# A STUDY ON UNDERSTANDING OF DEFINITE INTEGRAL AND RIEMANN SUM

• Oh, Hyeyoung (Department of Mathematics Education Incheon National University)
• Accepted : 2019.08.01
• Published : 2019.09.30

#### Abstract

Conceptual and procedural knowledge of integration is necessary not only in calculus but also in real analysis, complex analysis, and differential geometry. However, students show not only focused understanding of procedural knowledge but also limited understanding on conceptual knowledge of integration. So they are good at computation but don't recognize link between several concepts. In particular, Riemann sum is helpful in solving applied problem, but students are poor at understanding structure of Riemann sum. In this study, we try to investigate understanding on conceptual and procedural knowledge of integration and to analyze errors. Conducting experimental class of Riemann sum, we investigate the understanding of Riemann sum structure and so present the implications about improvement of integration teaching.

#### Acknowledgement

Supported by : Incheon National University

#### References

1. M. Artigue, Analysis in Advanced Mathematical Thinking, D. Tall, ed., Kluwer, Boston, 167-198, 1991.
2. D. M. Bressoud, Historical reflections on Teaching the fundamental theorem of integral calculus, The American Mathematical Monthly 118 (2) (2011), 99-115. https://doi.org/10.4169/amer.math.monthly.118.02.099
3. Jeongsun Byun, Understanding on concept of definite integral through measuration by division, graduate school of education at Ehwa woman's University, 1992.
4. A. Cauchy, Resume des Logons donne"es 1'Ecolo royale Polyteehniquo sur le Galcul infinitesimal, publiees sous la direction scientifique de l'Academie des sciences et sous les auspices de M. le ministre de 1'Instruction publique, Oeuvres completes d'Augustin Cauchy, 1899.
5. K.K. Chapell and K. Killpatrick, Effects of concept-based instruction on Students' conceptual understanding and procedural knowledge of calculus, Primus 13 (1) (2003), 17-37. https://doi.org/10.1080/10511970308984043
6. Jeonghyun Choi, Characteristic of understanding on definite integral symbol and didactical strategy, Korean journal of mathematical history, 24 (3) (2011), 77-94.
7. K. Devlin, MATHEMATICS: The Science of Patterns, Scientific American Library, 1994.
8. J. Ferrini-Mundy and M. Guardard, Preparation of pitfall in the study of college calculus , J. Res. Math. Educ. 23 (1) (1992), 56-71. https://doi.org/10.2307/749164
9. J. Grabiner, The Origins of Cauchy's Rigorous Calculus, Dover Publications, 2005.
10. J. Hiebert and P. Lefevre, Conceptual and procedural knowledge in mathematics: An introductory analysis, in Conceptual and Procedural Knowledge: The Case of Mathematics, J. Hiebert, ed., Lawrence Erlbaum, Hilllsdale (NJ) (1986), 1-27.
11. S. Jones, Areas, anti-derivatives, and adding up pieces: Definite integrals in pure mathematics and applied science contexts, Journal of Mathematical Behavior 38 (2015), 9-28. https://doi.org/10.1016/j.jmathb.2015.01.001
12. Youn Joon Joung and Kyeong Hwa Lee, A study on the Relationship between indefinite integral and definite integral, Journal of Korea Society of Educational Studies in Mathematics, School Mathematics 11 (2) (2009), 301-316.
13. Seah Eng Kiat, Analysis of Students' Difficulties in Solving Integration Problem, The Mathematics Educator 9 (1) (2005), 39-59.
14. Ohsung Kwon, Inclination understanding on integration chapter, master thesis Korea University, 2011.
15. N. Mahir, Conceptual and procedural performance of undergraduate students in integration.,International Journal of Mathematical Education in Science and Technology 40 (2) (2009), 201-211. https://doi.org/10.1080/00207390802213591
16. F. A. Medvedev, Floer homology of Lagrangian foliation and noncommutative mirror symmetry, Preprint 98-08, Kyoto Univ, 1998.
17. A. Orton, Students' understanding of integration, Educational Studies in Mathematics 14 (1) (1983), 1-18. https://doi.org/10.1007/BF00704699
18. A.B. Oaks, The effects of the interaction of conception of mathematics and affective constructs on college students in remedial mathematics, Ph.D.diss., University of Rochester, 1987, Dissertation Abstracts International, 49, 54A.
19. S. Rasslan and D. Tall, Definitions and images for the definite integral concept, Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education. Norwich, UK, 2002.
20. Bomi Sin, High school students' understanding on concept of definite integral, Journal of Korea Society of Educational Studies in Mathematics, School Mathematics 11 (1) (2009), 93-100.
21. J. Stewart, Calculus, 6th ed, Belmont: Brooks Cole, 2008.
22. Joseph F. Wagner, Students' Obstacles to Using Riemann Sum Interpretations of the Definite Integral, Springer International Publishing, 2017.