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Research Article

Backward transfer, the relationship between new learning and prior ways of reasoning, and action versus process views of linear functions

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Pages 71-89 | Received 31 Oct 2020, Accepted 29 Jan 2022, Published online: 16 Feb 2022

References

  • Arzi, H. J., Ben-Zvi, R., & Ganiel, U. (1985). Proactive and retroactive facilitation of long-term retention by curriculum continuity. American Educational Research Journal, 22(3), 369–388. https://doi.org/10.3102/00028312022003369
  • Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education II, CBMS issues in mathematics education (pp. 1–32). American Mathematical Society.
  • Ayan, R., Isiksal-Bostan, M., & Stephan, M. (2000). A math teacher’s participation in a classroom design research: Teaching of ratio and proportion. In U. T. Jankvist, M. Van der Heuvel-panhuizen, and M. Veldhuis (Eds.) Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (pp. 3580–3587). Freudenthal Group & Freudenthal Institute, Utrecht University, Utrecht, Netherlands and ERME. http://www.mathematik.uni-dortmund.de/~prediger/ERME/CERME11_Proceedings_2019.pdf
  • Bagley, S., Rasmussen, C., & Zandieh, M. (2015). Inverse, composition, and identity: The case of function and linear transformation. Journal of Mathematical Behavior, 37, 36–47. http://dx.doi.org/10.1016/j.jmathb.2014.11.003
  • Barnett, S. M., & Ceci, S. J. (2002). When and where do we apply what we learn? A taxonomy for far transfer. Psychological Bulletin, 128(4), 612–637. https://doi.org/10.1037/0033-2909.128.4.612
  • Beach, K. (1999). Consequential transitions: A sociocultural expedition beyond transfer in education. In A. Iran-Nejad & P. D. Pearson (Eds.), Review of research in education (Vol. 24, pp. 101–139). American Educational Research Association.
  • Bransford, J. D., & Schwartz, D. L. (1999). Rethinking transfer: A simple proposal with multiple implications. In A. Iran-Nejad & P. D. Pearson (Eds.), Review of research in education (Vol. 24, pp. 61–100). American Educational Research Association. https://doi.org/10.3102/0091732x024001061
  • Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23(3), 247–285. https://doi.org/10.1007/BF02309532
  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378. https://doi.org/10.2307/4149958
  • Cedillo, T. E. (2001). Toward an algebra acquisition support system: A study based on using graphic calculators in the classroom. Mathematical Thinking and Learning, 3(4), 221–259. https://doi.org/10.1207/S15327833MTL0304_01
  • Childers, A. B., & Vidakovic, D. (2014). Students’ understanding of the concept of vertex of quadratic functions in relation to their personal meaning of the concept of vertex. International Journal for Mathematics Teaching and Learning, 1–33. https://www.cimt.org.uk/journal/childers.pdf
  • Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86. https://doi.org/10.2307/749228
  • Cook, V. (2003). Effects of the second language on the first. Multilingual Matters.
  • Detterman, D. K. (1993). The case for the prosecution: Transfer as an epiphenomenon. In D. K. Detterman & R. J. Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction (pp. 1–24). Ablex Publishing.
  • Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229. https://doi.org/10.2307/30034866
  • Ellis, A. B., & Grinstead, P. (2008). Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. Journal of Mathematical Behavior, 27(4), 277–296. https://doi.org/10.1016/j.jmathb.2008.11.002
  • Gentner, D., Loewenstein, J., & Thompson, L. (2004). Analogical encoding: Facilitating knowledge transfer and integration. In K. Forbus, D. Gentner, & T. Regier (Eds.), Proceedings of the Twenty-Sixth Annual Conference of the Cognitive Science Society (pp. 452–457). Mahwah, NJ: Erlbaum.
  • Greer, B. (2012). Inversion in mathematical thinking and learning. Educational Studies in Mathematics, 79(3), 429–438. https://doi.org/10.1007/s10649-011-9317-2
  • Hines, E. (2002). Developing the concept of linear function: One student’s experiences with dynamic physical models. Journal of Mathematical Behavior, 20(3), 337–361. https://doi.org/10.1016/S0732-3123(02)00074-3
  • Hohensee, C. (2014). Backward transfer: An investigation of the influence of quadratic functions instruction on students’ prior ways of reasoning about linear functions. Mathematical Thinking and Learning, 16(2), 135–174. https://doi.org/10.1080/10986065.2014.889503
  • Hohensee, C., Gartland, S., Willoughby, L., & Melville, M. (2021). Backward transfer influences from quadratic functions instruction on students’ prior ways of covariational reasoning about linear functions. Journal of Mathematical Behavior, 61, 100834. https://doi.org/10.1016/j.jmathb.2020.100834
  • Hunt, J. H. (2015). Notions of equivalence through ratios: Students with and without learning disabilities. Journal of Mathematical Behavior, 37, 94–105. http://dx.doi.org/10.1016/j.jmathb.2014.12.002
  • Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235–287). State University of New York Press.
  • Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life. Cambridge University Press.
  • Lima, R. N., & Tall, D. O. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3–18. https://doi.org/10.1007/s10649-007-9086-0
  • Lobato, J. (2008). Research methods for alternate approaches to transfer: Implications for design experiments. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 167–194). Routledge.
  • Lobato, J. (2012). The actor-oriented transfer perspective and its contributions to educational research and practice. Educational Psychologist, 47(3), 232–247. https://doi.org/10.1080/00461520.2012.693353
  • Macgregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19. https://doi.org/10.1023/A:1002970913563
  • Marton, F. (2006). Sameness and difference in transfer. The Journal of the Learning Sciences, 15(4), 499–535. https://doi.org/10.1207/s15327809jls1504_3
  • Maxwell, J. A. (2004). Causal explanation, qualitative research, and scientific inquiry in education. Educational Researcher, 33(2), 3–11. https://doi.org/10.3102/0013189X033002003
  • Melhuish, K., & Fagan, J. (2018). Connecting the group theory concept assessment to core concepts at the secondary level. In N. H. Wasserman (Ed.), Connecting abstract algebra to secondary mathematics, for secondary mathematics teachers (pp. 19–45). Springer Natural. https://doi.org/10.1007/978-3-319-99214-3_2
  • Miles, M. B., Huberman, A. M., & Saldaña, J. (2014). Qualitative data analysis: A methods sourcebook (4th ed.). Sage.
  • Moore, T. (2012). What calculus do students learn after calculus? [Unpublised doctoral dissertation]. Available from ProQuest Dissertations and Theses database. (UMI No. 3526217). Retreived from http://krex.k-state.edu/dspace/handle/2097/14090
  • Moschkovich, J. N. (1998). Students’ use of the x-intercept as an instance of a transitional conception. Educational Studies in Mathematics, 37(2), 169–197. https://doi.org/10.1023/A:1003539828299
  • National Research Council. (2002). Scientific research in education. Washington, DC: National Academy Press.
  • Roschelle, J. (1995). Learning in interactive environments: Prior knowledge and new experience. In J. Falk & L. Dierking (Eds.), Public institutions for personal learning (pp. 37–51). American Association of Museums.
  • 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. https://doi.org/10.1007/BF00302715
  • Singley, M., & Anderson, J. (1989). The transfer of cognitive skill. Harvard University Press.
  • Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 33(3), 259–281. https://doi.org/10.1023/A:100293703
  • Sloane, F. C. (2008). Randomized trials in mathematics education: Recalibrating the proposed high watermark. Educational Researcher, 37(9), 624–630. https://doi.org/10.3102/0013189X08328879
  • Smith, J. P., diSessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115–163. https://doi.org/10.1207/s15327809jls0302_1
  • Strauss, A., & Corbin, J. (1994). Grounded theory methodology: An overview. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (pp. 273–285). Sage Publications.
  • Thompson, P. W. (1994). Students, functions, and the undergraduate mathematics curriculum. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in Collegiate Mathematics Education, 1 (Vol. 4, pp. 21–44). American Mathematical Society.
  • Thorndike, E. L., & Woodworth, R. S. (1901). The influence of improvement in one mental function upon the efficiency of other functions (I). Psychological Review, 8(3), 247–261. https://doi.org/10.1037/h0071363
  • Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2004). Remedying secondary school students’ illusions of linearity: A teaching experiment aiming at conceptual change. Learning and Instruction, 14(5), 485–501. https://doi.org/10.1016/j.learninstruc.2004.06.019
  • Weber, K. (2002). Students’ understanding of exponential and logarithmic functions. In I. Vakalis et al. (Eds.), Second conference on the Teaching of Mathematics (pp. 1–10). Crete: University of Crete. http://www.math.uoc.gr/~ictm2/Proceedings/pap145.pdf
  • Zaslavsky, O. (1997). Conceptual obstacles in the learning of quadratic functions. Focus on Learning Problems in Mathematics, 19(1), 20–44.

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