Understanding the definite integral with the help of Riemann sums
Abstract
Students encounter difficulties in understanding integral because the concept of integral requires the usage of theorems, formulas, daily life practices, and interdisciplinary approaches. From this point of view, in this study we examine the effects of a teaching process consisting of modelling activities on understanding the definite integral with the help of Riemann sums. The research is designed according to the case study based on a qualitative research method. Participants consist of 28 pre-service mathematics teachers who have limited understanding of integral although they have completed a Calculus course. The modelling activities were prepared in accordance with the emergent modelling approach. Data were collected through integral test and semi-structured interviews conducted before and after the teaching process. Before the teaching process consisting of modelling activities, pre-service mathematics teachers’ knowledge about the definite integral was included in the area under a curve, inverse of derivative and integral with known boundaries. In addition, participants did not refer to Riemann sums or cumulative sums. After the teaching process consisting of modelling activities, it was seen that almost all participants could explain the following equality (lim)┬(n→∞)〖∑_(k=1)^n▒f(c_k ) ∆x_k 〗=∫_a^b▒f(x)dx. At the end of the teaching process consisting of modelling activities, this process has enabled pre-service teachers to establish the relationship between Riemann sum and definite integral. Findings revealed that teaching process of the definite integral enhanced the understanding the definite integral.
Keywords
definite integral, Riemann sums, modelling activities, teaching process
Thanks
The manuscript was produced from the doctoral dissertation the first author completed under the supervision of the second author.
References
- Adams, R., & Essex, C. (2010). Calculus a complete course. Toronto, Ontario: Pearson.
- Artigue, M. (1991). Analysis, in Advanced Mathematical thinking, edited by D. Tall. Kluwer, Boston, pp. 167–198
- Bajracharya, R., & Thompson, J. (2014). Student understanding of the fundamental theorem of calculus at the mathematics-physics interface. Proceedings of the 17th special interest group of the Mathematical Association of America on research in undergraduate mathematics education. Denver (CO).
- Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479-495.
- Carlson, M. P., Smith, N., & Persson, J. (2003). Developing and Connecting Calculus Students' Notions of Rate-of Change and Accumulation: The Fundamental Theorem of Calculus. International Group for the Psychology of Mathematics Education, 2, 165-172.
- Chapell, K. K., & Kilpartrick, K. (2003). Effects of concept-based instruction on students’ Conceptual understanding and procedural knowledge of calculus. PRIMUS: problems, resources, and issues in mathematics undergraduate studies, 13(1), 17-37.
- Chhetri, K., & Oehrtman, M. (2015). The Equation Has Particles! How Calculus Students Construct Definite Integral Models. In Proceedings of the 18th Annual Conference on Research in Undergraduate Mathematics Education (pp. 19-25).
- Cohen, L., Manion, L., & Morrisson, K. (2000). Research methods in education (5th Ed.). London: Routlenge Falmer.
- Creswell, J. W. (2003). Research design: Qualitative, quantitative, and mixed methods approaches (2nd ed.). Thousand Oaks, CA: Sage
- Darvishzadeh, M., Shahvarani-Semnani, A., Alamolhodaei, H., & Behzadi, M. (2018). Analysis of student´s challenges and performances in solving integral´s problems. Journal for Educators, Teachers and Trainers, 9(1), 164 – 177.
