Abstract

This study investigates sixth-grade Turkish students’ pattern-generalization approaches among arithmetical generalization, algebraic generalization, and naïve induction. A qualitative case study design was employed. The data was collected from four sixth-grade students through the Pattern Questionnaire (PQ) and individual interviews based on the questionnaire. The findings revealed that all students generalized near terms using arithmetical generalization as the first step and then they mostly looked for a general rule through memorized procedures by skipping far term generalization. When they found the general rule, far terms were calculated by rote. In other words, students did not generalize the pattern to far terms using an algebraic generalization. The current study's findings would give valuable information to the mathematics educators regarding the necessity of avoiding creating a procedural instructional environment by focusing on the rote procedure of finding the general rule of a pattern. These findings would also expand the horizons of curriculum developers regarding the importance of objectives about both near terms and far term generalization by progressing from arithmetical generalization to algebraic generalization.

Keywords

References

• Aké, L. P., Godino, J. D., Gonzato, M., & Wilhelmi, M. R. (2013). Proto-algebraic levels of mathematical thinking. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 1-8). Leibniz-Institut.
• Akkan, Y., & Çakıroğlu, Ü. (2012). Generalization strategies of linear and quadratic pattern: a comparison of 6th-8th grade students. Education and Science, 37(165), 104-120.
• Amit, M., & Neria, D. (2008). “Rising to the challenge”: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40(1), 111-129. https://doi.org/10.1007/s11858-007-0069-5
• Ayber, G. (2017). An analysis of secondary school mathematics textbooks from the perspective of fostering algebraic thinking through generalization (Publication no. 463446). [Master’s thesis, Anadolu University]. Council of Higher Education Thesis Center.
• 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). ERME.
• Barbosa, A., & Vale, I. (2015). Visualization in pattern generalization: Potential and challenges. Journal of the European Teacher Education Network, 10, 57-70.
• Becker, J. R., & Rivera, F. (2006, July). Establishing and justifying algebraic generalization at the sixth grade level. In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 465-472). Charles University.
• Becker, J.R. & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. Chick & J.L. Vincent (Eds.), Proceedings of the twenty-ninth Conference of the International Group for the Psychology of Mathematics Education (pp. 121-128). The University of Melbourne.
• Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Heinemann.
• Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115. https://doi.org/0.2307/30034843
• Cooper, T. J., & Warren, E. (2011). Years 2 to 6 students’ ability to generalise: Models, representations, and theory for teaching and learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives. Advances in Mathematics Education (pp. 187–214). Springer-Verlag. https://doi.org/10.1007/978-3-642-17735-4_12
• Cresswell, J. W. (2013). Qualitative inquiry and research design: Choosing among five approaches. Sage.
• Demonty, I., Vlassis, J., & Fagnant, A. (2018). Algebraic thinking, pattern activities and knowledge for teaching at the transition between primary and secondary school. Educational Studies in Mathematics, 99(1), 1-19. https://doi.org/10.1007/s10649-018-9820-9
• Dörfler, W. (2008). En route from patterns to algebra: Comments and reflections. ZDM Mathematics Education, 40(1), 143–160. https://doi.org/10.1007/s11858-007-0071-y
• 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, 197 - 215. https://doi.org/10.1080/0020739X.2015.1068391
• Firdaus, A. M., Juniati, D., & Pradnyo, P. (2019, February). The characteristics of junior high school students in pattern generalization. Journal of Physics: Conference Series, 1157(4), 042080. https://doi.org/10.1088/1742-6596/1157/4/042080
• Garcia-Cruz, J., & Martinón, A. (1998). Levels of generalization in linear patterns. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 2, pp. 329–336). PME.
• Girit, D., & Akyüz, D. (2016). Algebraic thinking in Middle School students at different grades: Conceptions about generalization of patterns. Necatibey Faculty of Education, Electronic Journal of Science and Mathematics Education, 10(2), 243-272. https://doi.org/10.17522/balikesirnef.277815
• Godino, J. D., Neto, T., Wilhelmi, M., Aké, L., Etchegaray, S., & Lasa, A. (2015, February). Algebraic reasoning levels in primary and secondary education. In K. Krainer & N. Vondrova (Eds.), CERME 9-Ninth Congress of the European Society for Research in Mathematics Education (pp. 426-432). HAL Open Science.
• Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–546). Erlbaum.
• Kamol, N. & Har, Y. B. (2010). Upper Primary School students’ algebraic thinking. In L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education: Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia (pp. 289-296). MERGA.
• Knuth, E., Alibali, M. W., Weinberg, A., McNeil, N., & Stephens, A. (2005). Middle school students’ understanding of core algebraic concepts: Equality & variable. International Reviews on Mathematical Education, 37(1), 68–76. https://doi.org/10.1007/BF02655899
• Lannin, J. K. (2004). Developing mathematical power using explicit and recursive reasoning. Mathematics Teacher, 98(4), 216-223. https://doi.org/10.5951/MT.98.4.0216
• Lannin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding?. The Journal of Mathematical Behavior, 25(4), 299-317. https://doi.org/10.1016/j.jmathb.2006.11.004
• Lee, L., & Wheeler, D. (1987). Algebraic thinking in high school students: Their conceptions of generalisation and justification [Research report]. Concordia University.
• Lin, F. L., & Yang, K. L. (2004). Differentiation of Students' Reasoning on Linear and Quadratic Geometric Number Patterns. In M. J. Høines, & A. B. Fuglestad (Eds.), International Group for the Psychology of Mathematics Education (Vol. 4, pp. 457–464). Bergen University College.
• Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational studies in mathematics, 67(3), 255-276. https://doi.org/10.1007/s10649-007-9104-2
• Mason, J., Graham, A., & Johnston-Wilder, S. (2005). Developing thinking in algebra. Sage.
• Maudy, S. Y., Didi, S., & Endang, M. (2018). Students’ algebraic thinking level. International Journal of Information and Education Technology, 8(9), 672-676. https://doi.org/10.18178/ijiet.2018.8.9.1120
• Miller, J., & Warren, E. (2012). An exploration into growing patterns with young Australian Indigenous students. In J. Dindyal, L. P. Cheng, & S. F. Ng (Eds.), Proceedings of the 35th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 505-512). MERGA.
• Ministry of National Education [MONE]. (2018), Matematik Dersi Öğretim Programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar) [Mathematics curricula program for primary and middle school grades from 1st to 8th grade level]. Author.
• Moss, J., Beatty, R., McNab, S. L., & Eisenband, J. (2006, April). The potential of geometric sequences to foster young students’ ability to generalize in mathematics [Paper presentation]. Annual meeting of the American Educational Research Association, San Francisco.
• National Council of Teachers of Mathematics [NCTM]. (1997). A framework for constructing a vision of algebra. Algebra working group document. Author.
• National Council of Teachers of Mathematics [NCTM]. (2010). Making it happen: a guide to interpreting and implementing common core state standards for mathematics. Author.
• Noss, R., Healy, L., & Hoyles, C. (1997). The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, 33, 203-233. https://doi.org/10.1023/A:1002943821419
• Ontario Ministry of Education. (2013). Paying attention to algebraic reasoning: K-12. Author.
• Orton, A. & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (pp. 104-120). Cassell.
• Ozdemir, E., Dikici, R., & Kultur, M. N. (2015). Students’ pattern generalization process: the 7th grade sample. Kastamonu Education Journal, 23(2), 523-548.
• Radford, L. (1996). Some reflections on teaching algebra through generalization. In N. Bednarz, C. Kieran & L. Lee (Eds.), Approaches to algebra (pp. 107-114). Kluwer. https://doi.org/10.1007/978-94-009-1732-3_7
• Radford, L. (2000). Students’ processes of symbolizing in algebra: A semiotic analysis of the production of signs in generalizing tasks. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81–88). Hiroshima University.
• 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. (2006). Algebraic Thinking and the Generalization of Patterns: A Semiotic Perspective. In J. L. C. S. Alatorre, M. Sáiz & A. Méndez (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, North American Chapter (Vol. 1, pp. 2-21). Mexico: Mérida.
• 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
• Radford, L. (2010a). Layers of generality and types of generalization in pattern activities. PNA, 4(2), 37-62. https://doi.org/10.30827/pna.v4i2.6169
• Radford, L. (2010b). Signs, gestures, meanings: Algebraic thinking from a cultural semiotic perspective. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello, F. (Eds.), Proceedings of the Sixth Conference of European Research in Mathematics Education (pp. 33–53). Université Claude Bernard, CERME.
• Radford, L., Bardini, C., & Sabena, C. (2006, July). Rhythm and the grasping of the general. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 393-400). IGPME.
• Rivera, F. D. (2013). Teaching and learning patterns in school mathematics. Springer.
• Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. The Journal of Mathematical Behavior, 14(1), 15-39. https://doi.org/10.1016/0732-3123(95)90022-5
• Somasundram, P., Akmar, S. N., & Eu, L. K. (2019). Pattern generalisation by year five pupils. International Electronic Journal of Mathematics Education, 14(2), 353-362. https://doi.org/10.29333/iejme/5719
• Spangenberg, E. D., & Pithmajor, A. K. (2020). Grade 9 mathematics learners’ strategies in solving number-pattern problems. EURASIA Journal of Mathematics, Science and Technology Education, 16(7), em1862. https://doi.org/10.29333/ejmste/8252
• Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147-164. https://doi.org/10.1007/BF00579460
• Stacey, K., & MacGregor, M. (2001). Curriculum reform and approaches to algebra. In R. Lins (Ed.), Perspectives on school algebra (pp. 141–153). Kluwer Academic. https://doi.org/10.1007/0-306-47223-6_8
• Vale, I., & Cabrita, I. (2008, April). Learning through patterns: a powerful approach to algebraic thinking [Paper presentation]. European Teacher Education Network Annual Conference, Liverpool.
• Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2007). Elementary and Middle School Mathematics: Teaching Developmentally. Allyn & Bacon.
• Varhol, A., Drageset, O. G., & Hansen, M. N. (2021). Discovering key interactions. How student interactions relate to progress in mathematical generalization. Mathematics Education Research Journal, 33, 365-382. https://doi.org/10.1007/s13394-020-00308-z
• Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra. Mathematics Education Research Journal, 15(2), 122-137. https://doi.org/10.1007/BF03217374

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