Abstrakt
Cieľom článku je zistiť, akým spôsobom zovšeobecňujú a akými argumentami svoje zistenia podkladajú žiaci vo veku 12–13 rokov. To isté sme zisťovali aj u budúcich učiteľov matematiky, ktorý v tom čase študovali na našej fakulte. Hlavné zameranie článku je na algebrické uvažovanie, špeciálne na skúmanie vzorov a vyjadrenie zákonitosti pomocou číselných hodnôt. V tomto článku vysvetľujeme dôležité koncepty a značenia použité vo výskume, stručne charakterizujeme úrovne kognitívneho rozvoja a využité pojmy z Teórie didaktických situácii. Stanovili sme si tri výskumné otázky. Na získanie výskumných dát sme pracovali so skupinou 32 žiakov vo veku 12–13 rokov a s 19 študentmi učiteľstva matematiky. Obom skupinám sme zadali rovnaké úlohy na riešenie a požiadali sme ich o vysvetlenie riešenia, resp. vzťahu, ku ktorému sa dopracovali. Okrem toho sme krátkym dotazníkom zozbierali podporné dáta ohľadom skúmaných skupín. Výsledky sú diskutované v kontexte podobných výskumov, sú identifikované limitácie opísaného výskumu a vyvodené závery. Na základe
diskusie a limitov sme sformulovali odporučenia pre zmenu v príprave budúcich učiteľov matematiky.
Reference
Almossawi, A. (2016). An illustrated book of bad arguments. Scribe Publications.
Becker, J.R., & Rivera, F. (2006). Establishing and justifying algebraic generalization at the sixth grade level. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (vol. 4, pp. 465–472). PME, Charles University.
Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Heinemann.
Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–446. https://doi.org/10.2307/30034944
Breiteig, T., & Grevholm, B. (2006). The transition from arithmetic to algebra: to reason, explain, argue, generalize and justify. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 225–232). PME, Charles University.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer.
Eisenmann, P., Přibyl, J., Novotná, J., Břehovský, J., & Cihlář, J. (2017). Volba řešitelských strategií v závislosti na věku [Choice of heuristic strategies with respect to age]. Scientia in educatione, 8(2), 21–38. https://doi.org/10.14712/18047106.432
Friel, S.N., & Markworth, K.A. (2009). A framework for analyzing geometric pattern tasks. Mathematics Teaching in the Middle School, 15(1), 24–33. https://doi.org/10.5951/mtms.15.1.0024
Hallagan, J. E., Rule, A.C., & Carlson, L. F. (2009). Elementary school pre-service teachers’ understanding of algebraic generalizations. The Mathematics Enthusiast, 6(1), 201–206. https://scholarworks.umt.edu/tme/vol6/iss1/16
Kaput, J. (1995). A research based supporting long term algebra reform? Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21–24, 1995). Retrieved May 25, 2020 from https://files.eric.ed.gov/fulltext/ED389539.pdf
Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema, & T.A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 134–156). Routlege.
Lannin, J. (2004). Developing mathematical power by using explicit and recursive reasoning. The Mathematics Teacher, 98(4), 216–223. https://www.jstor.org/stable/27971686
Lee, L., & Wheeler, D. (1987). Algebraic thinking in high school students: Their conceptions of generalization and justification [Research Report]. Concordia Universtiy, Departmenf ot Mathematics.
Makowski, M.B. (2020). The written and oral justifications of mathematical claims of middle school pre-service teachers. Research in Mathematics Education, advance online publication https://doi.org/10.1080/14794802.2020.1777190
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra. Perspectives for research and teaching (vol. 18, pp. 65–86). Springer. https://doi.org/10.1007/978-94-009-1732-3 5
Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49. https://doi.org/10.1007/BF03217544
Novotná, J., Eisenmann, P., & Přibyl, J. (2015). Impact of heuristic strategies on pupils’ attitudes to problem solving. Journal on Efficiency and Responsibility in Education and Science, 8(1), 15–23. https://doi.org/10.7160/eriesj.2015.080103
Piaget, J. (1980). Adaptation and Intelligence: Organic Selection and Phenocopy. University of Chicago Press. Richardson, K., Berenson, S., & Staley, K. (2009). Prospective elementary teachers use of representation to reason algebraically. Journal of Mathematical Behavior, 28(2), 188–199. https://doi.org/10.1016/j.jmathb.2009.09.002
Rivera, F.D., & Becker, J.R. (2005). Figural and numerical models of generalization in algebra. Mathematics Teaching in the Middle School, 11(4), 198–203.
Rivera, F.D., & Becker, J.R. (2007). Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra. Journal of Mathematical Behavior, 26(2), 140–155. https://doi.org/10.1016/J.JMATHB.2007.05.001
Rivera, F.D., & Becker, J.R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM Mathematics Education, 40(1), 65–82. https://doi.org/10.1007/s11858-007-0062-z
Russel, S. J., Schifter, D., & Bastable, V. (2011). Developing algebraic thinking in the context of arithmetic. In J. Cai, & E. Knuth (Eds.), Early algebraization, advances in mathematics education (pp. 43–69). Springer. https://doi.org/10.1007/978-3-642-17735-4 4
Slavíčková, M., & Vargová, M. (2019). Pupils stage of a cognitive development in correlation with solving algebraic tasks. In ICERI 2019 proceeding (pp. 7260–7268). IATED. https://doi.org/10.21125/iceri.2019.1726
Strand, K., & Mills, B. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on algebra. The Mathematics Enthusiast, 11, 385–432. https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1307&context=tme
Van de Walle, J.A., Karp, K. S., & Bay-Williams, J.M. (2012). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Pearson.
Van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of preservice teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319–351. https://doi.org/10.2307/4149957
Van Dooren, W., Verschaffel, L., & Onghena, P. (2003). Pre-service teachers’ preferred strategies for solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6, 27–52. https://doi.org/10.1023/a:1022109006658
Watson, A., & Geest, E.D. (2005). Principled teaching for deep progress: Improving mathematical learning beyond methods and materials. Educational Studies in Mathematics, 58(2), 209–234.