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differentiation limit procedural knowledge conceptual knowledge

A Study of Pre-Service Teachers’ Performance on Two Calculus Tasks on Differentiation and Limit

Tin Lam Toh , Pee Choon Toh , Kok Ming Teo , Ying Zhu

The purpose of this paper is to report a part of a calculus research project, about the performance of a group of pre-service mathematics teachers on .

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The purpose of this paper is to report a part of a calculus research project, about the performance of a group of pre-service mathematics teachers on two tasks on limit and differentiation of the trigonometric sine function in which the unit of angle measurement was in degrees. Most of the pre-service teachers were not cognizant of the unit of angle measurement in the typical differentiation formula, and a number of participants recognized the condition on the unit of angle measurement but did not translate this to the correct procedure for performing differentiation. The result also shows that most of the participants were not able to associate the derivative formula with the process of deriving it from the first principle. Consequently, they did not associate it with finding  . In the process of evaluating this limit, the pre-service teachers exhibited further misconceptions about division of a number by zero.

Keywords: Differentiation; limit; procedural knowledge; conceptual knowledge.

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References

Aspinwell, L., & Miller, D. (1997). Students’ positive reliance on writing as a process to learn first semester calculus. Journal of Institutional Psychology, 24(4), 253 – 261.

Axtell, M. (2006). A two-semester precalculus/calculus I sequence: A case study. Mathematics and Computer Education, 40(2), 130-137.

Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2007). How can we assess mathematical understanding? In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 41 – 48). PME.

Brown, S., Seidelmann, A., & Zimmermann, G. (2002). In the trenches: Three teachers’ perspectives on moving beyond the math wars. Mathematically Sane Website. http://mathematicallysane.com/analysis/trenches.asp

Burn, B. (2005). The vice: Some historically inspired and proof-generated steps to limits of sequences. Educational Studies in Mathematics, 60(3), 269 – 295.

Carlson, M. P., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352 – 378.

Cho, P., & Nagle, C. (2017). Procedural and conceptual difficulties with slope: An analysis of students’ mistakes on routine tasks. International Journal of Research in Education and Science, 3(1), 135 – 150.

Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behaviour, 5(3), 281 – 303.

Engelbrecht, J., Bergsten, C., & Kagesten, O. (2009). Undergraduate students’ preference for procedural to conceptual solutions to mathematical problems. International Journal of Mathematical Education in Science and Technology, 40(7), 927 – 940.

Gagné, R. M. (1968). Learning hierarchies. Educational Psychologists, 6(1), 1 – 9.

Gordon, S. P. (2005). Discovering the Chain Rule Graphically. Mathematics and Computer Education, 39(3), 195-197.

Haladyna, T. M. (2004). Developing and validating multiple-choice test items. Erlbaum.

Hiebert, J., & Lefevre, P. (1988). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1 – 28). Lawrence Erlbaum Associates, Inc.

Huillet, D. (2005). Mozambican teachers’ professional knowledge about limits of functions. In H. L. hick, & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 169 – 176). PME.

Judson, T. W., & Nishimori, T. (2005). Concepts and skills in high school calculus: An examination of a special case in Japan and the United States. Journal for Research in Mathematics Education, 36(1), 24 – 43.

Lasut, M. (2015). Application of information computer-based learning in calculus package learning. International Journal of Scientific and Research Publications, 5(2), 1 – 4.

Lee, P. Y. (1993). Calculus. National University of Singapore.

Lim, K. F. (2008). Differentiation from first principles using spreadsheets. Australian Senior Mathematics Journal, 22(2), 41 – 48.

Lim-Teo, S. K., Ahuja, O. P., & Lee, P. Y. (2000). Attitude of junior college and tertiary students to calculus. PRIMUS, 10(2), 123 – 142.

Lin, C., Yang, D., Becker, J., Huang, T., & Byun, M. (2013). Preservice teachers’ conceptual and procedural knowledge of fraction operations: A comparative study of the United States and Taiwan. School Science and Mathematics, 113(1), 41 – 51.

Maciejewski, W., & Star, J. R. (2016). Developing flexible procedural knowledge in undergraduate calculus. Research in Mathematics Education, 18(3), 299 – 316.

Mahir, N. (2009). Conceptual and procedural performance of undergraduate students in integration. International Journal of Mathematical Education in Science and Technology, 40(2), 201 – 211.

Masteroides, E., & Zachariades, T. (2004). Secondary mathematics teachers’ knowledge concerning the concept of limit and continuity. In M. J. Hoines, & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 481-488). PME.

Mendezabal, M. J. N., & Tindowen, D. J. C. (2018). Improving students’ attitude, conceptual understanding and procedural skills in differential calculus through Microsoft mathematics. Journal of Technology and Science Education, 8(4), 385 – 397.

Ministry of Education. (2018a). Additional mathematics: Singapore-Cambridge General Certificate of Education Ordinary Level (2020) (Syllabus 4047). Singapore Examinations and Assessment Board. https://cutt.ly/5nsKnWQ

Ministry of Education. (2018b). Mathematics: Singapore-Cambridge General Certificate of Education Advanced Level Higher 2 (2020) (Syllabus 9758). Singapore Examinations and Assessment Board. https://cutt.ly/KnsKEpj

Muzangwa, J., & Chifamba, P. (2012). Analysis of errors and misconceptions in the learning of calculus by undergraduate students. Acta Didactica Napocensia, 5(2), 1-10.

Ng, K. Y., & Toh, T. L. (2008). Pre-university students’ errors in integration of rational functions and implications for classroom teaching. Journal of Science and Mathematics Education in Southeast Asia, 31(2), 100 – 116.

Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14(3), 235 – 250.

Parke, C. S., Lane, S., & Stone, C. A. (2006). Impact of a state performance assessment program in reading and writing. Educational Research and Evaluation, 12(3), 239 – 269.

Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skills in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346 – 362.

Robert, N., & Speer, N. (2001). Research in the teaching and learning of calculus / elementary analysis. In D. Holton (Ed.), The teaching and learning of mathematics at University Level: An ICMI study. Kluwer Academic Publisher.

Scheibling-Seve, C., Pasquinelli, E., & Sander, E. (2020). Assessing conceptual knowledge through solving arithmetic word problems. Educational Studies in Mathematics, 103(3), 293 – 311.

Sebsibe, A. S., & Feza, N. N. (2020). Assessment of students’ conceptual knowledge in limit of functions. International Electronic Journal of Mathematics Education, 15 (2). https://doi.org/10.29333/iejme/6294.

Selden, A., Selden, J., Hauk, S., & Mason, A. (1999). Do calculus students eventually learn to solve non-routine problems? (Tech. Rep. No. 1999:5). Tennessee Technological University. https://cutt.ly/wb3wlX4

Serhan, D. (2015). Students’ understanding of the definite integral concept. International Journal of Research in Education and Science, 1(1), 84 – 88.

Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15 (2), 4 – 14.

Siyepu, S. W (2015). Analysis of errors in derivatives of trigonometric functions. International Journal of STEM Education, 2, 1- 16. https://doi.org/10.1186/s40594-015-0029-5

Stewart, J. (2016). Calculus: Early transcendentals (8th ed). Cengage Learning.

Tall, D. (1992). Students difficulties in calculus. In K-D. Graf, N. Malara, N. Zehavi, & J. Ziegenbalg (Eds.), Proceedings of Working Group 3 at ICME-7, Quebec 1992 (pp. 13-28). Freie Universitat Berlin.

Tall, D. (2010, September). A sensible approach to the calculus. Paper presented at the National and International Meeting on the Teaching of Calculus, Pueblo, Mexico.

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.

Tatar, E., & Zengin, Y. (2016). Conceptual understanding of definite integral with Geogebra. Interdisciplinary Journal of Practice, Theory, and Applied Research, 33(2), 120 – 132.

Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127 – 146). NCTM.

Thompson, P. W., Byerley, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools, 30, 124 – 147.

Toh, T. L. (2007a). Calculus for secondary school teachers. McGraw-Hill.

Toh, T. L. (2007b). Mathematical reasoning from O-Level to A-Level. Mathematical Medley, 33(2), 34 – 40.

Toh, T. L. (2009). On in-service mathematics teachers’ content knowledge of calculus and related concepts. The Mathematics Educator, 12(1), 69 – 86.

Toh, T. L. (2021). School calculus curriculum and the Singapore mathematics curriculum framework. ZDM – Mathematics Education. https://doi.org/10.1007/s11858-021-01225-6.

Törner, G., Potari, D., & Zachariades, T. (2014). Calculus in European classrooms: Curriculum and teaching in different educational and cultural contexts. ZDM - The International Journal on Mathematics Education, 46, 549–560.

Wearne, D., & Hiebert, J. (1988). Constructing ad using meaning for mathematical symbols: The case of decimal fractions. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 220 – 235). NCTM.

Weller, K., Arnon, I., & Dubinsky, E. (2009). Preservice teachers’ understanding of the relation between a fraction or integer and its decimal expansion. Canadian Journal of Science, Mathematics and Technology Education, 9, 5 – 28.

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