How are the Concepts of Permutation and Combination Conceived in the Minds of High School Students?
Keywords:
Permutation, Combination, Abstraction, Mental Representation, Reflection, Mathematics EducationAbstract
Counting problems are fundamental to the development and formation of mathematical thinking, but they can be challenging for students. In order to understand the nature of the errors students make while solving counting problems, to identify the difficulties they experience, and to determine and interpret the variables that may affect the difficulties they experience, it is necessary to understand the conceptualizations of the concepts of permutation and combination that form the basis of the subject of counting (combinatorics) more deeply. In this study, it was examined how an 11th grade student who studied counting in the 10th grade made sense of the concepts of permutation and combination in his mind. The research was conducted with a case study, which is one of the qualitative research methods. The study group of the research consisted of an 11th grade student studying at a formal science high school. In the research, data were collected through a worksheet containing problem situations developed by the researcher, semi-structured interviews, and interview audio recordings. The data obtained from the worksheet and interviews were evaluated with content analysis. It was observed that the student's knowledge of the conceptual justifications of the multiplication principle and questioning his actions were effective in the abstraction of the concepts of permutation and combination and in the effective display of a successful counting operation. The study concluded that the schemas that characterize mental processes and the mental actions that form these schemas play important roles in the individual's abstraction, meaning and development of new mathematical concepts from their existing knowledge. In addition, the study suggests that teachers design activities that will use students' mental representations and reflection actions so that students can make conceptual progress, and that they support students in determining and following their own solutions in these activities.
References
Akbulut, Y. (2012). Veri çözümleme teknikleri. A. Şimşek (Ed.), Sosyal Bilimlerde araştırma yöntemleri içinde (s. 162-195). T.C. Anadolu Üniversitesi Yayını No: 2750, Açıköğretim Fakültesi Yayını No: 1708 (132-164), Eskişehir.
Akbulut, Y. (2013). Verilerin analizi. A. A. Kurt (Ed.), Bilimsel araştırma yöntemleri içinde (s. 132-164). T.C. Anadolu Üniversitesi Yayını No: 2750, Açıköğretim Fakültesi Yayını No: 1708, Eskişehir.
Aydın, S. ve Beyazıt, İ.(2021). Nitel araştırmalarda geçerlilik ve güvenirlik. M. Çelebi (Ed.), Nitel araştırma yöntemleri (1. Baskı) içinde (s.182-209). Ankara: Pegem Akademi
Batanero, C., Navarro-Pelayo, V., & Godino, J. D. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32(2), 181-199. https://doi.org/10.1023/A:1002954428327.
Cifarelli, V. V., & Sevim, V. (2014). Examining the role of re-presentation in mathematical problem solving: An application of Ernst von Glasersfeld’s conceptual analysis. Constructivist Foundations, 9(3), 360-369.
Clore, G. L. (1992). Cognitive phenomenology: Feelings and the construction of judgment. In L. L. Martin & A. Tesser (Eds.), The construction of social judgments (pp. 133-163). Hillsdale, N.J.: Erlbaum.
Çepni, S. (2021). Araştırma ve proje çalışmalarına giriş. (9. Baskı). Trabzon: Celepler basım yayın ve dağıtım.
Eizenberg, M., & Zaslavsky, O. (2004). Students' verification strategies for combinatorial problems. Mathematical Thinking and Learning, 6(1), 15-36. https://doi.org/10.1207/s15327833mtl0601_2
English, L. D. (2005). Combinatorics and the development of children's combinatorial reasoning. In Exploring probability in school: Challenges for teaching and learning (pp. 121-141). Boston, MA: Springer US. https://doi.org/10.1007/0-387-24530-8_6
Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. ZDM 5, 193–198.
Glasersfeld, E. von (1987). Learning as a constructive activity. Problems Of Representation İn The Teaching and Learning of Mathematics, 3(17), 83-90. https://vonglasersfeld.com/papers/083.pdf sayfasından erişilmiştir.
Glasersfeld, E. von (1995). Radical constructivism: a way of knowing and learning. London and Washington, DC: The Falmer Press.
Kapur, J. N. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 111-127. http://www.jstor.org/stable/3481871
Kavousian, S. (2008). Enquiries into undergraduate students' understanding of combinatorial structures (Unpublished doctoral dissertation). Vancouver, BC: Simon Fraser University.
Lamanna, L., Gea, M. M., & Batanero, C. (2022). Do Secondary School Students’ Strategies in Solving Permutation and Combination Problems Change with Instruction?. Canadian Journal of Science, Mathematics and Technology Education, 22(3), 602-616. https://doi.org/10.1007/s42330-022-00228-z
Lockwood, E. (2013). A model of students’ combinatorial thinking. The Journal of Mathematical Behavior, 32(2), 251-265. https://doi.org/10.1016/j.jmathb.2013.02.008
Lockwood, E., & Gibson, B. R. (2016). Combinatorial tasks and outcome listing: Examining productive listing among undergraduate students. Educational Studies in Mathematics, 91, 247-270.
Lockwood, E., A Swinyard, C., & S Caughman, J. (2015). Patterns, sets of outcomes, and combinatorial justification: Two students’ reinvention of counting formulas. International Journal of Research in Undergraduate Mathematics Education, 1, 27-62. https://doi.org/10.1007/s40753-015-0001-2
Lockwood, E., Wasserman, N. H., & McGuffey, W. (2018). Classifying combinations: Investigating undergraduate students’ responses to different categories of combination problems. International Journal of Research in Undergraduate Mathematics Education, 4, 305-322. https://doi.org/10.1007/s40753-018-0073-x
Maher, C. A., & Uptegrove, E. B. (2010). Methodology. In Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.), Combinatorics and reasoning: Representing, justifying and building isomorphisms (Vol. 47) (pp.9-16). London: Springer Science & Business Media
Melusova, J., & Vidermanova, K. (2015). Upper-secondary students’ strategies for solving combinatorial problems. Procedia-Social and Behavioral Sciences, 197, 1703-1709. https://doi.org/10.1016/j.sbspro.2015.07.223
Özdemir, M. (2012). Matematik olimpiyatlarına hazırlık-2. İzmir: Altın Nokta Yayınları.
Patton, M. Q. (2013). Nitel araştırma ve değerlendirme yöntemleri (3. Baskı).( M. Bütün ve S. B. Demir, Çev.). Ankara: Pegem Akademi.
Şahan, H. H. ve Uyangör, N. (2021). Bilimsel araştırmalarda örneklem seçimi. M. Çelebi (Ed.), Nitel araştırma yöntemleri (1.Baskı) içinde (s. 112-137). Ankara: Pegem Akademi.
Thanheiser, E. (2023). What is the mathematics in mathematics education? . The Journal of Mathematical Behavior, 70, 101033.
Yıldırım, A. ve Şimşek, H. (2021). Sosyal bilimlerde nitel araştırma yöntemleri (12. Baskı). Ankara: Seçkin Yayıncılık.
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