Report

Calculus students’ understanding of area and volume in calculus- and non-calculus contexts f(x) Abstract Researchers have documented difficulties that elementary school students have in understanding area and volume. We know very little about older students’ understanding of these concepts. This study develops descriptions of calculus students’ understanding of area and volume concepts in non-calculus contexts. Knowing what calculus students understand about area and volume will help us understand student difficulties with calculus applications such as integrals and Volumes of Revolution. Research questions: What do calculus students understand about area in non-calc contexts? What do calculus students understand about volume in non-calc contexts? x=a Calculus students… Allison Dorko Masters of Science in Teaching University of Maine •Not all students excel at calculus. •Calculus courses have low enrollment rates; low retention rates; and some students demonstrate a lessthan-ideal understanding of calculus topics (Steen 1987), sometimes getting by with just memorization. •There are a few core ideas of integration that some students do not understand, among them (1) taking a limit of Riemann sums to find area under a curve and (2) volume of revolution (Orton 1993). •Elementary school students struggle with computations, conceptual understanding, and units associated with various spatial measures (Lehrer 2003). Tasks & Subjects Results •Multivariable calculus students do calculations correctly and have expert-level conceptions of area and volume What is the volume of the object? •198 introductory calc students •43 multivariate calc students •Written tasks: •Computation •Short answer b Area under the curve = a f (x)dx •Introductory calculus students are relatively adept at area calculations but struggle with both volume computations and the conceptual aspects •Consistent with findings from about elementary school students’ understanding of area and volume x= b References Battista, M. T.(2007).The development of geometric and spatial thinking. In Lester, F. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 843-908). NCTM. Reston, VA: National Council of Teachers of Mathematics. Battista, M. and Clements, D. (1998). Students' understanding of three-dimensional cube arrays: findings from a research and curriculum development project. . In Lester, F. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 843-908). NCTM. Reston, VA: National Council of Teachers of Mathematics. Additional Selected Tasks Compute the area of the rectangle. 12 cm Dick, B. (2005) Grounded theory: a thumbnail sketch. Retrieved from http://www.uq.net.au/action_research/arp/grounded.html Ferrini-Mundy, J., & Graham, K. (1994). Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals. In J. J. Kaput & E. Dubinsky (Eds.), Research Issues in Undergraduate Mathematics Learning: Preliminary analysis and results (Vol. 33, pp. 29-45). Washington, DC: The Mathematical Association of America. Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. E. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. Ferrini-Mundy, J., & Graham, K. (1994). Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals. In J. J. Kaput & E. Dubinsky (Eds.), Research Issues in Undergraduate Mathematics Learning: Preliminary Analysis and Results (Vol. 33, pp. 29-45). Washington, DC: The Mathematical Association of America 4 cm Compute the area of the circle. 5 in Monk, S. (1987). Students' understanding of function in calculus courses. Unpublished manuscript. [This is very similar to what was published as “Monk, S. (1994). Students' understanding of functions in calculus courses. Humanistic Mathematics Network journal, 27, 21-27.” Orton, A. (1983). Students’ understanding of integration. Educational Studies in Mathematics. 14, 1-18. Siegler, R. (2003). Implications of Cognitive Science Research for Mathematics Education. In Kilpatrick, J., Martin, G., and Schifter, D., Editors, A Research Companion to Principles and Standards. What types of things can we use area to measure? Standards for School Mathematics, pages 289-303. Reston, Va.: National Council of Teachers of Mathematics. What types of things can we use volume to measure? Strauss, A. & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage. What units are areas measured in? White, P. and Mitchelmore, M. (1996). Conceptual Knowledge in Introductory Calculus. Journal for Research in Mathematics Education Vol. 27, No. I, 79-95 Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld & J. Kaput (Eds.), CBMS Issues in Mathematics: Research in Collegiate Mathematics Education (Vol. IV(8), pp. 103–127). What units are volumes measured in? What does area mean? What does volume mean?