pp. 621-632 | Article Number: iejme.2017.036
Published Online: September 11, 2017
Article Views: 183 | Article Download: 256
This study is qualitative in nature, aims to describe students’ reflective abstraction in solving problems related to number sequence. As subjects of this study are 6 students, composed of 3 students (2 female and one male) from State Senior High School (SMAN) 1 Pasuruan, and 3 students (2 male and 1 female) from SMAN 1 Pandaan, East Java, Indonesia. In this study, the researchers were acted as human instruments. Narrative description was compiled by assembling descriptions of reflective abstraction stages associated with the criteria of high, intermediate, or low thinking processes, and the cognitive levels including knowledge application and reasoning. The results suggest that students’ strategies of problem solving are not directly proportional to their level of reflective abstraction. It was also found that students’ reflective abstraction furnish students’ individually unique solution, simply complicated, while interventions are important in its attainments.
Keywords: Stages and levels of reflective abstraction, high school students, number sequence
Abrahamson, D. (2006). Exposing Piaget’s Scheme: Empirical Evidence for the Ontogenesis of Coordination in Learning a Mathematical Concept. University of California, Berkeley.
Belbase, S. (2010). A Reflective Journey through Theory and Research in Mathematical Learning and Development. Retrieved from http://files.eric.ed.gov/fulltext/ED514497.pdf
Beth, E. W., & Piaget, J. (1966). Mathematical Epistemology and Psychology. Dordrecht, The Netherlands: D. Reidel.
Bowie, L. 1998. A Learning Theory Approach to Students’ Misconceptions in Calculus. Thesis. University of Cape Town, South Africa. Retrieved from https://open.uct.ac.za/bitstream/ handle/11427/9556/thesis_sci_1998_bowie_l.pdf?sequence=1
Carter, P. & Russel, K. (2004). The Complete Book of Fun Maths, 250 Confidence Boosting Tricks, Test and Puzzles. New York, NY: John Wiley & Sons. Retrieved from http://as.wiley.com/WileyCDA/WileyTitle/productCd-0470870915.html
Clark, D.A. (2014). The Wiley Handbook of Cognitive Behavioral Therapy. First Edition. New York, NY: Guilford Press.
Copley, J. (2013). Mathematical Thinking. Retrieved from http://images.pearsonclinical.com/images/ Assets/WSS_5/Research Summary_Mathematical_Thinking_FNL.pdf
Creswell, J.W. (2014). Research Design: Qualitative, Quantitative, and Mixed Methods Approaches. Fourth Edition. Sage Publications.
Chrisopher, J.C. & Campbell, R.L. (2008). An Interactivist-Hermeneutic Metatheory for Positive Psychology. Theory & Psychology. Vol. 18(5), 675–697. Sage Publications. DOI: 10.1177/ 0959354308093401 Retrieved from http://tap.sagepub.com.
Dubinsky, Ed. (1992). Reflective Abstraction in Advanced Mathematical. In Advanced Mathematical Thinking. David Tall (ed), 95-123. Dordrecht, The Netherlands: Kluwer Academic Publisher.
Ferrari, P.L. (2003). Abstraction in Mathematics. Philosophical Transactions of the Royal Society B: Biological Sciences. Vol. 358(1435): 1225–1230. doi: 10.1098/rstb.2003.1316
Glasersfeld, Ernst von. (1991). Abstraction, Re-Presentation, and Reflection. In Epistemological foundations of mathematical experience. L.P. Steffe (Ed). New York, NY: Springer.
Goedecke, J. (2013). Abstraction in Mathematics. A course material on powerpoint file. Queen’s College. Retrieved from https://www.dpmms.cam.ac.uk/~jg352/pdf/TMSTalk.pdf
Gray, E. & Tall, D. (2001). Relationships between Embodied Objects and Symbolic Procepts: An Explanatory Theory of Success and Failure in Mathematics. Retrieved from http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot-pme25-pinto-tall.pdf
Hazzan, O & Zazkis, R. (2005). Reducing Abstraction: The Case of School Mathematics. Retrieved from http://www.sfu.ca/~zazkis/publications/Reducing%20Abstraction.pdf
Kasali, R. (2006). Change! Cetakan ke delapan. Jakarta: PT Gramedia Pustaka Utama.
Kumar, R. (2011). Research Methodology: a step-by-step guide for beginners. Third Edition. Sage Publications, Inc.
Marlow, E. (1990). Psychological Foundations in Teaching Mathematics. Retrieved from http://files.eric.ed.gov/fulltext/ED431606.pdf
Mason, J., Burton, L. & Stacey, K. (2010), Thinking Mathematically. Second Edition. England: Pearson Education Limited.
Michelmore, M & White, P. (2004). Abstraction in Mathematics and Mathematics Learning. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education. Vol 3, 329–336. Retrieved from https://www.emis.de/ proceedings/PME28/RR/RR031_Mitchelmore.pdf
Mitchelmore, M. & White, P. (2007). Abstraction in Mathematics Learning. In Mathematics Education Research Journal. Vol 19(2), 1-9.
Mousoulides, N. & Gagatsis, A. (2004). Algebraic and Geometry. Approach in Function Problem Solving. Retrieved from http:// files.eric.ed.gov/fulltext/ED489596.pdf
Ozmantar, F. M. & Monaghan, J. (2007). A Dialectical Approach to Formation of Mathematical Abstractions. Mathematics Education Research Journal,Vol.19 (2), 89-112.
Panasuk, R. M. (2011). Taxonomy for Assessing Conceptual Understanding in Algebra Using Multiple Representation. College Student Journal, Vol. 45 (2), 219-232. Spring Hill Station, Mobile, AL. Retrieved from http://jasonadair.wiki.westga.edu/file/view/Taxonomy+for+ assessing+conceptual+understanding+in+Algebra+using+multiple+representations.pdf
Paschos, T. & Farmaki, V. (2006). The Reflective Abstraction in the Construction of the Concept of the Definite Integral: A Case Study. Retrieved from ftp://ftp.math.ethz.ch/EMIS/proceedings/ PME30/4/337.pdf),
Ruch, F.L. (1967). Psychology and Life. Glenview, IL: Scott Foresman.
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, NY: MacMillan.
Silver, H.F., Brunsting, J.R., Walsh, T. & Thomas, E.J. (2012). Math Tools, Grades 3–12. 60+ Ways to Build Mathematical Practices, Differentiate Instruction, and Increase Student Engagement. Second Edition. Sage Publishing.
Solso, R.L, MacLin, O.H. & MacLin, M.K. (2008) Cognitive Psychology. Eighth Edition. Pearson.
Stacey, K. (2014) What is mathematical thinking and why is it important? Retrieved from https://www.researchgate.net/publication/254408829
Tall, D. (2002) Advanced Mathematical Thinking. New York, NY: Kluwer Academic Publishers.
Tall, D. (2009) The Development of Mathematical Thinking: Problem-Solving and Proof. Retrieved from http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2009d-paper-for-john mason.pdf
Turnau, S. (Ed) (2008). Handbook of Mathematics Teaching Improvement:Professional Practices that Address PISA. Output of the Krygowska Project. “Professional Development of Teacher-Researchers” 2005-2008. University of Rzeszów. KSERKOP, Kraków, Poland: Drukarnia Cyfrowa.
Walle, J. A.V. (2007). Elementary and Middle School Mathematics. Cetakan ke-tujuh. Jakarta: Penerbit Erlangga.
Zimbardo, P.G. & Ruch, F.L. (1977). Psychology and Life. Ninth Edition. Chicago, Illinois: Pearson Scott Foresman.
Zull, J. E. (2002). The Art of Changing the Brain. Sterling, VA: Stylush Publishing.