ANALISIS PROSES TERJADINYA PENALARAN REVERSIBEL UNTUK MASALAH INVERS
(1) Universitas Cokroaminoto Palopo
(2) Universitas Cokroaminoto Palopo
(3) Universitas Muhammadiyah Makassar
(*) Corresponding Author
Abstract
ujuan utama dari penelitian ini yaitu untuk menyelidiki proses terjadinya penalaran reversibel mahasiswa untuk masalah invers. Metode penelitian yang digunakan untuk mengungkapkan penalaran reversibel menggunakan metode penelitian kualitatif dengan pendekatan studi kasus. Pengambilan sampel dilakukan dengan menggunakan teknik purposive sampling di mana sampel penelitian dipilih berdasarkan kriteria penalaran reversibel. Pengambilan data dalam penelitian ini menggunakan hasil karya matematika mahasiswa, berpikir keras, wawancara, dan komponen yang menyebabkan penalaran reversibel. Hasil penelitian menunjukkan bahwa proses terjadinya penalaran riversibel diawali dengan adanya hambatan yang menyebabkan partisipan tidak mampu melanjutkan proses penyelesaian, sehingga terjadi proses metakognisi dengan menganalisa kembali masalah secara analitik dan mengembangkan strategi heuristik lainnya. Partisipan menunjukkan perubahan sudut pandang di mana ia awalnya memaknai invers sebagai tindakan menukar variabel independen dan dependen, dan beralih dengan memaknai invers sebagai kebalikan dari proses fungsi yang melibatkan analogi dan representasi gambar. Kontribusi penelitian ini yaitu memberikan pengetahuan bahwa penalaran reversibel dapat terjadi dalam memahami dan menyelesaikan masalah matematika pada materi invers.
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Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology: qualitative reaserch in psychology (Vol. 3). University of the West of England. https://doi.org/10.1191/1478088706qp063oa
Chun, J. (2017). Construction of the Sum of Two Covarying Oriented Quantities. Potential Analysis. University of Georgia. https://doi.org/10.1007/s11118-013-9365-6
Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating quantitative and qualitative research. Educational Research (Vol. 4). https://doi.org/10.1017/CBO9781107415324.004
Di Stefano, M., Litster, K., & MacDonald, B. L. (2017). Mathematics Intervention Supporting Allen, a Latino EL: A Case Study. Education Sciences, 7(2), 57. https://doi.org/10.3390/educsci7020057
Dougherty, B., Bryant, D. P., Bryant, B. R., & Shin, M. (2016). Helping Students With Mathematics Difficulties Understand Ratios and Proportions. TEACHING Exceptional Children, 49(2), 96–105. https://doi.org/10.1177/0040059916674897
Dougherty, B. J., Bryant, D. P., Bryant, B. R., Darrough, R. L., & Pfannenstiel, K. H. (2015). Developing Concepts and Generalizations to Build Algebraic Thinking: The Reversibility, Flexibility, and Generalization Approach. Intervention in School and Clinic, 50(5), 273–281. https://doi.org/10.1177/1053451214560892
Flanders, S. T. (2014). Investigating Flexibility, Reversibility, and Multiple Representation in a Calculus Enviroment. University of Pittsburg.
Goldin, G. A. (2000). A Scientific Perspective on Structured, Task-Based Interviews in Mathematics Education Research. Handbook of Research Design in Mathematics and Science Education. https://doi.org/10.4324/9781410602725.ch19
Haciomeroglu, E. S. (2007). Calculus Student’s Understanding of Derivative Graphs: Problem of Representation in Calculus. The Florida State University.
Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. (2009). The role of reversibility in the learning of the calculus derivative and antiderivative graphs. Proceedings of the 31st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 5, 5, 81–88.
Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. C. (2010). Contrasting cases of calculus students’ understanding of derivative graphs. Mathematical Thinking and Learning, 12(2), 152–176. https://doi.org/10.1080/10986060903480300
Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432. https://doi.org/10.1080/07370008.2010.511565
Hackenberg, A. J. (2013). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior, 32(3), 538–563. https://doi.org/10.1016/j.jmathb.2013.06.007
Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students’ fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196–243. https://doi.org/10.5951/jresematheduc.46.2.0196
Hackenberg, A. J., & Lee, M. Y. (2016). Students’ distributive reasoning with fractions and unknowns. Educational Studies in Mathematics, 93(2), 245–263. https://doi.org/10.1007/s10649-016-9704-9
Inhelder, & Piaget. (1958). The Growth of Logical Thinking From Child to Adolecence. New York: Basic Books, Inc.
Kim, Y. R., Park, M. S., Moore, T. J., & Varma, S. (2013). Multiple levels of metacognition and their elicitation through complex problem-solving tasks. Journal of Mathematical Behavior, 32(3), 377–396. https://doi.org/10.1016/j.jmathb.2013.04.002
Krutetskii, V. A. (1976). The Psychology of Mathematical Abilities in Schoolchildren. Journal for Research in Mathematics Education (Vol. 8). https://doi.org/10.2307/748528
Lee, M. Y., & Hackenberg, A. J. (2014). Relationships Between Fractional Knowledge and Algebraic Reasoning: the Case of Willa. International Journal of Science and Mathematics Education, 12(4), 975–1000. https://doi.org/10.1007/s10763-013-9442-8
Leron, U., & Hazzan, O. (2009). Intuitive vs analytical thinking: Four perspectives. Educational Studies in Mathematics, 71(3), 263–278. https://doi.org/10.1007/s10649-008-9175-8
Leron, U., & Paz, T. (2014). Functions via everyday actions: Support or obstacle? Journal of Mathematical Behavior, 36, 126–134. https://doi.org/10.1016/j.jmathb.2014.09.005
Linchevski, L., & Herscovics, N. (1996). Crossing the cognitive gap between arithmetic and algebra: Operating on the unknown in the context of equations. Educational Studies in Mathematics, 30(1), 39–65. https://doi.org/10.1007/BF00163752
Lubin, A., Simon, G., Houdé, O., & De Neys, W. (2015). Inhibition, conflict detection, and number conservation. ZDM - Mathematics Education, 47(5), 793–800. https://doi.org/10.1007/s11858-014-0649-0
Mackrell, K. (2011). Design decisions in interactive geometry software. ZDM - International Journal on Mathematics Education, 43(3), 373–387. https://doi.org/10.1007/s11858-011-0327-4
Miles, M. B., Huberman, A. M., & Saldana, J. (2014). Qualitative Data Analysis: A Methods Sourcebook (Third Edit). United States of America: SAGE Publications, Inc.
Nathan, M. J., & Koedinger, K. R. (2000). Teachers’ and researchers’ beliefs about the development of algebraic reasoning. Journal for Research in Mathematics Education, 31(2), 168–190. https://doi.org/10.2307/749750
Nolte, M., & Pamperien, K. (2017). Challenging problems in a regular classroom setting and in a special foster programme. ZDM - Mathematics Education, 49(1), 121–136. https://doi.org/10.1007/s11858-016-0825-5
Paoletti, T. (2015). Pre-Service Teachers’ Development of Bidirectional Reasoning.
Paoletti, T., Stevens, I. E., Hobson, N. L. F., Moore, K. C., & LaForest, K. R. (2018). Inverse function: Pre-service teachers’ techniques and meanings. Educational Studies in Mathematics, 97(1), 93–109. https://doi.org/10.1007/s10649-017-9787-y
Ramful, A. (2009). Reversible Reasoning In Multiplicative Situations: Conceptual Analysis, Affordances And Constra1nts. Dissertation.
Ramful, A. (2014). Reversible reasoning in fractional situations: Theorems-in-action and constraints. Journal of Mathematical Behavior, 33, 119–130. https://doi.org/10.1016/j.jmathb.2013.11.002
Ramful, A. (2015). Reversible reasoning and the working backwards problem solving strategy. Australian Mathematics Teacher, 71(4), 28–32.
Ramful, A., & Olive, J. (2008). Reversibility of thought: An instance in multiplicative tasks. Journal of Mathematical Behavior, 27(2), 138–151. https://doi.org/10.1016/j.jmathb.2008.07.005
Ron Tzur. (2011). Can Dual Processing Theories of Thinking Inform Conceptual Learning in Mathematics? The Mathematics Enthusiast, 8(3), 597–636. Retrieved from http://scholarworks.umt.edu/tme/vol8/iss3/7%5Cnhttp://jasonadair.wiki.westga.edu/file/view/Can+Dual+Processing+Theories+of+Thinking+Inform+Conceptual+Learning+in+Mathematics.pdf/349220092/Can Dual Processing Theories of Thinking Inform Conceptual Learning
Sangwin, C. J., & Jones, I. (2017). Asymmetry in student achievement on multiple-choice and constructed-response items in reversible mathematics processes. Educational Studies in Mathematics, 94(2), 205–222. https://doi.org/10.1007/s10649-016-9725-4
Simon, M. A., Kara, M., Placa, N., & Sandir, H. (2016). Categorizing and promoting reversibility of mathematical concepts. Educational Studies in Mathematics, 93(2), 137–153. https://doi.org/10.1007/s10649-016-9697-4
Simon, M. A., Placa, N., & Avitzur, A. (2016). Participatory and anticipatory stages of mathematical concept learning: Further empirical and theoretical development. Journal for Research in Mathematics Education, 47(1), 63–93. https://doi.org/10.5951/jresematheduc.47.1.0063
Sriraman, B. (2015). Problem Solving , and the Ability to Formulate Generalizations :, XIV(3), 151–165.
Steffe, L. P. (2002). Chapter 1 A New Hypothesis Concerning Children ’ s Fractional Knowledge. The Journal of Mathematical Behavior, 20(3), 1–12. https://doi.org/10.1007/978-1-4419-0591-8
Steffe, L. P., & Olive, J. (2009). Children’s fractional knowledge. Springer Science & Business Media.
Tzur, R. (2004). Teacher and students’ joint production of a reversible fraction conception. Journal of Mathematical Behavior, 23(1), 93–114. https://doi.org/10.1016/j.jmathb.2003.12.006
Tzur, R. (2007). Fine grain assessment of students’ mathematical understanding: Participatory and anticipatory stagesin learning a new mathematical conception. Educational Studies in Mathematics, 66(3), 273–291. https://doi.org/10.1007/s10649-007-9082-4
Tzur, R., & Simon, M. (2004). Distinguishing two stages of mathematics conceptual learning. International Journal of Science and Mathematics Education, 2(2), 287–304. https://doi.org/10.1007/s10763-004-7479-4
Vergnaud, G. (1998). A comprehensive theory of representation for mathematics education. Journal of Mathematical Behavior, 17(2), 167–181. https://doi.org/10.1016/S0364-0213(99)80057-3
Vilkomir, T., & O’Donoghue, J. (2009). Using components of mathematical ability for initial development and identification of mathematically promising students. International Journal of Mathematical Education in Science and Technology, 40(2), 183–199. https://doi.org/10.1080/00207390802276200
Yimer, A., & Ellerton, N. F. (2010). A five-phase model for mathematical problem solving: Identifying synergies in pre-service-teachers’ metacognitive and cognitive actions. ZDM - International Journal on Mathematics Education, 42(2), 245–261. https://doi.org/10.1007/s11858-009-0223-3
DOI: http://dx.doi.org/10.24127/ajpm.v10i2.3450
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