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BY 4.0 license Open Access Published by De Gruyter Open Access December 13, 2019

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

  • Zdeněk Halas EMAIL logo , Jarmila Robová , Vlasta Moravcová and Jana Hromadová
From the journal Open Education Studies

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.

References

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

Budínová, I. (2017a). Vytváření představ základních geometrických pojmů u žáků prvního stupně základní školy, Učitel matematiky, 25(2), 65–82.Search in 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.Search in 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.32789810.21890/ijres.327898Search in Google 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), 197–221.10.1007/s10857-005-0852-6Search in 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), 192–212.10.2307/749610Search in Google Scholar

Davies, M. (2011). Concept Mapping, Mind Mapping, Argument Mapping: What are the differences and do they matter? Higher Education, 62(3), 279–301.10.1007/s10734-010-9387-6Search in Google 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.Search in 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.Search in 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), 3–20.10.1080/14794800008520167Search in 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.Search in 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.2017.3.49710.13187/ejced.2017.3.497Search in Google 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), 295–311.10.1080/03075070802049251Search in Google Scholar

Hejný, M. (2012). Exploring the cognitive dimension of teaching mathematics through scheme-oriented approach to education. Orbis scholae, 6(2), 41–55.10.14712/23363177.2015.39Search in Google Scholar

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

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

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

Mareš, J. (2011). Učení a subjektivní mapy pojmů. Pedagogika, 61(3), 215–247.Search in 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.020300210.1371/journal.pone.0203002Search in Google Scholar

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

Molnár, J. (2011). Matematika pro střední odborné školy. Planimetrie. Praha: Prometheus.Search in 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), 179–196.10.1023/A:1004175020394Search in 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.Search in 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.Search in Google Scholar

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

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

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

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

Robová, J., Moravcová, V., Halas, Z., & Hromadová, J. (2019). Žákovské koncepty trojúhelníku na začátku druhého stupně vzdělávání. Scientia in Educatione, 10(1), 1–22.10.14712/18047106.1211Search in 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.Search in Google Scholar

Šarounová, A., Růžičková, J., & Väterová, V. (1998). Matematika 7, II. díl. Praha: Prometheus.Search in 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/BF00305619.10.1007/BF00305619Search in Google 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.Search in Google Scholar

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

Van Hiele, P. M. (1999). Developing geometric thinking through activities that begin with play. Teaching Children Mathematics, 5, 310–316.10.5951/TCM.5.6.0310Search in Google Scholar

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

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

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

Received: 2019-04-04
Accepted: 2019-07-17
Published Online: 2019-12-13

© 2019 Zdeněk Halas et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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