Best practices for teaching the concept of the derivative: Lessons from experienced calculus instructors

Thembinkosi Peter Mkhatshwa

doi:10.29333/ejmste/14380

Full Text (PDF)

Thembinkosi Peter Mkhatshwa ¹ ^*

More Detail

¹ Department of Mathematical and Physical Sciences, Miami University, Middletown, OH, USA^* Corresponding Author

Abstract

Much research has reported on difficulties exhibited by students when working with the concept of the derivative in first-semester university calculus. In an effort to generate and share a resource on effective instructional practices related to the teaching of the concept of the derivative, a 12-item questionnaire was administered to experienced calculus instructors in the United States. Most of the 10 experts who participated in this study considered providing ample examples and practice problems, using graphing utilities such as Desmos and GeoGebra, and employing problem solving strategies in the teaching of the concept of the derivative to be effective in supporting students learn about the concept of the derivative, among other things. These experts further remarked on representations of the concept of the derivative and real-world contexts they typically use in their teaching of the concept, in addition to providing rationales for the choice of real-world contexts they typically use in their teaching of the concept. Recommendations for instruction are included.

Keywords

calculus
derivatives
technology
teaching strategies

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2024, Volume 20, Issue 4, Article No: em2426

https://doi.org/10.29333/ejmste/14380

Publication date: 01 Apr 2024

Online publication date: 22 Mar 2024

Article Views: 515

Article Downloads: 360

Open Access References How to cite this article

References

Agyei, E., Darko Agyei, D., & Benning, I. (2022). In-service mathematics teachers’ preparedness, knowledge, skills, and self-efficacy beliefs of using technology in lesson delivery. Cogent Education, 9(1), 2135851. https://doi.org/10.1080/2331186X.2022.2135851
Aryal, H. P. (2022). The effect of inquiry-based learning on Calculus I students’ math anxiety [Doctoral dissertation, Ohio University].
Ashraf, A. L. A. M. (2020). Challenges and possibilities in teaching and learning of calculus: A case study of India. Journal for the Education of Gifted Young Scientists, 8(1), 407-433. https://doi.org/10.17478/jegys.660201
Begle, E. G. (1973). Some lessons learned by SMSG. Mathematics Teacher, 66(3), 207-214. https://doi.org/10.5951/MT.66.3.0207
Beichner, R. (1994). Testing student interpretation of kinematics graphs. American Journal of Physics, 62(8), 750-755. https://doi.org/10.1119/1.17449
Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479-495. https://doi.org/10.1016/j.jmathb.2003.09.006
Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE theory to improve students’ graphical understanding of derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(7), 2947-2967. https://doi.org/10.29333/ejmste/91451
Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77-101. https://doi.org/10.1191/1478088706qp063oa
Buddo, C., Sherriffe, D., & McCarthy-Curvin, A. (2019). Realistic mathematics education (RME) approach: Exploring solving differential equations in the learning of calculus at the University of the West Indies, Mona. Journal of Education & Development in the Caribbean, 18(1), 183-225. https://doi.org/10.46425/j071801f578
Carlson, M., 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. https://doi.org/10.2307/4149958
Cetin, N. (2009). The ability of students to comprehend the function-derivative relationship with regard to problems from their real life. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 19(3), 232-244. https://doi.org/10.1080/10511970701686987
Charalambous, C. Y., Delaney, S., Hsu, H. Y., & Mesa, V. (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Mathematical Thinking and Learning, 12(2), 117-151. https://doi.org/10.1080/10986060903460070
Clark, J. M., Cordero, F., Cottrill, J., Czamocha, B., Devries, D. J., St. John, D., Tolias, G., & Vidakovic, D. (1997). Constructing a schema: The case of the chain rule? Journal of Mathematical Behavior, 16(4), 345-364. https://doi.org/10.1016/S0732-3123(97)90012-2
Code, W., Piccolo, C., Kohler, D., & MacLean, M. (2014). Teaching methods comparison in a large calculus class. ZDM-Mathematics Education, 46, 589-601. https://doi.org/10.1007/s11858-014-0582-2
Cronhjort, M., Filipsson, L., & Weurlander, M. (2018). Improved engagement and learning in flipped-classroom calculus. Teaching Mathematics and its Applications: An International Journal of the IMA, 37(3), 113-121. https://doi.org/10.1093/teamat/hrx007
Dawkins, P. C., & Epperson, J. A. M. (2014). The development and nature of problem-solving among first-semester calculus students. International Journal of Mathematical Education in Science and Technology, 45(6), 839-862. https://doi.org/10.1080/0020739X.2014.884645
Drill, K., Miller, S., & Behrstock-Sherratt, E. (2013). Teachers’ perspectives on educational research. Brock Education Journal, 23(1), 16-33. https://doi.org/10.26522/brocked.v23i1.350
Duran, P. A., Castillo, A. J., Watson, C., Fuller, E., Potvin, G., & Kramer, L. H. (2022). Student attitudes and achievement in active learning calculus. International Journal of Mathematical Education in Science and Technology, 55(3), 759-780. https://doi.org/10.1080/0020739X.2022.2150902
Engelbrecht, J., Bergsten, C., & Kågesten, 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. https://doi.org/10.1080/00207390903200968
Engelke, N. (2004). Related rates problems: Identifying conceptual barriers. In D. McDougall (Ed.), Proceedings of the 26^th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 455-462).
Feudel, F., & Biehler, R. (2022). Students’ understanding of the economic interpretation of the derivative in the context of marginal cost. International Journal of Research in Undergraduate Mathematics Education, 8(3), 437-468. https://doi.org/10.1007/s40753-021-00144-x
Flynn, C. D., Davidson, C. I., & Dotger, S. (2018). Development and psychometric testing of the rate and accumulation concept inventory. Journal of Engineering Education, 107(3), 491-520. https://doi.org/10.1002/jee.20226
Fuchs, S., & Sahmbi, G. (2024). Investigating the impact of active learning in large coordinated calculus courses. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-024-00234-6
García-García, J., & Dolores-Flores, C. (2021). Pre-university students’ mathematical connections when sketching the graph of derivative and antiderivative functions. Mathematics Education Research Journal, 33, 1-22. https://doi.org/10.1007/s13394-019-00286-x
Habre, S., & Abboud, M. (2006). Students’ conceptual understanding of a function and its derivative in an experimental calculus course. The Journal of Mathematical Behavior, 25(1), 57-72. https://doi.org/10.1016/j.jmathb.2005.11.004
Haciomeroglu, E. S., & Andreasen, J. B. (2013). Exploring calculus with dynamic mathematics software. Mathematics and Computer Education, 47(1), 6.
Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. C. (2010). Contrasting cases of calculus students’ understanding of derivative graphs. Mathematical Thinking and Learning, 12(2), 152-176. https://doi.org/10.1080/10986060903480300
Hegedus, S. J., Dalton, S., & Tapper, J. R. (2015). The impact of technology-enhanced curriculum on learning advanced algebra in US high school classrooms. Educational Technology Research and Development, 63, 203-228. https://doi.org/10.1007/s11423-015-9371-z
Hiyam, B., Zoubi, A., & Khataybeh, A. (2019). Utilizing MATHEMATICA software to improve students’ problem solving skills of derivative and its applications. International Journal of Education and Research, 7(11), 57-70.
Ibrahim, B., & Rebello, N. S. (2012). Representational task formats and problem solving strategies in kinematics and work. Physical Review Special Topics-Physics Education Research, 8(1), 010126. https://doi.org/10.1103/PhysRevSTPER.8.010126
Illanes, M. K. G., Breda, A., Manríquez, D. D. C., & Martínez, H. A. A. (2022). Analysis of a teaching learning process of the derivative with the use of ICT oriented to engineering students in Chile. EURASIA Journal of Mathematics, Science and Technology Education, 18(7), em2130. https://doi.org/10.29333/ejmste/12162
Jones, S. R. (2017). An exploratory study on student understandings of derivatives in real-world, non-kinematics contexts. The Journal of Mathematical Behavior, 45, 95-110. https://doi.org/10.1016/j.jmathb.2016.11.002
Kamau, L. M. (2014). Technology adoption in secondary mathematics teaching in Kenya: An explanatory mixed methods study [Doctoral dissertation, Syracuse University].
Kertil, M., Erbas, A. K., & Cetinkaya, B. (2023). Pre-service mathematics teachers’ understanding of rate of change throughout a model development sequence. International Journal of Science and Mathematics Education, 21(6), 1769-1796. https://doi.org/10.1007/s10763-022-10324-z
Klymchuk, S., Zverkova, T., Gruenwald, N., & Sauerbier, G. (2010). University students’ difficulties in solving application problems in calculus: Student perspectives. Mathematics Education Research Journal, 22(2), 81-91. https://doi.org/10.1007/BF03217567
Leng, N. W. (2011). Using an advanced graphing calculator in the teaching and learning of calculus. International Journal of Mathematical Education in Science and Technology, 42(7), 925-938. https://doi.org/10.1080/0020739X.2011.616914
Liang, S. (2016). Teaching the concept of limit by using conceptual conflict strategy and Desmos graphing calculator. International Journal of Research in Education and Science, 2(1), 35-48. https://doi.org/10.21890/ijres.62743
Maharaj, A., & Ntuli, M. (2018). Students' ability to correctly apply differentiation rules to structurally different functions. South African Journal of Science, 114(11-12), 1-7. https://doi.org/10.17159/sajs.2018/5008
Marshall, N., Buteau, C., Jarvis, D., & Lavicza, Z. (2012). Do mathematicians integrate computer algebra systems in university teaching? Comparing a literature review to an international survey study. Computers & Education, 58, 423-434. https://doi.org/10.1016/j.compedu.2011.08.020
Martinez, L. S. (2017). Validity, face and content. In M. Allen (Ed.), The SAGE Encyclopedia of Communication Research Methods (pp. 1823-1824). SAGE Publications.
McLoughlin, M. (2009). Incorporating inquiry-based learning in the calculus sequence: A most challenging endeavor [Paper presentation]. The Annual Joint Meetings of the American Mathematical Society & the Mathematical Association of America.
Mkhatshwa, T. (2018). Business calculus students’ interpretations of marginal change in economic contexts. In Hodges, T. E., Roy, G. J., & Tyminski, A. M. (Eds.), Proceedings of the 40^th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 564-571). Greenville, South Carolina.
Mkhatshwa, T. (2020). Calculus students’ quantitative reasoning in the context of solving related rates of change problems. Mathematical Thinking and Learning, 22(2), 139-161.
Mkhatshwa, T. (2021). An investigation of students’ content understanding, perception changes, and experiences in a flipped precalculus course. Association for University Regional Campuses of Ohio Journal, 27, 41-73.
Mkhatshwa, T. (2022). Quantitative and covariational reasoning opportunities provided by calculus textbooks: The case of the derivative. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2022.2129497
Mkhatshwa, T. (2023a). A quantitative and covariational reasoning investigation of students’ interpretations of partial derivatives in different contexts. International Journal of Mathematical Education in Science and Technology, 54(4), 511-533. https://doi.org/10.1080/0020739X.2021.1958941
Mkhatshwa, T. (2023b). Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems. Eurasia Journal of Mathematics, Science and Technology Education, 19(11), 1-15. https://doi.org/10.29333/ejmste/13658
Mkhatshwa, T. P. (2016). Business calculus students’ reasoning about optimization problems: A study of quantitative reasoning in an economic context. [Doctoral dissertation, Syracuse University, New York].
Mkhatshwa, T., & Jones, S. R. (2018). A study of calculus students’ solution strategies when solving related rates of change problems. In Weinberg, Rasmussen, Rabin, Wawro, & Brown (Eds.), Proceedings of the 21^st Annual Conference on Research in Undergraduate Mathematics Education (pp. 408-415). San Diego, California.
Nabb, K. A. (2010). CAS as a restructuring tool in mathematics education. In Proceedings of the 22^nd International Conference on Technology in Collegiate Mathematics (pp. 247-259).
Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14(3), 235-250. https://doi.org/10.1007/BF00410540
Peters, T., Johnston, E., Bolles, H., Ogilvie, C., Knaub, A., & Holme, T. (2020). Benefits to students of team-based learning in large enrollment calculus. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 30(2), 211-229. https://doi.org/10.1080/10511970.2018.1542417
Piccolo, C., & Code, W. J. (2013). Assessment of students’ understanding of related rates problems. In S. Brown, G. Karakok, K. H. Roh, & M. Oehrtman (Eds.), Proceedings of the 16^th Meeting of the MAA Special Interest Group on Research in Undergraduate Mathematics Education (pp. 607-609).
Prince, M., Vigeant,M., & Nottis, K. (2012). Development of the heat and energy concept inventory: Preliminary results on the prevalence and persistence of engineering students’ misconceptions. Journal of Engineering Education, 101(3), 412-438. https://doi.org/10.1002/j.2168-9830.2012.tb00056.x
Rasmussen, C., & Marrongelle, K. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Journal for Research in Mathematics Education, 37, 388-420.
Reys, B. J., Reys, R. E., & Chavez, O. (2004). Why mathematics textbooks matter. Educational Leadership, 61(5), 61-66.
Robitaille, D. F., & Travers, K. J. (1992). International studies of achievement in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 687-709). Macmillan.
Rosly, W. N. S. W. M., Abdullah, S. S. S., & Shukri, F. N. A. (2020). The uses of WolframAlpha in mathematics. Articles of Teaching and Learning in Higher Education, 1, 96-103.
Sammartino, N. (2023). Teachers’ perspectives of the effects of educational technology tools on high school Algebra I skills [Doctoral dissertation, Saint Peter’s University].
Sari, P., Hadiyan, A., & Antari, D. (2018). Exploring derivatives by means of GeoGebra. International Journal on Emerging Mathematics Education, 2(1), 65-78. https://doi.org/10.12928/ijeme.v2i1.8670
Serkan, C. G. (2013). The good, the bad, and the ugly: The integration of computing technology into calculus classes [Doctoral dissertation, The University of Georgia].
Sevimli, E. (2016). Do calculus students demand technology integration into learning environment? case of instructional differences. International Journal of Educational Technology in Higher Education, 13, 37. https://doi.org/10.1186/s41239-016-0038-6
Siyepu, S. W. (2013). An exploration of students’ errors in derivatives in a university of technology. The Journal of Mathematical Behavior, 32(3), 577-592. https://doi.org/10.1016/j.jmathb.2013.05.001
Speer, N. M., Smith III, J. P., & Horvath, A. (2010). Collegiate mathematics teaching: An unexamined practice. The Journal of Mathematical Behavior, 29(2), 99-114. https://doi.org/10.1016/j.jmathb.2010.02.001
Stanberry, M. L. (2018). Active learning: A case study of student engagement in college calculus. International Journal of Mathematical Education in Science and Technology, 49(6), 959-969. https://doi.org/10.1080/0020739X.2018.1440328
Stroup, W. (2002). Understanding qualitative calculus: A structural synthesis of learning research. International Journal of Computers for Mathematical Learning, 7, 167-215. https://doi.org/10.1023/A:1021147132127
Tall, D., Smith, D., & Piez, C. (2008). Technology and calculus. Research on Technology and the Teaching and Learning of Mathematics, 1, 207-258.
Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2), 229-274. https://doi.org/10.1007/BF01273664
Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education (pp. 33-57). College of Education, University of Wyoming.
Thompson, P. W., Byerley, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools, 30(1-2), 124-147. https://doi.org/10.1080/07380569.2013.768941
Thurm, D., & Barzel, B. (2022). Teaching mathematics with technology: A multidimensional analysis of teacher beliefs. Educational Studies in Mathematics, 109(1), 41-63. https://doi.org/10.1007/s10649-021-10072-x
Ubuz, B. (2007). Interpreting a graph and constructing its derivative graph: Stability and change in students’ conceptions. International Journal of Mathematical Education in Science and Technology, 38(5), 609-637. https://doi.org/10.1080/00207390701359313
Viirman, O., & Pettersson, I. (2022). A small-scale implementation of inquiry-based teaching in a single-variable calculus course for first-year engineering students. Hiroshima Journal of Mathematics Education, 15(2), 129-139.
Weber, E., & Thompson, P. W. (2014). Students’ images of two-variable functions and their graphs. Educational Studies in Mathematics, 87(1), 67-85. https://doi.org/10.1007/s10649-014-9548-0
Wijaya, A., van den Heuvel-Panhuizen, M., & Doorman, M. (2015). Opportunity-to-learn context-based tasks provided by mathematics textbooks. Educational Studies in Mathematics, 89(1), 41-65. https://doi.org/10.1007/s10649-015-9595-1
Wu, L., & Li, Y. (2017). Project-based learning in calculus on the use of Maple software technology. Journal of Mathematics and System Science, 7, 142-47. https://doi.org/10.17265/2159-5291/2017.05.002
Zandieh, M., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. Journal of Mathematical Behavior, 25(1), 1-17. https://doi.org/10.1016/j.jmathb.2005.11.002
Zbarsky, E. S., Simundza, G., & Henriksen, M. (2021). Teaching calculus as a tool in the twenty-first century. In R. Buckmire, & J. M. Libertini (Eds.), Improving applied mathematics education (pp. 13-25). Springer. https://doi.org/10.1007/978-3-030-61717-2_2

How to cite this article

APA

Mkhatshwa, T. P. (2024). Best practices for teaching the concept of the derivative: Lessons from experienced calculus instructors. Eurasia Journal of Mathematics, Science and Technology Education, 20(4), em2426. https://doi.org/10.29333/ejmste/14380

Vancouver

Mkhatshwa TP. Best practices for teaching the concept of the derivative: Lessons from experienced calculus instructors. EURASIA J Math Sci Tech Ed. 2024;20(4):em2426. https://doi.org/10.29333/ejmste/14380

AMA

Mkhatshwa TP. Best practices for teaching the concept of the derivative: Lessons from experienced calculus instructors. EURASIA J Math Sci Tech Ed. 2024;20(4), em2426. https://doi.org/10.29333/ejmste/14380

Chicago

Mkhatshwa, Thembinkosi Peter. "Best practices for teaching the concept of the derivative: Lessons from experienced calculus instructors". Eurasia Journal of Mathematics, Science and Technology Education 2024 20 no. 4 (2024): em2426. https://doi.org/10.29333/ejmste/14380

Harvard

Mkhatshwa, T. P. (2024). Best practices for teaching the concept of the derivative: Lessons from experienced calculus instructors. Eurasia Journal of Mathematics, Science and Technology Education, 20(4), em2426. https://doi.org/10.29333/ejmste/14380

MLA

Mkhatshwa, Thembinkosi Peter "Best practices for teaching the concept of the derivative: Lessons from experienced calculus instructors". Eurasia Journal of Mathematics, Science and Technology Education, vol. 20, no. 4, 2024, em2426. https://doi.org/10.29333/ejmste/14380