The integral as accumulation function approach: A proposal of a learning sequence for collaborative reasoning

Sonia Palha 1, Jeroen Spandaw 2 *
More Detail
1 Centre for Applied Research on Education, Amsterdam University of Applied Sciences, Amsterdam, The Netherlands
2 Engineering, Mathematics and Computer Science, Delft University of Technology, Delft, The Netherlands
* Corresponding Author
EUR J SCI MATH ED, Volume 7, Issue 3, pp. 109-136. https://doi.org/10.30935/scimath/9538
OPEN ACCESS   692 Views   476 Downloads
Download Full Text (PDF)

ABSTRACT

Learning mathematical thinking and reasoning is a main goal in mathematical education. Instructional tasks have an important role in fostering this learning. We introduce a learning sequence to approach the topic of integrals in secondary education to support students mathematical reasoning while participating in collaborative dialogue about the integral-as-accumulation-function. This is based on the notion of accumulation in general and the notion of accumulative distance function in particular. Through a case-study methodology we investigate how this approach elicits 11th grade students’ mathematical thinking and reasoning. The results show that the integral-as-accumulation-function has potential, since the notions of accumulation and accumulative function can provide a strong intuition for mathematical reasoning and engage students in mathematical dialogue. Implications of these results for task design and further research are discussed.

CITATION

Palha, S., & Spandaw, J. (2019). The integral as accumulation function approach: A proposal of a learning sequence for collaborative reasoning. European Journal of Science and Mathematics Education, 7(3), 109-136. https://doi.org/10.30935/scimath/9538

REFERENCES

  • Bakker, A. and van Eerde, D. (2013). An introduction to design-based research with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Doing qualitative research: methodology and methods in mathematics education. New York: Springer.
  • Barron, B. (2000). Achieving coordination in collaborative problem-solving groups. The Journal of the Learning Sciences, 9(4), 403-436.
  • Bergqvist, T., Lithner, J. and Sumpter, L. (2008). Upper secondary students' task reasoning. International Journal of Mathematical Education in Science and Technology, 39(1), 1-12.
  • 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.
  • Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American educational research journal, 29(3), 573-604.
  • Cobb, P., & Whitenack, J. W. (1996). A method for conducting longitudinal analyses of classroom video-recordings and transcripts. Educational studies in mathematics, 30(3), 213-228.
  • Dekker, R., & Elshout-Mohr, M. (1998). A process model for interaction and mathematical level raising. Educational Studies in Mathematics, 35(3), 303–314.
  • Dekker, R., & Elshout-Mohr, M. (2004). Teacher interventions aimed at mathematical level raising during collaborative learning. Educational Studies in Mathematics, 56(1), 39–65.
  • Dekker, R., Elshout-Mohr, M. & Wood, T. (2004). Working together on assignments: multiple analysis of learning events. In J. v. d. Linden & P. Renshaw (Eds.), Dialogic Learning (pp. 145-170). Dordrecht: Kluwer Academic Publishers.
  • Dillenbourg, P., Baker P., Blaye M., O’Malley, C. (1995). The evolution of research on collaborative learning. In E. Spada & P. Reiman (Eds.), Learning in Humans and Machine: Towards an interdisciplinary learning science. pp. 189 – 211. Oxford: Elsevier.
  • Fischer, F., Bruhn, J., Gräsel, C., & Mandl, H. (2002). Fostering collaborative knowledge construction with visualization tools. Learning and Instruction, 12(2), 213-232.
  • Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: A "proceptual" view of simple arithmetic. Journal for research in Mathematics Education, 116-140.
  • Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. Second handbook of research on mathematics teaching and learning, 2, 805-842.
  • Jones, K. (2000). Providing a foundation for deductive reasoning: students' interpretations when using Dynamic Geometry software and their evolving mathematical explanations. Educational studies in mathematics, 44(1-2), 55-85
  • Kouropatov, A., & Dreyfus, T. (2013). Constructing the integral concept on the basis of the idea of accumulation: suggestion for a high school curriculum. International Journal of Mathematical Education in Science and Technology, 44(5), 641-651.
  • Lithner, J. (2003). Students' mathematical reasoning in university textbook exercises. Educational studies in mathematics, 52(1), 29-55.
  • Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67(3), 255-276
  • Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. The Journal of Mathematical Behavior, 31(1), 48-59.
  • Mueller, M., & Yankelewitz, D. (2014). Fallacious Argumentation in Student Reasoning: Are There Benefits?. European Journal of Science and Mathematics Education, 2(1), 27-38.
  • Nguyen, D. H., & Rebello, N. S. (2011). Students’ understanding and application of the area under the curve concept in physics problems. Physical Review Special Topics-Physics Education Research, 7(1)
  • 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.
  • Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 33(3), 259-281.
  • Tall, D. (1996). Functions and Calculus. In A. J. B. e. al (Ed.), International Handbook of Mathematics Education (pp. 289 - 325): Kluwer Academic Publishers.
  • Tall, D. O. (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM, 41(4), 481-492.
  • Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. In Learning Mathematics (pp. 125-170). Springer Netherlands.
  • Thompson, P. W., & Silverman, J. (2008). The concept of accumulation in calculus. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 43-52). Washington, DC: Mathematical Association of America.
  • Rasmussen, C., Marrongelle, K., & Borba, M. C. (2014). Research on calculus: what do we know and where do we need to go? ZDM, 46(4), 507-515.
  • Webb, N. M., Nemer, K. M., & Ing, M. (2006). Small-group reflections: Parallels between teacher discourse and student behavior in peer-directed groups. The Journal of the Learning Sciences, 15(1), 63-119.
  • Webb, N. M., Franke, M. L., Wong, J., Fernandez, C. H., Shin, N., & Turrou, A. C. (2014). Engaging with others’ mathematical ideas: Interrelationships among student participation, teachers’ instructional practices, and learning. International Journal of Educational Research, 63, 79-93.
  • Yackel, E. (2001). Explanation, Justification and Argumentation in Mathematics Classrooms.
  • Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. The Journal of Mathematical Behavior, 21(4), 423-440.