# The mediating role of a teacher’s use of semiotic resources in pupils’ early algebraic reasoning

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## Abstract

This paper focuses on the semiotic resources used by an experienced sixth-grade teacher when her pupils are working on a mathematical task involving written text and the two inscriptions of figure and diagram. Socio-cultural analytical constructs such as semiotic bundle, space of joint action and togethering are applied in order to enable and frame the collective activity of the teacher and pupils. Four extracts from different situations in the classroom illustrate the important role of both teacher gestures and pupil gestures, interacting with other modalities such as speech and inscription, in the process of making sense of pupils’ appropriation of coordinating two dimensions in a diagram. It is argued that the nature of the mathematical task is an important entry point into early algebraic reasoning. The study emphasises the mediating role of the dynamics of semiotic bundles produced in teacher–pupil dialogues as a promising way to address the fundamental relationships between mathematics, pupil and teacher in a classroom context in order to provoke pupil involvement and engagement when experiencing mathematics.

## Keywords

Pointing gestures Semiotic bundle Early algebraic reasoning Space of joint action Didactic triangle Mediation## Notes

### Acknowledgments

The LCM research project was supported by the Research Council of Norway (Norges Forskningsråd). Special thanks are due to the teacher who made this case study possible by giving me the opportunity to analyse episodes of her classroom. I also acknowledge the anonymous reviewers for their valuable comments which improved the paper.

## References

- Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom.
*Educational Studies in Mathematics,**70*, 97–109.CrossRefGoogle Scholar - Bell, A., Brekke, G., & Swan, M. (1986). Diagnostic teaching: 5 graphical interpretation teaching styles and their effects.
*Mathematics Teaching,**120*, 50–57.Google Scholar - Bjuland, R., Cestari, M. L., & Borgersen, H. E. (2008a). The interplay between gesture and discourse as mediating devices in collaborative mathematical reasoning. A multimodal approach.
*Mathematical Thinking and Learning,**10*(3), 271–292.CrossRefGoogle Scholar - Bjuland, R., Cestari, M. L., & Borgersen, H. E. (2008b). A teacher’s use of gesture and discourse as communicative strategies in the presentation of a mathematical task. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepúlveda (Eds.),
*Proceedings of the 32nd conference of the international group for the psychology of mathematics education*(PME 32) (Vol. 2, pp. 185–192). Morelia: Universidad Michoacana de san Nicolás de Hidalgo.Google Scholar - Bjuland, R., Cestari, M. L., & Borgersen, H. E. (2010). A teacher’s use of gesture and discourse as communicative strategies in concluding a mathematical task. In V. Durrand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.),
*Proceedings of the sixth congress of the European society for research in mathematics**education*(CERME 6, Lyon, France) (pp. 884–893). Université de Lyon.Google Scholar - Bjuland, R., & Jaworski, B. (2009). Teachers’ perspectives on collaboration with didacticians to create an inquiry community.
*Research in Mathematics Education,**11*(1), 21–38.CrossRefGoogle Scholar - Blanton, M. L., & Kaput, J. J. (2004). Elementary grades students’ capacity for functional thinking. In M. J. Høines & A. B. Fuglestad (Eds.),
*Proceedings of the international group for the psychological of mathematics education*(Vol. 2, pp. 135–142). Bergen: Bergen University College.Google Scholar - Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.),
*Early algebraization. A global dialogue from multiple perspectives*(pp. 5–23). Dordrecht: Springer.Google Scholar - Cai, J., & Knuth, E. (2011).
*Early algebraization. A global dialogue from multiple perspectives*. Dordrecht: Springer.Google Scholar - Carlsen, M. (2008). Appropriating mathematical tools through problem solving in collaborative small-group settings. Doctoral dissertation, University of Agder, Kristiansand, Norway.Google Scholar
- Carlsen, M. (2009). Reasoning with paper and pencil: the role of inscriptions in student learning of geometric series.
*Mathematics Education Research Journal,**21*(1), 54–84.CrossRefGoogle Scholar - Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester Jr (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 669–705). Charlotte, NC: Information Age Publishing.Google Scholar - Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics.
*Educational Studies in Mathematics,**61*, 103–131.CrossRefGoogle Scholar - Ernest, P. (2006). A semiotic perspective of mathematical activity: the case of number.
*Educational Studies in Mathematics,**61*, 67–101.CrossRefGoogle Scholar - Gjone, G. (1997).
*Kartlegging av matematikkforståing. Rettleiing til funksjonar*. Oslo: Nasjonalt Læremiddelsenter.Google Scholar - Hansson, Å. (2010). Instructional responsibility in mathematics education: modeling classroom teaching using Swedish data.
*Educational Studies in Mathematics,**75*, 171–189.CrossRefGoogle Scholar - Jaworski, B. (1994).
*Investigating mathematics teaching. A constructivist enquiry*. London: Falmer Press.Google Scholar - Jaworski, B., Fuglestad, A. B., Bjuland, R., Breiteig, T., Goodchild, S., & Grevholm, B. (2007).
*Learning communities in mathematics*. Bergen: Caspar Forlag As.Google Scholar - Linell, P. (1998).
*Approaching dialogue. Talk, interaction and contexts in dialogical perspectives*. Amsterdam: John Benjamins.Google Scholar - McNeill, D. (1992).
*Hand and mind: What gestures reveal about thought*. Chicago, IL: Chicago University Press.Google Scholar - Mehan, H. (1979).
*Learning lessons: Social organization in the classroom*. Cambridge, MA: Harvard University Press.Google Scholar - Radford, L. (2003). Gestures, speech and the sprouting of signs.
*Mathematical Thinking and Learning,**5*(1), 37–70.CrossRefGoogle Scholar - Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings.
*Educational Studies of Mathematics,**70*, 111–126.CrossRefGoogle Scholar - Radford, L., Edwards, L., & Arzarello, F. (2009). Introduction: beyond words.
*Educational Studies in Mathematics,**70*, 91–95.CrossRefGoogle Scholar - Radford, L., & Roth, W. M. (2011). Intercorporeality and ethical commitment: an activity perspective on classroom interaction.
*Educational Studies of Mathematics,**77*, 227–245.CrossRefGoogle Scholar - Sinclair, J., & Coulthard, R. (1975).
*Towards an analysis of discourse. The English used by teachers and pupils*. London: Oxford University Press.Google Scholar - Steinbring, H. (2005). Analyzing mathematical teaching–learning situations—the interplay of communicational and epistemological constraints.
*Educational Studies of Mathematics,**59*, 313–324.CrossRefGoogle Scholar - Stigler, J. W., & Hiebert, J. (1999).
*The teaching gap: Best ideas from the world’s teachers for improving education in the classroom*. New York, NY: The Free Press.Google Scholar