Metacognitive Similarities and Differences between Teachers and Students on Mathematical Problem Solving Skills

Author(s):

Sinem Sozen Ozdogan(presenting / submitting)Bilal Özcakır Burcu Orhan

Conference:

ECER 2015

Network:

Emerging Researchers' Group (for presentation at Emerging Researchers' Conference)

Format:

Paper

Session Information

ERG SES G 10, Mathematics and Education

Paper Session

Time:

2015-09-08

09:00-10:30

Room:

334. [Main]

Chair:

Carol Taylor

Contribution

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.

Method

This study aims to investigate the problem solving processes of the eighth grade mathematics teacher and the students chosen by teachers and by academic achievement levels in a public elementary school through cognitive-metacognitive framework.Therefore, the focus of the study is appropriate for the instrumental case study research design (Stake, 1995). Eighth grade mathematics teacher and six eighth grade students from public elementary school were the participants of this study. Participants were purposefully selected from the eighth grade levels due to the fact that they have completed three years’ mathematics lessons with the same teacher. The participants were in a public middle school in Elmalı - Antalya in Turkey.The school has students from a wide variety of different socio-cultural levels. Three students among six students were determined by the teacher according to his understanding of “successful student in problem solving” (Teacher selected group). Other three students were determined by the researchers based on their rankings of academic achievement of national examination in May 2013-2014 spring term (Academic achievement group). There was no common student between two groups. Each group included students from successful, novice and poor level. For AAG successful student was on the top in the ranking, novice was in the middle, poor level student was the last person in the ranking. On the other hand, the selection of TSG was determined by the teacher. The teacher and his six students were asked to solve one for practice and three for problem analysis procedure. The problems were selected from PISA 2012 mathematics survey items aimed to evaluate problem solving skills and the knowledge of 15 year-old students, which corresponds to eighth grade in Turkey. Selection of the problems to be used in the study was guided by the criteria of difficulty levels and open-ended having many alternative problem solving strategies but having one right answer. The translated version of the selected PISA 2012 problems was given to the participants. While solving the problems, think aloud process was used and all the process were video and audio recorded. The teacher was also interviewed to gather detailed information about his experiences related to problem solving. While analyzing the data, transcriptions of the teachers and students were encoded based on the cognitive and metacognitive framework of Artzt and Armour-Thomas.The framework was adapted to the individual problem solving process for this study.

Expected Outcomes

Analysis of the transcriptions indicates that, the students with different problem solving skill levels are found to exhibit different frequencies of cognitive and metacognitive behaviors against PISA problems. In fact, students who were selected as successful problem solvers had more frequencies of metacognitive behaviors and the frequency and types of the metacognitive behaviors of both TSG and AAG were found similar. Consequently, this study has revealed that there is a relationship between academic achievement of the students and frequency of metacognitive behavior and the teacher and successful problem solvers showed similar cognitive and metacognitive behaviors but not as much as teachers’ demonstrated. As it was assumed that metacognitive knowledge is transferrable by the teachers, there should be found some ways to promote this transfer in the classroom environment. Ader (2013) proposed a framework (FAMTAM) to encourage students reflect metacognitive behaviors based on the observations on teachers’ different emphasis on metacognition in the classrooms. It may not only due to the teachers. In the interview, according to the teacher, some teachers cannot apply or plan their lessons accurately as stated in curriculum. At this point, he claimed that application of problem solving in lessons is not only about curriculum or books but also about perspective of teachers to the problem solving. Therefore, teaching problem-solving processes in pre-service education programs or even as in-service training is more important to provide an insight to problem solving to teachers than expecting from them to apply problem-solving process by changing curriculum. Because, whether curriculum changes or not, teachers have still planned their lessons what their perceptions about teaching concepts.

References

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.

Author Information

Sinem Sozen Ozdogan (presenting / submitting)

TED University

Elementary Education

Ankara

Bilal Özcakır

Ahi Evran University, Turkey

Burcu Orhan

OSYM, Turkey

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