Abstract
Researchers have been interested in students’ transition to calculus since the early 1900s. One line of inquiry highlights students’ understandings of high school mathematics as impeding or supporting their successful transition to university mathematics. This paper addresses an underlying question in this line of inquiry: does school mathematics provide opportunities for students to develop productive meanings for calculus? This article reports on U.S. calculus students’ responses to items that assessed students’ variational reasoning, meanings for average rate of change, and representational use of notation—ideas ostensibly addressed in school mathematics. To make sense of students’ difficulty on these items we sought to understand the opportunities students had to reason with these ideas prior to calculus. We use two data sources to understand the likelihood that students have opportunities to construct productive meanings for function notation, variation, and average rate of change in their secondary mathematics education: meanings for these ideas supported by precalculus textbooks and meanings secondary teachers demonstrated. Our analysis revealed a disconnect between meanings productive for learning calculus and the meanings conveyed by textbooks and held by U.S. high school teachers. We include a comparison of meanings held by U.S. and Korean teachers to highlight that these meanings are culturally embedded in the U.S. educational system.
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Notes
The C1CI was developed within Project DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus, NSF Grant DUE-1625678, http://patthompson.net/ThoompsonCalc.) This instrument consists of 43 multiple choice items designed to reveal students’ meanings for function, function notation, rate of change, accumulation, and the fundamental theorem of calculus.
Students were volunteers and recruited from students currently enrolled in calculus 1 or calculus 2 at a large U.S. public university. There were two iterations of this instrument. Questions that were on both versions were seen by 356 students while questions that were modified between versions were seen by 224 students in the second iteration.
Thompson (2016) elaborates on instrument development and validation.
Houghton Mifflin Harcourt, Pearson, and McGraw-Hill account for a combined 83% of secondary math market share Fulkerson, W. O., Campbell, K., & Hudson, S. B. (2013). 2012 National survey of science and mathematics education: Compendium of tables.. These three publishers have seven programs available in the secondary market: Saxon Math, AGA, Integrated Math, Envision, CME Project, Illustrative Math, and Glencoe. Out of these seven series only one, Glencoe Math by McGraw-Hill, offers a precalculus textbook. All other series offer a three course sequence: Algebra I, Geometry, and Algebra II.
This item was not scored for Korean teachers because of a mistranslation from English to Korean.
References
Ärlebäck, J. B., Doerr, H. M., & O’Neil, A. H. (2013). A modeling perspective on interpreting rates of change in context. Mathematical Thinking and Learning, 15(4), 314–336.
Artigue, M. (1992). Cognitive difficulties and teaching practices. In G. Harel & E. Dubinsky (Eds.), The concept of function: aspects of epistemology and pedagogy, MAA Notes (Vol. 25, pp. 109–132). Washington: Mathematical Association of America.
Byerley, C. (2019). Calculus students’ fraction and measure schemes and implications for teaching rate of change functions conceptually. Journal of Mathematical Behavior. https://doi.org/10.1016/j.jmathb.2019.03.001.
Carter, J. A., Cuevas, G. J., Day, R., & Malloy, C. (2011). Glencoe Precalculus. Bothell: McGraw-Hill.
Cassidy, C., & Trew, K. (2001). Assessing identity change: A Longitudinal study of the transition from school to college. Group Processes & Intergroup Relations, 4(1), 49–60.
Castillo-Garsow, C. C. (2010). Teaching the Verhulst Model: A teaching experiment in covariational reasoning and exponential growth. Unpublished Ph.D. dissertation. School of Mathematical and Statistical Sciences, Arizona State University.
Castillo-Garsow, C. C. (2012). Continuous quantitative reasoning. In R. Mayes, L. Bonillia, L. Hatfield, & S. Belbase (Eds.), Quantitative reasoning: Current state of understanding, WISDOMe Monographs (Vol. 2, pp. 55–73). Laramie: University of Wyoming.
OpenStax College. (2017). Precalculus: OpenStax. https://openstax.org/details/books/precalculus.
Connally, E., Hughes-Hallet, D., & Gleason, A. M. (2019). Functions Modeling Change: A Preparation for Calculus (6th ed.). Hoboken: Wiley.
Copur-Gencturk, T. (2015). The effects of changes in mathematical knowledge on teaching: A longitudinal study of teachers’ knowledge and instruction. Journal for Research in Mathematics Education, 46(3), 280–330.
Dorko, A., & Weber, E. (2013). Students’ concept images of average rate of change. In M. Martinez & A.C. Superfne (Eds.), Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 386–393). Chicago, IL: University of Illinois at Chicago.
Fulkerson, W. O., Campbell, K., & Hudson, S. B. (2013). 2012 National survey of science and mathematics education: Compendium of tables. Chapel Hill: Horizon Research Inc.
Hoyles, C., Newman, K., & Noss, R. (2001). Changing patterns of transition from school to university mathematics. International Journal of Mathematical Education in Science and Technology, 32(6), 829–845.
Kilpatrick, J. (2019). A double discontinuity and a triple approach: Felix Klein’s perspective on mathematics teacher education. In H. G. Weigand, W. McCallum, M. Menghini, M. Neubrand, & G. Schubring (Eds.), The Legacy of Felix Klein (pp. 215–225). New York: Springer.
Klein, F. (1932). Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis (E. R. Hedrick & C. A. Noble, Trans.). New York: Macmillan.
Lobato, J., & Thanheiser, E. (2001). Developing understanding of ratio as measure as a foundation for slope. In B. Litwiller (Ed.), Making sense of fractions, ratios, and proportions: 2002 Yearbook (pp. 162–175). Reston: National Council of Teachers of Mathematics.
Mkhastwha, T. P. (2020). Calculus students’ quantitative reasoning in the context of solving related rates of change problems. Mathematical Thinking and Learning, 22(2), 139–161.
Nagle, C., & Moore-Russo, D. (2013). The concept of slope: Comparing teachers’ concept images and instructional content. Investigations in Mathematics Learning, 6(2), 1–18.
Parker, J. D. A., Summerfeldt, L. J., Hogan, M. J., & Majeski, S. A. (2004). Emotional intelligence and academic success: Examining the transition from high school to university. Personality and Individual Differences, 36(1), 163–172.
Stewart, S., & Reeder, S. (2017). Algebra underperformances at college level: What are the consequences? In S. Stewart (Ed.), And the Rest is Just Algebra (pp. 3–18). New York: Springer.
Sullivan, M., & Sullivan, M. I. (2013). Precalculus: Enhanced with Graphing Utilities (6th ed.). LOCATION: Pearson.
Teuscher, D., Reys, R. E., Evitts, T. A., & Heinz, K. (2010). slope, rate of change, and steepness: Do students understand these concepts? The Mathematics Teacher, 103(7), 519–524.
Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179–234). Albany: SUNY Press.
Thompson, P. W. (2013). In the absence of meaning. In K. Leaham (Ed.), Vital directions for research in mathematics education (pp. 57–93). New York: Springer.
Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 435–461). New York: Taylor & Francis.
Thompson, P. W., & Milner, F. A. (2019). Teachers’ meanings for function and function notation in South Korea and the United States. In H. G. Weigand, W. McCallum, M. Menghini, M. Neubrand, & G. Schubring (Eds.), The legacy of Felix Klein (pp. 55–66). New York: Springer.
Zaslavsky, O., Sela, H., & Leron, U. (2002). Being sloppy about slope: The effect of changing the scale. Educational Studies in Mathematics, 49, 119–140.
Acknowledgements
Research reported in this article was supported by NSF Grants No. MSP-1050595 and DUE-1625678, and IES Grant No. R305A160300. Any recommendations or conclusions stated here are the authors’ and do not necessarily reflect official positions of the NSF or IES.
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Frank, K., Thompson, P.W. School students’ preparation for calculus in the United States. ZDM Mathematics Education 53, 549–562 (2021). https://doi.org/10.1007/s11858-021-01231-8
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DOI: https://doi.org/10.1007/s11858-021-01231-8