Eunice Kolitsoe Moru, Anthony A Essien


The idea of limit is central to both differential and integral calculus. It is also applicable in other disciplines such as physics, engineering, economics, etc.  Because of this, conducting a study to further improve teachers’ knowledge about how social science students (whose major is economics) understand limits is of utmost importance. The reported study sought to find out how students understand the idea of limit with regard to the use of its symbolism. Sixty first year university students in the social sciences acted as the sample of the study. An adapted procept theory was used to analyse data obtained from these students through their solution to tasks on limit and explanations on their thinking and solution processes. Qualitative analysis of data indicated that some students understood the limit symbolism  to be a procept while others did not. When solving the mathematical tasks, students’ difficulties emanated from: (i) their inability to coordinate the two processes,  and , or  and  (ii) the proper use of the limit operator,  and (iii) inability to realise that the simplification has led to the same response as they could not see the relationship between the results. This resulted in misalignment between their reasoning and their choice of answers where justification was required. The results also show that limits at infinity were more problematic than those of the form  as where a is a constant. Students’ choice of method used depended mostly on how much efficient the method was in terms of saving time and not really on promoting understanding. The lesson learnt from the study is that when using the adjusted procept theory, the yes or no answers do not qualify to be used in concluding the level of thinking at which students are at. It is recommended that students be asked to show their working and also explain their answers so that the type of understanding that leads to their choices come to fore


procedure; process; concept; procept; limits


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Cornu, B. (1991). Limits. In D. O. Tall (Ed.), Advanced Mathematical Thinking (pp. 153 -166). Dordrecht, NL: Kluwer Academic Publishers.

Cottrill, J., Dubinsky, E., Nicholas, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme. Journal of Mathematical Behavior, 15(2), 167–192.

Denbel, D. G. (2014). Students’ misconceptions of the limit concept in a first calculus course. Journal of Education and Practice, 5(34), 24-40. Retrieved from

Essien, A. A. (2021). Rethinking exemplification in mathematics teacher education multilingual classrooms. Retrieved from

Güçler, B. (2014). The role of symbols in mathematical communication: The case of the limit notation. Research in Mathematics Education, 16(3), 251-268.

Gray E., & Tall, D. (1992). Success and failure in mathematics: Procept and procedure - A primary perspective. Retrieved from

Gray E., & Tall, D. (1994). Duality, ambiguity, and flexibility: A proceptual view of simple arithmetic. The Journal for Research in Mathematics Education, 26(2), 115–141. Retrieved from

Gray, E., & Tall, D. (2001). Relationships between embodied objects and symbolic procepts: An explanatory theory of success and failure in mathematics. Proceedings of PME25, Utrecht, 65–72. Retrieved from

Jones, S. R. (2015) Calculus limits involving infinity: The role of students’ informal dynamic reasoning. International Journal of Mathematical Education in Science and Technology, 46(1), 105-126.

Maharaj, A. (2010). An APOS analysis of students’ understanding of the concept of a limit of a function. Pythagoras: Journal of the Association for Mathematics Education of South Africa, 71, 41-52. Retrieved from

Mellor, K., Clark, R., & Essien, A. A. (2018). Affordances for learning linear functions: A comparative study of two textbooks from South Africa and Germany. Pythagoras: Journal of the Association for Mathematics Education of South Africa, 39(1).

Monaghan, J. (1991). Problems with language of limits. For the Learning of Mathematics, 11(3), 20 – 24. Retrieved from

Moru, E. K. (2009). Epistemological obstacles in coming to understand the limit of a function at undergraduate level: A case from the National University of Lesotho. International Journal of Science and Mathematics Education, 7, 431-454.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36.

Starman, A. B. (2013). The case study as a type of qualitative research. Journal of Contemporary Educational Studies, 1, 28-43. Retrieved from

Skemp, R. A. (1971). The psychology of learning mathematics. Harmondsworth, UK: Penguin Books.

Skemp, R. A. (1976). Relational understanding and instrumental understanding.
Mathematics Teaching, 77, 20–26. Retrieved from

Taback, S. (1975). The child’s concept of limit. In M.F. Rosskopf (Ed.), Children’s mathematical concepts (pp. 111 - 144). New York, NY: Teachers College Press.

Tall, D.O. (n.d.). The theory of PROCEPTs: Flexible use of symbols as both PROcess and conCEPT in arithmetic, algebra, calculus. Retrieved from

Tall, D. O., & Thomas, M. O. J. (1991). Encouraging versatile thinking in algebra using the computer. Educational Studies in Mathematics, 22(2), 125–147.

Tall, D. O. (1992). Mathematical processes and symbols in the mind. Retrieved from

Tall, D. O., Gray, E., Ali, M. B., Crowley, L., DeMarois, P., McGowen, M., Pitta. D., Pinto. M., Thomas, M., & Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and Technology Education, 1, 80-104.

Thomas, M. O. J., & Hong, Y. Y. (2001). Representations as conceptual tools: Process and structural perspectives. Retrieved from

Viirman, O., Vivier, L., & Monaghan, J. (2022). The limit notion at three educational levels in three countries. International Journal of Research in Undergraduate Mathematics Education, 8(2), 222–244.

Williams, S. R. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22(3), 219 -236.


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