Conceptual understanding of calculus is crucial in the fields of applied sciences, business, and engineering and technology subjects. However, the current status indicates that students possess only procedural knowledge developed from rote learning of procedures in calculus without insight of core ideas. Hence, this paper aims to assess students’ challenges and to get insight on common barriers towards attaining conceptual knowledge of calculus. A purposive sample of 238 grade 12 natural sciences students from four different schools in one administrative Zone of Ethiopia were selected to participate in this study. An open ended test about limit of functions at a point and at infinity was administered and analyzed quantitatively and qualitatively. The findings reveal a number of factors about students’ knowledge such as: lack of conceptual knowledge in limit of functions, knowledge characterized by a static view of dynamic processes, over generalization, inconsistent cognitive structure, over dependence on procedural learning, lack of coherent and flexibility of reasoning, lack of procedural fluency and wrong interpretation of symbolic notations. Students’ thinking strategies influencing these challenges originate from their arithmetic thinking rather than algebraic, linguistic ambiguities, compartmentalized learning, dependence on concept image than concept definition, focus in obtaining correct answers for wrong reasons, and attention given to lower level cognitive demanding exercises.

• Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1997). A framework for research and curriculum development in undergraduate mathematics education. Retrieved from https://www.researchgate.net
• Aspinwall, L., & Miller, L. D. (2001). Diagnosing conflict factors in calculus through students’ writings: One teacher’s reflections. The Journal of Mathematical Behavior, 20(1), 89-107. https://doi.org/10.1016/S0732-3123(01)00063-3
• Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 481-497. https://doi.org/10.1016/j.jmathb.2003.09.006
• Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32(4), 487-500. https://doi.org/10.1080/00207390010022590
• Carlson, M., & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Mathematical Association of America Research Sampler, (9). Retrieved from https://www.maa.org
• Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment: A tool for assessing students’ reasoning abilities and understandings. Cognition and Instruction, 28(2), 113-145. https://doi.org/10.1080/07370001003676587
• Çetin, I. (2009). Students’ understanding of limit concept: An APOS perspective (Doctoral dissertation), Middle East Technical University. Retrieved from https://etd.lib.metu.edu.tr
• Cottrill, J., Dubinsky, E., Nichols, 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. https://doi.org/10.1016/S0732-3123(96)90015-2
• Dubinsky, E. (2002). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (11^th ed., pp. 95-123). London: Kluwer Academic Publisher. https://doi.org/10.1007/0-306-47203-1_7
• Dubinsky, E., & McDonald, M. A. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research. In The teaching and learning of mathematics at university level (pp. 275-282). Springer, Dordrecht. https://doi.org/10.1007/0-306-47231-7_25
• Fernández-Plaza, J. A., Rico, L., & Ruiz-Hidalgo, J. F. (2013). Concept of finite limit of a function at a point: Meanings and specific terms. International Journal of Mathematical Education in Science and Technology, 44(5), 699-710. https://doi.org/10.1080/0020739X.2013.805887
• Glasersfeld, V. E. (1995). Radical constructivism: A way of learning and knowing. London: The Falmer Press.
• Idris, N. (2009). Enhancing students’ understanding in calculus trough writing. International Electronic Journal of Mathematics Education, 4(1), 36-55.
• Jaffar, S., & Dindyal, J. (2011). Language-related misconceptions in the study of limits. Mathematics: Traditions and [New] Practices, 390-397. Retrieved from https://www.merga.net.au
• 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 https://doi.org/10.1080/0020739X.2014.941427
• Jordaan, T. (2005). Misconceptions of the limit concept in a mathematics course for engineering students (Master’s thesis), University of South Africa. Retrieved from http://uir.unisa.ac.za
• Juter, K. (2006). Limits of functions-students solving tasks. Australian Senior Mathematics Journal, 20(1), 15-30. Retrieved from UNISA e-journal Database: Education Source
• Luneta, K., & Makonye, P. J. (2010). Learner errors and misconceptions in elementary analysis: A case study of a grade 12 class in South Africa. Acta Didactica Napocensia, 3(3), 35-46.
• Maharaj, A. (2010). An APOS analysis of students’ understanding of the concept of a limit of a function. Pythagoras, 71, 41-52. https://doi.org/10.4102/pythagoras.v0i71.6
• Maharaj, A. (2013). An APOS analysis of natural science students’ understanding of derivatives. South African Journal of Education, 33(1), 1-19. https://doi.org/10.15700/saje.v33n1a458
• Makonye, J. P. (2012). Learner errors on calculus tasks in the NSC examinations: Towards an analytical protocol for learner perturbable concepts in introductory differentiation. International Journal of Learning, 18(6), 339-357. https://doi.org/10.18848/1447-9494/CGP/v18i06/47634
• Messick, S. (1988). Validity. In R. L. Linn (Ed.), Educational measurement (3^rd ed. pp. 13-104). New York: Macmillan.
• Moru, E. K. (2006). Epistemological obstacles in coming to understand the limit concept at undergraduate level: a case of the National University of Lesotho (Doctoral dissertation), University of the Western Cape. Retrieved from http://etd.uwc.ac.za
• Muzangwa, J., & Chifamba, P. (2012). Analysis of Errors and Misconceptions in the Learning of Calculus by Undergraduate Students. Educational Studies in Mathematics, 80(3), 389-412. https://doi.org/10.1007/s10649-011-9357-7
• Pillay, E. (2008). Grade twelve learners’ understanding of the concept of derivative (Masters dissertation), University of KwaZulu-Natal. Retrieved from https://researchspace.ukzn.ac.za
• Pritchard, A., & Woollard, J. (2010). Psychology for the Classroom: Constructivism and Social Learning. New York: Routledge.
• Rabadi, N. (2015). Overcoming difficulties and misconceptions in calculus (Doctoral dissertation, Teachers College, Columbia University). Available from ProQuest dissertations and theses. (UMI 3683397).
• Roble, D. B. (2017). Communicating and valuing students’ productive struggle and creativity in calculus. Turkish Online Journal of Design Art and Communication, 7(2), 255-263. https://doi.org/10.7456/10702100/009
• Sadler, P. M., & Sonnert, G. (2017). Factors influencing success in introductory college calculus. The role of calculus in the transition from high school to college mathematics. Retrieved from https://www.maa.org
• Salas, S., Hille, E., & Etgen, G. (2007). Calculus: one and several variables (10^th ed.). United States of America: John Wiley & Sons, Inc.
• Siyepu, S. W. (2015). Analysis of errors in derivatives of trigonometric functions. International Journal of STEM Education, 2(1), 1-16. https://doi.org/10.1186/s40594-015-0029-5
• Stewart, S. (2008). Understanding linear algebra concepts through the embodied, symbolic and formal worlds of mathematical thinking (Doctoral dissertation), The University of Auckland. Retrieved from https://researchspace.auckland.ac.nz
• Tall, D. (1993). Students’ difficulties in calculus. Proceedings of Working Group 3 on Students’ Difficulties in Calculus, (August 1992), 3, 13-28.
• Tall, D. (2002). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical thinking (11^th ed., pp. 3-21). London: Kluwer academic publisher. https://doi.org/10.1007/0-306-47203-1_1
• Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational studies in Mathematics, 12(2), 151-169. https://doi.org/10.1007/BF00305619
• Zollman, A. (2014). University students’ limited knowledge of limits from calculus through differential equations. The mathematics education for the future project: Proceedings of the 12th International Conference, (pp. 693-698). Retrieved from https://www.researchgate.net

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