Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems
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Department of Mathematical and Physical Sciences, Miami University, Middletown, OH, USA
Online publication date: 2023-09-09
Publication date: 2023-11-01
EURASIA J. Math., Sci Tech. Ed 2023;19(11):em2346
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.
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.
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).
Begle, E. G. (1973). Some lessons learned by SMSG. Mathematics Teacher, 66(3), 207-214.
Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77-101.
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.
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.
Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. CBMS Issues in Mathematics Education, 7, 114-162.
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.
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.
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.
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.
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.
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.
Engelke, N. (2004). Related rates problems: Identifying conceptual barriers. In D. McDougall (Ed.), Proceedings of the 26th 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.
Freudenthal, H. (1993). Thoughts in teaching mechanics didactical phenomenology of the concept of force. Educational Studies in Mathematics, 25(1&2), 71-87.
Hare, A., & Phillippy, D. (2004). Building mathematical maturity in calculus: Teaching implicit differentiation through a review of functions. Mathematics Teacher, 98(1), 6.
Hausknecht, A. O., & Kowalczyk, R. E. (2008). Exploring calculus using innovative technology. In J. Foster (Ed.), Proceedings of the 19th 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://carnegieclassification....
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.
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 22nd 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 21st 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.
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.
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 16th 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.
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 19th 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.
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.
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.
White, P., & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus. Journal for Research in Mathematics Education, 27(1), 79-95.
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.
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.
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.