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Research Article

Relating students’ units coordinating and calculus readiness

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Pages 187-208 | Received 30 May 2019, Accepted 17 May 2020, Published online: 05 Jun 2020

References

  • Agresti, A. (2018). An introduction to categorical data analysis. Wiley.
  • Boyce, S., & Wyld, K. (2017). Supporting students’ understanding of calculus concepts: Insights from middle-grades mathematics education research. In a. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 20th annual conference on research in undergraduate mathematics education (pp. 1152–1157). San Diego, CA: SIGMAA on RUME.
  • Boyce, S., Byerley, C., Darling, A., Grabhorn, J. A., & Tyburski, B. (2019). Relationships between calculus students’ structures for static and dynamic reasoning. In S. Otten, a. Candela, Z. de Araujo, C. Haines, & C. Munter (Eds.), Proceedings of the 41st annual meeting of the North American chapter of the international group for the psychology of mathematics education. (pp. 991–1000). St. Louis, MO: University of Missouri.
  • Boyce, S., Grabhorn, J., & Byerley, C. (2018). Relationships between calculus students’ ways of coordinating units and their ways of understanding integration. Poster session presented at the 22nd annual conference on research in undergraduate mathematics education. San Diego, CA.
  • Boyce, S., & Norton, A. (2016). Co-construction of fractions schemes and units coordinating structures. Journal of Mathematical Behavior, 41, 10–25. https://doi.org/10.1016/j.jmathb.2015.11.003
  • Bressoud, D. M., Carlson, M. P., Mesa, V., & Rasmussen, C. (2013). The calculus student: Insights from the mathematical association of America national study. International Journal of Mathematical Education in Science and Technology, 44(5), 685–698. https://doi.org/10.1080/0020739X.2013.798874
  • Byerley, C., & Thompson, P. W. (2014). Secondary teaches’ relative size schemes. In P. Liljedal & C. C. Nicol (Eds.), Proceedings of the 38th meeting of the international group for the psychology of mathematics education (Vol 2, pp. 217–224). Vancouver, BC: PME.
  • Byerley, C. (2019). Calculus students’ fraction and measure schemes and implications for teaching rate of change functions conceptually. The Journal of Mathematical Behavior, 55, 100694. Retrieved from https://doi.org/10.1016/j.jmathb.2019.03.001
  • Byerley, C., & Thompson, P. W. (2017). Secondary teachers’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48, 168–193. https://doi.org/10.1016/j.jmathb.2017.09.003
  • Carlson, M., Larsen, S., & Lesh, R. (2003). Integrating a models and modeling perspective with existing research and practice. In R. Lesh & H. M. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 465–478). NJ: Lawrence Erlbaum Associates.
  • 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
  • Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment: A tool for assessing students’ reasoning abilities and understandings. Cognition and Instruction, 28(2), 113–145. https://doi.org/10.1080/07370001003676587
  • Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In J. Kaput, A. H. Shoenfeld, & E. Dubinsky (Eds.), Research in collegiate mathematics education. III. CBMS issues in mathematics education (pp. 114–162). American Mathematical Society.
  • Collingridge, D. S. (2013). A primer on quantitized data analysis and permutation testing. Journal of Mixed Methods Research, 7(1), 81–97. https://doi.org/10.1177/1558689812454457
  • Corder, G. W., & Foreman, D. I. (2014). Nonparametric statistics: A step-by-step approach. John Wiley & Sons.
  • Cromley, J. G., Booth, J. L., Wills, T. W., Chang, B. L., Tran, N., Madeja, M., Shipley, T. F., & Zahner, W. (2017). Relation of spatial skills to calculus proficiency: A brief report. Mathematical Thinking and Learning, 19(1), 55–68. https://doi.org/10.1080/10986065.2017.1258614
  • Frank, K. M. (2017). Examining the development of students’ covariational reasoning in the context of graphing [ProQuest Dissertations & Theses Global].
  • Haciomeroglu, E. S., Aspinwall, L., & Presmeg, N. C. (2010). Contrasting cases of calculus students’ understanding of derivative graphs. Mathematical Thinking and Learning, 12(2), 152–176. https://doi.org/10.1080/10986060903480300
  • Hackenberg, A., & Lee, M. (2015). Relationships between students’ fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196–243. https://doi.org/10.5951/jresematheduc.46.2.0196
  • Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432. https://doi.org/10.1080/07370008.2010.511565
  • Hackenberg, A. J., & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. The Journal of Mathematical Behavior, 28(1), 1–18. https://doi.org/10.1016/j.jmathb.2009.04.004
  • Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Information Age Publishing.
  • Neuhäuser, M. (2011). Wilcoxon–Mann–Whitney Test. In M. Lovric (Ed.), International encyclopedia of statistical science (pp. 1656–1658). Springer.
  • Norton, A., & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fractions schemes. Journal of Mathematical Behavior, 28(2–3), 150–161. https://doi.org/10.1016/j.jmathb.2009.06.002
  • Norton, A., & Wilkins, J. L. M. (2012). The splitting group. Journal for Research in Mathematics Education, 43(5), 557–583. https://doi.org/10.5951/jresematheduc.43.5.0557
  • Norton, A., & Wilkins, J. L. M. (2013). Supporting students’ constructions of the splitting operation. Cognition and Instruction, 31(1), 2–28. https://doi.org/10.1080/07370008.2012.742085
  • Norton, A., & Boyce, S. (2013). A cognitive core for common state standards. Journal of Mathematical Behavior, 32(2), 266–279. https://doi.org/10.1016/j.jmathb.2013.01.001
  • Norton, A., & Boyce, S. (2015). Provoking the construction of a structure for coordinating n+1 levels of units. Journal of Mathematical Behavior, 40(Part B), 211–232. https://doi.org/10.1016/j.jmathb.2015.10.006
  • Norton, A., Boyce, S., Phillips, N., Anwyll, T., Ulrich, C., & Wilkins, J. (2015). A written instrument for assessing students’ units coordinating structures. Mathematics Education, 10(2), 111–136. https://doi.org/10.12973/mathedu.2015.108a
  • Norton, A., & D’Ambrosio, B. (2008). ZPC and ZPD: Zones of teaching and learning. Journal for Research in Mathematics Education, 39(3), 220–246.
  • Nuzzo, R. (2017). Randomization test: An alternative analysis for the difference of two means. PM&R: The Journal of Injury, Function, and Rehabilitation, 9(3), 306–310. https://doi.org/10.1016/j.pmrj.2017.02.001
  • Piaget, J. (1970a). Genetic epistemology. (E. Duckworth, Trans.). Columbia University Press.
  • Piaget, J. (1970b). Structuralism. (C. Maschler, Trans.). Basic Books (Original work published 1968).
  • Saldanha, L., & Thompson, P. (1998). Re-thinking covariation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berensah & W. N. Coulombe (Eds.), Proceedings of the annual meeting of the psychology of mathematics education - North America. Raleigh, NC: North Carolina State University.
  • Simon, M. A., Placa, N., & Avitzur, A. (2016). Participatory and anticipatory states of mathematical concept learning: Further empirical and theoretical development. Journal for Research in Mathematics Education, 47(1), 63–93. https://doi.org/10.5951/jresematheduc.47.1.0063
  • Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. Springer.
  • Steffe, L. P. (1991). The learning paradox: A plausible counterexample. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 26–44). Springer.
  • Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309. https://doi.org/10.1016/1041-6080(92)90005-Y
  • Steffe, L. P. (2001). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20(3), 267–307. https://doi.org/10.1016/S0732–3123(02)00075-5
  • Steffe, L. P., Liss, D. R., II, & Lee, H. Y. (2014). On the operations that generate intensive quantity. In L. P. Steffe, K. C. Moore, L. L. Hatfield, & S. Belbase (Eds.), Epistemic algebraic students: Emerging models of students’ algebraic knowing (Vol. 4, pp. 49–79). University of Wyoming.
  • Steffe, L. P. (2017). Psychology in mathematics education: Past, present, and future. In E. Galindo & J. Newton (Eds). Proceedings of the 39th annual meeting of the north american chapter of the international group for the psychology of mathematics education (pp. 27–56). Indianapolis, IN.
  • Steffe, L. P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies. Springer.
  • Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & a. Sepulveda (Eds.), Plenary paper presented at the annual meeting of the international group for the psychology of mathematics education, (Vol 1, pp. 31–49). Morelia, Mexico: PME.
  • Thompson, P. W., & Carlson, M. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics.
  • Tillema, E. S. (2013). Relating one and two-dimensional quantities: Students’ multiplicative reasoning in combinatorial and spatial contexts. The Journal of Mathematical Behavior, 32(3), 331–352.
  • Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research in Mathematics Education, 30(4), 390–416. https://doi.org/10.2307/749707
  • Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics, 34(2), 97–129. https://doi.org/10.1023/a:1002998529016
  • Von Glaserfeld, E. (1995). Radical constructivism: A way of knowing and learning. The Falmer.
  • Von Glasersfeld, E. (1981). An attentional model for the conceptual construction of units and number. Journal for Research in Mathematics Education, 12(2), 83–94. https://doi.org/10.2307/748704
  • Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80–83. https://doi.org/10.2307/3001968
  • Wilkins, J. L. M., & Norton, A. (2018). Learning progression toward a measurement conception of fractions. International Journal of STEM Education, 27(2018). https://doi.org/10.1186/s40594-018-0119-2

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