Araştırma Makalesi
Ortaokul Öğrencilerinin Aritmetiksel İfadelere Yönelik Problem Kurma Becerilerinin İşlem Önceliği Bağlamında İncelenmesi
Öz
Bu araştırmada, ortaokul altıncı sınıf öğrencilerinin aritmetiksel
ifadelere yönelik problem kurma becerilerinin incelenmesi ve bu bağlamda düşük
başarıya neden olabilecek etkenlerden biri olarak işlem önceliği kuralının
rolünün belirlenmesi hedeflenmiştir. Araştırma, Doğu Karadeniz bölgesindeki bir
ilde öğrenim gören 96 ortaokul altıncı sınıf öğrencisiyle yürütülmüştür.
Öğrencilerden doğal sayılarla toplama, çıkarma, çarpma ve bölme işlemlerinden
ikisini içeren aritmetiksel ifadelere yönelik günlük yaşam durumlarıyla
ilişkili hikayeler oluşturmaları istenmiştir. Oluşturulan hikâyeler; günlük
yaşam durumlarıyla ilişkilendirme, işlemlerin/sayıların ifade edilmesi ve işlem
önceliği dikkate alınarak analiz edilmiştir. Elde edilen bulgulara göre, bütün
maddeler için işlem önceliği kuralının ihmal edilmesinden kaynaklı hatalı
cevapların oranlarının %11’in altında olduğu tespit edilmiştir. Öğrencilerin
yaşadıkları güçlüklerin aritmetiksel ifadelerdeki sayı ve işlemleri günlük
yaşam içerisinde sözel olarak ifade etmede daha fazla yoğunlaştığı
belirlenmiştir. Aritmetiksel ifadelerdeki işlem ve sayıların ifade edilemediği
cevapların oransal olarak yüksek olması, işlem önceliği kuralının problem kurma
bağlamında değerlendirilmesinin etkili bir yaklaşım olmadığını ortaya
koymuştur.
Anahtar Kelimeler
Kaynakça
- Abu-Elwan, R. (2002). Effectiveness of problem posing strategies on prospective mathematics teachers’ problem solving performance. Journal of Science and Mathematics Education in S.E. Asia, 25(1), 56-69.
- An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school, mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 7, 145–172.
- 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 & Seo, D. Y. (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 41-48). Seoul: PME.
- Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors. Journal of Research in Mathematics Education, 20(3), 301-308. Boulton-Lewis, G., Cooper, T.J., Atweh, B., Pillay, H., & Wilss, L. (1998). Pre-algebra: A cognitive perspective. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd International Conference for Psychology of Mathematics Education (Vol. 2, pp. 144–151). Stellenbosch, South Africa: Program Committee.
- Cai, J., & Jiang, C. (2017). An analysis of problem-posing tasks in Chinese and US elementary mathematics textbooks. International Journal of Science and Mathematics Education, 15(8), 1521-1540. Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83, 57-69.
- Capraro, M. M., & Joffrion, H. (2006). Algebraic equations: Can middle-school students meaningfully translate from words to mathematical symbols? Reading Psychology, 27, 147–164.
- Carlo, M., & Ioannis, P. (2011, July). Are useless brackets useful tools for teaching? Paper presented at the 35th Conference of the International Group for the Psychology of Mathematics Education, Ankara.
- Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., & Sriraman, B. (2005). An empirical taxonomy of problem posing processes. ZDM, 37(3), 149 – 158.
- Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16–30. Cohen, J. (1988). Statistical power analysis for the behavioral science. New Jersey: Lawrence Erlbaum Associates.
- Common Core State Standards Initiative [CCSSI]. (2014). Mathematics standards. Retrieved December 19, 2016 from www.corestandards.org/wp-content/uploads/Math_Standards1.pdf
- Doğan, M. ve Karakaya, V. (2016). Ortaokul matematik 6. sınıf ders kitabı. Ankara: Millî Eğitim Bakanlığı Yayınları.
- English, L. D. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83-106.
- Ev-Çimen, E. ve Yıldız, Ş. (2017). Ortaokul matematik ders kitaplarında yer verilen problem kurma etkinliklerinin incelenmesi. Türk Bilgisayar ve Matematik Eğitim Dergisi, 8(3), 378-407.
- Friedlander, A., & Tabach, M. (2001). Promoting multiple representations in algebra. In A. Cuoco (Ed.), The roles of representation in school mathematics (pp.173-184). Reston, VA: National Council of Teachers of Mathematics.
- Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations? Educational Studies in Mathematics, 92, 91–105.
- Işık, C. (2011). İlköğretim matematik öğretmeni adaylarının kesirlerde çarpma ve bölmeye yönelik kurdukları problemlerin kavramsal analizi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 41, 231-243
- Kar, T. (2016). Prospective middle school mathematics teachers’ knowledge of linear graphs in context of problem-posing. International Electronic Journal of Elementary Education, 8(4), 643-658.
- Kılıç, Ç. (2013). Pre-Service primary teachers’ free problem-posing performances in the context of fractions: An example from Turkey. The Asia-Pacific Education Researcher, 22(4), 677-686.
- Kılıç, Ç. (2017). A new problem-posing approach based on problem-solving strategy: Analyzing pre-service primary school teachers’ performance. Kuram ve Uygulamada Eğitim Bilimleri, 17(3), 771–789.
- Landy D., & Goldstone R. L. (2007, August). The alignment of ordering and space in arithmetic computation. Paper presented at the 29th Annual Meeting of the Cognitive Science Society, Nashville, Tennessee, USA.
- Lee, M. A., & Messner, S. J. (2000). Analysis of concatenations and order of operations in written mathematics. School Science and Mathematics, 100(4), 173-180.
- Leung, S. S. (1993). The relation of mathematical knowledge and creative things to the mathematical problems posing of prospective elementary school teachers on tasks differing in numerical information content (Unpublished doctoral dissertation). University of Pittsburg, USA.
- Leung, S. S. (2013). Teachers implementing mathematical problem posing in the classroom: Challenges and strategies. Educational Studies in Mathematics, 83(1), 103-116. Leung, S. S., & Silver, E. A. (1997). The role of task format, mathematics knowledge, and creative thinking on the arithmetic problem posing of prospective elementary school teachers. Mathematics Education Research Journal, 9, 5-24.
- Liebenberg, R., Linchevski, L., Olivier, A., & Sasman, M. (1998, July). Laying the foundation for algebra: Developing an understanding of structure. Paper presented at the 4th Annual Congress of the Association for Mathematics Education of South Africa (AMESA), Pietersburg.
- Linchevski, L., & Livneh, D. (1996). The competition between numbers and structure. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th International Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 257-264). Valencia: Encuademaciones Artesanas.
- Linchevski, L., & Livneh, D. (1999). Structure sense: the relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40(2), 173–196.
- Luo, F. (2009). Evaluating the effectiveness and insights of pre-service elementary teachers’ abilities to construct word problems for fraction multiplication. Journal of Mathematics Education, 2(1), 83–98.
- MacGregor, M., & Stacey, K. (1993). Cognitive models underlying students' formulation of simple linear equations. Journal for Research in Mathematics Education, 24(3), 217-232.
- McAllister, C. J., & Beaver, C. (2012). Identification of error types in preservice teachers' attempts to create fraction story problems for specified operations. School Science and Mathematics, 112(2), 88-98.
- Mcmillan, H. J., & Schumacher, S. (2010). Research in education. Boston, USA: Pearson Education.
- Merlin, E. M. (2008). Beyond PEMDAS: Teaching students to perceive algebraic structure (Unpublished master’s thesis). University of Maryland, USA.
- Milli Eğitim Bakanlığı [MEB]. (2013). Ortaokul matematik dersi (5, 6, 7 ve 8. sınıflar) öğretim programı. Ankara: MEB Basımevi.
- National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author.
- Öksüz, C. (2009). İşlem sırasının kavratılması. İlköğretim Online, 8(2), 306-312.
- Özgen, K., Aydın, M., Geçici, M. E. ve Bayram, B. (2017). Sekizinci sınıf öğrencilerinin problem kurma becerilerinin bazı değişkenler açısından incelenmesi. Türk Bilgisayar ve Matematik Eğitim Dergisi, 8(2), 323-351.
- Papadopoulos, I. (2015, February). The rules for the order of operations: The case of an inservice teacher. Paper presented at the Ninth Congress of the European Society for Research in Mathematics Education, Prague, Czech Republic.
- Pappanastos, E., Hall, M. A., & Honan, A. S. (2002). Order of operations: Do business students understand the correct order? Journal of Education for Business, 78(2), 81-84.
- Pehkonen, E. (1995). Introduction: Use of open-ended problems. International Reviews on Mathematical Education, 27(2), 55-57.
- Peterson, D. (2000). History of the order of operations. Retrieved February 10, 2017 from http://mathforum.org/library/drmath/view/52582.html
- Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
- Silver, E. A. (1995). The nature and use of open problems in mathematics education: Mathematical and pedagogical perspectives. International Reviews on Mathematical Education, 27(2), 67-72.
- Silver, E. A., & Cai, J. (2005). Assessing students’ mathematical problem posing. Teaching Children Mathematics, 12(3), 129-135.
- Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing. In P. Clarkson (Ed.), Technology in mathematics education (pp.518–525). Melbourne: Mathematics Education Research Group of Australasia.
- Ticha, M., & Hospesova, A. (2009, February). Problem posing and development of pedagogical content knowledge in preservice teacher training. Paper presented at the meeting of CERME 6, Lyon.
- Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166-175.
- Uça, S. (2010). Matematik öğretiminde işlem sırasının kavratılmasında yeni bir yaklaşım: Mnemoni (Yayınlanmamış yüksek lisans tezi). Adnan Menderes Üniversitesi, Sosyal Bilimler Enstitüsü, Aydın.
- Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2009). Elementary and middle school mathematics: Teaching develop-mentally (7th ed.). Boston: Pearson Education.
- Vanderbeek, G. (2007). Order of operations and RPN. Retrieved February 13, 2017 from https://digitalcommons.unl.edu/cgi/viewcontent.cgi?referer=https://www.google.com.tr/&httpsredir=1&article=1045&context=mathmidexppap
- Wu, H. (2007). “Order of operations” and other oddities in school mathematics. Retrieved February 1, 2017 from https://math.berkeley.edu/~wu/order5.pdf
Investigation of Elementary School Students’ Problem Posing Abilities for Arithmetic Expressions in the Context of Order of Operations
Öz
In this study, the aim is two folds; investigating sixth grade
elementary school students’ problem posing abilities for arithmetic expression
and, in this context, determining the role of order of operations rules, one of
the factors that may result in low achievement among students. The study is
conducted to 96 sixth grade enrolled in a public elementary school located in a
city at East Black Sea Region. Students were expected to create stories related
to daily life situations for arithmetic expressions involving any two of
addition, subtraction, multiplication, and division of natural numbers. The
stories created were analyzed according to the criteria of being associated
with daily life, whether the operations/numbers were expressed in them and
order of operations. The findings gathered revealed that the percentages of
incorrect answers reasoning from ignoring the order of operations were less
than 11% for all items. It was determined that the difficulties experienced by
students were more commonly observed in the attempts to express the numbers and
operations found in the arithmetic operations verbally in daily life context.
Considering the high percentage of students’ responses which stories they
created did not signify the numbers and operations found in the arithmetic
expressions, it was not an effective approach to evaluate the process of order
of operations in the context of problem posing.
Anahtar Kelimeler
Kaynakça
- Abu-Elwan, R. (2002). Effectiveness of problem posing strategies on prospective mathematics teachers’ problem solving performance. Journal of Science and Mathematics Education in S.E. Asia, 25(1), 56-69.
- An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school, mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 7, 145–172.
- 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 & Seo, D. Y. (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 41-48). Seoul: PME.
- Blando, J. A., Kelly, A. E., Schneider, B. R., & Sleeman, D. (1989). Analyzing and modeling arithmetic errors. Journal of Research in Mathematics Education, 20(3), 301-308. Boulton-Lewis, G., Cooper, T.J., Atweh, B., Pillay, H., & Wilss, L. (1998). Pre-algebra: A cognitive perspective. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd International Conference for Psychology of Mathematics Education (Vol. 2, pp. 144–151). Stellenbosch, South Africa: Program Committee.
- Cai, J., & Jiang, C. (2017). An analysis of problem-posing tasks in Chinese and US elementary mathematics textbooks. International Journal of Science and Mathematics Education, 15(8), 1521-1540. Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83, 57-69.
- Capraro, M. M., & Joffrion, H. (2006). Algebraic equations: Can middle-school students meaningfully translate from words to mathematical symbols? Reading Psychology, 27, 147–164.
- Carlo, M., & Ioannis, P. (2011, July). Are useless brackets useful tools for teaching? Paper presented at the 35th Conference of the International Group for the Psychology of Mathematics Education, Ankara.
- Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., & Sriraman, B. (2005). An empirical taxonomy of problem posing processes. ZDM, 37(3), 149 – 158.
- Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16–30. Cohen, J. (1988). Statistical power analysis for the behavioral science. New Jersey: Lawrence Erlbaum Associates.
- Common Core State Standards Initiative [CCSSI]. (2014). Mathematics standards. Retrieved December 19, 2016 from www.corestandards.org/wp-content/uploads/Math_Standards1.pdf
- Doğan, M. ve Karakaya, V. (2016). Ortaokul matematik 6. sınıf ders kitabı. Ankara: Millî Eğitim Bakanlığı Yayınları.
- English, L. D. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83-106.
- Ev-Çimen, E. ve Yıldız, Ş. (2017). Ortaokul matematik ders kitaplarında yer verilen problem kurma etkinliklerinin incelenmesi. Türk Bilgisayar ve Matematik Eğitim Dergisi, 8(3), 378-407.
- Friedlander, A., & Tabach, M. (2001). Promoting multiple representations in algebra. In A. Cuoco (Ed.), The roles of representation in school mathematics (pp.173-184). Reston, VA: National Council of Teachers of Mathematics.
- Gunnarsson, R., Sönnerhed, W. W., & Hernell, B. (2016). Does it help to use mathematically superfluous brackets when teaching the rules for the order of operations? Educational Studies in Mathematics, 92, 91–105.
- Işık, C. (2011). İlköğretim matematik öğretmeni adaylarının kesirlerde çarpma ve bölmeye yönelik kurdukları problemlerin kavramsal analizi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 41, 231-243
- Kar, T. (2016). Prospective middle school mathematics teachers’ knowledge of linear graphs in context of problem-posing. International Electronic Journal of Elementary Education, 8(4), 643-658.
- Kılıç, Ç. (2013). Pre-Service primary teachers’ free problem-posing performances in the context of fractions: An example from Turkey. The Asia-Pacific Education Researcher, 22(4), 677-686.
- Kılıç, Ç. (2017). A new problem-posing approach based on problem-solving strategy: Analyzing pre-service primary school teachers’ performance. Kuram ve Uygulamada Eğitim Bilimleri, 17(3), 771–789.
- Landy D., & Goldstone R. L. (2007, August). The alignment of ordering and space in arithmetic computation. Paper presented at the 29th Annual Meeting of the Cognitive Science Society, Nashville, Tennessee, USA.
- Lee, M. A., & Messner, S. J. (2000). Analysis of concatenations and order of operations in written mathematics. School Science and Mathematics, 100(4), 173-180.
- Leung, S. S. (1993). The relation of mathematical knowledge and creative things to the mathematical problems posing of prospective elementary school teachers on tasks differing in numerical information content (Unpublished doctoral dissertation). University of Pittsburg, USA.
- Leung, S. S. (2013). Teachers implementing mathematical problem posing in the classroom: Challenges and strategies. Educational Studies in Mathematics, 83(1), 103-116. Leung, S. S., & Silver, E. A. (1997). The role of task format, mathematics knowledge, and creative thinking on the arithmetic problem posing of prospective elementary school teachers. Mathematics Education Research Journal, 9, 5-24.
- Liebenberg, R., Linchevski, L., Olivier, A., & Sasman, M. (1998, July). Laying the foundation for algebra: Developing an understanding of structure. Paper presented at the 4th Annual Congress of the Association for Mathematics Education of South Africa (AMESA), Pietersburg.
- Linchevski, L., & Livneh, D. (1996). The competition between numbers and structure. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th International Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 257-264). Valencia: Encuademaciones Artesanas.
- Linchevski, L., & Livneh, D. (1999). Structure sense: the relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40(2), 173–196.
- Luo, F. (2009). Evaluating the effectiveness and insights of pre-service elementary teachers’ abilities to construct word problems for fraction multiplication. Journal of Mathematics Education, 2(1), 83–98.
- MacGregor, M., & Stacey, K. (1993). Cognitive models underlying students' formulation of simple linear equations. Journal for Research in Mathematics Education, 24(3), 217-232.
- McAllister, C. J., & Beaver, C. (2012). Identification of error types in preservice teachers' attempts to create fraction story problems for specified operations. School Science and Mathematics, 112(2), 88-98.
- Mcmillan, H. J., & Schumacher, S. (2010). Research in education. Boston, USA: Pearson Education.
- Merlin, E. M. (2008). Beyond PEMDAS: Teaching students to perceive algebraic structure (Unpublished master’s thesis). University of Maryland, USA.
- Milli Eğitim Bakanlığı [MEB]. (2013). Ortaokul matematik dersi (5, 6, 7 ve 8. sınıflar) öğretim programı. Ankara: MEB Basımevi.
- National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author.
- Öksüz, C. (2009). İşlem sırasının kavratılması. İlköğretim Online, 8(2), 306-312.
- Özgen, K., Aydın, M., Geçici, M. E. ve Bayram, B. (2017). Sekizinci sınıf öğrencilerinin problem kurma becerilerinin bazı değişkenler açısından incelenmesi. Türk Bilgisayar ve Matematik Eğitim Dergisi, 8(2), 323-351.
- Papadopoulos, I. (2015, February). The rules for the order of operations: The case of an inservice teacher. Paper presented at the Ninth Congress of the European Society for Research in Mathematics Education, Prague, Czech Republic.
- Pappanastos, E., Hall, M. A., & Honan, A. S. (2002). Order of operations: Do business students understand the correct order? Journal of Education for Business, 78(2), 81-84.
- Pehkonen, E. (1995). Introduction: Use of open-ended problems. International Reviews on Mathematical Education, 27(2), 55-57.
- Peterson, D. (2000). History of the order of operations. Retrieved February 10, 2017 from http://mathforum.org/library/drmath/view/52582.html
- Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
- Silver, E. A. (1995). The nature and use of open problems in mathematics education: Mathematical and pedagogical perspectives. International Reviews on Mathematical Education, 27(2), 67-72.
- Silver, E. A., & Cai, J. (2005). Assessing students’ mathematical problem posing. Teaching Children Mathematics, 12(3), 129-135.
- Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing. In P. Clarkson (Ed.), Technology in mathematics education (pp.518–525). Melbourne: Mathematics Education Research Group of Australasia.
- Ticha, M., & Hospesova, A. (2009, February). Problem posing and development of pedagogical content knowledge in preservice teacher training. Paper presented at the meeting of CERME 6, Lyon.
- Toluk-Uçar, Z. (2009). Developing pre-service teachers understanding of fractions through problem posing. Teaching and Teacher Education, 25(1), 166-175.
- Uça, S. (2010). Matematik öğretiminde işlem sırasının kavratılmasında yeni bir yaklaşım: Mnemoni (Yayınlanmamış yüksek lisans tezi). Adnan Menderes Üniversitesi, Sosyal Bilimler Enstitüsü, Aydın.
- Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2009). Elementary and middle school mathematics: Teaching develop-mentally (7th ed.). Boston: Pearson Education.
- Vanderbeek, G. (2007). Order of operations and RPN. Retrieved February 13, 2017 from https://digitalcommons.unl.edu/cgi/viewcontent.cgi?referer=https://www.google.com.tr/&httpsredir=1&article=1045&context=mathmidexppap
- Wu, H. (2007). “Order of operations” and other oddities in school mathematics. Retrieved February 1, 2017 from https://math.berkeley.edu/~wu/order5.pdf
Toplam 49 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | Türkçe |
|---|---|
| Bölüm | Araştırma Makaleleri |
| Yazarlar | |
| Yayımlanma Tarihi | 29 Ağustos 2018 |
| Yayımlandığı Sayı | Yıl 2018 Cilt: 9 Sayı: 2 |