Teacher-assisted open problem solving
DOI:
https://doi.org/10.7146/nomad.v18i2.148503Abstract
Previous research has developed several problem-solving models and suggested that the teacher plays a crucial role in guiding students’ problem solving. However, less is known about the particularities of problem solving and teacher guidance when dealing with open problems which include multiple possible solution pathways. The aim of this study is to understand students’ open problem-solving processes and teachers’ ways of supporting them. Data collection involved videotaping one 9th grade mathematics lesson with two video cameras and capturing the screens of the students’ computers. Seven student pairs worked on an open problem using GeoGebra under the guidance of a teacher trainee. We found that students had various kinds of problem-solving processes and that the teacher had a crucial role in guiding them. We elaborate on 9 ways how the teacher guided students to change between phases in open problem solving.
References
Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning. Journal of Mathematics Teacher Education, 9, 33-52. https://doi.org/10.1007/s10857-006-9005-9
Arzarello, F., Olivero, F., Paola, D. & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments. ZDM - The International Journal on Mathematics Education, 34 (3), 66-72. https://doi.org/10.1007/BF02655708
Boaler, J. (1998). Open and closed mathematics: student experiences and understandings. Journal for Research in Mathematics Education, 29 (1), 41-62. https://doi.org/10.2307/749717
Borwein, J. & Bailey, D. (2003), Mathematics by experiment: plausible reasoning in the 21st century. Natick: AK Peters.
Cai, J. & Lester, F. (2010). Why is teaching with problem solving important to student learning? Retrieved April 15, 2012 from http://www.nctm.org/uploadedFiles/Research_News_and_Advocacy/Research/Clips_and_Briefs/Research_brief_14_-_Problem_Solving.pdf.
Christou, C., Mousoulides, N., Pittalis, M. & Pitta-Pantazi, D. (2004). Proofs through exploration in dynamic geometry environments. International Journal of Science and Mathematics Education, 3 (2), 339-352. https://doi.org/10.1007/s10763-004-6785-1
Christou, C., Mousoulides, N., Pittalis, M. & Pitta-Pantazi, D. (2005). Problem solving and problem posing in a dynamic geometry environment. The Montana Mathematics Enthusiast, 2 (2), 125-143. https://doi.org/10.54870/1551-3440.1029
Cifarelli, V., & Cai, J. (2005). The evolution of mathematical explorations in open-ended problem-solving situations. Journal of Mathematical Behavior, 24, 302-324. https://doi.org/10.1016/j.jmathb.2005.09.007
Davis, B. (1997). Listening for differences: an evolving conception on mathematics education. Journal for Research in Mathematics Education, 28 (3), 355-376. https://doi.org/10.2307/749785
Davis, R., & Maher, C. (1990). What do we do when we "do mathematics"? Journal for Research in Mathematics Education. Monograph, Vol. 4, Constructivist views on the teaching and learning of mathematics, 65-78, 195-210. https://doi.org/10.2307/749913
Doerr, H. & English, L. (2006). Middle grade teachers' learning through students' engagement with modeling task. Journal of Mathematics Teacher Education, 9 (1), 5-32 https://doi.org/10.1007/s10857-006-9004-x
Francisco, J. & Maher, C. (2011). Teachers attending to students' mathematical reasoning: lessons from an after-school research program. Journal of Mathematics Teacher Education, 14, 49-66. https://doi.org/10.1007/s10857-010-9144-x
Healy, L. & Hoyles, C. (2001). Software tools for geometrical problem solving: potentials and pitfalls. International Journal of Computers for Mathematical Learning, 6, 235-256. https://doi.org/10.1023/A:1013305627916
Hähkiöniemi, M. & Leppäaho, H. (2012). Prospective mathematics teachers' ways of guiding high school students in GeoGebra-supported inquiry tasks. The International Journal for Technology in Mathematics Education, 19 (2), 45-58.
Hölzl, R. (2001). Using dynamic geometry software to add contrast to geometric situations - a case study. International Journal of Computers for Mathematical Learning, 6, 63-86. https://doi.org/10.1023/A:1011464425023
Jones, K. (2000). Providing a foundation for deductive reasoning: students' interpretations when using dynamic geometry software and their evolving mathematical explanations. Educational Studies in Mathematics, 44 (1-3), 55-85. https://doi.org/10.1023/A:1012789201736
Kwon, O. N., Park, J. S. & Park, J. H. (2006). Cultivating divergent thinking in mathematics through an open-ended approach. Asia Pacific Educational Review, 7 (1), 51-61. https://doi.org/10.1007/BF03036784
Lesh, R. & Doerr, H. (2003). Foundation of a models and modeling perspective on mathematics teaching, learning, and problem solving. In R. Lesh & H. Doerr (Eds.), Beyond constructivism: models and modeling perspective on mathematics teaching, learning, and problem solving (pp. 3-33). Mahwah: Erlbaum.
Lobato, J., Clarke, D. & Ellis, A. B. (2005). Initiating and eliciting in teaching: a reformulation of telling. Journal for Research in Mathematics Education, 36 (2), 101-136.
Martino, A. & Maher, C. (1999). Teacher questioning to promote justification and generalization in mathematics: what research practice has taught us. Journal of Mathematical Behavior, 18 (1), 53-78. https://doi.org/10.1016/S0732-3123(99)00017-6
Mason, J., Burton, L. & Stacey, K. (1982). Thinking mathematically. Bristol: Addison-Wesley.
Mousoulides, N. (2011). GeoGebra as a conceptual tool for modelling real world problems. In L. Bu & R. Schoen (Eds.), Model-centered learning: pathways to mathematical understanding using GeoGebra (pp. 105-118). Rotterdam: Sense. https://doi.org/10.1007/978-94-6091-618-2_8
Nohda, N. (2000). Teaching by open-approach method in Japanese mathematics classroom. In T. Nakahara and M. Koyama (Eds.), Proceedings of the 24th conference of the international group for the psychology of mathematics education (Vol. 1, pp. 39-53). Hiroshima: PME.
Nunokawa, K. (2005). Mathematical problem solving and learning mathematics: what we expect students to obtain. Journal of Mathematical Behavior, 24, 325-340. https://doi.org/10.1016/j.jmathb.2005.09.002
Pea, R. (2004). The social and technological dimensions of scaffolding and related theoretical concepts for learning, education and human activity. Journal of the Learning Sciences, 13 (3), 423-451. https://doi.org/10.1207/s15327809jls1303_6
Pehkonen, E. (1995). On pupils' reactions to the use of open-ended problems in mathematics. Nordic Studies in Mathematics Education, 3 (4), 43-57.
Pehkonen, E. (1997). Introduction to the concept "open-ended problem". In E. Pehkonen (Ed.), Use of open-ended problems on mathematics classroom (Research report 176 ) (pp. 7-11). Department of teacher education, University of Helsinki.
Pólya, G. (1945). How to solve it? A new aspect of mathematical method. Princeton University Press. https://doi.org/10.1515/9781400828678
Sahin, A. & Kulm, G. (2006). Sixth grade mathematics teachers' intentions and use of probing, quiding, and factual questions. Journal of Mathematics Teacher Education, 11 (3), 221-241. https://doi.org/10.1007/s10857-008-9071-2
Schoenfeld, A. (1985). Mathematical problem solving. London: Academic Press.
Stein, M., Engle, R., Smith, M. & Hughes, E. (2008). Orchestrating productive mathematical discussions: five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10, 313-340. https://doi.org/10.1080/10986060802229675
Sullivan, P., Mousley, J. & Zevenberger, R. (2006). Teacher actions to maximize mathematics learning opportunities in heterogeneous classrooms. International Journal of Science and Mathematics Education, 4, 117-143. https://doi.org/10.1007/s10763-005-9002-y
Sullivan, P., Warren, E. & White, P. (2000). Students' responses to content specific open-ended mathematical tasks. Mathematics Education Research Journal, 12 (1), 2-17. https://doi.org/10.1007/BF03217071
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