# Reification in the Learning of Square Roots in a Ninth Grade Classroom: Combining Semiotic and Discursive Approaches

## Abstract

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.

### Keywords

Discursive approach Reification Semiotic chaining Square roots## Notes

### Acknowledgments

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).

### References

- 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 - Caspi, S. & Sfard, A. (2012). Spontaneous meta-arithmetic as a first step toward school algebra.
*International Journal of Educational Research, 51–52*, 45–65.CrossRefGoogle Scholar - Dörfler, W. (2002). Formation of mathematical objects as decision making.
*Mathematical Thinking and Learning, 4*(4), 337–350.CrossRefGoogle 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 - Fischbein, E., Jehiam, R. & Cohen, D. (1995). The concept of irrational numbers in high school students and prospective teachers.
*Educational Studies in Mathematics, 29*, 29–44.CrossRefGoogle Scholar - Font, V., Godino, J. D. & Gallardo, J. (2013). The emergence of objects from mathematical practice.
*Educational Studies in Mathematics, 82*, 97–124.CrossRefGoogle Scholar - Gray, E. & Tall, D. (1994). Duality, ambiguity, and flexibility: A proceptual view of simple arithmetic.
*Journal for Research in Mathematics Education, 25*, 116–140.CrossRefGoogle Scholar - Güçler, B. (2014). The role of symbols in mathematical communication: The case of limit notation.
*Journal for Research in Mathematics Education, 16*(3), 251–268.CrossRefGoogle 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. - Herscovics, N. & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra.
*Educational Studies in Mathematics, 27*, 59–78.CrossRefGoogle Scholar - 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 - Maher, C. & Martino, M. (1996). The development of the idea of mathematical proof: A 5-year case study.
*Journal for Research in Mathematics Education, 27*(2), 194–214.CrossRefGoogle 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 - Nachilieli, T. & Tabach, M. (2012). Growing mathematical objects in the classroom—the case of function.
*International Journal of Educational Research, 51–52*, 10–27.CrossRefGoogle Scholar - Nardi, E., Ryve, A., Stadler, E. & Viirman, O. (2014). Commognitive analyses of the learning and teaching of mathematics at university level: The case of discursive shifts in the study of Calculus.
*Journal for Research in Mathematics Education, 16*(2), 182–198.CrossRefGoogle 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 - Presmeg, N. (2006). Semiotics and the “connections” standard: Significance of semiotics for teachers of mathematics.
*Educational Studies in Mathematics, 61*, 163–182.CrossRefGoogle 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 - Ryve, A. (2011). Discourse research in mathematics education: A critical evaluation of 108 journal articles.
*Journal for Research in Mathematics Education, 42*(2), 167–198.CrossRefGoogle Scholar - Sfard, A. (1991). On the dual nature of mathematics conception: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*, 1–36.CrossRefGoogle 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. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint.
*The Journal of The Learning Science, 16*(4), 567–615.CrossRefGoogle Scholar - Sfard, A. (2008).
*Thinking as communicating: Human development, the growth of discourse, and mathematizing*. NY: Cambridge University.CrossRefGoogle 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 - Sfard, A. & Linchevski, L. (1994). The gains and pitfalls of reification: The case of algebra.
*Educational Studies in Mathematics, 26*, 191–228.CrossRefGoogle Scholar - Sinclair, N. & Moss, J. (2012). The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment.
*International Journal of Educational Research, 51–52*, 28–44.CrossRefGoogle Scholar - Sirotic, N. & Zazkis, R. (2007a). Irrational numbers on the number line—where are they?
*International Journal of Mathematical Education in Science and Technology, 38*(4), 477–488.CrossRefGoogle Scholar - Sirotic, N. & Zazkis, R. (2007b). Irrational numbers: The gap between formal and intuitive knowledge.
*Educational Studies in Mathematics, 65*, 49–76.CrossRefGoogle 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 - Tall, D., Thomas, M., Davis, G., Gray, E. & Simpson, A. (2000). What is the object of the encapsulation of a process?
*Journal of Mathematical Behaviour, 18*(2), 223–241.CrossRefGoogle 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 - Zandieh, M., Roh, K. H. & Knapp, J. (2014). Conceptual blending: Student reasoning when proving “conditional implies conditional” statements.
*Journal of Mathematical Behaviour, 33*, 209–229.CrossRefGoogle Scholar - Zazkis, R. (2005). Representing numbers: Prime and irrational.
*International Journal of Mathematical Education in Science and Technology, 36*(2–3), 207–218.CrossRefGoogle Scholar