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What algebra do Calculus students need to know?


What algebra do Calculus students need to know?

Giovanniello, Sabrina (2017) What algebra do Calculus students need to know? Masters thesis, Concordia University.

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Students taking a Calculus course for the first time at Concordia University are mature students returning to school after an extended period of time away from formal education, or students lacking the prerequisites to enter into a science, technology, engineering, or mathematics (STEM) related field. Thus, an introductory Calculus course is the gateway for many STEM programs, inhibiting students’ academic progression if not passed. Calculus tends to be construed as a very difficult subject. This impression may be due to the fact that this course is taught in a condensed form, with limited class time, new knowledge (concept, type of problem, technique or method) introduced every week, and little practice time. Calculus requires higher order thinking in mathematics, compared to what students have previously encountered, as well as many algebraic techniques. As will be shown in this thesis, algebra plays an important role in solving problems that usually make up the final examination in this course. Through detailed theoretical analysis of problems in one typical final examination, and solutions produced by 63 students, we have identified the prerequisite algebraic knowledge for the course and the specific difficulties, misconceptions and false rules experienced and developed by students lacking this knowledge. We have also shown how the results of our analyses can be used in the construction of a “placement test” for the course – an instrument that could serve the goal of lessening the failure rate in the course, and attrition in STEM programs, by avoiding having underprepared students.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Giovanniello, Sabrina
Institution:Concordia University
Degree Name:M.T.M.
Program:Teaching of Mathematics
Date:July 2017
Thesis Supervisor(s):Sierpinska, Anna
ID Code:982929
Deposited On:16 Nov 2017 17:39
Last Modified:18 Jan 2018 17:56


Barbé, Q., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher's practice: The case of limits of functions in Spanish highschools. Educational Studies in Mathematics, 59, 235-268.
Booth, J. L., & Koedinger, K. R. (2008). Key misconceptions in algebraic problem solving. In B.C. Love, K. McRae, & V.M. Sloutsky (Eds), Proceedings of the 30th Annual Cognitive Science Society (pp. 571-576). Austin, TX: Cognitive Science Society.
Booth, J. L., Barbieri, C., Eyer, F., & Paré-Blagoev, J. (2014). Persistent and pernicious errors in algebraic problem solving. Journal of Problem Solving, 7, 10-21.
Bosch, M., & Gascón, J. (2014). Chapter 5: Introduction to the Anthropological Theory of the Didactic (ATD). In A. Bikner-Ahsbahs, & S. Prediger, Networking of Theories as a Research Practice in Mathematics Education, Advances in Mathematics Education (pp. 67-83). Springer Cham Heidelberg New York Dordrecht London: Springer International Publishing Switzerland. doi:10.1007/978-3-319-05389-9
Byers, V., & Erlwanger, S. (1984). Content and form in mathematics. Educational Studies in Mathematics, 15, 259-275.
Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In J. Kaput, A. H. Schoenfeld, & E. Dubinsky, Research in Collegiate Mathematics Education (Vol. 3, pp. 114-162). Washington, DC: Mathematical Association of America.
Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment: A tool for assessing students' reasoning abilities and understandings. Cognition And Instruction, 28, 113-145.
Chartrand, G., Polimeni, A. D., & Zhang, P. (2013). Mathematical proofs. A transition to advanced mathematics. Third edition. Boston: Pearson.
Chevallard, Y. (1999). L'analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathematiques, 19(2), 221-266.
Drijvers, P. (2011). Secondary Algebra Education: Revisiting Topics and Themes and Exploring Unknown. Rotterdam/Boston/Taipei: Sense Publishers.
Gleason, A., & Hughes-Hallet, D. (1998). Calculus Single Variable (2nd ed.). New York: John Wiley & Sons.
Hagman, J., Johnson, E., & Fosdick, B. (2017). Factors contributing to students and instructors experiencing a lack of time in college calculus. International Journal of STEM Education, 4, 1-15.
Hardy, N. (2009). Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses. Educational Studies in Mathematics, 72, 341-358.
Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students' understanding of core algebraic concepts: Equivalence & Variable. Zentralblatt für Didaktik der Mathematik, 37, 68-76. doi:doi:10.1007/BF02655899
Krussel, L. (1998). Teaching the language of mathematics. The Mathematics Teacher, 91, 436-441.
Küchemann, D. (1981). Algebra. In K. Hart, Children's understanding of mathematics: 11-16 (pp. 102-119). London: John Murray.
Lave, J., & Wenger, E. (1991). Situated learning. Legitimate peripheral participation. Cambridge: Cambridge University Press.
MacGregor, S., & Stacey, K. (1997). Students' understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 33, 1-19.
Marsden, J. E., & Weinstein, A. J. (1985). Calculus I. New York: Springer-Verlag. Retrieved from http://resolver.caltech.edu/CaltechBOOK:1985.001
McNeil, N., Fyfe, E., Petersen, L., Dunwiddie, A., & Brletic-Shipley, H. (2011). Benefits of practicing 4=2+2: Nontraditional problem formats facilitate children's understanding of mathematical equivalence. Child Development, 82, 1620-1633.
Palmiter, J. R. (1991). Effects of computer algebra systems on concept and skill acquisition in calculus. Journal for Research in Mathematics Education, 22, 151-156.
Payne, S. J., & Squibb, H. R. (1990). Algebra mal-rules and cognitive accounts of error. Cognitive Science, 14, 445-481.
Peters, B. (1999). Institutional theory in political science. London, New York: Continuum.
Ratti, J., & McWaters, M. (2014). Precalculus Essentials. Pearson Education.
Selden, J., Mason, A., & Selden, A. (1989). Can average calculus students solve nonroutine problems? Journal of Mathematical Behavior, 8, 45-50.
Sfard, A. (1987). Two conceptions of mathematical notions: Operational and structural. Proceedings of the 11th Annual Conference of the International Group for the Psychology of Mathematics Education (pp. 162-169). Montreal, Canada: Université de Montréal.
Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky, & G. Harel, The concept of function. Elements of pedagogy and epistemology. (Vol. 25, pp. 25-58). Boston: Notes and Reports Series of the Mathematical Association of America. Retrieved from https://www.researchgate.net/publication/238287243_On_understanding_the_notion_of_function
Sierpinska, A., & Hardy, N. (2010). ‘Unterrichten wir noch Mathematik?’ In C. Böttinger, M. Nührenbörger, R. Schwartzkopf, E. Söbbeke & K. Bräuning (Eds.),. In Mathematik im Denken der Kinder (pp. 94-100). Seelze: Friedrich Verlag GmbH.
Sierpinska, A., & Osana, H. (2012). Analysis of tasks in pre-service elementary teacher education courses. Research in Mathematics Education, 14, 109-135.
Sierpinska, A., Bobos, G., & Knipping, C. (2008). Sources of students' frustration in pre-university level, prerequisite mathematics courses. Instructional Science, 36, 289-320.
Sierpinska, A., Bobos, G., & Pruncut, A. (2011). Teaching absolute value inequalities to mature students. Educational Studies in Mathematics, 78, 275-305.
Star, J. R. (2004, April). The development of flexible procedural knowledge in equation solving. American Educational Research Association, (pp. 1-27). San Diego.
Stewart, J. (2016). Single Variable Calculus: Early Transcendentals. Math 203 and Math 205 Concordia University Department of Mathematics and Statistics. Toronto: Nelson Education.
Strømholm, P. (1968). Fermat's method of maxima and minima and of tangents. Archive for History of Exact Sciences, 5(1), 47-69.
Sullivan, M. (2016). College Algebra: Third Custom Edition for Concordia University. New York: Pearson Education.
Tallman, M. A., Carlson, M. P., Bressoud, D. M., & Pearson, M. (2016). A characterization of Calculus I final exams in U.S. colleges and universities. International Journal of Research in Undergraduate Mathematics Education, 2, 105-133.
The College Board. (n.d.). College Algebra. Retrieved from CLEP College Board: https://clep.collegeboard.org/science-and-mathematics/college-algebra
Thomas, G. (2008). Thomas' Calculus. Early transcendentals. Eleventh edition. Boston: Pearson.
Thompson, P., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai, Compendium for research in mathematics education (pp. 421- 456). Reston, VA: NCTM.
Thompson, P., Byerley, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools, 30, 124-147.
Tziritas, M. (2011). APOS theory as a framework to study the conceptual stages of Related Rates problems. Montreal: Concordia University. Retrieved from http://spectrum.library.concordia.ca/view/creators/Tziritas=3AMathew=3A=3A.html
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