ERG SES G 10, Mathematics and Education
Everyone often have to solve problems in their daily life and professional life. Therefore, the main aim of the education is to teach people to think, to use their powers and to become better problem solvers (Gagne, 1980). By learning problem solving, we learn not only to solve mathematical problems but also learn how to work and approach to overcome problems that we may face throughout our life. This perspective gives a clue about why problem solving in mathematics has become one of the main components of teaching and learning mathematics in many countries. International organizations and authorities emphasized the role of being a competent problem solver in academic and daily life (NCTM, 2000; Schoenfeld, 1992). Problem solving in most of the mathematics curricula were frequently associated with behaviors such as decisions about which operations and problem-solving strategies to use, organizing and checking their own work (Department for Education, 2013; MoNE, 2013). Those behaviors are called managerial decisions or metacognitive skills that are required for problem solving skills (Artzt & Armour- Thomas, 1992; Flavell, 1976; Garofalo & Lester, 1985; Mayer, 1998; Schoenfeld, 1985). These metacognitive behaviors affect students’ solving problems. In previous studies, students with learning disabilities and average achievers are found to be less strategic in their approach to mathematical problem solving than high achievers. In addition, perception of the difficulty level of mathematical problems may influence students' cognitive activity and their persistence in problem solving. For example, problems that appear easy to solve but actually are cognitively demanding may cause students to persist until they become frustrated and abandon the task. In the same way, problems that appear difficult may cause students to drop out the problems without attempting to solve them. (Montague & Applegate, 1993).
Schoenfeld (1992) stated that mathematical thinking and understanding were socially constructed and socially transmitted through experiences with mathematics. In classroom environment, this transmission occurs between students, students themselves and between students and teacher. The similar transmission is valid for problem solving processes as well. Students encounter different mathematical problems every day. While solving those problems, the role of the teacher is to be emphasized in terms of metacognitive aspect of problem solving. Teachers can transfer their metacognitive knowledge to their students and make them both check and self-regulate their cognitive tasks. There are experimental studies demonstrating the significant changes of students’ problem solving tasks in terms of metacognitive behaviors after the teachers instructed accordingly (Curwen et al, 2010; Desoete, 2007; van Dooren‚ Verschaffel‚ & Onghena‚ 2002). In order to examine problem-solving behaviors and cognitive processes of individuals, Artzt and Armour (1992) developed a framework which was a synthesis of cognitive and metacognitive levels of problem solving behaviors studied seperately by Garofalo and Lester, Polya, and Schoenfeld although they include similar stages or episodes.However, there is still a need to know about teachers’ and students’ daily problem solving interactions based on cognitive and metacognitive perspectives. Therefore, present study aims to shed light onto the metacognitive similarities and differences between teachers and students on mathematical problem solving by determining the contribution of teachers’ practices to the promotion of students thinking skills and to students’ metacognitive functioning. For this purpose, the behaviors of six eighth grade students and a mathematics teacher were recorded and encoded during solving PISA items based on the cognitive and metacognitive framework, and a semi-structured interview was used to obtain information regarding to the mathematics teachers' perceptions of mathematical problem solving processes.
Artzt, A. F. Armour, E. T. (1992). Development of a Cognitive-Metacognitive Framework for Protocol Analysis of Mathematical Problem Solving in Small Groups. Cognition and Instruction, 9(2), 137-175. Curwen, M. S., Miller, R. G., White-Smith, K. A., Calfee, R. C. (2010). Teachers’ Metacognition Develops Students’ Higher Learning during Content Area Literacy Instruction: Findings from the Read-Write Cycle Project. Issues in Teacher Education, 19, 2. Department for Education (2000). Mathematics programmes of study: Key stages 1 and 2 National curriculum in England – Statutory guidance to July 2015. Retrieved in June 11, 2014 fromhttps://www.gov.uk/government/uploads/system/uploads/attachment_data/file/286343/Primary_maths_curriculum_to_July_2015_RS.pdf Desoete A. (2007). Evaluating and improving the mathematics teaching-learning process through metacognition. ISSN: 1696-2095 (pp. 705-730). Electronic Journal of Research in Educational Psychology, N. 13 Vol 5 (3), 2007. Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. Resnick (Ed.), The nature of intelligence (pp. 231-236). Hillsdale, NJ: Erlbaum. Gagne, R. M. (1980) Learnable aspects of problem solving Educational Psychologist 15(2) 84-92 Garofalo, J., & Lester, F. K. (1985). 'Metacognition, cognitive monitoring, and mathematical performance', Journal for Research in Mathematics Education, 16, 163-176. Lester, F.K.Jr., Masingila, J.O., Mau, S.T., Lambdin, D.V., dos Santon, V.M. and Raymond, A.M. (1994). 'Learning how to teach via problem solving'. in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics, pp. 152-166. Reston, Virginia: NCTM. Mayer, R. E. (1998). Cognitive, metacognitive and motivational aspects of problem solving. Instructional Science, 26, 49-63. Montague, M., & Applegate, B. (1993). Middle school students' mathematical problem solving: An analysis of think-aloud protocols. Learning Disability Quarterly, 16(1), 19-32. Olkin, I. & Schoenfeld, A. (1994). A discussion of Bruce Reznick's chapter. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 39-51). Hillsdale, NJ: Lawrence Erlbaum Associates. Papaleontiou-Louca, E. (2003). The concept and instruction of metacognition. Teacher Development, 7(1), 9-30. Schoenfeld, A. H. (1985). Making sense of "out loud" problem-solving protocols. The Journal of Mathematical Behavior, 4, 171-191. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning. New York: Macmillan. Stake, R E. (1995). The art of case study research. Thousand Oaks, CA: Sage Publications.
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