Geometric error analysis in applied calculus problem solving

Ahmed Ibrahim Usman 1 *
More Detail
1 Department of General Studies (Mathematics), Jubail University College, Jubail Industrial City, Kingdom of Saudi Arabia
* Corresponding Author
EUR J SCI MATH ED, Volume 5, Issue 2, pp. 119-133. https://doi.org/10.30935/scimath/9502
OPEN ACCESS   1829 Views   1075 Downloads
Download Full Text (PDF)

ABSTRACT

The paper investigates geometric errors students made as they tried to use their basic geometric knowledge in the solution of the Applied Calculus Optimization Problem (ACOP). Inaccuracies related to the drawing of geometric diagrams (visualization skills) and those associated with the application of basic differentiation concepts into ACOP solution were reported. A test instrument was used to collect quantitative data, while qualitative data were generated using follow- up interviews (stimulated recall). The targeted samples were freshmen students who registered for Calculus I in the department of Mathematics at a University in south eastern region of the United States of America, USA. The study indicated that students had achieved a very low success rate on the ACOP solution process, immediately after receiving/completing instruction on the optimization in their calculus I class. In general, they failed to integrate basic geometric competences required in the ACOP solution. Qualitative evidence from students’ test performance indicated that failure to visualize geometric diagrams from word problems tended to preclude them getting the required formula. The overall finding of the research was that students face structural and procedural setbacks that ultimately led to a worsening of the ACOP solution process.
Keywords: applied calculus optimization problem, visualization, geometric, stimulated recall, optimization

CITATION

Usman, A. I. (2017). Geometric error analysis in applied calculus problem solving. European Journal of Science and Mathematics Education, 5(2), 119-133. https://doi.org/10.30935/scimath/9502

REFERENCES

  • Arcavi, A. (2003). The role of visual representations in the learning of mathematics.Educational Studies in Mathematics, 52, 215-241.
  • Barret, J. E., & Clements, D. H. (2003). Qualifying length: fourth-grade children’s developing abstractions for measures of linear quantity. Cognition and Instruction, 21(4), 475 – 520.
  • Battista, M. (2007).The development of geometric and spatial thinking.In F.K. Lester(Ed), Second handbook of research on mathematics teaching and learning (843 – 908). Greenwich, CT: Information Age Publishing.
  • Battista, M.T. (2001a). A research-based perspective on teaching school geometry. In J. Brophy (Ed), Advances in research on teaching: Subject-specific instructional methods and activities (pp. 145 – 185). New York: JAI Press.
  • Bishop, A. J. (1989). A Review of research on visualization in mathematics education.Focus on Learning Problems in Mathematics, 11(1&2), 7-16.
  • Bremigan, E. G. (2005).An analysis of Diagram Modification and Construction in Students’ Solution to Applied Calculus Problem.Journal for research in Mathematics Education, 36 (3), 248 – 277.
  • Brijlall, D., &Ndlovu, Z. (2013). High school learners’ mental construction during solving optimization problem in Calculus: a South African case study. South African Journal of Education; 33(2).
  • Burton, M. B. (1989). The Effect of Prior Calculus Experience on “Introductory” College Calculus. The American Mathematical Monthly, 96(4), 350 – 354.
  • Chappell, M. F., & Thompson, D. R. (1999).Perimeter or Area? Which measure is it? Mathematic Teaching in the Middle School, 5(1), 20 – 23.
  • Clements, D. H.., Battista, M. T., Sarama, J., Swaminathan, S., &McMillens, S. (1997). Students’ development of length concepts in a Logo-based unit on geometric paths.Journal for Research in Mathematics Education, 28(1), 70 – 95.
  • Davis, R. B. (1986). Calculus at University High School. In Douglas, R.G. (ed). Toward a Lean and Lively Calculus.MAA Notes Number 6, Mathematical Association of America.
  • Donaldson, M. (1963).A study of children’s thinking (pp.183-185). Tavistock publications: London.
  • Dreyfus, T. (1991).On the status of visual reasoning in mathematics and mathematics education.In F. Furinghetti (Ed.), Proceedings of the 15th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 33-48).Assisi, Italy: PME.
  • Duval, R. (1995). Geometrical pictures: kinds of representation and specific processings. In R. Sutherland & J. Mason (Eds), Exploring Mental Imagery with Computers in Mathematics Educations (pp. 142 – 157). Berlin: Springer-Verlag.
  • Duval, R. (2006).A cognitive analysis of problems of comprehension in the learning of mathematics.Educational Studies in Mathematics, 61, 103-131.
  • Ferrini-Mundy, J., & Graham, K. G. (1991). An Overview of the Calculus Curriculum Reform Effort: Issues for Learning, Teaching, and Curriculum Development. The American Mathematical Monthly, 98(7), 627 – 635.
  • Gutrierrez, A. (1997). Visualization in 3 – Dimensional Geometry: in search of a Framework. Retrievd from: http://www.uv.es/angel.gutierrez/archivos1/textospdf/Gut96c.pdf, on 16/1/14.
  • Hershkowitz, R., et al (1990). Psychological Aspects of Learning Geometry (Nesher, P., & Kilpatrick, J, Eds).Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education. (p. 69 – 95). Cambridge: Cambridge University Press.
  • Hohenwarter, M. & Jones, K., (2007). Ways of Linking Geometry and Algebra: The Case of Geogebra. In D. Kuchemann (Ed), Proceedings of the British Society for Research into Learning Mathematics, 27(3), 126 – 131.
  • Hurley, J. F., Koehn, U., &Ganter, S. L. (1999). Effects of Calculus reform: Local and National. The American Mathematical Monthly, 109(9), 800 – 811.
  • Jones, K. (2002). Issues in the Teaching and Learning of Geometry. In Linda Haggarty, (Ed), Aspects of Teaching Secondary Mathematics: perspectives on practice. London: RoutledgeFalmer. (p. 121 – 139).
  • Jones, K. (2000). Teacher Knowledge and Professional Development in Geometry. Proceedings of the British Society for Research into Learning Mathematics, 20(3), 109 – 114.
  • Jones, K. & Mooney, C. (2003).Making space for geometry in primary mathematics. In I. Thompson (Ed), Enhancing primary mathematics teaching. London: open press, 3 – 15.
  • Koç, Yusuf, et al. (n.d) "An Investigation on Students’ degree of Acquisition Related to Van Hiele Level of Geometric Reasoning: A Case of 6-8 Th Graders in Turkey." http://cerme8.metu.edu.tr/wgpapers/WG4/WG4_Koc.pdf
  • Martin, G., &Structchens, M. (2000).Geometry and measurement.In E. A. Silver & P. A. Kenny (Eds), Results from the seventh mathematics assessment of the National Assessment of Educational Progress (p. 193 – 234). Reston, VA: NCTM.
  • Marchett, P., et al. (2005). Comparing Perimeters and Areas Childrens’ Pre-conceptions and spontaneous procedures.CERME 4. 766 – 775.
  • Matta, C, Feinberg G, Cardetti, F. (2012).Exploring Learning difficulties in multivariable Calculus. Retrieved from: http://mathreu.uconn.edu/Math%20ED%20REU%20Poster.pdf on 29/09/2015
  • Muzangwa, J., &Chifamba, P. (2012).Analysis of Errors and Misconceptions in the Learning of Calculus by undergraduate students.ActaDidacticaNapocensia, Vol. 5 (2).
  • National Council of Teachers of Mathematics, (2000).Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
  • Nunes, T., Light, P., & Mason, J. (1993). Tools for thought: the measurement of length and area. Learning and Instruction, 3, 39 – 54.
  • Panaoura, G., &Gagatsis, A. (2009).The Geometrical Reasoning of Primary and Secondary School students.Proceedings of CERME 6,Working Group 5. January 28th – February 1st, 2009. Lyon France © INRP 2010.
  • Pesek, D. D., &Kirshner, D. (2000).Interference of instrumental instruction in subsequent relational learning.Journal for Research in Mathematics Education, 31(5), 524 – 540.
  • Pittalis, M., & Christou, C. (2010).Types of reasoning in 3D geometry thinking and their relation with spatial ability.Educational Studies in Mathematics, 75(2), 191-212.
  • Presmeg, N. C., & Balderas-Canas, P. E. (n.d.). Graduate Students’ Visualizations in Two Rate of Change Problems. Retrieved from: http://www.academia.edu/6665583/Graduate_Students_Visualizations_in_two_Rate_of_Change_Problems.
  • Stylianou, D. A., & Pitta-Pantazi, D. (2002). Visualization and High Achievement in Mathematics: A Critical Look at Successful Visualization Strategies. In Fernando Hitt (ed). Representations and Mathematics Visualization, North American Chapter of the International Group for the Psychology of Mathematics Education, Working group (1998 – 2002).
  • vanHiele, P.M. (1986). Structure and insight: A theory of mathematics education. Orlando, Fla.: Academic Press
  • Vinner, S. (1983).Concept definition, concept image and the notion of function.International Journal of mathematical Education in Science and Technology, 14, 239-305.
  • Woodward, E., & Byrd, F.)1983). Area: Included topic, neglected concept. School Science and Mathematics, 83(4), 343 -347.
  • Yasin, S., &Enver, T. (2007). Students’ Difficulties with Application of Definite Integration. EducaţiaMatematică Vol. 3, Nr. 1-2, 15-27
  • Young, G. S. (1986). Present Problems and Future Prospects. In Steen, L.A. (ed). Calculus for a New Century. Mathematical Association of America Notes Number 8, 172 – 175.
  • Zimmermann, W., & Cunningham, S., (1991).Visualization in Teaching and Learning Mathematics. MAA Notes No. 19. Washington D.C.: Mathematical Association of America.