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

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 898 Views 599 Downloads

## 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.