Examining Students’ Mathematical Understanding of Patterns by Pirie-Kieren Model

Year 2020, Volume: 35 Issue: 3, 644 - 661, 31.07.2020

Pınar Güner Tuğba Uygun

Abstract

The purpose of this study is to investigate students’ mathematical understanding of patterns. Three 7th grade students who enrolled in a public school solved the questions regarding the patterns. Semi-structured interviews were conducted with them about their solutions. The data were analyzed by using the Pirie-Kieren theory of mathematical understanding. The findings of this study revealed that students’ mathematical understanding varied between first six levels from primitive knowing to observing and their mathematical understanding mostly occurred between Image Making and Formalising layers. In terms of theory, students were able to pass the first and second “Don’t Need” boundaries but they could not progress their understanding over the third “Don’t Need” boundary. The results also illustrated that all of the students had knowledge about the patterns. In order to find the general rule of the pattern, they mostly endeavored to determine a formula and check its correctness by writing initial three steps.

Keywords

mathematical understanding, Pirie-Kieren model, patterns, middle school students

Öğrencilerin Örüntülere İlişkin Matematiksel Anlamalarının Pirie-Kieren Modeli ile İncelenmesi

Abstract

Bu çalışmanın amacı öğrencilerin örüntülere ilişkin matematiksel anlamalarını araştırmaktır. Bir devlet okulunda öğrenim gören üç yedinci sınıf öğrencisi çalışmaya katılmış ve örüntülerle ilgili soruları çözmüştür. Bu öğrencilerle çözümlerine yönelik yarı yapılandırılmış görüşmeler gerçekleştirilmiştir. Veriler Pirie-Kieren teorisi kullanılarak analiz edilmiştir. Elde edilen bulgular, öğrencilerin matematiksel anlamalarının ön bilgiden gözlem yapmaya kadar ilk altı düzey arasında çeşitlilik gösterdiğini ve genellikle imaj oluşturma ile formülleştirme aşamalarında gerçekleştiğini ortaya koymaktadır. Teoriye göre, öğrencilerin ilk ve ikinci ihtiyaç duyulmayan sınırların ilerisine gidebildikleri fakat üçüncü sınırı geçemedikleri görülmüştür. Sonuçlar öğrencilerin örüntülere ilişkin bilgilerinin olduğunu ve örüntünün genel formülünü bulmak için çoğunlukla bir kural belirlemeye ve bunun doğrulunu ilk üç terim için kontrol etmeye çalıştıklarını göstermektedir.

Keywords

matematiksel anlama, Pirie-Kieren model, örüntüler, ortaokul öğrencileri

References

• Alajmi, A. H. (2016). Algebraic generalization strategies used by Kuwaiti pre-service teachers. International journal of science and mathematics education, 14(8), 1517 1534.
• Argat, A. (2012). Pirie-Kieren dinamik modeli ile öğrencilerde matematiksel anlamanın gelişiminin incelenmesi. Unpublished master’s thesis. University of Marmara Education Science Institute: İstanbul.
• Arslan, E. (2013). Ortaokul öğrencilerinin “Pirie ve Kieren modeli” ne göre matematiksel anlama seviyelerinin belirlenmesi. Unpublished master thesis, Erzincan Üniversitesi Fen Bilimleri Enstitüsü, Erzincan.
• Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2007). How can we assess mathematical understanding? In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Ed.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education. 2, pp. 41-48. Seoul: PME.
• Borgen, K. L. (2006). From mathematics learner to mathematics teacher: Preservice teachers’ growth of understanding of teaching and learning mathematics. Doctoral dissertation, University of British Columbia.
• Burns, M. (2000). About teaching mathematics. A-K 8 research (2nd ed.) Sausaluto, California. CA: Math Solutions Publication.
• Cavey, L. O., & Berenson, S. B. (2005). Learning to teach high school mathematics: Patterns of growth in understanding right triangle trigonometry during lesson plan study. Journal of Mathematical Behavior, 24, 171–190.
• Codes, M., González Astudillo, M. T., Delgado Martín, M. L., & Monterrubio Pérez, M.C. (2013). Growth in the understanding of infinite numerical series: A glance through the Pirie and Kieren theory. International Journal of Mathematical Education in Science and Technology, 44 (5), 652-662.
• Dole, S. (2000). Promoting percent as a proportion in eighth-grade mathematics. School Science and Mathematics, 100(7), 380–389.
• English, L. D., & Warren, E. A. (1998). Introducing the variable through pattern exploration. The mathematics teacher, 91(2), 166.
• Fox, J. (2005). Child-initiated mathematical patterning in the pre-compulsory years. In H. L. Chick & J. L. Vincent (Eds.), Proceeding of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 313-320). Melbourne: PME.
• George, L. (2017). Children's learning of the partitive quotient fraction sub-construct and the elaboration of the don't need boundary feature of the Pirie-Kieren theory. Unpublished Doctoral dissertation, University of Southampton.
• Grinevitch, O. A. (2004). Student understanding of abstract algebra: a theoretical examination. Unpublished PhD Thesis, Bowling Green State University.
• Gökalp, N. D. (2012). A study on sixth grade students’ understanding of multiplication of fractions using Pirie and Kieren model. Unpublished doctoral dissertation, Middle East Technical University, Ankara, Turkey.
• Gülkilik, H., Ugurlu, H. H., & Yürük, N. (2015). Examining Students' Mathematical Understanding of Geometric Transformations Using the Pirie-Kieren Model. Educational Sciences: Theory and Practice, 15(6), 1531-1548.
• Hallagan, J., Rule, A., & Carlson, L. (2009). Elementary school pre-service teachers’ understanding of algebraic generalizations. The Montana Mathematics Enthusiast, 6(1), 201–206.
• Hargreaves, M., Threlfall, J., Frobisher, L., & Shorrocks-Taylor, D. (1999). Children’s strategies with linear and quadratic sequences. Pattern in the teaching and learning of mathematics, 67-83.
• Higgins, J. & Parsons, R. (2009). A successful professional development model in mathematics a system-wide New Zealand case. Journal of Teacher Education, 60 (3), 231-242.
• Hollebrands, K. F. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. The Journal of Mathematical Behavior, 22(1), 55–72.
• Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and learning, 7(3), 231-258.
• 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.
• Lawan, A. (2011). Growth of students’ understanding of part-whole sub-construct of rational number on the layers of Pirie-Kierien theory. The Association for Mathematics Education of South Africa, 69.
• Lester, F. K. (2005). On the theoretical, conceptual and philosophical foundations for research in mathematics education. ZDM, 37(6), 457-467.
• Li, X., Ding M., Capraro, M. M., & Capraro, R. M. (2008). Sources of differences in children’s understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and in the United States. Cognition and Instruction, 26, 195–217.
• Lyndon, C. M. (2008). Folding back and the dynamical growth of mathematical understanding: Elaborating the Pirie-Kieren theory. Journal of Mathematical Behavior, 27, 64-85.
• Mabotja, K. S. (2017). An exploration of folding back in improving grade 10 students’ reasoning in geometry. Unpublished Master dissertation, University of Limpopo.
• MacCullough, D. L. (2007). A study of experts' understanding of arithmetic mean. Unpublished doctoral dissertation, The Pennsylvania State University, Pennsylvania.
• Manu, S. S. (2005). Growth of mathematical understanding in a bilingual context: analysis and implications. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 3, pp.289-296). Melbourne, Australia: University of Melbourne.
• Martin, L. C. (2008). Folding back and the dynamical growth of mathematical understanding Elaborating the Pirie-Kieren theory. Journal of Mathematical Behavior, 27(1), 64-85.
• Martin, L., Towers, J., & Pirie, S. (2006). Collective mathematical understanding as improvisation. Mathematical Thinking and Learning, 8(2), 149–183.
• Martin, L., & Towers, J. (2016). Folding back and growing mathematical understanding: a longitudinal study of learning. International Journal for Lesson and Learning Studies, 5(4), 281-294.
• Miles, M. B. & Huberman, A. M. (1994). An expanded sourcebook qualitative data analysis. United States of America: Sage Publicatons.
• Muir, T., Beswick, K., & Williamson, J. (2008). "I'm not very good at solving problems": An exploration of students' problem solving behaviours. Journal of Mathematical Behavior, 27(3), 228-241.
• NCTM. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
• Nillas, L. A. (2010) Characterizing pre-service teachers’ mathematical understanding of algebraic relationships. International Journal for Mathematics Teaching and Learning, 1-24.
• Phillips, E. (1995) Issues surrounding algebra. In C. B. Lacampagne, W. Blair, & J. Kaput (Eds.), The algebra initiative colloquium, 2, (pp. 69-81). Washington, DC: U. S. Dept. of Education.
• Pirie, S. E. B., & Kieren, T. (1989). A recursive theory of mathematical understanding. For the Learning of Mathematics, 9(3), 7–11.
• Pirie, S. E. B., & Kieren, T. E. (1991). Folding back: Dynamics in the growth of mathematical understanding. In F. Furinghetti (Ed.), Proceedings of the 15th Annual Meeting of the International Group for the Psychology of Mathematics Education, 3, (pp. 169–176). Assisi, Italy.
• Pirie, S. E. B., & Kieren, T. E. (1992). Watching Sandy's understanding grow. Journal of Mathematical Behavior, 11, 243-257.
• Pirie, S. E., & Kieren, T. (1994). Growth in mathematical understanding: how can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165–190.
• Reys, R. E., Suydam, M. N., Lindquist M. M., & Smith. N. L. (1998). Helping children learn mathematics (5th ed.). Boston, MA: Allyn and Bacon.
• Stake, R. (1995). The art of case study research. Thousand Oaks, CA: Sage.
• Thom, J. S., & Pirie S. E.B. (2006). Looking at the complexity of two young children’s understanding of number. Journal of Mathematical Behavior 25, 185–195.
• Torbeyns, J., De Smedt, B., Stassens, N., Ghesquie `re, P., & Verschaffel, L. (2009). Solving subtraction problems by means of indirect addition. Mathematical Thinking and Learning, 11, 79–91.
• Towers, J., & Martin, L. C. (2006). Improvisational coactions and the growth of collective mathematical understanding IN: Alatorre, J.L., Saiz, C.M. and Mendez, A. (eds.) Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Merida, Mexico. 631-638.
• Towers, J., & Martin, L. C. (2014). Building mathematical understanding through collective property noticing. Canadian Journal of Science, Mathematics and Technology Education, 14 (1), 58-75.
• Trends in International Mathematics and Science Study (2007). Highlights from TIMSS 2007: Mathematics and Science Achievement of US Fourth and Eighth-Grade Students in an International Context. Retrieved from http://nces.ed.gov/pubs2009/2009001.pdf.
• Valcarce, M. C., Astudillo, M. T. G., Martín, M. L. D., & Pérez, M. C. M. (2012). Growth in the understanding of the concept of infinite numerical series: A glance through Pirie and Kieren theory.
• Van de Walle J. A. (2004). Elementary and Middle School Mathematics. Teaching Developmentally. (5th ed.). Boston: Allyn &Bacon.
• Warner, L. B. (2008). How do students’ behaviours relate to the growth of their mathematical ideas? The Journal of Mathematical Behaviour, 27(3), 206-227.
• Wilson, P. H., & Stein, C. C. (2007). The role of representations in growth of understanding in pattern–finding tasks. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.550.2395Wright, V. (2014). Frequencies as proportions: Using a teaching model based on Pirie and Kieren’s model of mathematical understanding. Mathematics Education Research Journal, 26(1), 101-128.
• Yin, R. K. (2003). Case study research: Design and method (3rd ed.). Thousand Oaks, CA: Sage.