Abstract
This study assessed the ability of university students enrolled in an introductory calculus course to solve related-rates problems set in geometric contexts. Students completed a problem-solving test and a test of performance on the individual steps involved in solving such problems. Each step was characterised as primarily relying on procedural knowledge or conceptual understanding. Results indicated that overall performance on the geometric related-rates problems was poor. The poorest performance was on steps linked to conceptual understanding, specifically steps involving the translation of prose to geometric and symbolic representations. Overall performance was most strongly related to performance on the procedural steps.
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Balomenos, R. H., Ferrini-Mundy, J., & Dick, T. (1987). Geometry for calculus readiness. In M. M. Lindquist (Ed.),Learning and teaching geometry, K-12 (pp. 195–209). Reston, VA: National Council of Teachers of Mathematics.
Berkey, D. D. (1988).Calculus (2nd ed.). New York, NY: Holt, Rinehart and Winston.
Bookman, J. & Friedman, C. P. (1994). A comparison of the problem solving performance of students in lab based and traditional courses.Conference Board of the Mathematical Sciences Issues in Mathematics Education, 4, 101–116.
Cipra, B. A. (1988). Calculus: Crisis looms in mathematics’ future.Science, 239, 1491–1492.
Cooney, T. J., Davis, E. J., & Henderson, K. B. (1975).Dynamics of teaching secondary school mathematics. Boston: Houghton Mifflin Company.
Culotta, E. (1992). The calculus of education reform.Science, 225, 1060–1062.
Davis, B., Porta, H., & Uhl, J. (1994).Calculus and Mathematica. Reading, MA: Addison-Wesley.
Dick, T. P. & Patton, C. M. (1992).Calculus (Vol. 1). Boston: PWS-KENT.
Douglas, R. G. (1986).Toward a lean and lively calculus. Washington, DC: Mathematical Association of America.
Dreyfus, T. (1991). Advanced mathematical thinking. In P. Nesher & J. Kilpatrick (Eds.),Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education (pp. 113–134). Cambridge, England: Cambridge University Press.
Feroe, J. & Steinhoffi, C. (1991).Single-variable calculus with discrete mathematics. San Diego, CA: Harcourt Brace Jovanovich.
Ferrini-Mundy, J. & Gaudard, M. (1992). Secondary school calculus: Preparation or pitfall in the study of college calculus?Journal for Research in Mathematics Education, 23, 56–71.
Ferrini-Mundy, J. & Graham, K. G. (1991). An overview of the calculus curriculum reform effort: Issues for learning, teaching, and curriculum development.American Mathematical Monthly, 98, 627–635.
Harel, G. & Kaput, J. (1991). The role of conceptual entities and their symbols in building advanced mathematical concepts. In D. Tall (Ed.),Advanced mathematical thinking (pp. 82–94). Dordrecht, Netherlands: Kluwer.
Hiebert, J. & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.),Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum.
Hughes-Hallett, D., Gleason, A. M., Flath, D. E., Gordon, S. P., Lomen, D.O., Lovelock, D., McCallum, W. G., Osgood, B. G., Pasquale, A., Tecosky-Feldman, J., Thrash, J. B., Thrash, K. R., & Tucker, T. W. (1994).Calculus. New York, NY: Wiley.
Lial, M. L., Miller, C. D. & Greenwell, R. N. (1993).Calculus with applications (5th ed.). New York, NY: Harper Collins.
Martin, T. S. (1997).Calculus students’ abilities to solve geometric related rate problems and their understanding of related geometric growth factors. Unpublished doctoral dissertation, Boston University.
Meel, D. E. (1998). Honors students’ calculus understandings: Comparing Calculus á Mathematica and traditional calculus students.Conference Board of the Mathematical Sciences Issues in Mathematics Education, 7, 163–215.
National Assessment of Educational Progress. (1988).Mathematics objectives: 1990 Assessment. Princeton, NJ: Educational Testing Service.
Orton, A. (1983a). Students’ understanding of integration.Educational Studies in Mathematics, 14, 1–18.
Orton, A. (1983b). Students’ understanding of differentiation.Educational Studies in Mathematics, 14, 235–250.
Park, K. & Travers, K. J. (1996). A comparative study of a computer-based and a standard college first-year calculus course.Conference Board of the Mathematical Sciences Issues in Mathematics Education, 6, 155–176.
Peterson, I. (1986). The troubled state of calculus: A push to revitalise college calculus teaching has begun.Science News, 129, 220–221.
Schneider, M. (1992). A propos de l’apprentissage du taux de variation instantane [On the learning of instantaneous rate of change].Educational Studies in Mathematics, 23, 317350.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.Educational Studies in Mathematics, 22, 136.
Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. H. Schoenfeld (Ed.),Mathematical thinking and problem solving (pp. 53–70). Hillsdale, NJ: Lawrence Erlbaum.
Silver, E. A. & Marshall, S. P. (1990). Mathematical and scientific problem solving: Findings, issues and instructional implications. In B. F. Jones and L. Idol (Eds.),Dimensions of thinking and cognitive instruction (Vol. I, pp. 265–290). Hillsdale, NJ: Lawrence Erlbaum.
Smith, D. A. & Moore, L. C. (1996).Calculus: Modeling and application. Lexington, MA: Heath.
Steen, L. A. (1986). Twenty questions for calculus reformers. In R. G. Douglas (Ed.),Toward a lean and lively calculus. MAA Notes No.6 (pp. 157-165). Washington, DC: Mathematical Association of America.
Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus.Educational Studies in Mathematics, 26, 229–274.
Tucker, A. C. & Leitzel, J. R. C. (Eds.). (1995).Assessing calculus reform efforts: A report to the community (MAA Report No.6). Washington, DC: Mathematical Association of America.
White, P. & Mitchelmore, M. (1996). Conceptual knowledge in introductory calculus.Journal for Research in Mathematics Education, 27, 79–95.
Wilson, R. (1997). A decade of teaching ‘reform calculus’ has been a disaster, critics charge: Mathematicians divide over a curricular movement that some say has cheated students.The Chronicle of Higher Education, 43 (22), A12-A13.
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Martin, T. Calculus students’ ability to solve geometric related-rates problems. Math Ed Res J 12, 74–91 (2000). https://doi.org/10.1007/BF03217077
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DOI: https://doi.org/10.1007/BF03217077