Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems

Thembinkosi Peter Mkhatshwa

doi:10.29333/ejmste/13658

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Thembinkosi Peter Mkhatshwa ¹ ^*

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

Abstract

Much research has reported on students’ difficulties with solving related rates problems in calculus. In an effort to generate a resource that could potentially address some of these difficulties from a teaching standpoint, a questionnaire about effective instructional approaches related to the teaching of related rates problems, among other things, was administered to 14 veteran calculus instructors. Analysis of the responses provided by the instructors revealed that all the instructors considered the use of diagrams to be helpful when solving related rates problems. Furthermore, a majority of these instructors noted that introducing a set of steps (i.e., a guideline), during classroom instruction, that students could follow when solving related rates problems is helpful for students when working with this type of problems. These instructors further identified strengths and weaknesses in the way related rates problems are typically presented in calculus textbooks. Implications for instruction are included.

Keywords

related rates problems
calculus learning
calculus teaching

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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, Volume 19, Issue 11, November 2023, Article No: em2346

https://doi.org/10.29333/ejmste/13658

Publication date: 01 Nov 2023

Online publication date: 09 Sep 2023

Article Views: 1360

Article Downloads: 938

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References

Alajmi, A. H. (2012). How do elementary textbooks address fractions? A review of mathematics textbooks in the USA, Japan, and Kuwait. Educational Studies in Mathematics, 79(2), 239-261. https://doi.org/10.1007/s10649-011-9342-1
Azzam, N. A., Eusebio, M., & Miqdadi, R. (2019). Students’ difficulties with related rates problem in calculus. In Proceedings of the 2019 Advances in Science and Engineering Technology International Conferences (pp. 1-5). https://doi.org/10.1109/ICASET.2019.8714489
Begle, E. G. (1973). Some lessons learned by SMSG. Mathematics Teacher, 66(3), 207-214. https://doi.org/10.5951/MT.66.3.0207
Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77-101. https://doi.org/10.1191/1478088706qp063oa
Bressoud, D. M., Carlson, M. P., Mesa, V., & Rasmussen, C. (2013). The calculus student: Insights from the Mathematical Association of America national study. International Journal of Mathematical Education in Science and Technology, 44(5), 685-698. https://doi.org/10.1080/0020739X.2013.798874
Caridade, C. M., Encinas, A. H., Martín‐Vaquero, J., Queiruga‐Dios, A., & Rasteiro, D. M. (2018). Project‐based teaching in calculus courses: Estimation of the surface and perimeter of the Iberian Peninsula. Computer Applications in Engineering Education, 26(5), 1350-1361. https://doi.org/10.1002/cae.22032
Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. CBMS Issues in Mathematics Education, 7, 114-162. https://doi.org/10.1090/cbmath/007/04
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
Chen, C. L., & Wu, C. C. (2020). Students’ behavioral intention to use and achievements in ICT-Integrated mathematics remedial instruction: Case study of a calculus course. Computers & Education, 145, 103740. https://doi.org/10.1016/j.compedu.2019.103740
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-International Journal on Mathematics Education, 46(4), 589-601. https://doi.org/10.1007/s11858-014-0582-2
Ekici, C., & Gard, A. (2017). Inquiry-based learning of transcendental functions in calculus. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 27(7), 681-692. https://doi.org/10.1080/10511970.2016.1214654
Ellis, J., Hanson, K., Nuñez, G., & Rasmussen, C. (2015). Beyond plug and chug: An analysis of calculus I homework. International Journal of Research in Undergraduate Mathematics Education, 1(2), 268-287. https://doi.org/10.1007/s40753-015-0012-z
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).
Engelke, N. (2007). Students’ understanding of related rates problems in calculus [Doctoral dissertation, Arizona State University].
Engelke-Infante, N. (2021). Helping students think like mathematicians: Modeling-related rates with 2 diagrams. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 31(7), 749-759. https://doi.org/10.1080/10511970.2020.1712669
Freudenthal, H. (1993). Thoughts in teaching mechanics didactical phenomenology of the concept of force. Educational Studies in Mathematics, 25(1&2), 71-87. https://doi.org/10.1007/BF01274103
Hare, A., & Phillippy, D. (2004). Building mathematical maturity in calculus: Teaching implicit differentiation through a review of functions. Mathematics Teacher, 98(1), 6. https://doi.org/10.5951/MT.98.1.0006
Hausknecht, A. O., & Kowalczyk, R. E. (2008). Exploring calculus using innovative technology. In J. Foster (Ed.), Proceedings of the 19^th Annual International Conference on Technology in Collegiate Mathematics (pp. 75-79).
Indiana University Center for Postsecondary Research (n. d.). The Carnegie classification of institutions of higher education. https://carnegieclassifications.acenet.edu/
Jeppson, H. P. (2019). Developing understanding of the chain rule, implicit differentiation, and related rates: Towards a hypothetical learning trajectory rooted in nested multi-variation [Unpublished master’s thesis]. Brigham Young University.
Kolovou, A., van den Heuvel-Panhuizen, M., & Bakker, A. (2009). Non-routine problem solving tasks in primary school mathematics textbooks–A needle in a haystack. Mediterranean Journal for Research in Mathematics Education, 8(2), 31-68.
Kottath, A. (2021). An investigation of students’ application of critical thinking to solving related rates problems [Unpublished master’s thesis]. Oklahoma State University.
Martin, T. (2000). Calculus students’ ability to solve geometric related-rates problems. Mathematics Education Research Journal, 12(2), 74-91. https://doi.org/10.1007/BF03217077
Martinez, L. S. (2017). Validity, face and content. In M. Allen (Ed.), The SAGE encyclopedia of communication research methods (pp. 1823-1824). SAGE.
Mirin, A. C., & Zazkis, D. (2019). Making implicit differentiation explicit. In A. Weinberg, D. Moore-Russo, H. Soto, & M. Wawro (Eds.), Proceedings of the 22^nd Annual Conference on Research in Undergraduate Mathematics Education (pp. 792-800).
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.
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.
Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. Harel, & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 175-193). Mathematical Association of America.
Oktaviyanthi, R., & Supriani, Y. (2015). Utilizing Microsoft mathematics in teaching and learning calculus. Indonesian Mathematical Society Journal on Mathematics Education, 6(1), 63-76. https://doi.org/10.22342/jme.6.1.1902.63-76
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).
Reys, B. J., Reys, R. E., & Chavez, O. (2004). Why mathematics textbooks matter. Educational Leadership, 61(5), 61-66.
Sahin, A., Cavlazoglu, B., & Zeytuncu, Y. E. (2015). Flipping a college calculus course: A case study. Journal of Educational Technology & Society, 18(3), 142-152.
Salleh, T. S., & Zakaria, E. (2016). The effects of maple integrated strategy on engineering technology students’ understanding of integral calculus. Turkish Online Journal of Educational Technology, 15(3), 183-194.
Shelton, T. (2017). Injecting inquiry-oriented modules into calculus. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 27(7), 669-680. https://doi.org/10.1080/10511970.2016.1211205
Speer, N., & Kung, D. (2016). The complement of RUME: What’s missing from our research? In T. Fukawa-Connelly, N. Infante, M. Wawro, & S. Brown (Eds.), Proceedings of the 19^th Meeting of the MAA Special Interest Group on Research in Undergraduate Mathematics Education (pp. 1288-1295).
Stratton, S. J. (2021). Population research: convenience sampling strategies. Prehospital and Disaster Medicine, 36(4), 373-374. https://doi.org/10.1017/S1049023X21000649
Taylor, A. V. (2014). Investigating the difficulties of first year mainstream mathematics students at the University of Western Cape with “related rates” problems [Unpublished master’s thesis]. University of Western Cape.
Törnroos, J. (2005). Mathematics textbooks, opportunity to learn and student achievement. Studies in Educational Evaluation, 31(4), 315–327. https://doi.org/10.1016/j.stueduc.2005.11.005
Wasserman, N. H., Quint, C., Norris, S. A., & Carr, T. (2017). Exploring flipped classroom instruction in calculus III. International Journal of Science and Mathematics Education, 15, 545-568. https://doi.org/10.1007/s10763-015-9704-8
White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27(1), 79-95. https://doi.org/10.2307/749199
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, 142-147. https://doi.org/10.17265/2159-5291/2017.05.002
Yimer, S. T. (2022). Effective instruction for calculus learning outcomes through blending co-operative learning and GeoGebra. Mathematics Teaching Research Journal, 14(3), 170-189.

How to cite this article

APA

Mkhatshwa, T. P. (2023). Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems. Eurasia Journal of Mathematics, Science and Technology Education, 19(11), em2346. https://doi.org/10.29333/ejmste/13658

Vancouver

Mkhatshwa TP. Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems. EURASIA J Math Sci Tech Ed. 2023;19(11):em2346. https://doi.org/10.29333/ejmste/13658

AMA

Mkhatshwa TP. Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems. EURASIA J Math Sci Tech Ed. 2023;19(11), em2346. https://doi.org/10.29333/ejmste/13658

Chicago

Mkhatshwa, Thembinkosi Peter. "Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems". Eurasia Journal of Mathematics, Science and Technology Education 2023 19 no. 11 (2023): em2346. https://doi.org/10.29333/ejmste/13658

Harvard

Mkhatshwa, T. P. (2023). Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems. Eurasia Journal of Mathematics, Science and Technology Education, 19(11), em2346. https://doi.org/10.29333/ejmste/13658

MLA

Mkhatshwa, Thembinkosi Peter "Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems". Eurasia Journal of Mathematics, Science and Technology Education, vol. 19, no. 11, 2023, em2346. https://doi.org/10.29333/ejmste/13658