# The interplay between language, gestures, dragging and diagrams in bilingual learners’ mathematical communications

## Abstract

This paper discusses the importance of considering bilingual learners’ non-linguistic forms of communication for understanding their mathematical thinking. In particular, I provide a detailed analysis of communication involving a pair of high school bilingual learners during an exploratory activity where a touchscreen-based dynamic geometry environment (DGE) was used. The paper focuses on the *word-use*, *gestures* and *dragging actions* in student-pair communication about calculus concepts as they interacted with a touchscreen-based DGE. Findings suggest that the students relied on gestures and dragging as non-linguistic features of the mathematical discourse to communicate dynamic aspects of calculus. Moreover, by examining the interplay between language, gestures, dragging and diagrams, it was possible to identify bilingual learners’ competence in mathematical communications. This paper raises questions about new forms of communication mobilised in dynamic, touchscreen environments, particularly for bilingual learners.

### Keywords

Thinking as communicating Non-linguistic communication Bilingual learners Dynamic geometry environment Touchscreen dragging### References

- Adler, J. (2001).
*Teaching mathematics in multilingual Classrooms*. Dordrecht: Kluwer.Google Scholar - Ainley, J. (1999). Who are you today? Complementary and conflicting roles in school-based research.
*For the Learning of Mathematics, 19*(1), 39–47.Google Scholar - Arzarello, F. (2006). Semiosis as a multimodal process,
*Relime, Numero Especial*, 267–299Google Scholar - Arzarello, F., Paola, D., Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom.
*Educational Studies in Mathematics, 70*(2), 97–109.CrossRefGoogle Scholar - Bartolini-Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini-Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.),
*Handbook of international research in mathematics education*(2nd Revised ed., pp. 746–805). Mahwah: Lawrence Erlbaum.Google Scholar - British Columbia Teachers’ Federation. (2012).
*2012 BC education facts.*Retrieved May 22, 2013, from http://www.bctf.ca/uploadedFiles/Public/Publications/2012EdFacts.pdf - Chen, C. L., & Herbst, P. (2012). The interplay among gestures, discourse, and diagrams in students’ geometrical reasoning.
*Educational Studies in Mathematics, 83*, 285–307.CrossRefGoogle Scholar - Clarkson, P. (2007). Australian Vietnamese students learning mathematics: High ability bilinguals and their use of their languages.
*Educational Studies in Mathematics, 64*, 191–215.CrossRefGoogle Scholar - de Freitas, E., & Sinclair, N. (2012). Diagram, gesture, agency: Theorizing embodiment in the mathematics classroom.
*Educational Studies in Mathematics, 80*, 133–152.CrossRefGoogle Scholar - Edwards, L. D., Ferrara, F., & Moore-Russo, D. (Eds.). (2014).
*Emerging perspectives on gesture and embodiment in mathematics*. Charlotte: Information Age Publishing.Google Scholar - Falcade, R., Laborde, C., & Mariotti, M. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation.
*Educational Studies in Mathematics, 66*, 317–333.CrossRefGoogle Scholar - Ferrara, F., Pratt, D., & Robutti, O. (2006). The role and uses of technologies for the teaching of algebra and calculus: Ideas discussed at PME over the last 30 years. In A. Gutierrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present and future*(pp. 237–273). Rotterdam: Sense Publishers.Google Scholar - Gol Tabaghi, S. (2012). Dynamic geometric representation of eigenvectors. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.),
*Proceedings of the 15th Research in Undergraduate Mathematics Education Conference*(pp. 53–58). Portland, Oregon: SIGMAA.Google Scholar - Grosjean, F. (1985). The bilingual as a competent but specific speaker-hearer.
*Journal of Multilingual and Multicultural Development, 6*, 467–477.Google Scholar - Gutierrez, K. T., Sengupta-Irving, D., & Dieckmann, J. (2010). Developing a mathematical vision: Mathematics as a discursive and embodied Practice. In J. Moschkovich (Ed.),
*Language and mathematics education: Multiple perspectives and directions for research*. Charlotte: Information Age Publishers.Google Scholar - Hong, Y., & Thomas, M. (2013). Graphical construction of a local perspective. In A. M. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 81–90). Kiel: PME.Google Scholar - Jackiw, N. (2001).
*The geometer’s sketchpad*[Computer program]. Key Curriculum Press.Google Scholar - Lave, J., & Wenger, E. (1991).
*Situated learning: Legitimate peripheral participation*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Moschkovich, J. (2007). Bilingual mathematics learners: How views of language, bilingual learners, and mathematical communication impact instruction. In N. Nasir & P. Cobb (Eds.),
*Diversity, equity, and access to mathematical ideas*(pp. 89–104). New York: Teachers College Press.Google Scholar - Moschkovich, J. (2009). How language and graphs support conversation in a bilingual mathematics classroom. In R. Barwell (Ed.),
*Multilingualism in mathematics classrooms: Global perspectives*(pp. 78–96). Bristol: Multilingual Matters.Google Scholar - Moschkovich, J. (Ed.). (2010).
*Language and mathematics education: Multiple perspectives and directions for research*. Charlotte: Information Age Publishers.Google Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston: Author.Google Scholar - Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition.
*Educational Studies in Mathematics, 70*(2), 159–174.CrossRefGoogle Scholar - Ng, O. (2014). The interplay between language, gestures, and diagrams in bilingual learners’ mathematical communications. In D. Allen, S. Oesterle, & P. Liljedahl (Eds.),
*Proceedings of the Joint Meeting of PME 38 and PMENA 36*(Vol. 4, pp. 289–296). Vancouver: PME.Google Scholar - Ng, O., & Sinclair, N. (2013). Gestures and temporality: Children’s use of gestures on spatial transformation tasks. In A. M. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 3, pp. 361–368). Kiel: PME.Google Scholar - Núñez, R. (2006). Do real numbers really move? Language, thought, and gesture: The embodied cognitive foundations of mathematics. In R. Hersh (Ed.),
*18 unconventional essays on the nature of mathematics*(pp. 160–181). New York: Springer.CrossRefGoogle Scholar - Peltenburg, M., Van den Heuvel-Panhuizen, M., & Robitzsch, A. (2012). Special education students’ use of indirect addition in solving subtraction problems up to 100 - A proof of the didactical potential of an ignored procedure.
*Educational Studies in Mathematics, 79*, 351–369.CrossRefGoogle Scholar - Planas, N. (2014). One speaker, two languages: Learning opportunities in the mathematics classroom.
*Educational Studies in Mathematics, 87*(1), 51–66.CrossRefGoogle Scholar - Planas, N., & Setati-Phakeng, M. (2014). On the process of gaining language as a resource in mathematics education.
*ZDM The International Journal on Mathematics Education, 46*(6), 883–893.CrossRefGoogle Scholar - Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings.
*Educational Studies in Mathematics, 70*(2), 111–126.CrossRefGoogle Scholar - Radford, L., Edwards, L., & Arzarello, F. (2009). Beyond words.
*Educational Studies in Mathematics, 70*(2), 91–95.Google Scholar - Setati, M. (2005). Power and access in multilingual mathematics classrooms. In M. Goos, C. Kanes, & R. Brown (Eds.),
*Proceedings of the fourth international mathematics education and society conference*(pp. 7–18). Brisbane: Centre for Learning Research, Griffith University.Google Scholar - Setati, M., & Moschkovich, J. (2011). Mathematics education and language diversity: A dialogue across settings [Special issue on Equity].
*Journal for Research in Mathematics Education, 42*(1), 119–128.Google Scholar - Sfard, A. (2008).
*Thinking as communicating: Human development, the growth of discourses, and mathematizing*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Sfard, A. (2009). What’s all the fuss about gestures? A commentary.
*Educational Studies in Mathematics, 70*, 191–200.CrossRefGoogle Scholar - Sinclair, N., & de Freitas, E. (2015). The haptic nature of gesture: Rethinking gesture with new multitouch digital technologies.
*Gestures*, in press.Google Scholar - Stewart, J. (2008).
*Calculus: Early transcendental*(6th ed.). Belmont: Brooks Cole.Google Scholar - van den Heuvel-Panhuizen, M. (2015, June).
*It’s time to reveal what students with MLD know, rather than what they do not know.*Plenary speech presented at Macau University, Macau SAR, China.Google Scholar - Wenger, E. (1998).
*Communities of practice. Learning, meaning and identity*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Yerushalmy, M., & Swidan, O. (2012). Signifying the accumulation graph in a dynamic and multi-representation environment.
*Educational Studies in Mathematics, 80*(3), 287–306.CrossRefGoogle Scholar - Yoon, C., Thomas, M. O., & Dreyfus, T. (2011). Gestures and insight in advanced mathematical thinking.
*International Journal of Mathematical Education in Science and Technology, 42*(7), 891–901.Google Scholar