Abstract
Measurement in geometry belongs among key areas of school mathematics, however, pupils make serious mistakes when solving problems on measurement and hold misconception. The article focuses on possible links between lower secondary pupils’ (n = 870) success in solving non-measurement and in calculations tasks on area and volume and on their problems when solving measurement tasks. The study uses a mixed research design. Statistical methods are used to find correlations between the two types of tasks and a qualitative analysis is carried out to identify mistakes and misconceptions. The results show that there are indeed relatively strong links between success in non-measurement tasks and in calculation tasks and consequently when teaching this topic, attention must be paid to development of both types of skills. The study identified pupils’ mistakes in tasks which are within the Czech curriculum but which proved to be difficult for them, such as a missing link between an algebraic and geometric representations, a tendency to linearize and/or to employ pseudo-analytical thinking. The study identified differences between individual classes which point to the significant role of the teacher and/or influence of the textbook used.References
Amthauer, R., Brocke, B., Liepmann, D., & Beauducel, A. (2001). Intelligenz-Struktur-Test 2000 R. Göttingen: Hogrefe, 2.
Battista, M. T. (2004). Applying cognition-based assessment to elementary school students’ development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185–204.
Battista, M. T. (2007). The development of geometric and spatial thinking. In Lester, F. K. Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843–908). Charlotte, NC: Information Age Publishing Inc.
Curry, M., Mitchelmore, M. & Outhred, L. (2006). Development of children’s understanding of length, area, and volume measurement principles. In J. Novotná, H. Moraová, M. Krátká, N. Stehlíková (Eds.), Proceedings of the 30th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 377–384). Prague: PME.
De Vauss, D. (2002). Analyzing Social Science Data. London: SAGE.
Divišová, B. (2012). Geometrické úlohy řešitelné bez výpočtu. [Geometric problems solvable without calculations.] Disertační práce. Praha: PedF UK v Praze. Školitelka N. Vondrová.
De Bock, D., Van Dooren, W., Janssens, D. & Verschaffel, L. (2007). The illusion of linearity: From analysis to improvement. New York, N.Y.: Springer.
Dembo, Y., Levin, I. & Siegler, R.S. (1997). A comparison of the geometric reasoning of students attending Israeli ultraorthodox and mainstream schools. Developmental Psychology, 33, 92–103.
Eames L.C., Clements & D. H., Sarama, J. et al. (2014). Interactions among Hypothetical Learning Trajectories for Length, Area, and Volume Measurement. In NCTM Research Conference, April 2014.
Friedman, L. (1995). The space factor in mathematics: Gender differences. Review of Educational Research, 65, 22–50.
Hannighofer, J., Van den Heuvel-Panhuizen, M., Weirich, S., & Robitzsch, A. (2011). Revealing German primary school students’ achievement in measurement. ZDM, 43(5), 651-665. doi:10.1007/s11858-011-0357-y
Hejný, M. (2007). Budování matematických schémat. [Building mathematical schemes.] In Hošpesová, A., Stehlíková N. & Tichá, M. (Eds.), Cesty zdokonalování kultury vyučování matematice (s. 81–122). České Budějovice: Jihočeská univerzita v Českých Budějovicích.
Heller, K. A., & Perleth, Ch. (2000). Kognitiver Fahigkeitstest für 4. - 12. Klassen, Revision (KFT 4 – 12+ R). [Cognitive ability test for grade 4-12, revision (KFT 4-12+ R).] Gottingen: Hogrefe.
Herendiné-Kónya, E. (2015). The level of understanding geometric measurement. In Krainer, K. & Vondrová, N. (Eds.), CERME9: Proceedings of the ninth congress of the European Society for Research in Mathematics Education (pp. 536–542). Prague: Charles University in Prague, Faculty of Education and ERME.
Huang, H.-M. E. & Witz, K. G. (2011). Developing children's conceptual understanding of area measurement: A curriculum and teaching experiment. Learning and Instruction, 21(1), 1–13.
Huang, H.-M. E. & Witz, K. G. (2013). Children’s conceptions of area measurement and their strategies for solving area measurement problems. Journal of Curriculum and Teaching, 2(1), 10–26.
Kamii, C. & Kysh, J. (2006). The difficulty of “length × width”: Is a square the unit of measurement? Journal of Mathematical Behavior, 25, 105–115.
Kordaki, M. (2003). The effect of tools of a computer microworld on students' strategies regarding the concept of conservation of area. Educational Studies in Mathematics, 52(2), 177–209.
Kordaki, M. & Potari, D. (2002). The effect of area measurement tools on student strategies: the role of a computer microworld. International Journal of Computers for Mathematical Learning, 7, 65–100.
Kospentaris, G., Spyrou, P. & Lappas, D. (2011). Exploring students’ strategies in area conservation geometrical tasks. Educational Studies in Mathematics, 77(1), 105–127.
Kuřina, F. (2011). Matematika a řešení úloh. [Mathematics and problem solving.] České Budějovice: Jihočeská univerzita v ČB.
McGee, M.G. (1979). Human spatial abilities: psychometric studies and environmental, genetic, hormonal, and neurological influences. Psychological bulletin, 86(5), 889–918.
Outhred, L. N. & Mitchelmore, M. C. (2000). Young children’s intuitive understanding of rectangular area measurement. Journal for Research in Mathematics Education, 31(2), 144–167.
Piaget, J., Inhelder, B. & Szeminska, A. (1960). The child's conception of geometry. New York: Norton.
Pittalis, M. & Christou, C. (2010). Types of reasoning in 3D geometry thinking and their relation with spatial ability. Educational Studies in Mathematics, 75(2), 191–212.
Plšková, Z. (2010). Rozvoj prostorové představivosti žáků ZŠ. [The development of three-dimensional imagination of elementary school students.] Disertační práce. Universita Palackého v Olomouci, Pedagogická fakulta. Školitel: A. Stopenová. Dostupné z: http://theses.cz/id/ht91sr/.
Rahim, M. H. & Siddo, R. A. (2012). High school student-teachers attempts to justify mathematical propositions utilizing spatial structuring on shape transform. Research in Mathematical Education, 16(2), 107–123.
Rendl, M. & Vondrová, N. (2014). Kritická místa v matematice u českých žáků na základě výsledků šetření TIMSS 2007. [Critical issues in mathematics of Czech pupils based on TIMSS 2007 results.] Pedagogická orientace, 24(1), 22–57.
Sarama, J.A. & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge.
Slezáková, J. (2011). Geometrická představivost v rovině. [Spatial imagination in a plane.] Disertační práce. Universita Palackého v Olomouci, Přírodovědecká fakulta. Školitel J. Molnár. Dostupné z http://theses.cz/id/op6350/?furl=%2Fid%2Fop6350%2F;so=nx;lang=en.
Sorby, S. A. (2009). Educational research in developing 3–D spatial skills for engineering students. International Journal of Science Education, 31(3), 459–480.
Tan Sisman, G., & Aksu, M. (2016). A Study on Sixth Grade Students’ Misconceptions and Errors in Spatial Measurement: Length, Area, and Volume. International Journal of Science and Mathematics Education, 14(7), 1293–1319. doi:10.1007/s10763-015-9642-5
Tartre, L. A. (1990). Spatial orientation skill and mathematical problem solving. Journal for Research in Mathematics Education, 216–229.
Tůmová, V. & Janda, D. (2014). Vliv používání vizualizace a matematických operací na úspěšnost žáků v úlohách týkajících se objemu a obsahu. [Influence of using visualisation and mathematical operations on pupils’ success in area and volume problems.] In Bastl, B., Lávička, M. (Eds.), Sborník z konference setkání učitelů matematiky všech typů a stupňů škol 2014, s. 225–230. Plzeň: Vydavatelský servis.
Tůmová, V. (2017a). How do pupils of the 5th and 6th grade structure space. In Novotná, J., Moraová, H. (Eds.), International Symposium Elementary Maths Teaching SEMT ‘17, Proceedings. Equity and diversity in elementary mathematics education (pp. 411–421). Prague: Charles University, Faculty of Education.
Tůmová, V. (2017b). Chápání pojmů obsah a objem u žáků základní školy. Disertační práce. [Conceptions of area and volume of pupils at the elementary school. Dissertation thesis.] Univerzita Karlova, Pedagogická fakulta.
Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics, 34(2), 97–129.
Vondrová, N. (2015). Obtíže žáků 2. stupně ve zjišťování obsahu útvarů a objemů těles. [Elementary school pupils‘ difficulties when finding area and volume.] In Vondrová, N., a Rendl, M. et al. Kritická místa matematiky základní školy v řešeních žáků (s. 253–318). Praha: Karolinum.
Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas. The Journal of Mathematical Behaviour, 25(3), 224–239.