Representation of Algebraic Reasoning in Sets through Argumentation

Year 2019, Volume: 6 Issue: 2, 215 - 229, 13.12.2019

Tuğba Uygun , Pinar Güner

https://doi.org/10.33200/ijcer.557781

Cited By: 1

Abstract

The purpose of the current study is to examine the
ways in which preservice middle school mathematics teachers (PMSMT) apply and
represent algebraic reasoning in their solution processes for the problems in
the concept of sets. This model provides detailed information about the
reasoning made through the process of the solution of set problems. The study
group of this case study was composed of 20 preservice mathematics teachers.
The data were collected through written documents and whole class discussions.
Based on the findings of the study, three ways to represent algebraic reasoning
in sets emerged; context-based representation of algebraic reasoning,
generalization-based representation of algebraic reasoning and
formulization-based representation of algebraic reasoning. These ways determined
based on the argumentations that they formed. They produced different warrants
since they reasoned differently.

Keywords

Algebraic reasoning, Argumentation, Representation of algebraic reasoning, sets

References

• Abi-El-Mona, I. & Abd-El-Khalick, F. (2011). Perceptions of the nature and goodness of argument among college students, science teachers and scientists. International Journal of Science Education, 33(4), 573-605.
• Alcock, L. & Weber, K. (2005). Proof validation in real analysis: Inferring and checking warrants. The Journal of Mathematical Behavior, 24, 125–134.
• Arzarello, F. (2006). Semiosis as a multimodal process. Revista Latinoamericana de Investigacio´n en Matema ´ tica Educativa, special issue on semiotics, culture, and mathematical reasoning: 267-99.
• Baykul, Y. (2012). Mathematics teaching in primary education [İlkokulda matematik öğretimi]. Pegem Academy.
• Binkley, R. W. (1995). Argumentation, education and reasoning. Informal Logic, 17(2), 127–143.
• Blanton, M. L. & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446.
• Brown, R. (2017). Using collective argumentation to engage students in a primary mathematics classroom. Mathematics Education Research Journal, 29(2), 183-199.
• Carraher, D.W. & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester, Jr., (Ed.), Second handbook of research on mathematics teaching and learning (Vol II., pp. 669-705). Charlotte, NC: Information Age Publishing.
• Chapman, O. (2007). Facilitating preservice teachers' development of mathematics knowledge for teaching arithmetic operations. Journal of Mathematics Teacher Education, 10(4), 341-349.
• Conner, A. (2012). Warrants as indications of reasoning patterns in secondary mathematics classes. In Proceedings of the 12 th International Congress on Mathematical Education (ICME-12), Topic Study Group 14 (pp. 2819–2827). Seoul, Korea.
• Conner, A. M., Singletary, L. M., Smith, R. C., Wagner, A. & Francisco, R. T. (2014). Identifying kinds of reasoning in collective argumentation. Mathematical Thinking and Learning, 16, 181-200.
• Courant, R.. Robbins, H., & Stewart, I. (1996). What is mathematics?: An elementary approach to ideas and methods. Oxford University Press US, 1996. ISBN 978-0-19-510519-3.
• Creswell, J. W. (2012). Qualitative inquiry and research design: Choosing among five traditions. Thousand Oaks, California: Sage Publications.
• Donmez, A. (2002). The history and adventure of mathematics “the encyclopedia of history of mathematics in the world”, mathematics dictionary. 1. İstanbul: Toplumsal Donusum Publications.
• Duschl, R., & Osborne, J. (2002). Supporting and promoting argumentation discourse. Studies in Science Education, 38, 39-72.
• Ernest, P. (2008). Towards a semiotics of mathematical text (part 1). For the Learning of Mathematics, 28 (1), 2-8.
• Gavalas, D. (2005). Conceptual mathematics: an application to education. International Journal Mathematics Education in Science and Technology. 36(5), 497-0516.
• Giordano, G. (1990). Strategies that help learning disabled students solve verbal mathematical problems. Preventing School Failure, 35, 24-28.
• Harel, G. (2008). DNR Perspective on Mathematics Curriculum and Instruction: Focus on Proving, Part I. Zentralblatt für Didaktik der Mathematik 40, 487–500.
• Hoch, M. & Dreyfus, T. (2006). Structure sense versus manipulation skills: An unexpected result. In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, ed. J. Novotna ´, H. Moraova ´, M.Kra´tka´,and N. Stehlı ´kova ´, Vol. 3, 305-12. Prague, Czech Republic: Charles University.
• Hodgson, T. (1996). Students’ ability to visualize set expressions: An initial investigation. Educational Studies in Mathematics, 30(2), 159-178. Husserl, E. (1970). Logical investigations. London: Routledge and Kegan Paul.
• Ibraim, S. S. & Justi, R. (2016). Teachers’ knowledge in argumentation: contributions from an explicit teaching in an initial teacher education programme. International Journal of Science Education 38(12), 1996–2025.
• Inglis, M., Meija-Ramos, J. P. & Simpson, A. (2007). Modeling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3-21. Doi: 10.1007/s10649-006-9059-8.
• Jonassen, D. & Kim, B. (2010). Arguing to learn and learning to argue: Design justifications and guidelines. Educational Technology Research and Development, 58, 439-457. Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema and T. Romberg (eds.), Mathematics Classrooms that Promote Understanding, Erlbaum, Mahwah, NJ, pp. 133–155.
• Kaput, J. & Blanton, M. (2005). Algebrafying the elementary mathematics experience in a teacher-centered, systemic way. In T. Romberg & T. Carpenter (Eds.), Understanding mathematics and science matters (pp. 99–125). Mahwah, NJ: Lawrence Erlbaum Associates.
• Kızıltoprak, A. & Yavuzsoy-Köse, N. (2017). Relational thinking: The bridge between arithmetic and algebra. International Electronic Journal of Elementary Education, 10(1), 131-145.
• Kieran, C. (1990). A procedural-structural perspective on algebra research. In Proceedings of the 15th Conference of the International Group for the Psychology of Mathematics Education, ed. F. Furinghetti, Vol. 2, 245-53. Assisi, Italy: Universita`di Genova.
• Kirshner, D. (2001). The structural algebra option revisited. In R. Sutherland, T. Rojano, R. Lins, and A. Bell (eds.) Perspectives on school algebra, (83-98). Dordrecht: Kluwer.
• Knipping, C. (2008). A method for revealing structures of argumentations in classroom proving processes. ZDM: The International Journal on Mathematics Education, 40(3), 427–441.
• Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures (pp. 229-269). Hillsdale, NJ: Erlbaum.
• Lew, H. C. (2004). Developing Algebraic Thinking in Early Grades: Case Study of Korean Elementary School Mathematics. The Mathematics Educator, 8 (1), 88-106.
• Lincoln, Y. S. & Guba, E. G. (1985). Naturalistic Inquiry. Thousand Oaks, California: Sage Publications.
• Merriam, S. B. (1998). Qualitative research and case study applications in education. San Francisco: Jossey-Bass.
• NCTM. (1998). Mathematics Teaching in the Middle School, 4.
• NCTM. (2006). Curricular focal points. Reston.
• Ohlsson, S. (1995). Learning to do and learning to understand? A lesson and a challenge for cognitive modelling. In P. Reimann & H. Spads (Eds.), Learning in humans and machines (pp. 37-62). Oxford: Elsevier.
• Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23-41.
• Prusak, N., Hershkowitz, R. & Schwarz, B. B. (2012). From visual reasoning to logical necessity through argumentative design. Educational Studies in Mathematics, 79(1), 19-40.
• Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. Routledge & Kegan Paul, London.
• Radford, L. (2002). The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22 (2), 14-23.
• Rasmussen, C. L. & Stephan, M. (2008). A methodology for documenting collective activity. In A. Kelly, R. Lesh, & J. Back (Eds.), Handbook of Design Research Methods in Education: Innovations in Science, Technology, Engineering, and Mathematics Teaching and Learning (pp. 195-215). New York, NY: Routledge.
• Rodd, M. (2000). On mathematical warrants. Mathematical Thinking and Learning, 3, 222-24
• Russell, B. (1976). An inquiry into meaning and truth. London: George Allen and Unwin Ltd.
• Sfard, A. & Linchevski, L. (1994). The gains and the pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26, 191-228.
• Siegel, H. (1995). Why should educators care about argumentation. Informal Logic, 17(2), 159–176.
• Staples, M., & Newton, J. (2016). Teachers' contextualization of argumentation in the mathematics classroom. Theory into Practice, 55(4), 294-301.
• Tall, D. & Thomas, M. (1989). Versatile learning and the computer. FOCUS on Learning Problems in Mathematics, 11, 117-125.
• Toulmin, S. E. (2003). The Uses of Argument (updated ed.). New York, NY: Cambridge University Press. (Original work published 1958).
• Toulmin, S., Rieke, R. & Janik, A. (1984). An Introduction to Reasoning. New York, NY: Macmillan.
• Usiskin, Z. (1995). Why is algebra important to learn?. In: B. Moses (Ed.), Algebraic reasoning in grades K-12: Readings from NCTM’s school-based journals and other publications. Reston, VA: NCTM, p. 16-21.
• Uygun, T. & Akyuz, D. (2019). Developing subject matter knowledge through argumentation. International Journal of Research in Education and Science (IJRES), 5(2), 532-547.
• Van de Walle, J. A., Karp, K. & Bay-Williams, J. (2011). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Allyn & Bacon.
• Van Eemeren, F. H. (1995). A world of difference: The rich state of argumentation theory. Informal Logic, 17(2), 144–158.
• Vile, A. (1997). From Peirce towards a Semiotic of Mathematical Meaning. In Quesada, J. F (Ed.) Logic, Semiotic, Social and Computational Perspectives on Mathematical Languages (Chapter-5: 64-76).
• Walkerdine, V.(1988). The Mastery of Reason. Routledge & Kegan Paul, London.
• Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. Journal of Mathematical Behavior, 21, 423-440.
• Yackel, E. & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458-477.