Man In India, 97 (12) : 213-220
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PROBLEM SOLVING, BASED ON METACOGNITIVE LEARNING ACTIVITIES, TO IMPROVE MATHEMATICAL REASONING SKILLS OF STUDENTS Muhammad Ammar Naufal1, Noor Azean Atan2, Abdul Halim Abdullah3 and Mohd Salleh Abu4 Many studies have linked individual weakness in Mathematical reasoning skills with difficulties in Mathematics problem solving. Some other studies found that, a metacognitive process, based on appropriate mathematical thinking, can help a person to solve Mathematical problems more effectively and meaningfully. This study aimed to improve Mathematical reasoning skills of students in a secondary school in Indonesia through a learning strategy, based on a metacognitive process, namely, PMMK. The design and development of PMMK were implemented in five phases. The first phase, the basic information of Mathematical reasoning skills and Metacognitive skills used Mathematical Reasoning Test (UPM) and Self-Monitoring Questionnaire (SMQ), respectively. The results of the basic information were then used in the design and construction phase of PMMK, based on Garofalo and Lester Metacognitive Thinking and Shipman Mathematical Reasoning. Content validity and construction of PMMK which covered probability topics were examined by three assessors of Mathematics education. Overall, the assessors found that, the content, activities, language and design of PMMK qualified for the objectives with a Kappa Agreement = 0.65. Moreover, in the implementation and verification phase, the impact of PMMK was conducted on a total of 32 students in grade 11. A comparative analysis, based on UPM score, showed that, PMMK had helped to improve Mathematical reasoning skills of the students highly significantly, with the effect size = 79.25%. Keywords: Problem Solving, Metacognitive Skills, Mathematical Reasoning Skills.
INTRODUCTION Nowadays, it is necessary for individuals to have thinking skills such as Mathematical reasoning skills (Arikan and Unal, 2014) and metacognitive skills, which are among the basic skills required in the 21st Century (Duran and Sendag, 2012). One of the ways to provide pupils with the opportunity to stimulate their thinking skills is to let them conduct an investigation, where they will learn to solve non-routine problems. In the latest development of Mathematics education, Mathematical reasoning skills are increasingly emphasized in teaching and learning Mathematics. National Council of Teachers of Mathematics (NCTM) (1989) in Curriculum and Evaluation Standards for School Mathematics had stressed the importance of Mathematical reasoning skills as the main vision of Mathematics education. 1 2 3 4
Universiti Teknologi Malaysia, E-mail:
[email protected] Universiti Teknologi Malaysia, E-mail:
[email protected] Universiti Teknologi Malaysia, E-mail:
[email protected] Universiti Teknologi Malaysia, E-mail:
[email protected]
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A previous research showed that, the application of metacognitive strategies was very important for the learning process (Zhang, 2008; Aydin, 2011). Teachers could incorporate and use a variety of teaching and learning strategies to enhance students’ development of metacognitive skills (Delvecchio, 2011), while the ability to develop students’ metacognitive skills, can inspire them to have creative and innovative minds (Ghasempour, et. al., 2013). However, Hartman (2001) reported that, many students experienced academic difficulties because they always focused on content without first studying the metacognitive strategies needed to support such initiatives. Furthermore, Everson and Tobias (2001) and Matanzo and Harris (1999) stated that, most students who enter colleges had not been taught about metacognitive at the school level to check their metacognitive skills. Matanzo and Harris (1999) also found that, many students did not know about metacognitive. According to Aydin (2011), metacognitive could be developed through teaching and learning. Therefore, metacognitive strategies to increase Mathematical reasoning skills were identified as being suitable to be adopted in classroom activities, deemed important from cognitive and metacognitive point of view. BACKGROUND OF THE PROBLEM Enhancement of Mathematical education is a matter that has been a concern in many countries. Mastery of Mathematical literacy is a key component that affects the successfulness of students in education and society in the future (Engle et al., 2007). The focus on learning Mathematics and Mathematical ability development has been a regularly discussed topic in the study of education and psychology for over 100 years (Geary, 2006). It is undeniable that the quality of Mathematics education, especially in Indonesia, is still low compared to that in other countries. So far, Indonesia has not been able to go out from the lowest level of the International examination such as TIMSS and PISA (TIMSS 2011, PISA 2012). The Indonesian students were found to have a less Higher Order Thinking Skills (HOTS) or lacking in the ability to solve non-routine problems, which could improve their thinking skills. This is really a disadvantage, because the main weighting in the TIMSS and PISA for the cognitive domain are questions on fractions, in the form of applications (39%) and reasoning (39%), while 40% more questions are on the level of understanding. The assessment showed that the competences of Indonesian students were at a level where they were able to answer arithmetic questions, but not questions involving passages, giving opinions and producing reasoning (TIMSS 2011 and PISA 2012). In general, it was found that, many students had limited Mathematical reasoning skills at a high level thinking (Lawson et al., 1991; Lawson, 1995; Melissa, 2007). This is because, the current practice of teaching still relies on traditional methods and does not emphasize on the development of thinking skills. Students only
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complete the tasks assigned by the teachers as found in textbooks, thinking that Mathematics is a procedure to obtain the correct answers (Ellerton, 2013). They are not accustomed to using metacognitive strategies in solving problems that exist in real life. These are supported by findings from previous studies, such as those by Haidar and Al Naqabi (2008), Fatin (2006) and Johari et al. (2008) which revealed that, students were unable to master metacognitive strategies such as planning, interpretation, execution, review, assessment, monitoring and good reflection. The students are weak in problem solving not because they do not have the knowledge to solve the problem, but mainly because they lack metacognitive knowledge and Mathematical reasoning skills. Schoenfeld (1987) and many other researchers had found that, students only had procedural knowledge to solve problems but were not able to use it properly because of lack of metacognitive skills. Garofalo and Lester (1985), Schoenfeld (1994), De Corte (2003), and many other researchers had explained that, metacognition was an important element in solving Mathematical problems to enable pupils to evaluate their ability to perform complex tasks and to think the alternative ways, if their work seemed useless or ineffective. Teaching methods, based on Metacognitive skills, can foster students’ ability to review their own cognitive process. In other words, metacognition enables students to work independently and helps them to be responsible for their own learning, particularly in Mathematics (Nelson, 2012). NCTM (2000) stated that, to prepare pupils for the 21st century, students should have the skills on the use of Mathematical knowledge to solve problems, Mathematical communication skills and Mathematical reasoning skills. Mathematical reasoning skills have a very important role in students’ thinking process. If these skills are not developed in the students, they will consider Mathematics merely as one set of Mathematical procedures, in accordance with sample questions and answers without knowing their meanings. RESEARCH OBJECTIVE This study aimed to design and develop a module of problem solving strategies based on metacognitive learning activities (namely PMMK) to increase the Mathematical reasoning skills of students in secondary schools in Makassar, Indonesia and also to evaluate the impact of PMMK to help students improve their Mathematical reasoning skills. METHODOLOGY This design and development study used ADDIE Model as a basic guidance to construct the PMMK. The quantitative method which was used, involved grade11 students in senior high school of 5 in Makassar, Indonesia. The PMMK was reviewed by three lecturers who were experts in the field of Mathematics for the
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content and face validity, prior to the actual study. Mathematical Reasoning Test (UPM), which comprised two non-routine questions and was used as an assessment instrument, was distributed to the 32 respondents, prior to and after intervention using PMMK. The data collected from the assessors were analyzed using analysis of Cohen Kappa by IBM SPSS Software version 23. Subsequently, the data collected from students were analyzed using statistical inference (paired sample t-test) while t2 2 the size-effect was analyzed by the formula of r � 2 . t � df RESULTS Based on Table 1, the Kappa agreement found among the evaluators against PMMK was at a good level with a mean value of 0.653. TABLE 1: THE KAPPA AGREEMENT VALUES AMONG THE ASSESSORS Assessor 1
Assessor 2
Assessor 3
Mean
0.765
0.765
0.429
0.653
The Kappa Agreement
Meanwhile, the result of the inference analysis, which assessed the impact of PMMK to enhance mathematical reasoning skills of students can be seen in Table 2. The results showed that the value of p < 0.001 < 0.05. Therefore, the null hypothesis was rejected, indicating that, there were significant differences in means, between pre-test and post-test. TABLE 2: COMPARATIVE ANALYSIS OF MATHEMATICAL REASONING SKILLS OF STUDENTS BEFORE AND AFTER INTERVENTION (PRE–POST T -TEST) Pre-test Post-test
N
df
Mean
Standard Deviation
Differences of Mean
t
Sig. Value
32 32
31 31
4.8 13.3
4.488 2.372
8.500
10.880
0.000
A result of the calculation based on the interpretation of Cohen (1988), found a very large effect size.
10.880 2 t2 r 2 � 2 � � 0.7925 (79.25%) t � df 10.8802 � 31 DISCUSSION The overall results showed that, the face and content validity of the PMMK conducted by the assessors was in a good level, indicating that, the constructed PMMK qualified for the conditions of use in the actual study and was also suitable with the objectives of the development of PMMK, based on metacognitive learning activities to improve Mathematical reasoning skills of the students. This was in
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agreement with the finding by Slavin (2011) which regarded the measurement of validity as being important to make sure that the instrument used really measured what was intended to be measured. The results of the inference analysis revealed that, the PMMK could help improve Mathematical reasoning skills of the students in the Secondary Schools in Makassar. The increased level of mathematical reasoning skills showed that, there were positive effects gained in the use of PMMK, based on metacognitive learning activities built in the ADDIE Model. The use of PMMK can improve Mathematical reasoning skills of the students as the activities designed enable them to have an enhancement in the effectiveness, non-routine problem solving skills and the number of maximum students in the learning process. Therefore, this study was able to support teaching and learning process by using PMMK. These findings were in line with those in previous studies, in terms of the use of metacognitive methods in teaching to improve Mathematical reasoning skills of students (Amsel et al., 2008; Kramarski and Mevarech, 2003) Based on the comparative analysis of t-test, Mathematical reasoning skills of the students had a mean score of 4.8 (low level) and 13.3 (good level), before the intervention and after the intervention, respectively. This showed that, the PMMK had contributed to the improvement in the level of Mathematical reasoning skills of the students. The null hypothesis, therefore, which stated that, there were no significant differences in the increased level of the students’ Mathematical reasoning skills before (pre-test) and after (post-test) intervention, was rejected. In other words, the PMMK, based on metacognitive learning activities and used as a learning tool had a positive impact on the improvement of Mathematical reasoning skills of the students. Using PMMK also resulted in a large effect size on Mathematical reasoning skills. This finding was in agreement with the finding by Kramarski and Mevarech (2003) which stated that, the use of metacognitive methods would be able to improve Mathematical reasoning skills of peers more successfully during learning sessions. CONCLUSION The whole study successfully met its objective in assessing the impact of PMMK, based on metacognitive learning activities to help enhance Mathematical reasoning skills of secondary school students in Makassar. The majority of the Mathematical reasoning skills of the students increased after the intervention using PMMK in the learning process and were at a good level. However, this study only involved 32 participants from senior high school of 5 in Makassar. As such, it is proposed that, the study be extended to the other schools. This study can be a reference to the importance of metacognitive skills to enhance students’ Mathematical reasoning
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skills, to the teachers, curriculum writers, lecturers and schools. Indirectly, it is expected to be able to improve the students’ performance at the international level through TIMSS and PISA. References Amsel, E., Klaczynski, P. A., Johnston, A., Bench, S., Close, J., Sadler, E., and Walker, R. (2008). A dual-process account of the development of scientific reasoning: The nature and development of metacognitive intercession skills. Cognitive Development, 23(4), 452-471. Arikan, E, E, dan Unal, H. (2014). An investigation of eight grade students’ problem posing skills (Turkey sample). International Journal of Research in Education and Science (IJRES), 1(1), 23-30. Aydin, F. (2011). Geography teaching and metacognitive: Educational Research and Reviews (Academic Journals). 6(3): 274-278. Bain, R. (2005). They thought the world was flat?: Applying the principles of how people learn in teaching high school history. In M.S. Donovan and J. D. Bransford (Eds), How students learn: history, Mathematics and Science in the classroom (pp 179-216). Washington, DC: The national Academies Press. Baker, L. dan Cerro, L. C. (2000). Assessing metacognitive in children and adults. In G. Schraw and J. C. Impara (Eds.), Issues in the measurement of metacognitive (pp 99-145). Lincoln: Buros Institute of Mental Measurements. Davidson, J. E., Deuser, R., dan Sternberg, R. J. (1994). The role of metacognitive in problem solving. In J. Metcalfe and A. P. Shimamura (Eds.), Metacognitive: knowing about knowing (ms. 207-226). Cambridge, MA: MIT Press. De Corte, E. (2003). Intervention Research: A Toll for Bridging the Theory-Practice Gap in Mathematics Education? In Rogerson, A. Proceeding of Mathematics Education: The Mathematics Education into the 21th Century Project. Czech Republic. September 19-25, 2003. Delvecchio, F. (2011). Students’ use of metacognitive skills while problem solving in high school chemistry. (Doctoral Dissertation of the Queen’s University, Canada). Dunlosky, J.dan Metcalfe, J. (2009). Metacognitive. Thousand Oaks, CA: Sage. Duran, M., dan Sendag, S. (2012). A Preliminary Investigation into Critical Thinking Skills of Urban High School Students: Role of an IT/STEM Program. Creative Education, 3(02), 241. Efklides, A. (2006). Metacognitive and affect: What can metacognitive experiences tell us about the learning proses? Educational Research Review, (1): 3-14. Ellerton, N. F. (2013). Engaging pre-service middle-school teacher-education students in mathematical problem posing: development of an active learning framework. Educational Studies in Mathematics, 83(1), 87-101. Engle, P., Grantham-McGregor, Black, M., Walker, S., dan Wachs, T. (2007). How to avoid the loss of potential in over 200 million young children in the developing world, Child Health and Education, 1(2), 68-87.
PROBLEM SOLVING, BASED ON METACOGNITIVE LEARNING...
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Everson, H. T. and Tobias, S. (2001). The ability to estimate knowledge and performance in college: A metacognitive analysis. In H. J. Hartman, Metacognition in Learning and Instruction: Theory, Research and Practice: (pp. 64-92). Boston: Kluwer. Fatin Aliah Phang Abdullah (2006). The Patterns of Physics Problem-Solving from the Perspective of Metacognition. Master Thesis. University of Cambridge, Cambridge. Fisher, R. (2007). Dialogic teaching: Developing thinking and metacognitive through philosophical discussion. Early Child Development and Care, 177(6-7): 615-631. Flavell, J. H. (1979). Metacognitive and Cognitive Monitoring: A New Area of Cognitive Developmental Inquiry. American Psychologist, 34, 906 – 911. Garofalo, J. and Lester, F.K. (1985). Metacognitive, Cognitive Monitoring and Mathematical Performance. Journals for Research in Mathematics Education. 16(3): 163-176. Geary, D. C. (2006). Development of mathematical understanding. In D. Kuhl and R. S. Siegler (Vol. Eds.), Cognition, perception, and language, Vol 2 (pp. 777-810). W. Damon (Gen. Ed.), Handbook of child psychology (6th Ed.). New York: John Wiley and Sons. Ghasempour, Z., Md Nor Bakar, and Jahanshahloo, G., R. (2013). Innovation in Teaching and Learning through Problem Posing Tasks and Metacognitive Strategies. International Journal of Pedagogical Innovations, 1(1): 57-66. Haidar, A. H. and Al Naqabi, A. K. (2008). Emirati High School Students’ Understandings of Stoichiometry and the Influence of Metacognition on Their Understanding. Research in Science and Technology Education. 26(2), 215-237. Hartman, H. J. (2001). Teaching metacognitively. In H. J. Hartman (Ed.), Metacognition in Learning and Instruction: Theory, Research and Practice (pp. 149-172). Boston: Kluwer. Johari bin Surif, Nor Hasniza binti Ibrahim dan Mohammad Yusof bin Arshad (2008). Kajian Pembinaan Konsep Sains berdasarkan Model Generatif Metakognitif di Kalangan Pelajar. Project Report. Faculty of Education, Skudai, Johor. Kramarski, B., and Mevarech, Z. R. (2003). Enhancing Mathematical Reasoning in the Classroom: The Effects of Cooperative Learning and Metacognitive Training. American Educational Research Journal, 40(1), 281–310. http://doi.org/10.3102/00028312040001281. Lawson, A. E., McElrath, C. B., Burton, M. S., James, B. D., Doyle, R. P., Woodward, S. L., et al. (1991). Hypothetico-deductive reasoning skill and concept acquisition: Testing a constructivist Hypothesis. Journal of Research in Science Teaching, 8(10), 953-970. Lawson, A. E. (1995). Science teaching and the development of thinking. Belmont, CA: Wadsworth Publishing Company. Matanzo, J. B. and Harris, D. L. (1999). Encouraging metacognitive awareness in pre-service literacy course: Advancing the world of literacy. Florida: Florida Atlantic University. Melissa, M. S. (2007). Scientific Mathematical Reasoning Skills Development in The Introductory Biology Courses for Undergraduates. United States: The Ohio State University. National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA.: Author. National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA.: Author.
220
MAN IN INDIA
Nelson, L. L. (2012). The effectiveness of metacognitive strategies on 8 th grade students in mathematical achievements and problem solving skills. Doctor of Philosophy Thesis. The Graduate School Southern University and A and M College Baton Rouge, Louisiana. Panaoura, A. (2004). Young pupils’ metacognitive ability in maths. Tesis master, Department of Education, University of Cyprus. Cyprus. Phang, F.A. (2009). The Patterns of Physics Problem-Solving from the Perspective of Metacognitive. Doctoral Dissertation. University of Cambridge. England. Schoenfeld, A. H. (1987). What’s All the Fuss About Metacognitive? In Cognitive Science and Mathematics Education. New Jersey: Lawrence Erlbaum Associated. 189-215. Schoenfeld, A. H. (1994). Reflection on Doing and Teaching Mathematics. In Mathematical Thinking and Problem Solving. New Jersey: Lawrence Associated. 53-71. Slavin, R. E. (2011). Educational Psychology: Theory and Practice (10th ed.). Boston: Pearson. Sperling, R., Howard, B., Staley, R. dan DuBois, N. (2004). Metacognitive and self-regulated learning constructs. Educational Research and Evaluation, 10(2): 117- 139. Veenman, M. V. J., Kok, R., dan Blote, A. W. (2005). The relation between intellectual and metacognitive skills at the onset of metacognitive skill development. Instructional Science, 33: 193–211 Zhang, L. J. (2008). Constructivist pedagogy in strategic reading instruction: Exploring pathways to learner development in English as a SL (ESL) classroom. Instruction Science, 36(2): 89-116.
