Reification in the Learning of Square Roots in a Ninth Grade Classroom: Combining Semiotic and Discursive Approaches
This paper reports on combining semiotic and discursive approaches to reification in classroom interactions. It focuses on the discursive characteristics and semiotic processes involved in the teaching and learning of square roots in a ninth grade classroom in Japan. The purpose of this study is to characterize the development of mathematical discourses in a series of mathematics lessons in terms of the commognitive framework and the model of semiotic chaining. To achieve this objective, the notion of reification is revisited from cognitive, semiotic, and discursive points of view, and classroom activities are observed and analysed in terms of the combined theoretical framework. The results suggest that the model of semiotic chaining is not incommensurable with a strong discursive approach and the changes of meta-discursive rules—an essential aspect of the reification of new signifiers—that take place in the phases of the learning of addition on square roots.
KeywordsDiscursive approach Reification Semiotic chaining Square roots
I am grateful to Nathalie Sinclair (Simon Fraser University, Canada) and Taro Fujita (University of Exeter, UK) for their valuable suggestions on earlier drafts of this paper. This study is supported by the Grant-in-Aid for Scientific Research, JSPS KAKENHI (No. 26780497).
- Abrahamson, D. & Sánchez-García, R. (2016). Learning is moving in new ways: The ecological dynamics of mathematics education. The Journal of the Learning Sciences, 25, 203–239.Google Scholar
- Douady, R. (1991). Tool, object, setting, window: Elements for analysing and constructing didactical situations in mathematics. In A. J. Bishop & S. Mellin-Olsen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 109–130). Dordrecht: Kluwer Academic Publishers.Google Scholar
- Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–126). Dordrecht: Kluwer Academic Publishers.Google Scholar
- Güçler, B. (2016). Making implicit metalevel rules of the discourse on function explicit topic of reflection in the classroom to foster student learning. Educational Studies in Mathematics. doi: 10.1007/s10649-015-9636-9.
- Kieran, C. (1990). Cognitive processes involved in learning school algebra. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education (pp. 96–112). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
- Kjeldsen, T.H. & Blomhoej, M. (2012). Beyond motivation: history as a method for learning meta-discursive rules in mathematics. Educational Studies in Mathematics, 80, 327–349.Google Scholar
- Ministry of Education, Culture, Sports, Science and Technology-Japan (2008). Chuugakko gakushuu shidou youryou kaisetsu suugaku-hen. Tokyo: Kyouiku Shuppan. [National course of study of lower secondary school mathematics in Japan] Retrieved from: http://www.mext.go.jp/a_menu/shotou/new-cs/youryou/eiyaku/1298356.htm.Google Scholar
- Presmeg, N. (1997). Reasoning with metaphors and metonymies in mathematics learning. In L. D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 267–279). Mahwah: LEA.Google Scholar
- Radford, L. (2002). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 43(3), 237–268.Google Scholar
- Radford, L. (2013). Three key concepts of the theory of objectification: Knowledge, knowing, and learning. Journal of Research in Mathematics Education, 2(1), 7–44.Google Scholar
- Roach, D., Gibson, D. & Weber, K. (2004). Why is √25 not ±5? Mathematics Teacher, 97(1), 12–13.Google Scholar
- Sfard, A. (2000). Symbolizing mathematical reality into being: Or how mathematical discourse and mathematical objects create each other. In P. Cobb, E. Yackel & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives in discourse, tools, and instructional design (pp. 37–98). Mahwah: LEA.Google Scholar
- Sfard, A. (2013). Discursive research in mathematics education: Conceptual and methodological issues. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 157–161). Kiel: PME.Google Scholar
- Sriraman, B. & Nardi, E. (2013). Theories in mathematic education: Some developments and ways forward. In M. A. Clements, A. Bishop, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds.), Third international handbook of mathematics education (pp. 303–325). NY: Springer.Google Scholar
- Winsløw, C. & Emori, H. (2006). Elements of a semiotic analysis of the secondary level classroom in Japan. In F. K. Leung (Ed.), Mathematics education in different cultural traditions-a comparative study of East Asia and the West. The 13th ICMI study (pp. 553–566). NY: Springer.CrossRefGoogle Scholar