Presentation of combinatorial concepts in mathematics textbooks and its compliance with a concept development theory

David Zenkl

doi:10.14712/18047106.1938

Vol. 12 No. 1 (2021): sciED Research Reports

Presentation of combinatorial concepts in mathematics textbooks and its compliance with a concept development theory

David Zenkl, ¹

Katedra matematiky a didaktiky matematiky, Pedagogická fakulta, Univerzita Karlova, Magdalény Rettigové 4, 116 39, Praha 1

DOI: https://doi.org/10.14712/18047106.1938

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Zenkl, D. (2021). Presentation of combinatorial concepts in mathematics textbooks and its compliance with a concept development theory. Scientia in Educatione, 12(1), 37-52. https://doi.org/10.14712/18047106.1938

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Abstract

Článek analyzuje přístupy k vybraným kombinatorickým konceptům v pěti českých středoškolských učebnicích matematiky s cílem zjistit, do jaké míry tyto učebnice poskytují podporu učitelům pro výuku založenou na teorii poznávacího procesu v matematice. Analýza učebnic byla vedena na pozadí teorie generických modelů, přičemž důraz byl položen na přítomnost a kvalitu a) izolovaných modelů a nemodelů budoucího poznatku, b) výzev k zobecňování jako nutnému předpokladu tvorby generického modelu, c) grafických reprezentací jako nezbytné podpory rozvoje kombinatorického uvažování. Bylo zjištěno, že učebnice jsou si v mnoha směrech dosti podobné, ale v jistých ohledech se odlišují. V učebnicích jsme identifikovali mj. nedostatečnou motivaci pro kombinatoriku, nízký počet izolovaných modelů budoucího poznatku, absenci explicitních výzev k zobecňování a z toho plynoucí absenci významného konceptu izomorfismus. Navzdory výzkumně prokázanému pozitivnímu vlivu přítomnosti a vlastní tvorby grafických reprezentací na rozvoj žákovského kombinatorického myšlení se v učebnicích vyskytují reprezentace zřídka, a navíc s malou rozmanitostí. Autoři učebnic až na výjimky nevyzývají čtenáře k tvorbě vlastních grafických reprezentací. Od analyzovaných učebnic se odlišuje jedna, která obsahuje časté výzvy k tvorbě vlastních reprezentací, pracuje záměrně s izomorfními úlohami a vyzývá čtenáře k zobecňování konkrétních postupů.

Keywords kombinatorika učebnice grafická reprezenatce teorie pojmotvorného procesu teorie generických modelů

References

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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 Open DOI Search in Google Scholar
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Corter, J.E., & Zahner, D. C. (2007). Use of external visual representations in probability problem solving. Statistics Education Research Journal, 6(1), 22–50. Search in Google Scholar
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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 Open DOI Search in Google Scholar
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 Open DOI Search in Google Scholar
Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. Zentralblatt für Didaktik der Mathematik, 5, 193–198. Search in Google Scholar
Grimaldi, R.P. (2003). Discrete and combinatorial mathematics: An applied introduction. Addison-Wesley. Search in Google Scholar
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. Search in Google Scholar
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 Open DOI Search in Google Scholar
Hejný, M. (2014). Vyučování matematice orientované na budování schémat: Aritmetika 1. stupně [Teaching schematic oriented mathematics: Grade 1 arithmetic]. Univerzita Karlova, Pedagogická fakulta. Search in Google Scholar
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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. Search in Google Scholar
Kapur, J.N. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3(1), 111–127. Search in Google Scholar
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 Open DOI Search in Google Scholar
Mason, J., & Johnston-Wilder, S. (2005). Fundamental constructs in mathematics education. RoutledgeFalmer. Search in Google Scholar
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/ Search in Google Scholar
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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 Open DOI Search in Google Scholar
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 Open DOI Search in Google Scholar
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 Open DOI Search in Google Scholar
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 Open DOI Search in Google Scholar
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 Open DOI Search in Google Scholar
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 Open DOI Search in Google Scholar
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 Open DOI Search in Google Scholar
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. Search in Google Scholar
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 Open DOI Search in Google Scholar

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

[2] 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 Open DOI Search in Google Scholar

[3] 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. Search in Google Scholar

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

[5] 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 Open DOI Search in Google Scholar

[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 Open DOI Search in Google Scholar

[7] 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 Open DOI Search in Google Scholar

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

[9] Grimaldi, R.P. (2003). Discrete and combinatorial mathematics: An applied introduction. Addison-Wesley. Search in Google Scholar

[10] 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. Search in Google Scholar

[11] 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 Open DOI Search in Google Scholar

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

[13] 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. Search in Google Scholar

[14] 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. Search in Google Scholar

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

[16] 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. Search in Google Scholar

[17] Kapur, J.N. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3(1), 111–127. Search in Google Scholar

[18] 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 Open DOI Search in Google Scholar

[19] Mason, J., & Johnston-Wilder, S. (2005). Fundamental constructs in mathematics education. RoutledgeFalmer. Search in Google Scholar

[20] 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/ Search in Google Scholar

[21] 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 Search in Google Scholar

[22] 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 Open DOI Search in Google Scholar

[23] 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 Open DOI Search in Google Scholar

[24] 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 Open DOI Search in Google Scholar

[25] 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 Open DOI Search in Google Scholar

[26] 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 Open DOI Search in Google Scholar

[27] 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 Open DOI Search in Google Scholar

[28] 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 Open DOI Search in Google Scholar

[29] 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. Search in Google Scholar

[30] 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 Open DOI Search in Google Scholar

Scientia in educatione

Presentation of combinatorial concepts in mathematics textbooks and its compliance with a concept development theory

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