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Analyzing the Structures of Figural Patterns Produced by Middle School Students Based on Number Patterns

Yıl 2017, Cilt: 13 Sayı: 1, 65 - 79, 16.04.2017
https://doi.org/10.17860/mersinefd.305756

Öz

Pattern
is one of the most important topics of mathematics education in all grades.
Some researchers asserted that pattern is the
heart and soul of mathematics. Many skills and mathematical knowledge of
students can be developed through pattern activities.
In this study analyzing the structures of figural
patterns/situations produced by middle school students based on number patterns
was investigated.
In
total, 474 middle school students (254 were
girls and 220 were boys) attended to
study. Data were collected from a pattern task including linear and quadratic
(non-linear) number patterns, in which participants were asked to generate
figural patterns based on those number patterns. The obtained data were
analysed at two levels. The results of the study indicated that participants
created 18 linear forms and 33 non-linear forms for linear number pattern and
13 linear forms and 20 non-linear forms for quadratic number pattern. During
the study participants preferred to use same geometrical shapes such as
circles, triangles and squares, etc. while creating figural patterns. Moreover,
participants 
created many more figural patterns for the
3,5,7,9,11,… number pattern than for the 2,6,12,20,30,… number pattern.
Some of the participants had problems while generating figural patterns
such as producing non pattern forms,
no answer, creating an irrelevant pattern and extending number patterns. 

Kaynakça

  • Akkan, Y. (2013). Comparison of 6th-8th graders’ efficiencies, strategies and representations regarding generalization patterns. Bolema, 27(47), 703-732. 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. Bassarear, T. (2008). Mathematics for elementary school teachers. Belmont, CA:Brooks/Cole. Bishop, J. (2000). Linear geometric number patterns: Middle school students' strategies. Mathematics Education Research Journal, 12(2), 107-126. Cathcart, W.G., Pothier, Y.M., Vance, J.H., & Bezuk, N.S. (2003). Learning mathematics in elementary and middle schools. Upper Saddle River, N.J.:Merrill/Prentice Hall. Fox, J. (2005). Child-initiated mathematical patterning in the pre-compulsory years. In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 313-320). Melbourne: PME. Frobisher, L., & Threlfall, J. (1999). Teaching and assessing patterns in number in the primary years. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics (pp.84-103). London and New York: Casse. Göktepe, S. ve Özdemir, A.Ş. (2013). İlköğretim matematik öğretmen adaylarının uzamsal görselleştirme becerilerinin solo modeli ile incelenmesi. Kalem Eğitim ve İnsan Bilimleri Dergisi,3(2), 91-146. Gregg, D.U. (2002). Building students’ sense of linear relationships by stacking cubes. Mathematics Teacher, 95(5), 330–333. Jurdak, M.E. & El Mouhayar, R.R. (2014). Trends in the development of student level of reasoning in pattern generalization tasks across grade level. Educational Studies in Mathematics, 85, 75–92. Lannin, J.K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3): 231-258. Lin, F-L., Yang, K-L., & Chen, C-Y.(2004). The features and relationships of reasoning, proving and understanding proof in number patterns. International Journal of Science and Mathematics Education, 2, 227–256. McGarvey, L.M. (2012). What is a pattern? Criteria used by teachers and young children. Mathematical Thinking and Learning, 14(4), 310-337. Ministry of National Education (2013). Ortaokul matematik dersi (5,6,7,8. Sınıflar) öğretim programı. [Middle School Mathematics Curriculum (5-8. grades)]. Ankara Devlet Kitapları Basımevi. Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33-49. Orton, J. Orton, A., & Roper, T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics (pp.18-30). London and New York: Casse. Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM Mathematics Education. DOI 10.1007/s11858-007-0061-0. Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2–7. Reys, R.E., Suydam, M.N., Lindquist, M.M. & Smith, N.L. (1998). Helping children learn mathematics. Needham Heights: Allyn&Bacon. Rivera, F.D., & Becker, J.R. (2007). Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra. Journal of Mathematical Behavior, 26,140–155. Rivera, F.D., & Becker, J.R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM Mathematics Education, 40, 65–82. Smith, S.P. (1997). Early Childhood Mathematics. Needham Heights: Ally&Bacon. Souviney, R.J. (1994). Learning to teach mathematics. Englewood Cliffs: Macmillan Publishing Company. Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164. Steele, D. (2008). Seventh-grade students’ representations for pictorial growth and change problems. ZDM Mathematics Education, 40, 97–110. Threlfall, J. (1999). Repeating patterns in the early primary years. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics (pp.18-30). London and New York: Casse. Van de Walle J.A. (2004). Elementary and middle school mathematics. teaching developmentally. Boston: Allyn &Bacon. Vogel, R.(2005). Patterns – a fundamental idea of mathematical thinking and learning. ZDM Mathematics Education, 37(5), 445-449. Walkowiak, T.A. (2014). Elementary and middle school students’ analyses of pictorial growth. Journal of Mathematical Behavior, 33, 56– 71. Waring, S., Orton, A., & Roper, T. (1999). Pattern and proof. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics (pp.18-30). London and New York: Casse. Warren, E. (2005). Young children’s ability to generalise the pattern rule for growing patterns. In Chick, H.L. & Vincent, J.L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 305-312. Melbourne: PME. Warren, E., & Cooper, T. (2006). Using repeating patterns to explore functional thinking. APMC, 11(1), 9-14. Wicket, M., Kharas, K., & Burns, M. (2002). Grades 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solution Publications. Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379–402.

Ortaokul Öğrencilerinin Sayı Örüntülerine Dayalı Olarak Oluşturdukları Şekil Örüntülerinin Yapılarının Analiz Edilmesi

Yıl 2017, Cilt: 13 Sayı: 1, 65 - 79, 16.04.2017
https://doi.org/10.17860/mersinefd.305756

Öz

Örüntü matematik eğitiminde her düzeyde en önemli konulardan
biridir. Bazı araştırmacılar örüntünün matematiğin kalbi ve ruhu olduğunu
belirtmektedirler. Öğrencilerin bir çok becerisi ve matematiksel bilgisi örüntü
etkinlikleri ile geliştirilebilir. Bu çalışmada öğrencilerin sayı örüntülerine
bağlı olarak oluşturdukları şekil örüntülerinin yapıları analiz edilmiştir.
Çalışmaya toplam 474 ortaokul öğrencisi (254 kız ve 220 erkek) katılmıştır.
Çalışmanın verileri lineer ve lineer olmayan sayı örüntülerinin bulunduğu
örüntü görevi ile toplanmıştır. Araştırmadan elde edilen veriler iki düzeyde
analiz edilmiştir. Araştırmadan elde edilen sonuçlara bakıldığında
katılımcıların 18 lineer yapı ve 33 lineer olmayan yapı lineer sayı örüntüsü
için, 13 lineer yapı ve 20 lineer olmayan yapı quadratik örüntüler için
üretmişlerdir. Örüntü oluşturma sırasında katılımcılar daire, üçgen ve kare
gibi geometrik şekiller kullanmayı tercih etmişlerdir. Ayrıca, katılımcıların 3, 5, 7, 9, 11,..  sayı örüntüsü için oluşturdukları şekiller,
2, 6, 12, 20, 30,..sayı örüntüsü için oluşturdukları şekillerden daha fazladır.  Katılımcılardan bazıları şekil örüntüsü
oluştururken örüntü olmayan yapılar oluşturma, yanıt verememe, ilgisiz örüntü
oluşturma ve örüntüyü devam ettirme gibi sorunlar yaşamışlardır. 

Kaynakça

  • Akkan, Y. (2013). Comparison of 6th-8th graders’ efficiencies, strategies and representations regarding generalization patterns. Bolema, 27(47), 703-732. 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. Bassarear, T. (2008). Mathematics for elementary school teachers. Belmont, CA:Brooks/Cole. Bishop, J. (2000). Linear geometric number patterns: Middle school students' strategies. Mathematics Education Research Journal, 12(2), 107-126. Cathcart, W.G., Pothier, Y.M., Vance, J.H., & Bezuk, N.S. (2003). Learning mathematics in elementary and middle schools. Upper Saddle River, N.J.:Merrill/Prentice Hall. Fox, J. (2005). Child-initiated mathematical patterning in the pre-compulsory years. In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 313-320). Melbourne: PME. Frobisher, L., & Threlfall, J. (1999). Teaching and assessing patterns in number in the primary years. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics (pp.84-103). London and New York: Casse. Göktepe, S. ve Özdemir, A.Ş. (2013). İlköğretim matematik öğretmen adaylarının uzamsal görselleştirme becerilerinin solo modeli ile incelenmesi. Kalem Eğitim ve İnsan Bilimleri Dergisi,3(2), 91-146. Gregg, D.U. (2002). Building students’ sense of linear relationships by stacking cubes. Mathematics Teacher, 95(5), 330–333. Jurdak, M.E. & El Mouhayar, R.R. (2014). Trends in the development of student level of reasoning in pattern generalization tasks across grade level. Educational Studies in Mathematics, 85, 75–92. Lannin, J.K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3): 231-258. Lin, F-L., Yang, K-L., & Chen, C-Y.(2004). The features and relationships of reasoning, proving and understanding proof in number patterns. International Journal of Science and Mathematics Education, 2, 227–256. McGarvey, L.M. (2012). What is a pattern? Criteria used by teachers and young children. Mathematical Thinking and Learning, 14(4), 310-337. Ministry of National Education (2013). Ortaokul matematik dersi (5,6,7,8. Sınıflar) öğretim programı. [Middle School Mathematics Curriculum (5-8. grades)]. Ankara Devlet Kitapları Basımevi. Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33-49. Orton, J. Orton, A., & Roper, T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics (pp.18-30). London and New York: Casse. Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM Mathematics Education. DOI 10.1007/s11858-007-0061-0. Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2–7. Reys, R.E., Suydam, M.N., Lindquist, M.M. & Smith, N.L. (1998). Helping children learn mathematics. Needham Heights: Allyn&Bacon. Rivera, F.D., & Becker, J.R. (2007). Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra. Journal of Mathematical Behavior, 26,140–155. Rivera, F.D., & Becker, J.R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM Mathematics Education, 40, 65–82. Smith, S.P. (1997). Early Childhood Mathematics. Needham Heights: Ally&Bacon. Souviney, R.J. (1994). Learning to teach mathematics. Englewood Cliffs: Macmillan Publishing Company. Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164. Steele, D. (2008). Seventh-grade students’ representations for pictorial growth and change problems. ZDM Mathematics Education, 40, 97–110. Threlfall, J. (1999). Repeating patterns in the early primary years. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics (pp.18-30). London and New York: Casse. Van de Walle J.A. (2004). Elementary and middle school mathematics. teaching developmentally. Boston: Allyn &Bacon. Vogel, R.(2005). Patterns – a fundamental idea of mathematical thinking and learning. ZDM Mathematics Education, 37(5), 445-449. Walkowiak, T.A. (2014). Elementary and middle school students’ analyses of pictorial growth. Journal of Mathematical Behavior, 33, 56– 71. Waring, S., Orton, A., & Roper, T. (1999). Pattern and proof. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics (pp.18-30). London and New York: Casse. Warren, E. (2005). Young children’s ability to generalise the pattern rule for growing patterns. In Chick, H.L. & Vincent, J.L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 305-312. Melbourne: PME. Warren, E., & Cooper, T. (2006). Using repeating patterns to explore functional thinking. APMC, 11(1), 9-14. Wicket, M., Kharas, K., & Burns, M. (2002). Grades 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solution Publications. Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379–402.
Toplam 1 adet kaynakça vardır.

Ayrıntılar

Konular Eğitim Üzerine Çalışmalar
Bölüm Makaleler
Yazarlar

Çiğdem Kılıç

Yayımlanma Tarihi 16 Nisan 2017
Yayımlandığı Sayı Yıl 2017 Cilt: 13 Sayı: 1

Kaynak Göster

APA Kılıç, Ç. (2017). Ortaokul Öğrencilerinin Sayı Örüntülerine Dayalı Olarak Oluşturdukları Şekil Örüntülerinin Yapılarının Analiz Edilmesi. Mersin Üniversitesi Eğitim Fakültesi Dergisi, 13(1), 65-79. https://doi.org/10.17860/mersinefd.305756

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