İlköğretim Matematik Öğretmeni Adaylarının Örüntüleri Genelleme Süreçleri: Stratejiler ve Gerekçelendirmeler

Yıl 2017, Cilt: 8 Sayı: 3, 513 - 550, 20.12.2017

Yaşar Akkan Mesut Öztürk Pınar Akkan

https://doi.org/10.16949/turkbilmat.323384

Öz

Bu çalışmanın amacı, ilköğretim matematik öğretmeni adaylarının farklı
örüntü problemleri ile ilgili genelleme stratejilerini incelemek,
genellemelerinin altında yatan gerekçelendirmeleri keşfetmek ve genelleme ile
gerekçelendirmeleri arasındaki ilişkileri belirlemektir. Çalışma nitel
araştırma desenlerinden olgubilim modeline göre tasarlanmıştır. Çalışma, Doğu
Karadeniz Bölgesindeki bir üniversitenin İlköğretim Matematik Öğretmenliği
programında öğrenim gören 4. sınıf öğretmen adayları ile yürütülmüştür. Veri
toplama araçları, hem literatür hem de öğretim üyesi desteğiyle hazırlanan ve
farklı çözüm stratejilerinin ve gerekçelendirme çeşitlerinin üretilebildiği lineer
ve kuadratik örüntü problemleridir. Mülakatlar sonucu toplanan veriler
araştırmanın kavramsal çerçevesi dâhilinde betimsel analiz tekniği kullanılarak
çözümlenmiştir. Elde edilen sonuçlardan, öğretmen adaylarının en yaygın
kullandığı strateji türü fonksiyonel strateji olmakla birlikte, içeriksel,
yinelemeli, tahmin-kontrol ve karma stratejileri de kullanmışlardır. Öğretmen
adaylarının çoğu gerekçelendirmelerini sayısal kontrol yoluyla doğrulama ile
yapmışken, açıklama ve dışsal bilgi kaynağı yoluyla gerekçelendirme yapan
öğretmen adayları da tespit edilmiştir.

Anahtar Kelimeler

Örüntü, gerekçelendirme, Örüntü, genelleme, stratejiler, öğretmen adayları

Pre-Service Elementary Mathematics Teachers’ Generalization Processes of Patterns: Strategies and Justifications

Öz

The aim of this study is to investigate the generalizations created by
pre-service elementary mathematic teachers, to explore the justifications
predicted for their generalizations, and to determine any relationships between
generalization and justification. We used phenomenology design from qualitative
research methods in the study. The study was conducted by the 4th grade
students/pre-service teachers who are studying in a department of Elementary Mathematics
Teaching at a university located in the Eastern Black Sea region. Data
collection tools are linear and quadratic pattern problems which are prepared
with the support of literature and teaching staff and in which different
solution strategies and justification types can be produced. Interviews were
analyzed using the descriptive analysis technique within the conceptual
framework of the research. The results show that the most common type of
strategy used by pre-service teachers was functional strategy, contextual,
recursive, guess-check and mixed strategies. While many of the pre-service
teachers have justified their verification by numerical control, pre-service
teachers who have justified through explanation and external knowledge sources
have also been identified.

Anahtar Kelimeler

Pattern, justification, generalization, strategies, pre-service teachers

Kaynakça

• Akkan, Y. ve Çakıroğlu, Ü. (2012). Generalization strategies of linear and quadratic pattern: A comparison of 6th-8th grade students. Education and Science, 37, 165, 104-120.
• Akkan, Y. (2013). Comparison of 6th-8th graders’ efficiencies, strategies and representations regarding generalization patterns. BOLEMA, 27, 47, 703-732.
• Akturan, U. ve Esen, A. (2008). Nitel Araştırma Yöntemleri, (ss. 83-98). Ankara: Seçkin Yayıncılık.
• 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, 111-129.
• Armstrong, B. E. (1995). Teaching patterns, relationships and multiplication as worthwhile mathematical tasks. Teaching Children Mathematics, 1, 446-450.
• Baki, A. (2008). Kuramdan Uygulamaya Matematik Eğitimi. Harf Eğitim Yayıncılığı, Ankara.
• Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.) Mathematics teachers and children (pp. 216–235). London: Holdder & Stoughton.
• Becker, J. & Rivera, F. (2003). Research on gender and mathematics from multiple perspectives. In N. A. Pateman, B. J. Dougherty & J. Zilliox (Eds.) Proceedings of the Joint Meeting of PME and PMENA, Vol.1 (p 190). Hawaii, University of Hawaii.
• Becker, J. R. & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. L. Chick, & J. L. Vincent (Eds.) Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4 (pp. 121–128). Melbourne: International Group for the Psychology of Mathematics Education.
• Becker, J. R. & Rivera, F. (2007). Factors affecting seventh graders’ cognitive perceptions of patterns involving constructive and deconstructive generalizations. In J. H. Woo, H. C. Lew, K. S. Park & D. Y. Seo (Eds.) Procedings of the 31th Conference of the International Group for the Psychology of Mathematics Education, Vol.4 (pp. 129-136). Seoul: PME.
• Becker, J. R. & Rivera, F. (2006). Sixth graders’ figural and numerical strategies for generalizing patterns in algebra. In S. Alatorre, J. L. Cortina, M. Saiz & A. Mendez (Eds.) Proceeding of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol.2 (pp.95-101). Merida, Mexico: Universidad Pedagogica Nacional.
• Bednarz, N., Kieran, C. & Lee, L. (1996). Approaches to Algebra. London: Kluwer Academic Publisher.
• Bell, A. W. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7, 23–40.
• Blanton, M. L. & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 5–23). Heidelberg, Germany: Springer.
• Chua, B. L. & Hoyles, C. (2010). Generalisation and perceptual agility: How did teachers fare in a quadratic generalsating problem? In M. Joubert (Ed.), Proceedings of the British Society for Research into Learning Mathematics, Vol. 29, No. 2 (pp. 13–18).
• Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics teachers’ beliefs structures and their influence on instructional practices. J Math Teacher Education, 12, 325 – 346.
• Dörfler, W. (1991). Forms and means of generalization in mathematics. In A. J. Bishop (Ed.), Mathematical knowledge: Its growth through teaching (pp. 63–85). Dordrecht, Netherlands: Kluwer.
• Ebersbach, M. & Wilkening, F. (2007). Children’s intuitive mathematics: The development of knowledge about nonlinear growth. Children Development, 78, 296-308.
• Ellis, A. B. (2007a). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
• Ellis, A. B. (2007b). A taxonomy for categorizing generalizations: Generalizing actions and reflective generalizations. The Journal of the Learning Sciences, 16(2), 221–262.
• English, L. D. & Warren, E. A. (1998). Introducing the variable through pattern exploration. Mathematics Teacher, 912, 166–170.
• Feifei, Y. (2005). Diagnostic assessment of urban middle school student learning of pre-algebra patterns. (Doctoral dissertation, Ohio State University, USA).
• Garcia-Cruz, J. A. & Martinon, A. (1997). Actions and invariant schemata in linear generalizing problems. In E. Pehkonen (Ed.) Proceedings of the 21th Conference of the International Group for the Psychology of Mathematics Education. (pp. 289-296). University of Helsinki.
• Harel, G. & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234–283). Providence, RI: American Mathematical Society.
• Harel, G. & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842). Charlotte, NC: Information Age Publishing.
• Herbert, K. & Brown, R. H. (1997). Patterns as tools for algebraic reasoning. Teaching Children Mathematics, 3, 123- 128.
• Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester, (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age Publishing
• Kirwan, J. V. (2013). Pre-service elementary teachers’ anchors for generalization. Poster session presented at the 35th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Chicago, IL.
• Kirwan, J. V. (2015). Preservice secondary mathematics teachers’ knowledge of generalization and justification on geometric numerical patterning tasks. (Doctoral dissertation, Illinois State University, USA).
• Krebs, A. S. ( 2003). Middle grade students‟ algebraic understanding in a reform curriculum. School Science and Mathematics, 103, 5, 233-243.
• Kümbetoğlu, B. (2005).Sosyolojide ve antropolojide niteliksel yöntem ve araştırma. Bağlam: İstanbul.
• Lannin, J. (2003). Developing algebraic reasoning through generalization. Mathematics Teaching in the Middle School, 8(7), 342-348.
• Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 73(7), 231-258.
• Lawshe, C. H. (1975). A quantitative approach to content validity. Personnel Psychology, 28, 563–575.
• Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran and L. Lee (Eds.) Approaches to algebra: Perspectives for research and teaching (pp. 87-106), Kluwer Academic Publishers.
• Ley, A. F. (2005). A cross-sectional investigation of elementary school students’ ability to work with linear generalizing patterns: The impact of format and age on accuracy and strategy choice. (Master dissertation, Toronto University, Canada).
• Maher, C. A. & Davis, R. B. (1990). Building representations of children’s meanings. In R. B. Davis, C. A. Maher, & N. Noddings (Eds.) Constructivist views on the teaching and learning of mathematics. Reston, VA: NCTM.
• Marrades, R. & Gutierrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics 44 (1/2), 87-125.
• Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht, The Netherlands: Kluwer Academic Publishers.
• Nathan, M. J. (2003). Confronting teachers’ beliefs about algebra development: Investigating an approach for professional development (Technical Report No. 03- 04). Boulder, CO: University of Colorado, Institute of Cognitive Science.
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
• 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, London.
• Papic, M. & Mulligan, J. (2005). Preschoolers’ mathematical patterning. P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce et al. (Eds.), Building connections: research, theory and practice. Mathematics Education Research Group of Australasia Conference Proceedings 28 (MERGA28).
• Radford, L. (1996). Some reflections on teaching algebra through generalization. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 107–111). Dordrecht, The Netherlands: Kluwer.
• Richardson, K., Berenson, S. & Staley, K. (2009). Prospective elementary teachers use of representation to reason algebraically. Journal of Mathematical Behavior, 28(2/3), 188–199.
• Rivera, F. & Becker, J. (2005). Figural and numerical modes of generalizing in algebra. In Mathematics Teaching in the Middle School, 11(4),198-203.
• Rivera, F. & Becker, J. R. (2003). The effects of numerical and figural cues on the induction processes of preservice elementary teachers. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 2003 Joint Meeting of PME and PMENA, Vol. 4 (pp. 63 – 70). Honolulu, HI: University of Hawaii.
• Rivera, F. (2007). Visualizing as a mathematical way of knowing: Understanding figural generalization. Mathematics Teacher, 101(1), 69-75.
• Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164.
• Steele, D. & Johanning D. I. (2004). A schematic–theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57, 65–90.
• Swafford, J. O. & Langrall, C. W. (2000). Grade 6 students’ pre-instructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89–112.
• Tanışlı, D. ve Olkun, S. (2009). Basitten karmaşığa örüntüler. Ankara: Maya Akademi.
• Tanışlı, D. ve Yavuzsoy Köse, N. (2011). Lineer şekil örüntülerine ilişkin genelleme stratejileri: görsel ve sayısal ipuçlarının etkisi. Eğitim ve Bilim, 36 (160), 184- 198. Tanışlı, D. ve Özdaş, A. (2009). İlköğretim beşinci sınıf öğrencilerinin örüntüleri genellemede kullandıkları stratejiler. Educational Sciences: Theory & Practice, 9(3), 1453-1497.
• Vaiyavutjamai, P. & Clements, M. A. (2006). Effects of classroom instruction on students’ understanding of quadratic equations. Mathematics Education Research Journal, 18(1), 47–77.
• Zazkis, R. & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402.