International Electronic Journal of Mathematics Education

International Electronic Journal of Mathematics Education
Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities
APA
In-text citation: (Yao & Elia, 2021)
Reference: Yao, X., & Elia, J. (2021). Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities. International Electronic Journal of Mathematics Education, 16(2), em0628. https://doi.org/10.29333/iejme/9770
AMA
In-text citation: (1), (2), (3), etc.
Reference: Yao X, Elia J. Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities. INT ELECT J MATH ED. 2021;16(2), em0628. https://doi.org/10.29333/iejme/9770
Chicago
In-text citation: (Yao and Elia, 2021)
Reference: Yao, Xiangquan, and John Elia. "Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities". International Electronic Journal of Mathematics Education 2021 16 no. 2 (2021): em0628. https://doi.org/10.29333/iejme/9770
Harvard
In-text citation: (Yao and Elia, 2021)
Reference: Yao, X., and Elia, J. (2021). Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities. International Electronic Journal of Mathematics Education, 16(2), em0628. https://doi.org/10.29333/iejme/9770
MLA
In-text citation: (Yao and Elia, 2021)
Reference: Yao, Xiangquan et al. "Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities". International Electronic Journal of Mathematics Education, vol. 16, no. 2, 2021, em0628. https://doi.org/10.29333/iejme/9770
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Yao X, Elia J. Connections between Empirical and Structural Reasoning in Technology-Aided Generalization Activities. INT ELECT J MATH ED. 2021;16(2):em0628. https://doi.org/10.29333/iejme/9770

Abstract

Mathematical generalization can take on different forms and be built upon different types of reasoning. Having utilized data from a series of task-based interviews, this study examined connections between empirical and structural reasoning as preservice mathematics teachers solved problems designed to engage them in constructing and generalizing mathematical ideas aided by digital tools. The study revealed closer connections between naïve empiricism and result pattern generalization, between naïve empiricism and recognizing a structure in thought, between reasoning by generic example and process pattern generalization, and between reasoning by generic example and reasoning in terms of general structures. Results from this study imply that the ability to generalize based on perception and numerical pattern does not necessarily lead learners to generalize based on mathematical structure.

References

  • Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environments. ZDM, 34(3), 66-72. https://doi.org/10.1007/BF02655708
  • Baccaglini-Frank, A. (2019). Dragging, instrumented abduction and evidence, in processes of conjecture generation in a dynamic geometry environment. ZDM, 51, 779-791. https://doi.org/10.1007/s11858-019-01046-8
  • Baccaglini-Frank, A., & Mariotti, M. A. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model. International Journal of Computers for Mathematical Learning, 15(3), 225-253. https://doi.org/10.1007/s10758-010-9169-3
  • Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216-235). London: Hodder and Stoughton.
  • Blanton, M. L., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understandings of algebraic thinking for teaching mathematics in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
  • Christou, C., Mousoulides, N., Pittalis, M., & Pitta-Pantazi, D. (2004). Proofs through exploration in dynamic geometry environments. International Journal of Science and Mathematics Education, 2(3), 339-352. https://doi.org/10.1007/s10763-004-6785-1
  • Ciosek, M. (2012). Generalization in the process of defining a concept and exploring it by students. In B. Maj-Tatsis & K. Tatsis (Eds.), Generalization in mathematics at all educational levels (pp. 38–56). Rzeszow: University of Rzeszow.
  • Dörfler, W. (1991). Forms and means of generalization in mathematics. In A. J. Bishop, S. Mellin-Olsen, & J. Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 61-85). Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-94-017-2195-0_4
  • El Mouhayar, R. (2018). Trends of progression of student level of reasoning and generalization in numerical and figural reasoning approaches in pattern generalization. Educational Studies in Mathematics, 99(1), 89-107. https://doi.org/10.1007/s10649-018-9821-8
  • El Mouhayar, R., & Jurdak, M. (2016). Variation of student numerical and figural reasoning approaches by pattern generalization type, strategy use and grade level. International Journal of Mathematical Education in Science and Technology, 47(2), 197-215. https://doi.org/10.1080/0020739X.2015.1068391 3
  • Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zaskis (Eds.), Learning and teaching number theory (pp. 185-212). New Jersey: Ablex Publishing Corporation.
  • Harel, G., & Soto, O. (2017). Structural reasoning. International Journal of Research in Undergraduate Mathematics Education, 3(1), 225-242. https://doi.org/10.1007/s40753-016-0041-2
  • Harel, G., & Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38-42. https://www.jstor.org/stable/40248005
  • Hawthorne, C., & Druken, B. K. (2019). Looking for and using structural reasoning. The Mathematics Teacher, 112(4), 294-301. https://doi.org/10.5951/mathteacher.112.4.0294
  • Hoch, M., & Dreyfus, T. (2004). Structure sense in high school algebra: The effects of brackets. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 49-56). Bergen, Norway: PME.
  • Hollebrands, K. F., Conner, A., & Smith, R. C. (2010). The nature of arguments provided by college geometry students with access to technology while solving problems. Journal for Research in Mathematics Education, 41(4), 324-350. https://www.jstor.org/stable/41103879
  • Komatsu, K., & Jones, K. (2019). Task Design Principles for Heuristic Refutation in Dynamic Geometry Environments. International Journal of Science and Mathematics Education, 17(4), 801-824. https://doi.org/10.1007/s10763-018-9892-0
  • Küchemann, D. (2010). Using patterns generically to see structure. Pedagogies: An International Journal, 5(3), 233-250. https://doi.org/10.1080/1554480X.2010.486147
  • Küchemann, D., & Hoyles, C. (2009). From empirical to structural reasoning in mathematics: Tracking changes over time. In D. A. Stylianou, M. L. Blanton, & E. J. Knuth (Eds.), Teaching and learning proof across the grades (pp. 171-191). New York, NY: Routledge.
  • Kuzle, A. (2017). Delving into the nature of problem solving processes in a dynamic geometry environment: Different technological effects on cognitive processing. Technology, Knowledge and Learning, 22(1), 37-64. https://doi.org/10.1007/s10758-016-9284-x
  • Lachmy, R., & Koichu, B. (2014). The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment. Journal of Mathematical Behavior, 36, 150-165. https://doi.org/10.1016/j.jmathb.2014.07.002
  • Leung, A. (2014). Principles of acquiring invariant in mathematics task design: A dynamic geometry example. In P. Liljedahl, C. Nical, S. Oesterle, & D. Allan (Eds.), Proceedings of the 38th Conference the International Group for the Psychology of Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 4, pp. 89-96). Vancouver, Canada.
  • Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd Edition). London: Pearson.
  • Mason, J., Graham, A., & Johnston-Wilder, S. (2005). Developing thinking in algebra. London: Sage.
  • Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structure for all. Mathematics Education Research Journal, 21(2), 10-32. https://doi.org/10.1007/BF03217543
  • Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33-49. https://doi.org/10.1007/BF03217544
  • Mulligan, J., & Mitchelmore, M. (2012). Developing pedagogical strategies to promote structural thinking in early mathematics. In J. Dindyal, L. P. Cheng, & S. F. Ng (Eds.), Mathematics education: Expanding horizons. Proceedings of the 35th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 529-536). Singapore: MERGA.
  • Olive, J., & Makar, K. (2010). Mathematical knowledge and practices resulting from access to digital technologies. In C. Hoyles & J.-B. Lagrange (Eds.), Mathematics education and technology—Rethinking the terrain (pp. 133-177). New York, NY: Springer. https://doi.org/10.1007/978-1-4419-0146-0_8
  • Pedemonte, B., & Balacheff, N. (2016). Establishing links between conceptions, argumentation and proof through the ck¢-enriched Toulmin model. Journal of Mathematical Behavior, 41, 104-122. https://doi.org/10.1016/j.jmathb.2015.10.008
  • Radford, L. (2003). Gestures, speech and the sprouting of signs. Mathematical Thinking and Learning, 5(1), 37-70. https://doi.org/10.1207/S15327833MTL0501_02
  • Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM, 40(1), 83-96. https://doi.org/10.1007/s11858-007-0061-0
  • Reid, D. A., & Knipping, C. (2011). Proof in mathematics education: Research, learning and teaching. Rotterdam, the Netherlands: Sense Publishers.
  • Richard, P., Venant, F., & Gagnon, M. (2019) Issues and challenges in instrumental proof. In G. Hanna, D. A. Reid, & M. de Villiers (Eds.), Proof technology in mathematics research and teaching (pp. 139-172). Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-030-28483-1_7
  • Sinclair N. & Robutti O. (2012). Technology and the role of proof: The case of dynamic geometry. In M. Clements, A. Bishop, C. Keitel, J. Kilpatrick, & F. Leung (eds), Third international handbook of mathematics education (pp. 571-596). New York, NY: Springer. https://doi.org/10.1007/978-1-4614-4684-2_19
  • Weber, K. (2013). On the sophistication of naïve empirical reasoning: factors influencing mathematicians’ persuasion ratings of empirical arguments. Research in Mathematics Education, 15(2), 100-114. https://doi.org/10.1080/14794802.2013.797743
  • Yao, X. (2020). Characterizing learners’ growth of geometric understanding in dynamic geometry environments: A perspective of the Pirie-Kieren theory. Digital Experiences in Mathematics Education, 6, 293-319. https://doi.org/10.1007/s40751-020-00069-1
  • Yao, X., & Manouchehri, A. (2019). Middle school students’ generalizations about properties of geometric transformations in a dynamic geometry environment. The Journal of Mathematical Behavior, 55, 1-19. https://doi.org/10.1016/j.jmathb.2019.04.002
  • Yerushalmy, M. (1993). Generalization in geometry. In J. L. Schwartz, M. Yerushalmy, & B. Wilson (Eds.), The geometric supposer: What is it a case of (pp. 57-84). Hillsdale, NJ: Lawrence Erlbaum Associates.

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.