Learning opportunities offered by a classical calculus textbook
DOI:
https://doi.org/10.7146/nomad.v15i2.148241Abstract
In this paper we present results of an analysis of what the textbook used by the first year engineering students offers the students, when they take a basic calculus course. The aim of this analysis is to examine as an entirety what students are offered by the book to learn about the concept of derivative. The results show that the presentation of the concept is formal and depends on students’ previous knowledge. The treatment of the concept emphasises procedural knowledge. It is not easy for students using the book to make connections between conceptual and procedural knowledge of the concept of derivative.
References
Adams, R. A. (2006 ). Calculus. A complete course. Toronto: Pearson.
Artigue, M. (2001). What can we learn from educational research at theuniversity level? In D. Holton (Ed.), The teaching and learning of mathematics at university level (pp. 207-220). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47231-7_21
Ausubel, D. (2000). The acquisition and retention of knowledge: a cognitive view. Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-015-9454-7
Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-165). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47203-1_10
Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced mathematical thinking (pp. 25-41). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47203-1_2
Fischbein, E. (1994). The interaction between the formal, the algorithmic, and the intuitive components in a mathematical activity. In R. Biehler, R. W. Scholtz, R. Strässer & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 231-245). Dordrecht: Kluwer Academic Publishers.
Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning. Educational Studies in Mathematics, 38, 11-50. https://doi.org/10.1023/A:1003488222875
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 (pp. 1-28). London: Lawrence Erlbaum.
Hiebert, J. (2003). What research says about the NTCM standards? In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 5-23). Reston: National Council of Teachers of Mathematics.
Hähkiöniemi, M. (2006). The role of representations in learning the derivative (Doctoral thesis). University of Jyväskylä.
Johansson, M. (2005). The mathematics textbook: from artefact to instrument. Nordic Studies in Mathematics Education, 10 (3-4), 43-64.
Johansson, M. (2006). Teaching mathematics with textbooks. A classroom and curricular perspective (PhD thesis). Department of Mathematics, Luleå University of Technology.
Juter, K. (2006). Limits of functions. University students' concept development (PhD thesis). Department of Mathematics, Luleå University of Technology.
Juter, K. & Grevholm, B. (2007). Limits and infinity - a study of university students' performance. In C. Bergsten, B. Grevholm, H. Måsøval & F. Rønning (Eds.), Relating practice and research in mathematics education. Proceedings of Norma05, fourth Nordic conference on mathematics education, (pp. 337-348). Trondheim: Tapir Akademisk Forlag.
Kvalitetsreformen (2000). St. melding nr 27. Oslo: Det kongelige kirke-, utdannings-og forskningsdepartement.
Kummerer, B. (2001). Trying the impossible. In D. Holton (Ed.), The teaching and learning of mathematics at university level (pp. 321-334). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47231-7_29
Lithner, J. (2003). Students' mathematical reasoning in university textbook exercises. Educational Studies in Mathematics, 52, 29-55. https://doi.org/10.1023/A:1023683716659
Lithner, J. (2004). Mathematical reasoning in calculus textbook exercises. Journal of Mathematical Behaviour, 23, 405-427. https://doi.org/10.1016/j.jmathb.2004.09.003
Martin Jr, E. W (1969). Mathematics for decision making. Homewood: Richard D. Irwin.
Novak, J. D. & Gowin, D. B. (1984). Learning how to learn. Cambridge University Press. https://doi.org/10.1017/CBO9781139173469
Oldervoll, T., Orskaug, O. & Vaaje, A. (2000). Sinus. Oslo: Cappelen Forlag.
Orton, A. (1983). Students' understanding of the differentiation. Educational Studies in Mathematics, 14, 235-250. https://doi.org/10.1007/BF00410540
Pepin, B. & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: a way to understand teaching and learning cultures. Zentralblatt fur Didaktik der Mathematik, 33 (5), 158-175. https://doi.org/10.1007/BF02656616
Pettersson, K. (2008). Växelverkan mellan intuitiva idéer och formella resonemang. Nordic Studies in Mathematics Education, 13 (1), 29-50.
Raman, M. (2002). Coordinating informal and formal aspects of mathematics: students behaviour and textbooks messages. Journal of Mathematical Behaviour, 21, 135-150. https://doi.org/10.1016/S0732-3123(02)00119-0
Randahl, M. (2010). Constraints in the process of using the calculus textbook by engineering students. Manuscript submitted for publication.
Robert, A. & Schwarzenberg, R. (1992). Research into teaching and learning. In D. Tall (Ed.), Advanced mathematical thinking (pp. 127-139). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47203-1_8
Selden, A. & Selden, J. (2001). Tertiary mathematics education research and its future. In D. Tall (Ed.), Advanced mathematical thinking (pp. 237-254). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47231-7_23
Silver, E. A. (1986). Using conceptual and procedural knowledge. In J. Hiebert (Ed.), Conceptual and procedural knowledge: the case of mathematics (pp. 1-28). London: Lawrence Erlbaum.
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 401-411.
Stewart, J. (2003). Calculus: early transcendentals. Belmont: Broks/Cole. Tall, D. (1991). The psychology of advanced mathematical thinking. In D.
Tall (Ed.), Advanced mathematical thinking (pp. 3-24). Dordrecht: Kluwer Academic Publishers.
Tall, D. (1992a). The transition to advanced mathematical thinking. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning. New York: Macmillan.
Tall, D. (1992b). Students' difficulties in calculus. Plenary presentation in working group 3, ICME-7, Quebec. Retrieved June 12, 2010 from http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1993k-calculus-wg3-icme.pdf.
Tall, D. (2005). A theory of mathematical growth through embodiment, symbolism and proof. Paper presented at the International Colloqium on Mathematical Learning from Early Childhood to Adulthood, Nivelles. Retrieved June 12, 2010 from http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2005e-crem-child-adult.pdf
Tall, D. & Maselan, A. bin. (1996). Procedural and conceptual aspects of standards algorithms. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th conference of the international group for the Psychology of Mathematics Education (pp. 19-26). Dep. de Didàctica de la Matemàtica, Universitat de València.
Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169. https://doi.org/10.1007/BF00305619
Viholainen, A. (2008). Prospective mathematics teachers' informal and formal reasoning about the concepts of derivative and differentiability (PhD thesis). University of Jyväskylä.
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47203-1_5
Vollrath, H. J. (1994). Reflections on mathematical concepts as starting points for didactical thinking. In R. Biehler, R. W. Scholtz, R. Strässer & B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 61-72). Dordrecht: Kluwer Academic Publishers.
Zazkis, R. & Leikin, R. (2007). Generating examples: from pedagogical tool to a research tool. For the learning of mathematics, 27 (2), 15-21.
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
