## 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.

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## Notes

- 1.
- 2.
An example in mathematics, provided by Sfard (2008), shows a situation that a student tries to solve the parametric equation “

*kx*–*x*= −2 for*x*”. In this case, the student “balked at the sight of the ‘unrealized’ expression*k −*1” (p. 184). - 3.
For instance, students’ tendency to simplify expression, though it is a well-known phenomenon, such as treating “3

*a*+ 5*b*” as “8*ab*” can be considered as a case of the anxiety.

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## 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).

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Shinno, Y. Reification in the Learning of Square Roots in a Ninth Grade Classroom: Combining Semiotic and Discursive Approaches.
*Int J of Sci and Math Educ* **16, **295–314 (2018). https://doi.org/10.1007/s10763-016-9765-3

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### Keywords

- Discursive approach
- Reification
- Semiotic chaining
- Square roots