Determining Ways of Thinking and Understanding Related To Generalization of Eighth Graders

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Gülçin Oflaz Handan Demircioğlu



The main purpose of this study is to determine ways of thinking and understanding of eight graders related to generalizing act. To carry out this aim, a DNR based teaching experiment was developed and applied to 9 eight graders. The design of the study consists of three stages; preparation process in which teaching experiment is prepared, teaching process in which teaching experiment is applied, and analysis process in which continuous and retrospective analyses are carried out. Analysing the data, it was found that students’ ways of thinking could be determined as relating, searching, and extending. Ways of understanding belonging to generalizing act could be determined as identification, definition, and influence. It was recommended to add two new categories “relating with an authority” and “searching the same piece” to the generalization taxonomy.


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OFLAZ, Gülçin; DEMIRCIOĞLU, Handan. Determining Ways of Thinking and Understanding Related To Generalization of Eighth Graders. International Electronic Journal of Elementary Education, [S.l.], v. 11, n. 2, p. 99-112, nov. 2018. ISSN 1307-9298. Available at: <>. Date accessed: 15 oct. 2019.


Baki, A., Kartal, T. (2004). Kavramsal ve işlemsel bilgi bağlamında lise öğrencilerinin cebir bilgilerinin karakterizasyonu. Türk Eğitim Bilimleri Dergisi, 2(1), 27-46.
Barbosa, A. (2011). Patterning problems: sixth graders' ability to generalize. In M. Pytlak, T. Rowland & E. Swoboda (Eds.), Proceedings Of The Seventh Congress Of The European Society For Research In Mathematics Education (pp. 420-428).
Becker, J. R., & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. Chick & J.L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (pp. 4-121 through 4- 128). Melbourne, Australia: University of Melbourne.
Becker, J. R., & Rivera, F. (2006). Sixth graders’ figural and numerical strategies for generalizing patterns in algebra'. In Alatorre, S., Cortina, J., Sâiz, M. & Méndez, A. (eds), Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Mérida, Mexico, Universidad Pedagôgica Nacional, 2, pp. 95-101.
Booker, G. (2009). Algebraic Thinking: Generalizing number and geometry to express patterns and properties succinctly. Griffith: Griffith University Brisbane.
Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades.(Report No. 002). Madison, WI: National Center For Improving Student Learning And Achievement In Mathematics And Science.
Carraher, D. W., Martinez, M. V., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM, 40(1), 3-22.
Chua, B., & Hoyles, C. (2011). Secondary school students’ perception of best help generalizing strategies. In Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (CERME).
Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A.E. Kelly, R. Lesh, & J. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 68-95). New York: Routledge.
Cobb, P., Jackson, K., & Dunlap, C. (2014). Design research: An analysis and critique. In L. English & D. Kirshner (Eds.), Handbook of international research in mathematics education, 481-503.
Cooper, R. (1991). The role of mathematical transformations and practice in mathematical development. In L. Steffe (Ed.), Epistemological Foundations of Mathematical Experience. New York: Springer-Verlag.
Elia, I., & Spyrou, P. (2006). How students conceive function: a triarchic conceptualsemiotic model of the understanding of a complex concept. The Montana Mathematics Ensthusiant, 3(2), 256–272.
Ellis, A. B. (2007). A taxonomy for categorizing generalizations: Generalizing actions and reflective generalizations. The Journal of the Learning Sciences, 16(2), 221-262.
Friel, S. N., & Markworth, K. A. (2009). A framework for analyzing geometric pattern tasks. Mathematics Teaching in the Middle School, 15(1), 24-33.
Gambrill, E. (1999). Evidence-based practice: An alternative to authority-based practice. Families in Society: The Journal of Contemporary Social Services, 80(4), 341-350.
Garcia-Cruz, J. A., & Martinon, A. (1997). Actions and invariant schemata in linear generalising problems. In E. Pehkonen (ed.) Proceedings of the 21th International Conference for the Psychology of Mathematics Education, University of Helsinki, Vol. 2, 289-296.
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. (2008a). DNR perspective on mathematics curriculum and instruction, Part I: focus on proving. ZDM 40(3), 487-500.
Harel, G. (2008b). A DNR perspective on mathematics curriculum and instruction, Part II: with reference to teacher’s knowledge base. ZDM, 40(5), 893-907.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and Procedural Knowledge: The Case of Mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum
Kieren, T. E., & Pirie, S. E. (1991). Recursion and the mathematical experience. In L. Steffe (ed.), The Epistemology of Mathematical Experience, Springer Verlag Psychology Series, New York, pp. 78-101.
Kirwan, J. V. (2015). Preservice secondary mathematics teachers' knowledge of generalization and justification on geometric-numerical patterning tasks. Doctoral Dissertation, Illınoıs State Unıversıty, USA.
Küchemann, D. (1978). Children's understanding of numerical variables. Mathematics In School, 7(4), 23-26.
Kuş, E. (2006). Sosyal bilimlerde bilgisayar destekli nitel veri analizi: Örnek program NVivo ile gösterimler. Ankara: Anı.
Lannin, J. K. (2003). Developing algebraic reasoning through generalizaton. Mathematics Teaching in the Middle School, 8(7), 342.
Lannin, J. K. (2004). Developing mathematical power by using explicit and recursive reasoning. Mathematics Teacher, 98(4), 216-223.
Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258.
Lobato, J. (2003). How Design Experiments Can Inform a Rethinking of Transfer and ViceVersa. Educational Researcher, 32(1), 17-20.
Molina, M., Castro, E., & Castro, E. (2007). Teaching experiments within design research. The International Journal of Interdisciplinary Social Sciences, 2(4), 435-440.
Orton, A., & Orton, J. (1999). Pattern and Approach to Algebra. In A. Orton (Ed.), Pattern in the Teaching and Learning of Mathematics (pp. 104-124). London: Cassel.
Piaget, J. (1964). Development and learning. In R. E. Ripple & V. N. Rockcastle (Eds.), Piaget Rediscovered, (pp. 7-20).
Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An analytical model for studying the development of learners’ mathematical ideas and reasoning using videotape data. The Journal of Mathematical Behavior, 22(4), 405-435.
Rivera, F. D. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73, 297-328.
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.
Simon, M. A. (2000). Research on the development of teachers: The teacher development experiment. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education (pp. 335–360). Mahwah, N.J.: Erlbaum.
Skemp, R. (1976). Instrumental understanding and relational understanding. Mathematics Teaching, 77, 20-26.
Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147-164.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research on design in mathematics and science education (pp. 267–307). Hillsdale, NJ: Lawrence Erlbaum Associates.
Tanışlı, D., & Özdaş, A. (2009). İlköğretim beşinci sınıf öğrencilerinin örüntüleri genellemede kullandıkları stratejiler. Educational Sciences: Theory & Practice, 9(3), 1453-1497.
Tanışlı, D., & Yavuzsoy Köse, N. (2011). Lineer şekil örüntülerine ilişkin genelleme stratejileri: Görsel ve sayısal ipuçlarının etkisi. Eğitim ve Bilim, 36(160).
Windsor, W. J. J. (2009). Algebraic thinking-more to do with why, than x and y. Proceedings of the 10th International Conference “Models in Developing Mathematics Education”. The Mathematics Education into the 21st Century Project. Saxony, Germany.
Yerushalmy, M. (1993). Generalization, induction, and conjecturing: A theoretical perspective. In L. Schwartz, M. Yerushalmy & B. Wilson (Eds.), The geometric suppose: What is it a case of? (pp. 57-84). Hillsdale, NJ: Lawrence Erlbaum.
Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379-402.