Soulad zavedení kombinatorických konceptů v učebnicích matematiky s teorií poznávacího procesu
PDF (English)

Jak citovat

Zenkl, D. (2021). Soulad zavedení kombinatorických konceptů v učebnicích matematiky s teorií poznávacího procesu. Scientia in Educatione, 12(1), 37-52. https://doi.org/10.14712/18047106.1938

Abstrakt

This article analyses the approach taken by five Czech secondary school mathematics textbooks to selected combinatorial concepts, in order to determine the extent to which they provide pedagogical support to teachers based on the theory of generic models, which is a theory of concept development. The analysis of textbooks in relation to this theory focused on the presence and quality of a) isolated models and non-models of future knowledge, b) prompts to generalise as a prerequisite for the creation of a generic model, and c) the supportive role of graphical representations in developing combinatorial thinking. Most notably, we identified insufficient motivation for combinatorial problems, few isolated models of future knowledge, the absence of explicit prompts to generalise and a consequent lack of a significant concept of isomorphism. Despite the research-proven positive influence of the creation of graphical representations on the development of pupils’ combinatorial thinking, they are rare in textbook chapters about combinatorics, and lack diversity. With a few exceptions, textbook authors do not encourage readers to create their own graphical representations. One textbook stood out in that it frequently prompts the creation of personalised representations, works purposefully with isomorphic problems and encourages the reader to generalise specific procedures.

https://doi.org/10.14712/18047106.1938
PDF (English)

Reference

Apple, M. (1986). Teachers and texts: A political economy of class and gender relations in education. Routledge & Kegan Paul.

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241. https://doi.org/10.1023/A:1024312321077

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.

Corter, J.E., & Zahner, D. C. (2007). Use of external visual representations in probability problem solving. Statistics Education Research Journal, 6(1), 22–50.

English, L.D. (2005). Combinatorics and the development of children’s combinatorial reasoning. In G. Jones. (Ed.) Exploring probability in school (pp. 121–141). Springer. https://doi.org/10.1007/0-387-24530-8 6

Fan, L. (2013). Textbook research as scientific research: towards a common ground on issues and methods of research on mathematics textbooks. ZDM, 45(5), 765–777. https://doi.org/10.1007/s11858-013-0530-6

Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: development status and directions. ZDM, 45(5), 633–646. https://doi.org/10.1007/s11858-013-0539-x

Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. Zentralblatt für Didaktik der Mathematik, 5, 193–198.

Grimaldi, R.P. (2003). Discrete and combinatorial mathematics: An applied introduction. Addison-Wesley.

Hejný, M. (2004). Mechanismus poznávacího procesu [Mechanism of the cognitive process]. In M. Hejný, J. Novotná, & N. Stehlíková (Eds.), Dvacet pět kapitol z didaktiky matematiky (s. 23–42). PedF UK v Praze.

Hejný, M. (2012). Exploring the cognitive dimension of teaching mathematics through scheme-oriented approach to education. Orbis scholae, 6(2), 41–55. https://doi.org/10.14712/23363177.2015.39

Hejný, M. (2014). Vyučování matematice orientované na budování schémat: Aritmetika 1. stupně [Teaching schematic oriented mathematics: Grade 1 arithmetic]. Univerzita Karlova, Pedagogická fakulta.

Hejný, M., Benešová, M., Bereková, H., Bero, P., Hrdina, L’., Repáš, V., & Vantuch, J. (1990). Teória vyučovania matematiky 2 [Theory of teaching mathematics 2]. SPN.

Hejný, M., Eichlerová, K., & Šalom, P. (2017). Matematika B: Pracovní sešit pro 2. stupeň ZŠ a víceletá gymnázia (první vydání) [Mathematics B: Workbook for the 2nd level of elementary school and multiyear grammar school (first edition)]. H-mat.

Hejný, M., & Kuřina, F. (2009). Dítě, škola a matematika. Konstruktivistické přístupy k vyučování [Child, school and math. Constructivist approaches to teaching.]. Portál.

Hejný, M., & Šalom, P. (2017). Matematika E: Učebnice pro 2. stupeň ZŠ a víceletá gymnázia (první vydání) [Mathematics E: Textbook for the 2nd grade of elementary school and multiyear grammar school (first edition)]. H-mat.

Kapur, J.N. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3(1), 111–127.

Lockwood, E., & Gibson, B.R. (2016). Combinatorial tasks and outcome listing: Examining productive listing among undergraduate students. Educational Studies in Mathematics, 91(2), 247–270. https://doi.org/10.1007/s10649-015-9664-5

Mason, J., & Johnston-Wilder, S. (2005). Fundamental constructs in mathematics education. RoutledgeFalmer.

MŠMT [Ministry of Education, Youth and Sports], (2017). Rámcový vzdělávací program pro základní vzdělávání [Framework Education Programme for Basic Education]. Praha. http://www.msmt.cz/file/43792/

MŠMT [Ministry of Education, Youth and Sports], (2016). Rámcový vzdělávací program pro gymnázia [Framework Education Programme for Secondary General Education]. [online] Praha. http://www.nuv.cz/t/rvp-pro-gymnazia

Salavatinejad, N., Alamolhodaei, H., & Radmehr, F. (2021). Toward a model for students’ combinatorial thinking. The Journal of Mathematical Behavior, 61, 100823. https://doi.org/10.1016/j.jmathb.2020.100823

Speiser, B., Walter, C., & Sullivan, C. (2007). From test cases to special cases: Four undergraduates unpack a formula for combinations. The Journal of Mathematical Behavior, 26(1), 11–26. https://doi.org/10.1016/j.jmathb.2007.03.003

Stofflett, R.T., & Baker, D.R. (2016). The effects of training in combinatorial reasoning and propositional logic on formal reasoning ability in junior high school students. Research in Middle Level Education, 16(1), 159–177. https://doi.org/10.1080/10825541.1992.11670007

Tarr, J. E., Chávez, O., Reys, R. E., & Reys, B. J. (2006). From the written to the enacted curricula: The intermediary role of middle school mathematics teachers in shaping students’ opportunity to learn. School Science and Mathematics, 106(4), 191–201. https://doi-org.ezproxy.is.cuni.cz/10.1111/j.1949-8594.2006.tb18075.x

Uptegrove, E. B. (2015). Shared communication in building mathematical ideas: A longitudinal study. The Journal of Mathematical Behavior, 40, 106–130. https://doi.org/10.1016/j.jmathb.2015.02.001

Van Steenbrugge, H., Valcke, M., & Desoete, A. (2013). Teachers’ views of mathematics textbook series in Flanders: Does it (not) matter which mathematics textbook series schools choose? Journal of Curriculum Studies, 45(3), 322–353. https://doi.org/10.1080/00220272.2012.713995

Vinner, S. (2014). Concept development in mathematics education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education (pp. 91–95). Springer. https://doi.org/10.1007/978-94-007-4978-8 147

Vondrová, N. (2019). Didaktika matematiky jako nástroj zvládání kritických míst v matematice [Didactics of mathematics as a tool for managing critical places in mathematics]. Univerzita Karlova, Pedagogická fakulta.

Zahner, D., & Corter, J.E. (2010). The process of probability problem solving: Use of external visual representations. Mathematical Thinking and Learning, 12(2), 177–204. https://doi.org/10.1080/10986061003654240