Drawings as External Representations of Children’s Fundamental Ideas and the Emotional Atmosphere in Geometry Lessons
Keywords:
drawings, emotional atmosphere, external representations, fundamental ideas, geometry
Abstract
The important role that geometry plays in the mathematics curriculum has been extensively documented. However, the reduction of geometry in school mathematics, and the focus on basic computation and procedures, raises the question of the competencies students acquire and the classroom atmosphere in geometry lessons. The goal of this multiple case study was to analyse four students’ conceptions of geometry
and the emotional atmosphere in geometry lessons on an individual level. Drawings were used as external representations of the students’ geometrical ideas and the emotional atmosphere. The results show that the participants have a narrow understanding of geometry, and that geometry teaching in their classrooms is reduced to frontal teaching with very limited communication. Nevertheless, the emotional atmosphere
in these four cases could be described as positive or ambivalent. Based on the data, the results are discussed not only with regard to the utility of drawings as a research method to gain insights into students’ conceptions
of geometry and emotional atmosphere in geometry lessons, but also with regard to their theoretical and practical implications.
Downloads
Download data is not yet available.
References
Anning, A., & Ring, K. (2004). Making sense of children’s drawings. Maidenhead, UK: Open University Press.
Berg, B. L. (2007). Qualitative research methods for the social sciences (6th ed.). Boston, MA: Pearson.
Blumer, H. (1986). Symbolic interactionism. Perspective and method. Berkeley, CA: University of California Press.
Carmichael, C., Callingham, R., & Watt, H. (2017). Classroom motivational environment influences on emotional and cognitive dimensions of student interest in mathematics. ZDM Mathematics Education, 49(3), 449–460.
Dörfler, W. (2006). Diagramme und Mathematikunterricht [Diagrams and mathematics teaching]. Journal für Mathematik-Didaktik, 27(3/4), 200–219.
Dohrmann, C., & Kuzle, A. (2014). Unpacking children’s angle “Grundvorstellungenâ€: The case of distance Grundvorstellung of 1° angle. In P. Liljedahl, C. Nicol, S. Oesterkle, & D. Allan (Eds.), Proceedings of the 38th conference of the international group for the psychology of mathematics education and the 36th conference of the north American chapter of the psychology of mathematics education (Vol. 2, pp. 409–416). Vancouver: PME.
Einarsdóttir, J. (2007). Research with children: Methodological and ethical challenges. European Early Childhood Education Research Journal, 15(2), 197–211.
Glasnović Gracin, D., & Domović, V. (2009). Upotreba matematiÄkih udžbenika u nastavi viÅ¡ih razreda osnovne Å¡kole [Using mathematics textbooks in lower secondary education]. Odgojne znanosti, 11(2), 297–317.
Halverscheid, S., & Rolka, K. (2006). Student beliefs about mathematics encoded in pictures and words. In J. Novotná, H. Moraová, M. Krátká, & N. StehlÃková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 233–240). Prague: PME.
Hannula, M. S. (2011). The structure and dynamics of affect in mathematical thinking and learning. In M. Pytlak, E. Swoboda, & T. Rowland (Eds.), Proceedings of the seventh congress of the European society for research in mathematics education (pp. 34–60). Poland: University of RzesoÌw.
Jones, K. (2000). Critical issues in the design of the geometry curriculum. In B. Barton (Ed.), Readings in mathematics education (pp. 75–90). Auckland: University of Auckland.
Laine, A., Näveri, L., Ahtee, M., Hannula, M. S., & Pehkonen, E. (2013). Emotional atmosphere in third-graders’ mathematics classroom – an analysis of pupils’ drawings. Nordic Studies in Mathematics Education, 17(3-4), 101–116.
Laine, A., Ahtee, M., Näveri, L., Pehkonen, E., Portaankorva-Koivisto, P., & Tuohilampi, L. (2015). Collective emotional atmosphere in mathematics lessons based on Finnish fifth graders’ drawings. LUMAT – Research and Practice in Math, Science and Technology Education, 3(1), 87–100.
Mammana, C., & Villani, V. (1998). Perspectives on the teaching of geometry for the 21st century: An ICMI study. Dordrecht: Kluwer.
MZOS [Ministry of Science, Education and Sports] (2006). Nastavni plan i program za osnovnu Å¡kolu [The educational plan and programme for elementary school]. Zagreb: Ministry of Science, Education and Sports.
Patton, M. Q. (2002). Qualitative research and evaluation methods. Thousand Oaks, CA: Sage.
Pehkonen, E., Ahtee, M., Tikkanen, P., & Laine, A. (2011). Pupils’ conception on mathematics lessons revealed via their drawings. In B. Roesken & M. Casper (Eds.), Proceedings of the MAVI-17 conference (pp. 182–191). Bochum: Ruhr-Universität Bochum.
Pehkonen, E., Ahtee, M., & Laine, A. (2016). Pupils’ drawings as a research tool in mathematical problem-solving lessons. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems. Advances and new perspectives (pp. 167–188). New York, NY: Springer.
Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 257–315). Charlotte, NC: Information Age.
Rolka, K., & Halverscheid, S. (2006). Pictures as a means for investigating mathematical beliefs. In S. Alatorre, J. L. Cortina, M. SaÌiz, & A. MeÌndez (Eds.), Proceedings of the 28th annual meeting of the North American chapter of the international group for the psychology of mathematics education. MeÌrida: Universidad PedagoÌgica Nacional.
Rolka, K., & Halverscheid, S. (2011). Researching young students’ mathematical world views. ZDM Mathematics Education, 43(4), 521–533.
Winter, H. (1976). Was soll Geometrie in der Grundschule [What should geometry be like in elementary school?]. Zentralblatt Didaktik der Mathematik, 8, 14–18.
Wittmann, E. Ch. (1999). Konstruktion eines Geometriecurriculums ausgehend von Grundideen der Elementargeometrie [The construction of a geometry curriculum based on the fundamental ideas of elementary geometry]. In H. Henning (Ed.), Mathematik lernen durch Handeln und Erfahrung. Festschrift zum 75. Geburtstag von Heinrich Besuden (pp. 205–223). Oldenburg: Bueltmann und Gerriets.
Zhang, J. (1997). The nature of external representations in problem solving. Cognitive Science, 21(2), 179–217.
Berg, B. L. (2007). Qualitative research methods for the social sciences (6th ed.). Boston, MA: Pearson.
Blumer, H. (1986). Symbolic interactionism. Perspective and method. Berkeley, CA: University of California Press.
Carmichael, C., Callingham, R., & Watt, H. (2017). Classroom motivational environment influences on emotional and cognitive dimensions of student interest in mathematics. ZDM Mathematics Education, 49(3), 449–460.
Dörfler, W. (2006). Diagramme und Mathematikunterricht [Diagrams and mathematics teaching]. Journal für Mathematik-Didaktik, 27(3/4), 200–219.
Dohrmann, C., & Kuzle, A. (2014). Unpacking children’s angle “Grundvorstellungenâ€: The case of distance Grundvorstellung of 1° angle. In P. Liljedahl, C. Nicol, S. Oesterkle, & D. Allan (Eds.), Proceedings of the 38th conference of the international group for the psychology of mathematics education and the 36th conference of the north American chapter of the psychology of mathematics education (Vol. 2, pp. 409–416). Vancouver: PME.
Einarsdóttir, J. (2007). Research with children: Methodological and ethical challenges. European Early Childhood Education Research Journal, 15(2), 197–211.
Glasnović Gracin, D., & Domović, V. (2009). Upotreba matematiÄkih udžbenika u nastavi viÅ¡ih razreda osnovne Å¡kole [Using mathematics textbooks in lower secondary education]. Odgojne znanosti, 11(2), 297–317.
Halverscheid, S., & Rolka, K. (2006). Student beliefs about mathematics encoded in pictures and words. In J. Novotná, H. Moraová, M. Krátká, & N. StehlÃková (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 3, pp. 233–240). Prague: PME.
Hannula, M. S. (2011). The structure and dynamics of affect in mathematical thinking and learning. In M. Pytlak, E. Swoboda, & T. Rowland (Eds.), Proceedings of the seventh congress of the European society for research in mathematics education (pp. 34–60). Poland: University of RzesoÌw.
Jones, K. (2000). Critical issues in the design of the geometry curriculum. In B. Barton (Ed.), Readings in mathematics education (pp. 75–90). Auckland: University of Auckland.
Laine, A., Näveri, L., Ahtee, M., Hannula, M. S., & Pehkonen, E. (2013). Emotional atmosphere in third-graders’ mathematics classroom – an analysis of pupils’ drawings. Nordic Studies in Mathematics Education, 17(3-4), 101–116.
Laine, A., Ahtee, M., Näveri, L., Pehkonen, E., Portaankorva-Koivisto, P., & Tuohilampi, L. (2015). Collective emotional atmosphere in mathematics lessons based on Finnish fifth graders’ drawings. LUMAT – Research and Practice in Math, Science and Technology Education, 3(1), 87–100.
Mammana, C., & Villani, V. (1998). Perspectives on the teaching of geometry for the 21st century: An ICMI study. Dordrecht: Kluwer.
MZOS [Ministry of Science, Education and Sports] (2006). Nastavni plan i program za osnovnu Å¡kolu [The educational plan and programme for elementary school]. Zagreb: Ministry of Science, Education and Sports.
Patton, M. Q. (2002). Qualitative research and evaluation methods. Thousand Oaks, CA: Sage.
Pehkonen, E., Ahtee, M., Tikkanen, P., & Laine, A. (2011). Pupils’ conception on mathematics lessons revealed via their drawings. In B. Roesken & M. Casper (Eds.), Proceedings of the MAVI-17 conference (pp. 182–191). Bochum: Ruhr-Universität Bochum.
Pehkonen, E., Ahtee, M., & Laine, A. (2016). Pupils’ drawings as a research tool in mathematical problem-solving lessons. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems. Advances and new perspectives (pp. 167–188). New York, NY: Springer.
Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 257–315). Charlotte, NC: Information Age.
Rolka, K., & Halverscheid, S. (2006). Pictures as a means for investigating mathematical beliefs. In S. Alatorre, J. L. Cortina, M. SaÌiz, & A. MeÌndez (Eds.), Proceedings of the 28th annual meeting of the North American chapter of the international group for the psychology of mathematics education. MeÌrida: Universidad PedagoÌgica Nacional.
Rolka, K., & Halverscheid, S. (2011). Researching young students’ mathematical world views. ZDM Mathematics Education, 43(4), 521–533.
Winter, H. (1976). Was soll Geometrie in der Grundschule [What should geometry be like in elementary school?]. Zentralblatt Didaktik der Mathematik, 8, 14–18.
Wittmann, E. Ch. (1999). Konstruktion eines Geometriecurriculums ausgehend von Grundideen der Elementargeometrie [The construction of a geometry curriculum based on the fundamental ideas of elementary geometry]. In H. Henning (Ed.), Mathematik lernen durch Handeln und Erfahrung. Festschrift zum 75. Geburtstag von Heinrich Besuden (pp. 205–223). Oldenburg: Bueltmann und Gerriets.
Zhang, J. (1997). The nature of external representations in problem solving. Cognitive Science, 21(2), 179–217.
Published
2018-06-27
How to Cite
Glasnović Gracin, D., & Kuzle, A. (2018). Drawings as External Representations of Children’s Fundamental Ideas and the Emotional Atmosphere in Geometry Lessons. Center for Educational Policy Studies Journal, 8(2), 31–53. https://doi.org/10.26529/cepsj.299
Section
FOCUS
Authors who publish with this journal agree to the following terms:
- Authors are confirming that they are the authors of the submitted article, which will be published online in the Center for Educational Policy Studies Journal (for short: CEPS Journal) by University of Ljubljana Press (University of Ljubljana, Faculty of Education, Kardeljeva ploščad 16, 1000 Ljubljana, Slovenia). The Author’s/Authors’ name(s) will be evident in the article in the journal. All decisions regarding layout and distribution of the work are in the hands of the publisher.
- The Authors guarantee that the work is their own original creation and does not infringe any statutory or common-law copyright or any proprietary right of any third party. In case of claims by third parties, authors commit themselves to defend the interests of the publisher, and shall cover any potential costs.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under https://creativecommons.org/licenses/by/4.0/deed.en that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
