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Articles

The impact of procedural and conceptual teaching on students' mathematical performance over time

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Pages 404-426 | Received 22 Apr 2019, Published online: 17 Nov 2019

References

  • Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Roa, S., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. New York: Springer.
  • Artigue, M. (1991). Analysis. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 167–198). Dordrecht: Kluwer.
  • Badillo, E., Azcárate, C., & Font, V. (2011). Analysis of mathematics teachers’ level of understanding of the objects f′(a) and f′(x). Enseñanza de las ciencias, 29(2), 191–206. doi: 10.5565/rev/ec/v29n2.546
  • Baker, B., Cooley, L., & Trigueros, M. (2000). A calculus graphing schema. Journal for Research in Mathematics Education, 31(5), 557–578. doi: 10.2307/749887
  • Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. Mahwah, NJ: Erlbaum.
  • Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38(2), 115–131.
  • Bögeholz, S., Böhm, M., Eggert, S., & Barkmann, J. (2014). Education for sustainable development in German science education: Past – Present – Future. EURASIA Journal of Mathematics, Science and Technology Education, 10(4), 231–248. doi: 10.12973/eurasia.2014.1079a
  • Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE theory to improve students’ graphical understanding of derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(7), 2947–2967. doi: 10.29333/ejmste/91451
  • Borji, V., Font, V., Alamolhodaei, H., & Sánchez, A. (2018). Application of the Complementarities of two theories, APOS and OSA, for the analysis of the university students’ understanding on the graph of the function and its derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(6), 2301–2315. doi: 10.29333/ejmste/89514
  • Borji, V., Font, V., Alamolhodaei, H., Sánchez, A., & Pino-Fan, L. (2018). On students’ understanding of implicit differentiation, based on two lenses: APOS and OSA. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.). Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education ( Vol. 5, p. 25). Umeå, Sweden: PME.
  • Borji, V., & Martínez-Planell, R. (2019). What does 'y is defined as an implicit function of x' mean?: An application of APOS-ACE. The Journal of Mathematical Behavior, 56, 100739. doi: 10.1016/j.jmathb.2019.100739
  • Bush, S. B., & Karp, K. S. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32(3), 613–632. doi: 10.1016/j.jmathb.2013.07.002
  • Canning, M. (2017). How do New Zealand Teachers like to be supported by psychologists? (Master's thesis).
  • Canobi, K. H. (2009). Concept-procedure interactions in children's addition and subtraction. Journal of Experimental Child Psychology, 102(2), 131–149. doi: 10.1016/j.jecp.2008.07.008
  • Chappell, K. K., & Killpatrick, K. (2003). Effects of concept-based instruction on students’ conceptual understanding and procedural knowledge of calculus. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 13(1), 17–37. doi: 10.1080/10511970308984043
  • Cilli-Turner, E. (2017). Impacts of inquiry pedagogy on undergraduate students conceptions of the function of proof. The Journal of Mathematical Behavior, 48, 14–21. doi: 10.1016/j.jmathb.2017.07.001
  • Clark, J. M., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D. J., St. John, D., … Vidakovic, D. (1997). Constructing a schema: The case of the chain rule? The Journal of Mathematical Behavior, 16(4), 345–364. doi: 10.1016/S0732-3123(97)90012-2
  • Cottrill, J. F. (1999). Students’ understanding of the concept of chain rule in first year calculus and the relation to their understanding of composition of functions. Unpublished doctoral dissertation, Purdue University, West Lafayette.
  • Edwards, D. M. (2015). Teaching fractions procedurally and conceptually to pre-service elementary education teachers (Doctoral dissertation in practice). Electronic Theses and Dissertations. 5004. Retrieved from http://stars.library.ucf.edu/etd/5004
  • Engelbrecht, J., Harding, A., & Potgieter, M. (2005). Undergraduate students’ performance and confidence in procedural and conceptual mathematics. International Journal of Mathematical Education in Science and Technology, 36(7), 701–712. doi: 10.1080/00207390500271107
  • Gelman, R. & Williams, E. M. (1998). Enabling constraints for cognitive development and learning: Domain specificity and epigenesis. In D. Kuhn & R. S. Siegler (Eds.), Handbook of child psychology: Cognition, perception, and language (5th ed., Vol. 2, pp. 575–630). New York: John Wiley.
  • Gerson, H., & Bateman, E. (2010). Authority in an agency-centered, inquiry-based university calculus classroom. The Journal of Mathematical Behavior, 29(4), 195–206. doi: 10.1016/j.jmathb.2010.10.003
  • Graham, E. L., Berman, J., & Bellert, A. (2015). Sustainable learning: Inclusive practices for 21st century classrooms. Port Melbourne: Cambridge University Press.
  • Haapasalo, L., & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation. JMD-Journal for Mathematic-Didaktik, 21, 139–157. doi: 10.1007/BF03338914
  • 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. 1–27). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
  • Hughes-Hallet, D., Gleason, A. M., & Mccallum, W. G. (2017). Calculus: Single and multivariable (6th ed.). New York: Wiley.
  • Joffrion, K. H. (2005). Conceptual and procedural understanding of algebra concepts in the middle grades (Master’s thesis). Retrieved from https://core.ac.uk/download/pdf/4271888.pdf
  • Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141. doi: 10.1016/j.jmathb.2012.12.004
  • Khoule, A., Bonsu, N., & El Houari, H. (2017). Impact of conceptual and procedural knowledge on students mathematics anxiety. International Journal of Educational Studies in Mathematics, 4(1), 8–18.
  • Kiat, S. E. (2005). Analysis of students’ difficulties in solving integration problems. The Mathematics Educator, 9(1), 39–59.
  • Krainer, K., Zehetmeier, S., Hanfstingl, B., Rauch, F., & Tscheinig, T. (2018). Insights into scaling up a nationwide learning and teaching initiative on various levels. Educational Studies in Mathematics. doi: 10.1007/s10649-018-9826-3
  • Maciejewski, W., & Star, J. R. (2019). Justifications for choices made in procedures. Educational Studies in Mathematics, 101(3), 325–340. doi: 10.1007/s10649-019-09886-7
  • Mahir, N. (2009). Conceptual and procedural performance of undergraduate students in integration. International Journal of Mathematical Education in Science and Technology, 40(2), 201–211. doi: 10.1080/00207390802213591
  • Martínez-Planell, R., Trigueros, M., & McGee, D. (2015). On students’ understanding of the differential calculus of functions of two variables. The Journal of Mathematical Behavior, 38, 57–86. doi: 10.1016/j.jmathb.2015.03.003
  • Mateus, E. (2016). Análisis Didáctico a un Proceso de Instrucción del Método de Integración por Partes [Teaching analysis to process integration method instruction by parties]. Bolema: Boletim de Educação Matemática, 30(55), 559–585. doi: 10.1590/1980-4415v30n55a13
  • Merriam-Webster’s Collegiate Dictionary. (2012). 11th ed. Springfield, MA: Merriam-Webster.
  • National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
  • National Research Council. (1997). Science teaching reconsidered: A handbook. Washington, DC: The National Academies Press.
  • NCTM. (2014). Procedural fluency in mathematics: A position of the National Council of Teachers of Mathematics. Retrieved from https://www.nctm.org/Standards-and-Positions/PositionStatements/Procedural-Fluency-in-Mathematics/
  • Oaks, A. B. (1990). Writing to learn mathematics: Why do we need it and how can it help us?. Paper presented at the Annual Meeting of the Association of Mathematics Teachers of New York State, Ellenville, NY.
  • Orton, A. (1983). Students’ understanding of integration. Educational Studies in Mathematics, 14(1), 1–18. doi: 10.1007/BF00704699
  • Park, J. (2015). Is the derivative a function? If so, how do we teach it? Educational Studies in Mathematics, 89(2), 233–250. doi: 10.1007/s10649-015-9601-7
  • Pesek, D. D., & Kirschner, D. (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education, 31, 524–540. doi: 10.2307/749885
  • Piaget, J., & García, R. (1983). Psychogenesis and the history of science. New York: Columbia University Press.
  • Pino-Fan, L., Font, V., Gordillo, W., Larios, V., & Breda, A. (2018). Analysis of the meanings of the antiderivative used by students of the first engineering courses. International Journal of Science and Mathematics Education, 16(6), 1091–1113. doi: 10.1007/s10763-017-9826-2
  • Radmehr, F. (2016). Exploring students’ learning of integral calculus using revised Bloom's taxonomy (Doctoral dissertation). Victoria University of Wellington, Wellington, New Zealand.
  • Radmehr, F., & Drake, M. (2017). Exploring students’ mathematical performance, metacognitive experiences and skills in relation to fundamental theorem of calculus. International Journal of Mathematical Education in Science and Technology, 48(7), 1043–1071. doi: 10.1080/0020739X.2017.1305129
  • Radmehr, F., & Drake, M. (2019). Students' mathematical performance, metacognitive experiences and metacognitive skills in relation to integral-area relationships. Teaching Mathematics and Its Application: An International Journal of the IMA, 38(2), 85–106. doi: 10.1093/teamat/hry006
  • Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology ( Vol. 3, pp. 41–95). Hillsdale, NJ: Erlbaum.
  • Rittle-Johnson, B., & Schneider, M. (2015). Developing conceptual and procedural knowledge of mathematics. In R. C. Kadosh, & A. Dowker (Eds.), Oxford library of psychology. The Oxford handbook of numerical cognition (pp. 1118–1134). New York, NY: Oxford University Press. doi: 10.1093/oxfordhb/9780199642342.013.014
  • Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27, 587–597. doi: 10.1007/s10648-015-9302-x
  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362. doi: 10.1037/0022-0663.93.2.346
  • Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2015). Developing pre-service teachers’ noticing of students’ understanding of the derivative concept. International Journal of Science and Mathematics Education, 13(6), 1305–1329. doi: 10.1007/s10763-014-9544-y
  • Selter, C., Prediger, S., Nührenbörger, M., & Hußmann, S. (2012). Taking away and determining the difference—A longitudinal perspective on two models of subtraction and the inverse relation to addition. Educational Studies in Mathematics, 79(3), 389–408. doi: 10.1007/s10649-011-9305-6
  • 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(1), 1–36. doi: 10.1007/BF00302715
  • Siegler, R. S., & Stern, E. (1998). Conscious and unconscious strategy discoveries: A microgenetic analysis. Journal of Experimental Psychology: General, 127, 377–397. doi:10.1037/0096–3445.127.4.377 doi: 10.1037/0096-3445.127.4.377
  • Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.
  • Skemp, R. (1987). The psychology of learning mathematics. Hillsdale, NJ: Erlbaum.
  • Southall, E. (2016). The formula triangle and other problems with procedural teaching in mathematics. School Science Review (SSR), 97(360), 49–53.
  • Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.
  • Stewart, J. (2010). Calculus. Mason, OH: Brooks/Cole Cengage Learning.
  • Thomas, G. B., Weir, M. D., Hass, J., & Giordano, R. F. (2010). Thomas’ calculus: Early transcendentals. Boston: Pearson Addison-Wesley.
  • Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2), 229–274. doi: 10.1007/BF01273664
  • Varsavsky, C. (2012). Use of CAS in secondary school: A factor influencing the transition to university-level mathematics? International Journal of Mathematical Education in Science and Technology, 43(1), 33–42. doi: 10.1080/0020739X.2011.582179
  • Yackel, E., Rasmussena, C., & King, K. (2000). Social and sociomathematical norms in an advanced undergraduate mathematics course. The Journal of Mathematical Behavior, 19(3), 275–287. doi: 10.1016/S0732-3123(00)00051-1
  • Yopp, D. (2014). Undergraduates’ use of examples in online discussions. The Journal of Mathematical Behavior, 33, 180–191. doi: 10.1016/j.jmathb.2013.11.004
  • Zandieh, M. (2000). A theoretical framework for analyzing students understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education ( Vol. IV, pp. 103–127). Providence, RI: American Mathematical Society.

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