American Journal of Educational Research
ISSN (Print): 2327-6126 ISSN (Online): 2327-6150 Website: http://www.sciepub.com/journal/education Editor-in-chief: Ratko Pavlović
Open Access
Journal Browser
Go
American Journal of Educational Research. 2015, 3(4), 476-482
DOI: 10.12691/education-3-4-14
Open AccessArticle

Calculus Students’ Visual Thinking of Definite Integral

Chih Hsien Huang1,

1Department of Electrical Engineering, Ming Chi University of Technology, New Taipei City, Taiwan, ROC

Pub. Date: April 08, 2015

Cite this paper:
Chih Hsien Huang. Calculus Students’ Visual Thinking of Definite Integral. American Journal of Educational Research. 2015; 3(4):476-482. doi: 10.12691/education-3-4-14

Abstract

Visualization as both the product and the process of creation, interpretation and reflection upon pictures and images, is gaining increased visibility in mathematics and mathematics education. The use of diagrams to visualize definite integral concept, however, is problematic for many students and may actually hinder their problem-solving efforts. The purpose of this study, not only extends our understanding of students’ difficulties and strengths associated with visualization but also identifies types of visual image they utilized while solve integral problems. Through the detailed analyses of students' work and verbal protocols, the students with high visualization ability use of imagination images in high percentages along with algebraic representations and linking these two representations lead to the success of problem solving. The students with low visualization ability use of memory images. It is discovered that students can produce imagination images that play a significant role in a problem solving process. As such, a process of visualization allows an articulation between representations to produce another representation that could help students solve given problems.

Keywords:
calculus students definite integral representation visual thinking visualization

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  Arcavi, A. “The role of visual representations in the learning of mathematics.” Educational Studies in Mathematics, 52(3). 215-241. March. 2003.
 
[2]  Aspinwall, L., and Shaw, K. “Representations in calculus: Two contrasting cases.” Mathematics Teacher, 95. 434-440. September. 2002.
 
[3]  Aspinwall, L., Shaw, K. and Presmeg, N. “Uncontrollable mental imagery: Graphical connections between a function and its derivative.” Educational Studies in Mathematics, 33. 301-317. September. 1997.
 
[4]  Berry, J., and Nyman, M. “Promoting students’ graphical understanding of the calculus.” Journal of Mathematical Behavior, 22(4). 479-495. October. 2003.
 
[5]  Dreyfus, T. “On the status of visual reasoning in mathematics and mathematics education.” In Proceedings of the Fifteenth Annual Meeting of the International Group for the Psychology of Mathematics Education, Assisi, Italy, 33-48. 1991.
 
[6]  Duval, R. “A cognitive analysis of problems of comprehension in a learning of mathematics.” Educational Studies in Mathematics, 61(1). 103-131. February. 2006.
 
[7]  Eisenberg, T., and Dreyfus, T. Visualization in Teaching and Learning Mathematics, Mathematical Association of America, Washington D.C., 1991, 25-37.
 
[8]  Goldin, G. “Representational systems, learning, and problem solving in mathematics.” Journal of mathematical behavior, 17 (2). 137-165. February. 1998.
 
[9]  Goldin, G. Handbook of Research Design in Mathematics and Science Education, Erlbaum, Mahwah, New Jersey, 2000, 517-545.
 
[10]  Haciomeroglu, E. S., Aspinwall, L. and Presmeg, N.C. “Contrasting Cases of Calculus Students' Understanding of Derivative Graphs.” Mathematical Thinking and Learning, 12(2). 152-176. March. 2009.
 
[11]  Hiebert, J., and Carpenter, T. P. Handbook of research on mathematics teaching and learning, National Council of Teachers of Mathematics, Reston, VA, 1992, 65-97.
 
[12]  Hitt, F. “Working Group on Representations and Mathematics Visualization. In Proceedings of the Twentieth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, North Carolina. USA. Columbus, 1-7. November. 1998.
 
[13]  Hughes-Hallett, D., McCallum, W. G., Gleason, A. M., Pasquale, A., Flath, D. E., Quinney, D., Lock, P. F., Raskind, W., Gordon, S. P., Rhea, K., Lomen, D. O., Tecosky-Feldman, J., Lovelock, D., Thrash, J. B., Osgood, B. G., and Tucker, T. W. (2002). Calculus: Single Variable. MA: John Wiley & Sons, Inc, Danvers, 2002.
 
[14]  Kaput, J. J. Problems of representation in teaching and learning mathematics, Erlbaum, Hillsdale, New Jersey, 1987, 19-26.
 
[15]  Mahir, N. “Conceptual and procedural performance of undergraduate students in integration.” International Journal of Mathematical Education in Science and Technology, 40(2). 201–211. January. 2009.
 
[16]  Nemirovsky, R., and Noble, T. “On Mathematical Visualisation and the Place Where We Live.” Educational Studies in Mathematics 33(2). 595-610. July. 1997.
 
[17]  Noss, R., Healy, L. and Hoyles, C. “The Construction of Mathematical Meanings: Connecting the Visual with the Symbolic.” Educational Studies in Mathematics, 33(2). 203-33. July. 1997.
 
[18]  Pólya, G. How to solve it, Princeton University Press, Princeton, New Jersey, 1945.
 
[19]  Presmeg, N.C. Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future. Sense Publishers, Rotterdam, 2006, 205-235.
 
[20]  Sealey, V. “Definite integrals, Riemann sums, and area under a curve: what is necessary and sufficient?” In Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mérida, México: Universidad Pedagógica Nacional, 46-53. November. 2006.
 
[21]  Stylianou, D.A., and Silver, E.A. “The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences.” Mathematical Thinking and Learning, 6(4). 353-387. November. 2004.
 
[22]  von Glasersfeld, E. Problems of representation in the teaching and learning of mathematics, Lawrence Erlbaum Associates, Inc, New Jersey, 1987, 215-225.
 
[23]  Yoon, C., Thomas, M. O. J., and Dreyfus, T. “Gestures and virtual space.” In Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Thessaloniki, Greece, 409-416. 2009.
 
[24]  Zazkis, R., Dubinsky, E. and Dautermann, J. “Coordinating visual and analytic strategies: a study of students’ understanding.” Journal for Research in Mathematics Education, 27(4). 435-437. July. 1996.
 
[25]  Zimmermann, W. Visualization in teaching and learning mathematics, Mathematical Association of America, Washington, D.C., 1991, 127-138.
 
[26]  Zimmermann, W., and Cunningham, S. Visualization in Teaching and Learning Mathematics, Mathematical Association of America, Washington, D.C., 1991, 1-8.