The role of symbols in mathematical communication: the case of the limit notation

Abstract

Symbols play crucial roles in advanced mathematical thinking by providing flexibility and reducing cognitive load but they often have a dual nature since they can signify both processes and objects of mathematics. The limit notation reflects such duality and presents challenges for students. This study uses a discursive approach to explore how one instructor and his students think about the limit notation. The findings indicate that the instructor flexibly differentiated between the process and product aspects of limit when using the limit notation. Yet, the distinction remained implicit for the students, who mainly realised limit as a process when using the limit notation. The results of the study suggest that it is important for teachers to unpack the meanings inherent in symbols to enhance mathematical communication in the classrooms.

Keywords:

• duality of symbols
• the limit notation
• teacher and student discourse
• undergraduate mathematics education

Notes

1. See Güçler (2012) for a detailed elaboration on the student difficulties about limit indicated by the literature.

2. A detailed analysis of the instructor's discourse, results of the survey, and interview sessions was presented in Güçler (2013).

3. See Güçler (2013) for the participants' uses of graphs, written words, and algebraic notation.

4. The categorisation of participants' word use as operational and objectified is compatible with the assumptions of Sfard's (2008) framework but is different from the model she presents due to the nature of the data in this study. Sfard mainly focuses on examining the development of discourse over time, whereas this study focuses on the description of participants' discourses in the context of the interviews, making no claim about development.

5. Throughout this section, the students' incorrect or incomplete utterances and notations are preserved to keep the originality of their word use and representations.

6. The entire task is not presented here but only the information relevant to the students' use of the limit notation.

7. See Güçler (2013) for the details of the task.