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Open Education Studies

Editor-in-Chief: Bastiaens, Theo

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Students’ Concepts of the Trapezoid at the End of Lower Secondary Level Education

Zdeněk Halas
  • Corresponding author
  • Department of Mathematics Education, Charles University, Faculty of Mathematics and Physics, Sokolovská 49/83 186 75 Prague Czech Republic
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jarmila Robová
  • Department of Mathematics Education, Charles University, Faculty of Mathematics and Physics, Sokolovská 49/83 186 75 Prague Czech Republic
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Vlasta Moravcová
  • Department of Mathematics Education, Charles University, Faculty of Mathematics and Physics, Sokolovská 49/83 186 75 Prague Czech Republic
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Jana Hromadová
  • Department of Mathematics Education, Charles University, Faculty of Mathematics and Physics, Sokolovská 49/83 186 75 Prague Czech Republic
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-12-13 | DOI: https://doi.org/10.1515/edu-2019-0013

Abstract

Understanding basic geometric concepts is important for the development of students’ thinking and geometric imagination, both of which facilitate their progress in mathematics. Factors that may affect students’ understanding include the teacher of mathematics (in particular his/her teaching style) and the textbooks used in mathematics lessons. In our research, we focused on the conceptual understanding of trapezoids among Czech students at the end of the 2nd level of education (ISCED 2). As part of the framework for testing the understanding of geometrical concepts, we assigned a task in which students had to choose trapezoids from six figures (four trapezoids and two other quadrilaterals). In total, 437 students from the 9th grade of lower secondary schools and corresponding years of grammar schools participated in the test. The gathered data were subjected to a qualitative analysis. We found that less than half of the students commanded an adequate conceptual understanding of trapezoids. It turned out that some students did not recognise the non-model of a trapezoid, or vice versa, considering a trapezoid in an untypical position to be a non-model. Our research was supplemented by analysis of geometry textbooks usually used in the Czech Republic which revealed that the trapezoid is predominantly presented in a prototypical position. Moreover, we investigated how pre-service teachers define a trapezoid. Finally, we present some recommendations for mathematics teaching and pre-service teacher training that could help.

Keywords: conceptual maps; conceptual understanding; hierarchy of quadrilaterals; non-models; trapezoid

References

  • Binterová, H., Fuchs, L., & Tlustý, P. (2008). Matematika 7. Geometrie, učebnice pro základní školy a víceletá gymnázia. Plzeň: Fraus.Google Scholar

  • Budínová, I. (2017a). Vytváření představ základních geometrických pojmů u žáků prvního stupně základní školy, Učitel matematiky, 25(2), 65–82.Google Scholar

  • Budínová, I. (2017b). Vytváření představ základních geometrických pojmů u žáků prvního stupně základní školy: čtverec a obdélník, Učitel matematiky, 25(5), 272–286.Google Scholar

  • Butuner, S. O., & Filiz, M. (2017). Exploring Turkish mathematics teachers’ content knowledge of quadrilaterals. International Journal of Research in Education and Science (IJRES), 3(2), 395–408. DOI: 10.21890/ijres.327898CrossrefGoogle Scholar

  • Chinnappan, M., & Lawson, M. J. (2005). A framework for analysis of teachers’ geometric content knowledge and geometric knowledge for teaching. Journal of Mathematics Teacher Education, 8(3), 197–221.Google Scholar

  • Clements, D. H., Swaminathan, S., Hannibal, M. A. Z., & Sarama, J. (1999). Young children’s concepts of shape. Journal for Research in Mathematics Education, 30(2), 192–212.CrossrefGoogle Scholar

  • Davies, M. (2011). Concept Mapping, Mind Mapping, Argument Mapping: What are the differences and do they matter? Higher Education, 62(3), 279–301.CrossrefGoogle Scholar

  • Erdogan, E. O., & Dur, Z. (2014). Preservice mathematics teachers’ personal figural concepts and classifications about quadrilaterals. Australian Journal of Teacher Education, 39(6), 107–133.Google Scholar

  • Fujita, T., & Jones, K. (2006). Primary trainee teachers’ understanding of basic geometrical figures in Scotland. Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague: PME, vol. 3, 129–136.Google Scholar

  • Fujita, T., & Jones K. (2007). Learners’ understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing. Research in Mathematics Education, 9(1 & 2), 3–20.Google Scholar

  • Gouli, E., Gogoulou, A, Papanikolaou, K., & Grigoriadou, M. (2004). COMPASS: an adaptive web-based concept map assessment tool. In Cañas, A. J., Novak, J. D., & Gonzáles, F. M. (eds.). Proceedings of the First International Conference on Concept Mapping. Pamplona: Universidad Pública de Navarra.Google Scholar

  • Gunčaga, J., Tkačik, Š., & Žilková, K. (2017). Understanding of selected geometric concepts by pupils of pre-primary and primary level education. European Journal of Contemporary Education, 6(3), 497–515. DOI: 10.13187/ejced.2017.3.497Web of ScienceCrossrefGoogle Scholar

  • Hay, D., Kinchin, I., & Lygo-Barker, S. (2008). Making learning visible: the role of concept mapping in higher education. Studies in Higher Education, 33(3), 295–311.Web of ScienceCrossrefGoogle Scholar

  • Hejný, M. (2012). Exploring the cognitive dimension of teaching mathematics through scheme-oriented approach to education. Orbis scholae, 6(2), 41–55.Google Scholar

  • Herman J., Chrápavá, V., Jančovičová, E., & Šimša, J. (2006). Matematika: Trojúhelníky a čtyřúhelníky. Sekunda. Praha: Prometheus.Google Scholar

  • Hershkowitz, R. (1989). Visualization in geometry – two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.Google Scholar

  • Hromadová, J., Halas, Z., Moravcová, V., & Robová, J. (2017). Variace na téma lichoběžník a jeho obsah. Matematika–fyzika– informatika, 26(5), 336–345.Google Scholar

  • Jirotková D. (2017). Ze života lichoběžníků. In Vondrová, N. (ed.). Dva dny s didaktikou matematiky. Praha: Univerzita Karlova, Pedagogická fakulta, 111–116.Google Scholar

  • Mareš, J. (2011). Učení a subjektivní mapy pojmů. Pedagogika, 61(3), 215–247.Google Scholar

  • Martínková, P., Goldhaber, D., & Erosheva, E. (2018). Disparities in ratings of internal and external applicants: A case for model-based inter-rater reliability. PLoS ONE 13(10): e0203002. https://doi.org/10.1371/journal.pone.0203002

  • Molnár, J., Lepík, L., Lišková, H., & Slouka, J. (1999). Matematika 7, učebnice s komentářem pro učitele. Olomouc: Prodos.Google Scholar

  • Molnár, J. (2011). Matematika pro střední odborné školy. Planimetrie. Praha: Prometheus.Google Scholar

  • Monaghan, F. (2000). What difference does it make? Children’s views of the differences between some quadrilaterals. Educational Studies in Mathematics, 42(2), 179–196.Google Scholar

  • Moravcová, V., Hromadová, J., Halas, Z., & Robová, J. (2018). Jak žáci znázorňují úsečku a polopřímku? In Bastl B., & Lávička, M. (eds.). Setkání učitelů matematiky všech typů a stupňů škol 2018. Plzeň: Vydavatelský servis, 105–108.Google Scholar

  • Moravcová, V., Robová, J., Hromadová, J., & Halas, Z. (2019). The development of the concept of axial symmetry in pupils and students. In Proceedings of the 16th international conference on efficiency and responsibility in education (ERIE 2019). Prague: Czech University of Life Sciences, 351–357.Google Scholar

  • MŠMT. (2017). Rámcový vzdělávací program pro základní vzdělávání. Praha: VÚP.Google Scholar

  • Odvárko, O., & Kadleček, J. (2012). Matematika pro 7. ročník základní školy. Shodnost. Středová souměrnost. Čtyřúhelníky. Hranoly. Praha: Prometheus.Google Scholar

  • Okazaki, M., & Fujita, T. (2007). Prototype phenomena and common cognitive paths in the understanding of the inclusion relations between quadrilaterals in Japan and Scotland. In Woo, J. H., Lew, H. C., Park, K. S., & Seo, D. Y. (eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Seoul: PME, 4, 41–48.Google Scholar

  • Pomykalová, E. (1993). Matematika pro gymnázia. Planimetrie. Praha: Prometheus.Google Scholar

  • Pomykalová, E., Gergelitsová, Š., & Schovancová, J. (2012). Matematika pro střední školy. Planimetrie I. Čtyřúhelníky. Plzeň: Fraus.Google Scholar

  • Robová, J., Moravcová, V., Halas, Z., & Hromadová, J. (2019). Žákovské koncepty trojúhelníku na začátku druhého stupně vzdělávání. Scientia in Educatione, 10(1), 1–22.Google Scholar

  • Rösken, B., & Rolka, K. (2007). Integrating Intuition: The role of concept image and concept definition for students’ learning of integral calculus. The Montana Mathematics Enthusiast, Monograph 3, 181–204.Google Scholar

  • Šarounová, A., Růžičková, J., & Väterová, V. (1998). Matematika 7, II. díl. Praha: Prometheus.Google Scholar

  • Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. DOI: 10.1007/BF00305619.CrossrefGoogle Scholar

  • Tirosh, D., Tsamir, P., Tabach, M., Levenson, E., & Barkai, R. (2011). Geometrical knowledge and geometrical self-efficacy among abused and neglected kindergarten children. Scientia in Educatione, 2(1), 23–36.Google Scholar

  • Türnüklü, E. (2014). Concept images of trapezoid: some cases from Turkey. Education Journal, 3(3), 179–185.Google Scholar

  • Van Hiele, P. M. (1999). Developing geometric thinking through activities that begin with play. Teaching Children Mathematics, 5, 310–316.Google Scholar

  • Vaňková, E. (2014). Pojmové mapy ve vzdělávání. Praha: UK.Google Scholar

  • Vinner, S., & Hershkowitz, R. (1980). Concept images and common cognitive paths in the development of some simple geometrical concepts. In Karplus, R. (ed.). Proceedings of the Fourth International Conference for the Psychology of Mathematics Education. Berkeley: California University, 177–184.Google Scholar

  • Vondra, J. (2013a). Matematika pro střední školy – 3. díl: Planimetrie – Učebnice. Brno: Didaktis.Google Scholar

  • Vondra, J. (2013b). Matematika pro střední školy – 3. díl: Planimetrie – Pracovní sešit. Brno: Didaktis.Google Scholar

  • Žilková, K. et al. (2018). Young children’s concepts of geometric shapes. Pearson. Retrieved from https://www.researchgate.net/publication/331114419_Young_children’s_concepts_of_geometric_shapes

About the article

Received: 2019-04-04

Accepted: 2019-07-17

Published Online: 2019-12-13


Citation Information: Open Education Studies, Volume 1, Issue 1, Pages 184–197, ISSN (Online) 2544-7831, DOI: https://doi.org/10.1515/edu-2019-0013.

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© 2019 Zdeněk Halas et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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