    Article  Google Scholar 

  • Roche, A., Clarke, D. M., Clarke, D. J., & Sullivan, P. (2014). Primary teachers’ written unit plans in mathematics and their perceptions of essential elements of these. Mathematics Education Research Journal, 26(4), 853–870. doi:10.1007/s13394-014-0130-y.

    Article  Google Scholar 

  • Schuster, K., Plumanns, L., Groß, K., Vossen, R., Richert, A., & Jeschke, S. (2015). Preparing for Industry 4.0—Testing collaborative virtual learning environments with students and professional trainers. International Journal of Advanced Corporate Learning, 8(4), 14–20. doi:10.3991/ijac.v8i4.4911.

    Article  Google Scholar 

  • Shimizu, Y., Kaur, B., Huang, R., & Clarke, D. (Eds.). (2010). Mathematical tasks in classrooms around the world. Rotterdam: Sense Publishers.

    Google Scholar 

  • Stacey, K. (2005). The place of problem solving in contemporary mathematics curriculum documents. The Journal of Mathematical Behavior, 24(3), 341–350. doi:10.1016/j.jmathb.2005.09.004.

    Article  Google Scholar 

  • Steffe, L. P., & Wiegel, H. G. (1994). Cognitive play and mathematical learning in computer microworlds. Educational Studies in Mathematics, 26(2/3), 111–134.

    Article  Google Scholar 

  • Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2(1), 50–80. doi:10.1080/1380361960020103.

    Article  Google Scholar 

  • Sullivan, P., & Clarke, D. J. (1988). Asking better questions. Journal of Science and Mathematics Education in South East Asia, 11, 14–19.

    Google Scholar 

  • Sullivan, P., & Clarke, D. J. (1991). Catering to all abilities through “good” questions. The Arithmetic Teacher, 39(2), 14–18. doi:10.2307/41194944.

    Google Scholar 

  • Sullivan, P., & Clarke, D. J. (1992). Problem solving with conventional mathematics content: Responses of pupils to open mathematical tasks. Mathematics Education Research Journal, 4(1), 42–60. doi:10.1007/bf03217231.

    Article  Google Scholar 

  • Sullivan, P., Knott, L., & Yang, Y. (2015). The relationships between task design, anticipated pedagogies, and student learning. In A. Watson & M. Ohtani (Eds.), Task Design In Mathematics Education: An ICMI study 22 (pp. 83–114). Cham: Springer International Publishing.

    Chapter  Google Scholar 

  • Sullivan, P., Warren, E., & White, P. (2000). Students’ responses to content specific open-ended mathematical tasks. Mathematics Education Research Journal, 12(1), 2–17. doi:10.1007/bf03217071.

    Article  Google Scholar 

  • Thomas, P. V., & Higbee, J. (1996). Enhancing mathematics achievement through collaborative problem solving. Learning Assistance Review, 1(1), 38–46.

    Google Scholar 

  • Victorian Curriculum Assessment Authority (2016). Australian Victorian Essential Learning Standards (AusVELS). http://pandora.nla.gov.au/pan/129125/20121206-0015/ausvels.vcaa.vic.edu.au/index.html. Accessed 10 June 2017.

  • Viseu, F., & Oliveira, I. B. (2012). Open-ended tasks in the promotion of classroom communication in mathematics. International Electronic Journal of Elementary Education, 4(2), 287–300.

    Google Scholar 

  • Webb, N. M. (1991). Task-related verbal interaction and mathematics learning in small groups. Journal for Research in Mathematics Education, 22(5), 366–389. doi:10.2307/749186.

    Article  Google Scholar 

  • Williams, G. (2000). Collaborative problem solving and discovered complexity. In J. Bana & A. Chapman (Eds.), Proceedings of the Mathematics Education Research Group of Australasia (pp. 656–663). Freemantle, Western Australia: Mathematics Education Research Group of Australasia.

    Google Scholar 

  • World Economic Forum (2014). Matching skills and labour market needs building social partnerships for better skills and better jobs. Cologny/Geneva: World Economic Forum.

    Google Scholar 

  • Wu, H. (1994). The role of open-ended problems in mathematics education. The Journal of Mathematical Behavior, 13(1), 115–128. doi:10.1016/0732-3123(94)90044-2.

    Article  Google Scholar 

  • Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. doi:10.2307/749877.

    Article  Google Scholar 

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Acknowledgements

This research was conducted with Science of Learning Research Centre funding provided by the Australian Research Council Special Initiatives Grant (SR120300015) and the Discovery Projects funding scheme (DP170102541). We would like to thank the students, parents, teachers, and school staff for their invaluable support of this project.

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  1. International Centre for Classroom Research, Melbourne Graduate School of Education, Level 3, 234 Queensberry Street, The University of Melbourne, VIC, 3010, Australia

    Man Ching Esther Chan & David Clarke

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  1. Man Ching Esther Chan
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Correspondence to Man Ching Esther Chan.

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Chan, M.C.E., Clarke, D. Structured affordances in the use of open-ended tasks to facilitate collaborative problem solving. ZDM Mathematics Education 49, 951–963 (2017). https://doi.org/10.1007/s11858-017-0876-2

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