Investigating the effectiveness of schematic representation-based instruction on the ability of solving non-routine problems in mathematics
Subject Areas : Research in Curriculum Planning
Mehran
Azizi Mahoodabad
1
(Ph.D Candidate of Curriculum Studies, University of Isfahan, Isfahan, Iran.)
Mohammad Javad
Liaghatdar
2
(Professor, Department of Education, University of Isfahan, Isfahan, Iran.)
Hamid Reza
Oreyzi
3
(Professor, Department of Psychology, University of Isfahan, Isfahan, Iran.)
Keywords: non-routine problems, Problem-Solving Ability, schematic representation, mathematics,
Abstract :
The aim of the present study is to investigate the effectiveness of schematic representation-based instruction on 6th grade students' ability of solving non-routine problems in mathematics. It is a content analysis and quasi-experimental design research of the pretest-posttest type, follow up by the control group. After studying and educational designing, 40 sixth-grade male students of an elementary school in the academic year of 1396-97 in Yasouj voluntarily participated in the study. They were randomly assigned to experimental and control group and were educated for 12 sessions (one session per week) using representation-based instruction method. Data collection instruments were developed in the content analysis section of the checklist and in the semi-experimental part of the test. Shannon Entropy technique was adopted in order to analyze data in content analysis, and variance analysis test with repeated measures was used for the quasi-experimental part. The results of the content analysis show that the greatest attention for the distribution and the presence of non-routine problems has been directed to hierarchical presentation, part-whole presentation, and network presentation problems, respectively. It was indicated that matrix presentation problems have no place in textbooks. In addition, the results of the variance analysis with repeated measures reveal that there is a significant difference between the averages of all the components of the ablitiy of solving non-routine problems of control group and experimental group (p<0.05). Therefore, it can be concluded that schematic representation instruction increases students' ability of solving non-routine problems in mathematics. Besides, the results indicate that using these methods repeatedly for solving non-routine problems will be consistent over time. The results of the study emphasize the need for special attention to the schematic representations in the sixth grade elementary mathematics textbook, and the necessity of applying these representations in solving non-routine problems by mathematic teachers.
Alkhateeb, M. (2018). Multiple Representations in 8th Grade Mathematics Textbook and the Extent to which Teachers Implement Them. International Electronic Journal of Mathematics Education, 14(1), 137-145.
Booth, R. D., & Thomas, M. O. (2000). Visualization in mathematics learning: Arithmetic problem-solving and student difficulties. The Journal of Mathematical Behavior, 18(2), 169-190.
Bibi, A., Ahmad, M., Shahid, W., Zamri, S. N. S., & Abedalaziz, N. A. M. (2019). An Evolving Research to Tackle Teaching and Learning Challenges during Differential Equations Course: A Combination of Non-routine Problems and Teacher Training. International Electronic Journal of Mathematics Education, 14(3), 647-656.
Deliyianni, E., Gagatsis, A., Elia, I., & Panaoura, A. (2016). Representational flexibility and problem-solving ability in fraction and decimal number addition: A structural model. International Journal of Science and Mathematics Education, 14(2), 397-417.
Diezmann, C. M. (2002). Enhancing students' problem solving through diagram use. Australian Primary Mathematics Classroom, 7(3), 4-8.
Diezmann, C. M., & English, L. D. (2001). Promoting the use of diagrams as tools for thinking. 2001 National Council of Teachers of Mathematics Yearbook: The Role of Representation in School Mathematics, 77-89.
Diezmann, C. M. (2005). Assessing primary students’ knowledge of networks, hierarchies and matrices using scenario-based tasks. MERGA.
DeWindt-King, A. M., & Goldin, G. A. (2003). Children’s visual imagery: Aspects of cognitive representation in solving problems with fractions. Mediterranean Journal for Research in Mathematics Education, 2(1), 1-42.
Fagnant, A., & Vlassis, J. (2013). Schematic representations in arithmetical problem solving: Analysis of their impact on grade 4 students. Educational Studies in Mathematics, 84(1), 149-168.
Flores, R., Koontz, E., Inan, F. A., & Alagic, M. (2015). Multiple representation instruction first versus traditional algorithmic instruction first: Impact in middle school mathematics classrooms. Educational Studies in Mathematics, 89(2), 267-281.
Gall, M. D., Borg, W. R., & Gall, J. P. (2004). Educational research: An introduction. translated by Ahmad Reza Nasr et all. Tehran. samt. [persian]
Hmelo-Silver, C. E., Jordan, R., Eberbach, C., & Sinha, S. (2017). Systems learning with a conceptual representation: a quasi-experimental study. Instructional Science, 45(1), 53-72.
Jitendra, A., DiPipi, C. M., & Perron-Jones, N. (2002). An Exploratory Study of Schema-Based Word-Problem—Solving Instruction for Middle School Students with Learning
Disabilities An Emphasis on Conceptual and Procedural Understanding. The Journal of Special Education, 36(1), 23-38.
Koichu, B., & Leron, U. (2015). Proving as problem solving: The role of cognitive decoupling. The Journal of Mathematical Behavior, 40, 233-244.
Kuzle, A. (2017). Delving into the Nature of Problem Solving Processes in a Dynamic Geometry Environment: Different Technological Effects on Cognitive Processing. Technology, Knowledge and Learning, 22(1), 37-64.
Kazemi, F., Rafiepour, A, & Fadaie, M. (2019). Investigate content knowledge and pedagogy content knowledge of the primary school teachers and its relation with the students’ problem-solving ability at mathematical fractions. Research in Curriculum Planning. 33(2), 104-120. [persian]
Levain, J. P., Le Borgne, P., & Simard, A. (2006). Apprentissage de schemas et resolution de problemes en SEGPA. Revue française de pedagogie. Recherches en education, (155), 95-109.
Montague, M., Warger, C., & Morgan, T. H. (2000). Solve it! Strategy instruction to improve mathematical problem solving. Learning Disabilities Research & Practice, 15(2), 110-116.
Mellone, M., Verschaffel, L., & Van Dooren, W. (2017). The effect of rewording and dyadic interaction on realistic reasoning in solving word problems. The Journal of Mathematical Behavior, 46, 1-12.
Monoyiou, A., Papageorgiou, P., & Gagatsis, A. (2007). Students’ and teachers’ representations in problem solving. In Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education: Working Group (Vol. 1, pp. 141-151).
Novick, L. R. (2006). Understanding spatial diagram structure: An analysis of hierarchies, matrices, and networks. The Quarterly Journal of Experimental Psychology, 59(10), 1826-1856.
Novick, L. R., & Hurley, S. M. (2001). To matrix, network, or hierarchy: That is the question. Cognitive Psychology, 42(2), 158-216.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (Vol. 1). Natl Council of Teachers of Mathematics.
Pantziara, M., Gagatsis, A., & Elia, I. (2009). Using diagrams as tools for the solution of non-routine mathematical problems. Educational Studies in Mathematics, 72(1), 39-60.
Panwalkar, S. S., & Koulamas, C. (2019). The evolution of schematic representations of flow shop scheduling problems. Journal of Scheduling, 1-13.
Shiakalli, M. A., & Zacharos, K. (2014). Building meaning through problem solving practices: the case of four-year olds. The Journal of Mathematical Behavior, 35, 58-73.
Salimi, M., Sadi Pour, E., Delavar, A., & Maleki, H. (2014). Comparison of training effects in imagery strategies of mental imagery of think-aloud of written representation and motor representation on the performance of elementary students in solving verbal math problems. Research in Curriculum Planning. 14(2). 12-22. [persian]
Uesaka, Y., Manalo, E., & Ichikawa, S. I. (2007). What kinds of perceptions and daily learning behaviors promote students' use of diagrams in mathematics problem solving?. Learning and Instruction, 17(3), 322-335.
Van Garderen, D. (2007). Teaching students with LD to use diagrams to solve mathematical word problems. Journal of Learning Disabilities, 40(6), 540-553.
Van Meter, P. (2001). Drawing construction as a strategy for learning from text. Journal of educational psychology, 93(1), 129.
Wu, H. K., Lin, Y. F., & Hsu, Y. S. (2013). Effects of representation sequences and spatial ability on students’ scientific understandings about the mechanism of breathing. Instructional Science, 41(3), 555-573.
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