Gifted Students’ repeating patterning skills and cognitive demand levels

Yıl 2023, Cilt: 6 Sayı: 1, 70 - 95, 31.05.2023

Fatma Erdoğan , Neslihan Gül

https://doi.org/10.33400/kuje.1221801

Cited By: 1

Öz

The ability to generalize, one of the key characters of mathematical giftedness, is related to mathematical patterns. Patterns and especially repeating patterns come to the fore as a context for the development of algebraic and functional thinking at an early age. In addition, determining the cognitive effort that students put forward in the process of working with repeating patterns is important for the development of patterning skills. In line with what has been stated, the aim of this study was to explore the repeating patterning skills of gifted students and their cognitive demand levels. In the study, case study design was used. Participants are five fifth grade students who were diagnosed as gifted through diagnostic tests. The data were collected with the "Repeating Number Pattern Task Form" consisting of open-ended problems. The data collection method is task-based interview. The data were analyzed by thematic analysis method. According to the findings, all students correctly determined the immediate, near, far term, and rule of the repeating number pattern task. According to the results of the study, students used “recursive”, “counting”, “division with remainder”, and “counting up/down from a multiple” strategies to find the immediate, near, and far term. All students explained the relationship in the arrangement of the numbers in the pattern by determining the unit of repeat. The results of the study show that students exhibit cognitive demand at the level of “procedures without connections” and “procedures with connections” to find the immediate and near term of the pattern task. In addition, students exhibited cognitive demand at the level of “procedures with connections” to find the far term and rule of the pattern.

Anahtar Kelimeler

özel yeteneklilik, matematiksel özel yeteneklilik, örüntüler, bilişsel istem, matematik eğitimi

Özel yetenekli öğrencilerin tekrarlanan örüntü becerileri ve bilişsel istem düzeyleri

Öz

Matematiksel özel yetenekliliğin kilit karakterlerinden biri olan genelleme becerisi, matematiksel örüntülerle ilişkilidir. Erken yaşlarda cebirsel ve fonksiyonel düşünmenin gelişimi için bir bağlam olarak örüntüler ve özellikle tekrarlanan örüntüler öne çıkmaktadır. Ayrıca, öğrencilerin tekrarlanan örüntülerle çalışma süreçlerinde ortaya koydukları bilişsel çabanın belirlenmesi, örüntü becerisinin gelişimi açısından önemlidir. Belirtilenler doğrultusunda, bu çalışmanın amacı, özel yetenekli öğrencilerin tekrarlanan örüntü becerilerini ve tekrarlanan örüntülerle çalışma sürecinde ortaya koydukları bilişsel istem düzeylerini keşfetmektir. Çalışmada, durum çalışması deseni kullanılmıştır. Katılımcılar, beşinci sınıf düzeyinde öğrenim gören, tanılama testleri aracılığıyla özel yetenekli tanısı konulan beş öğrencidir. Veriler, açık uçlu problemlerden oluşan “Tekrarlanan Sayı Örüntüsü Görev Formu”yla toplanmıştır. Veri toplama yöntemi, görev temelli görüşmedir. Veriler tematik analiz yöntemiyle çözümlenmiştir. Bulgulara göre, tüm öğrenciler, tekrarlanan sayı örüntüsü görevinin yakın, orta, uzak terimine ve kuralına doğru bir şekilde ulaşmıştır. Çalışma sonuçlarına göre, özel yetenekli öğrenciler tekrarlanan sayı örüntüsü görevinin yakın, orta ve uzak terimini bulmak için “yinelemeli”, “sayma”, “bölümden kalanı sayma” ve “çarpım üzerine sayma” stratejilerini kullanmışlardır. Örüntüde yer alan rakamların dizilişindeki ilişkiyi tüm öğrenciler tekrar birimini belirleyerek açıklamıştır. Çalışma sonuçları, özel yetenekli öğrencilerin örüntü görevinin yakın ve orta uzaklıktaki terimini bulmak için “bağlantısız işlemler” ve “bağlantılı işlemler” düzeyinde bilişsel istem sergilediklerini göstermiştir. Ayrıca, öğrenciler örüntünün uzak terimini ve kuralını bulmak için “bağlantılı işlemler” düzeyinde bilişsel istem sergilemişlerdir.

Anahtar Kelimeler

özel yeteneklilik, matematiksel özel yeteneklilik, örüntüler, bilişsel istem, matematik eğitimi

Kaynakça

• 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, 40(1), 111-129.
• Assmus, D. (2018). Characteristics of mathematical giftedness in early primary school age. F. M. Singer (Ed.) içinde, Mathematical creativity and mathematical giftedness: Enhancing creative capacities in mathematically promising students (ss. 145–167). Springer.
• Assmus, D., & Fritzlar, T. (2022). Mathematical creativity and mathematical giftedness in primary school age-An interview study on creating figural patterns. ZDM-Mathematics Education, 54, 113–131. https://doi.org/10.1007/s11858-022-01328-8
• Benedicto, C., Gutiérrez, A., & Jaime, A. (2017). When the theoretical model does not fit our data: a process of adaptation of the cognitive demand model. T. Dooley & G. Gueudet (Eds.) içinde, Proceedings of the CERME 10 (ss. 2791-2798). ERME.
• Benedicto, C., Jaime, A., & Gutiérrez, A. (2015). Análisis de la demanda cognitiva de problemas de patrones geométricos. C. Fernández, M. Molina, & N. Planas (Eds.) içinde, Investigación enEducación matemática XIX (ss. 153–162). SEIEM.
• Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa
• Cabral, J., Oliveira, H., & Mendes, F. (2021). Preservice teachers’ mathematical knowledge about repeating patterns and their ability to notice preschoolers algebraic thinking. Acta Scientiae Revista de Ensino de Ciências e Matemática, 23(7), 30-59.
• Collins, M. A., & Laski, E. V. (2015). Preschoolers’ strategies for solving visual pattern tasks. Early Childhood Research Quarterly, 32, 204–214. https://doi.org/10.1016/j.ecresq.2015.04.004
• Dayan, Ş. (2017). Üstün yetenekli ve normal öğrencilerin matematiksel örüntü başarılarının incelenmesi (Yayımlanmamış yüksek lisans tezi). Abant İzzet Baysal Üniversitesi.
• 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
• Diago, P. D., Yáñez, D. F., & Arnau, D. (2022). Relations between complexity and difficulty on repeating-pattern tasks in early childhood (Relaciones entre complejidad y dificultad en tareas con patrones reiterativos en la primera infancia). Journal for the Study of Education and Development, 45(2), 311-350. https://doi.org/10.1080/02103702.2021.2000127
• Eraky, A., Leikin, R., & Hadad, B. S. (2022). Relationships between general giftedness, expertise in mathematics, and mathematical creativity that associated with pattern generalization tasks in different representations. Asian Journal for Mathematics Education, 1(1), 36-51.
• Fritzlar, T., & Karpinski-Siebold, N. (2012). Continuing patterns as a component of algebraic thinking—An interview study with primary school students. Pre-proceedings of the 12th International Congress on Mathematical Education içinde (ss. 2022–2031). ICMI.
• Girit-Yildiz, D., & Durmaz, B. (2021). A gifted high school student’s generalization strategies of linear and nonlinear patterns via gauss’s approach. Journal for the Education of the Gifted, 44(1), 56-80. https://doi.org/10.1177/0162353220978295
• Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. A. E. Kelly & R. Lesh (Eds.) içinde, Handbook of research design in mathematics and science education (ss. 517–546). Lawrence Erlbaum.
• Gutiérrez, A., Benedicto, C., Jaime, A., & Arbona, E. (2018a). The cognitive demand of a gifted student’s answers to geometric pattern problems. F. M. Singer (Ed.) içinde, Mathematical creativity and mathematical giftedness (ss. 196-198). Springer International Publishing.
• Gutiérrez, A., Jaime, A., & Gutiérrez, P. (2021). Networked analysis of a teaching unit for primary school symmetries in the form of an e-book. Mathematics, 9(8), 832. https://doi.org/10.3390/math9080832
• Gutiérrez, A., Ramírez, R., Benedicto, C., Beltrán-Meneu, M. J., & Jaime, A. (2018b). Visualization abilities and complexity of reasoning in mathematically gifted students’ collaborative solutions to a visualization task. A networked analysis. K. S. S. Mix, & M. T. Battista (Eds.) içinde, Visualizing mathematics. The role of spatial reasoning in mathematical thought (ss. 309-337). Springer.
• Hadar, L. L., & Ruby, T. L. (2019). Cognitive opportunities in textbooks: the cases of grade four and eight textbooks in Israel. Mathematical Thinking and Learning, 21(1), 54-77.
• Kabael, T. U., & Tanışlı, D. (2010). Cebirsel düşünme sürecinde örüntüden fonksiyona öğretim. İlköğretim Online, 9(1), 213-228.
• Keleş, T., & Yazgan, Y. (2022). Indicators of gifted students’ strategic flexibility in non-routine problem solving. International Journal of Mathematical Education in Science and Technology, 53(10), 2797-2818. https://doi.org/10.1080/0020739X.2022.2105760
• Kidd, J. K., Pasnak, R., Gadzichowski, K. M., Gallington, D. A., McKnight, P., Boyer, C.E., et al. (2014). Instructing first-grade children on patterning improvesreading and mathematics. Early Education & Development, 25, 134–151. http://dx.doi.org/10.1080/10409289.2013.794448
• Kidd, J., Lyu, H., Peterson, M., Hassan, M., Gallington, D., Strauss, L., Patterson, A., & Pasnak, R. (2019). Patterns, mathematics, early literacy, and executive functions. Creative Education, 10(13), 3444–3468. https://doi.org/10.4236/ce.2019.1013266
• Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra: Research into its nature, its learning, its teaching. G. Kaiser (Ed.) içinde, ICME-13 Topical surveys. Springer Open. https://doi.org/10.1007/978-3-319- 32258-2
• Krutetskii, V. A. (1976). The psychology of mathematical abilities in school children. University of Chicago Press.
• Lannin, J.K., Barker, D., & Townsend, B. (2006). Algebraic generalization strategies: Factors inf¬luencing student strategy selection. Mathematics Education Research Journal, 18(3), 3-28.
• Larkin, K., Resnick, I., & Lowrie, T. (2022). Preschool children’s repeating patterning skills: Evidence of their capability from a large scale, naturalistic, Australia wide study. Mathematical Thinking and Learning, 1-16. https://doi.org/10.1080/10986065.2022.2056320
• Leikin R. (2018). Giftedness and high ability in mathematics. S. Lerman (Ed.) içinde, Encyclopedia of mathematics education (ss. 315-325). Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_65
• Leikin, R. (2021). When practice needs more research: The nature and nurture of mathematical giftedness. ZDM-Mathematics Education, 53, 1579–1589. https://doi.org/10.1007/s11858-021-01276-9
• Leikin, R., Koichu, B., Berman, A., & Dinur, S. (2017). How are questions that students ask in high level mathematics classes linked to general giftedness? ZDM-Mathematics Education, 49(1), 65-80. https://doi.org/10.1007/s11858-016-0815-7
• Leikin, R., & Sriraman, B. (2022). Empirical research on creativity in mathematics (education): From the wastelands of psychology to the current state of the art. ZDM-Mathematics Education, 54(1), 1–17. https://doi.org/10.1007/s11858-022-01340-y
• Liljedahl, P. (2004). Repeating pattern or number pattern: The distinction is blurred. Focus on Learning Problems in Mathematics, 26(3), 24–42.
• MacKay, K., & De Smedt, B. (2019). Patterning counts: Individual differences in children’s calculation are uniquely predicted by sequence patterning. Journal of Experimental Child Psychology, 177, 152–165. https://doi.org/10.1016/j. jecp.2018.07.016.
• Maher, C. A., & Sigley, R. (2014). Task-based interviews in mathematics education. S. Lerman (Ed.) içinde, Encyclopedia of Mathematics Education (ss. 579–582). Springer.
• Masingila, J. O., Olanoff, D., & Kimani, P. M. (2018). Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving. Journal of Mathematics Teacher Education, 21(5), 429-450.
• Merriam, S. B. (2018). Nitel araştırma desen ve uygulama için bir rehber (Çeviri Ed. S. Turan). Nobel Yayıncılık.
• Merriam, S. B., & Tisdell, E. J. (2015). Qualitative research: A guide to design and implementation. John Wiley & Sons.
• Miles, M. B., & Huberman, A. M. (1994). An expanded source book: Qualitative data analysis. Sage Publications.
• Miller, R. C. (1990). Discovering mathematical talent. Eric Clearinghouse on Handicapped and Gifted Children.
• 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
• Nguyen, T., Watts, T. W., Duncan, G. J., Clements, D. H., Sarama, J. S., Wolfe, C., et al. (2016). Which preschool mathematics competencies are most predictive of fifth grade achievement? Early Childhood Research Quarterly, 36, 550–560. http://dx.doi.org/10.1016/j.ecresq.2016.02.003
• Orton, A., & Orton, J. (1999). Pattern and the approach to algebra. A. Orton (Ed.) içinde, Pattern in the teaching and learning of mathematics (ss. 104-120). Cassell.
• Özdemir, E., Dikici, R., & Kültür, M. N. (2015). Öğrencilerin örüntüleri genelleme süreçleri: 7. sınıf örneği. Kastamonu Eğitim Dergisi, 23(2), 523-548.
• Öztürk, M., Akkan, Y., & Kaplan, A. (2018). 6-8. sınıf üstün yetenekli öğrencilerin problem çözerken sergiledikleri üst bilişsel beceriler: Gümüşhane örneği. Ege Eğitim Dergisi, 19(2), 446-469. https://doi.org/10.12984/egeefd.316662
• Papic, M. (2007). Promoting repeating patterns with young children-more than just alternating colours! Australian Primary Mathematics Classroom, 12(3), 8-13.
• Papic, M. M., Mulligan, J. T., & Mitchelmore, M. C. (2011). Assessing thedevelopment of preschoolers’ mathematical patterning. Journal for Research in Mathematics Education, 42, 237–268.
• Patton, M. Q. (1990). Qualitative evaluation and research methods (2. Baskı). Sage.
• Paz-Baruch, N., Leikin, M., & Leikin, R. (2022). Not any gifted is an expert in mathematics and not any expert in mathematics is gifted. Gifted and Talented International, 37(1), 25-41. https://doi.org/10.1080/15332276.2021.2010244
• Pinto, E., & Cañadas, M. C. (2021). Generalizations of third and fifth graders within a functional approach to early algebra. Mathematics Education Research Journal, 33(1), 113-134. https://doi.org/10.1007/s13394-019-00300-2
• Pitta-Pantazi, D. (2017). What have we learned about giftedness and creativity? an overview of a five years journey. Leikin, R., Sriraman, B. (Eds.) içinde, Creativity and giftedness. advances in mathematics education (ss. 201-223). Springer, Cham. https://doi.org/10.1007/978-3-319-38840-3_13
• Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26(2), 257–277. https://doi.org/10.1007/s13394-013-0087-2
• Renzulli, J. S. (1978). What makes giftedness? Reexamining a definition. Phi Delta Kappan, 60(3), 180–184
• Rittle-Johnson, B., Fyfe, E. R., McLean, L. E., & McEldoon, K. L. (2013). Emergingunderstanding of patterning in 4-year-olds. Journal of Cognition and Development, 14, 376–396. http://dx.doi.org/10.1080/15248372.2012.689897
• Rittle-Johnson, B., Fyfe, E. R., Hofer, K. G., & Farran, D. C. (2017). Early math trajectories: Low-income children’s mathematics knowledge from ages 4–11. Child Development, 88(5), 1727-1742. http://dx.doi.org/10.1111/cdev.12662
• Rittle-Johnson, B., Zippert, E., & Boice, K. (2019). The roles of patterning and spatial skills in early mathematics development. Early Childhood Research Quarterly, 46(1), 166–178. https://doi.org/10.1016/j.ecresq.2018.03.006
• Rivera, F. D. (2018). Pattern generalization processing of elementary students: Cognitive factors affecting the development of exact mathematical structures. EURASIA Journal of Mathematics, Science and Technology Education, 14(9), Article em1586. https://doi.org/10.29333/ejmste/92554
• Rivera, F. D., & Becker, J. R. (2011). Formation of pattern generalization involving linear figural patterns among middle school students: Results of a three-year study. J. Cai, & E. Knuth (Eds.) içinde, Early algebraization (ss. 323-366). Springer.
• Silver, E. A., & Mesa, V. (2011). Coordinating characterizations of high quality mathematics teaching: Probing the intersection. Y. Li & G. Kaiser (Eds.) içinde, Expertise in mathematics instruction (ss. 63–84). Springer.
• Singer, F. M., Sheffield, L. J., Freiman, V., & Brandl, M. (2016). Research on and activities for mathematically gifted students. Springer Nature.
• Smedsrud, J. (2018) Mathematically gifted accelerated students participating in an ability group: A qualitative interview study. Front. Psychol., 9, 1-12.
• Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem-solving experiences of four gifted students. Journal of Secondary Gifted Education, 14(3), 151-165.
• Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147-164.
• Steen, L. A. (1988). The Science of patterns. Science, 240(4852), 611–616. https://doi.org/10.1126/science.240.4852.611
• Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.
• Stein, M. K., & Smith, M. S. (1998). Mathematical task as a framework for reflection: from research to practice. Mathematics Teaching in the Middle School, 3(4), 268-277.
• Stephens, A. C., Fonger, N. L., Strachota, S., Isler, I., Blanton, M., Knuth, E., & Gardiner, A. (2017). A learning progression for elementary students’ functional thinking. Mathematical Thinking and Learning, 19(3), 143-166.
• Sternberg, R. J. (2003). A broad view of intelligence: The theory of successful intelligence. Consulting Psychology Journal: Practice and Research, 55(3), 139–154.
• Sternberg, R. J., Chowkase, A., Desmet, O., Karami, S., Landy, J., & Lu, J. (2021). Beyond transformational giftedness. Education Sciences, 11(5), 192. https://doi.org/10.3390/educsci11050192
• Sternberg, R. J., & Davidson, J. E. (Eds.). (1986). Conceptions of giftedness. Cambridge University Press.
• Sternberg, R. J., & Grigorenko, E. L. (2004). Successful intelligence in the classroom. Theory into Practice, 43(4), 274–280.
• Tanışlı, D. (2008). İlköğretim beşinci sınıf öğrencilerinin örüntülere ilişkin anlama ve kavrama biçimlerinin belirlenmesi (Yayımlanmamış doktora tezi). Anadolu Üniversitesi.
• Threlfall, J. (1999). Repeating patterns in the early primary years. A. Orton (Ed.) içinde, Pattern in the teaching and learning of mathematics (ss. 18-30). Cassell.
• Tirosh, D., Tsamir, P., Levenson, E., Barkai, R., & Tabach, M. (2019). Preschool teachers’ knowledge of repeating patterns: Focusing on structure and the unit of repeat. Journal of Mathematics Teacher Education, 22(3), 305–325. https://doi.org/10.1007/s10857-017-9395-x
• Tural-Sönmez, M. (2019). Ortaya çıkan modelleme yaklaşımıyla parantez kullanımının anlamlandırılma süreci. Journal of Computer and Education Research, 7(13), 62-89. https://doi.org/10.18009/jcer.499845
• Türkmen, H., & Tanışlı, D. (2019). Cebir öncesi: 3, 4 ve 5. sınıf öğrencilerinin fonksiyonel ilişkileri genelleme düzeyleri. Eğitimde Nitel Araştırmalar Dergisi, 7(1), 344-372. https://doi.org/10.14689/issn.2148-2624.1.7c1s.16m
• Warren, E., & Cooper, T. (2006). Using repeating patterns to explore functional thinking. Australian Primary Mathematics Classroom, 11(1), 9–14.
• Warren, E., & Cooper, T. (2007). Repeating patterns and multiplicative thinking: Analysis of classroom interactions with 9-year-old students that support thetransition from the known to the novel. Journal of Classroom Interaction, 41, 7–17.
• Wijns, N., Torbeyns, J., Bakker, M., De Smedt, B., & Verschaffel, L. (2019). Four-year olds’ understanding of repeating and growing patterns and its association with early numerical ability. Early Childhood Research Quarterly, 49, 152–163. https://doi.org/10.1016/j.ecresq.2019.06.004
• Yin, R. K. (2014). Case study research design and methods (5. Baskı). Sage Publication.
• Zaskis, R., & Liljedahil, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402.
• Zippert, E., Douglas, A., & Rittle-Johnson, B. (2020). Finding patterns in objects and numbers: Repeating patterning in pre-K predicts kindergarten mathematics knowledge. Journal of Experimental Child Psychology, 200, Article 104965. https://doi.org/10.1016/j.jecp.2020.104965