EXAMINATION OF THE ELEMENTARY SCHOOL STUDENTS` ALGEBRAIC PROBLEMS SOLVING ACHIEVEMENT WITH REGARD METACOGNITIVE KNOWLEDGE
Yıl 2013,
Cilt: 8 Sayı: 4, 411 - 427, 01.05.2013
Sare Şengül
Fatma Erdoğan
Öz
The purpose of this research was to investigate the elementary school students` algebraic problem solving achievement with regard metacognitive knowledge and to compare the levels of achievement based on grade level. Survey method was used in this study. The sample of the study were composed of 235 students which selected randomly from an elementary school on the European side Istanbul during the 2011-2012 academic year. The data was collected using "Mathematical Achievement Test Regarding Metacognitive Knowledge (MATRMK)".Frequency tables and analysis of variance were employed to analyze data. The result of the study indicated that, while there were not differences in elementary school students` algebraic problems solving achievement with regard declarative knowledge point of view grade level, there were differences in algebraic problems solving achievement with regard procedural and conditional knowledge point of view grade level. Finally suggestions were included for learning and teaching mathematics in elementary schools.
Kaynakça
- Akkan, Y., Baki, A., ve Çakıroğlu, Ü., (2012). 5-8. Sınıf Öğrencilerinin Aritmetikten Cebire Geçiş Süreçlerinin Problem Çözme Bağlamında İncelenmesi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 43, 1-13.
- Akkaya, R. ve Durmuş, S., (2006). İlköğretim 6-8. Sınıf Öğrencilerinin Cebir Öğrenme Alanındaki Kavram Yanılgıları. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 31, 1-12.
- Alibali, M.W., Knuth, E.J., Hattikudur, S., McNeil, N.M., and Stephens, A.C., (2007). Alongitudinal Examination of Middle School Students’ Understanding of the Equal Sign and Equivalent Equations. Mathematical Thinking and Learning, 9(3), 221-247. DOI:1080/10986060701360902
- Atılgan, H., Kan, A. ve Doğan, N., (2006). Eğitimde Ölçme ve Değerlendirme. Ankara: Anı Yayınları.
- Biryukov, P., (2004). Metacognitive Aspects of Solving Combinatorics Problems. International Journal for Mathematics Teaching and Learning, 1-19. DOI:10.1.1.178.3238
- Booth, L. and Herscovics, N., (1986). Algebra. In the Proceedings of the 5 th International Congress on Mathematical Education (pp. 150-155). Boston, USA.
- Büyüköztürk, Ş., (2012). Sosyal Bilimler İçin Veri Analizi El Kitabi. Ankara: PegemA Yayıncılık.
- Cooper, T.J., Boulton ‐Lewis, G., Athew, B., Willss, L., and Mutch, S., (1997). The Transition Arithmetic to Algebra: Initial Understandings of Equals, Operations and Variable. International Group for the Psychology of Matematics Education, 21(2), 89-96. Dede, Y. ve Peker, M., (2007). Öğrencilerin Cebire Yönelik Hata ve Yanlış Anlamaları: Matematik Öğretmen Adaylarının Bunları Tahmin Becerileri ve Çözüm Önerileri. İlköğretim Online, 6(1), 35Desoete, A., (2001). Off-Line Metacognition in Children with Mathematics Learning Disabilities. Unpublished doctoral dissertation. Belgium: Universiteit Gent.
- Desoete, A., Roeyers, H. and Buysse, A. (2001). Metacognition and Mathematical Problem Solving in Grade 3. Journal of Learning Disability, 34(5), 435–449. DOI: 10.1177/002221940103400505
- Erbaş, A.K., Çetinkaya, B. ve Ersoy, Y., (2009). Öğrencilerin Basit Doğrusal Denklemlerin Çözümünde Karşılaştıkları Güçlükler ve Kavram Yanılgıları. Eğitim ve Bilim, 34(152), 44-59.
- Falkner, K.P., Levi, L., and Carpenter, T.P., (1999). Children’s Understanding of Equality: A Foundation for Algebra. Teaching Children Mathematics, 6, 232-236.
- Flavell, J.H., (1979). Metacognition and Cognitive Monitoring: A New Area of Cognitive Developmental İnquiry. American Psychologist, 34(10), 906–911. DOI:10.1037/0003-066X.34.10.906 Flavell, J.H., (1992). Perspectives on Perspective Taking. In H. Beilin and P. Pufall (Eds.), Piaget’s theory: Prospects and Possibilities (pp. 107–139). Hillsdale, NJ: Erlbaum.
- Fortunato, I., Hecht, D., Title, C., and Alvarez, L., (1991). Metacognition and Problem Solving. Arithmetic Teacher, 39(4), 38Fouche, K.K., (1997). Algebra for Everyone: Start Early. Mathematics Teaching in the Middle School, 2(4), 226-229.
- Gürbüz, R. ve Akkan, Y., (2008). Farklı Öğrenim Seviyesindeki Öğrencilerin Aritmetikten Cebire Geçiş Düzeylerinin Karşılaştırılması: Denklem Örneği. Eğitim ve Bilim, 33(148), 64
- Hoch, M. and Dreyfus, T., (2004). Structure sense in high school algebra: The effect of brackets. In M.J. Hİines and A.B. Fuglestad (Eds.), In the Proceedings of the 28 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp.49-56). Bergen, Norway: PME.
- Ifenthaler, D., (2012). Determining the Effectiveness of Prompts for Self-Regulated Learning in Problem-Solving Scenarios. Educational Technology & Society, 15(1), 38–52.
- Karasar, N., (2009). Bilimsel Araştırma Yöntemi. Ankara: Nobel yayıncılık.
- Kieran, C., (1992). The Learning and Teaching of School Algebra. In D.A. Grouws (Eds.), Handbook of Research on Mathematics Teaching and Learning (pp. 390 ‐419). New York: Macmillan.
- Kieran, C., (2007). Learning and Teaching Algebra at the Middle School through College Levels. In Proceedings of the Second Handbook of Research on Teaching and Learning Mathematics (pp.707–762). Charlotte: Information Age.
- Kieran, C. and Chaloug, L., (1993). Prealgebra: The Transitions from Arithmetic to Algebra. In D. T. Owens (Eds.), Research İdeas for the Classroom: Middle Grades Mathematics (pp.179-198). New York: Macmillan.
- Kramarski, B., Mevarech, Z.R., and Liberman, A., (2001). The Effects of Multilevel Versus Unilevel-Metacognitive Training on Mathematical Reasoning. The Journal of Educational Research, 94(5), 292-300. DOI:10.1080/00220670109598765
- Kuchemann, D., (1981). Algebra. In K. Hart (Eds.), Children’s Understanding of Mathematics: 11-16 (pp. 102-119). London: Murray.
- Lane, S., (1993). The Conceptual Framework for the Development of a Mathematics Performance Assessment Instrument. Educational Measurement: Issues & Practices, 12(2), 16-23. DOI:1111/j.1745-3992.1993.tb00529.x
- Lesh, R., Post, T., and Behr, M., (1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. In C. Janvier (Eds.), Problems of Representation in the Teaching and Learning of Mathematics (pp.33-40). New Jersey: Lawrence Erlbaum Associates.
- MacGregor, M. and Stacey, K., (1997). Students’ Understanding of Algebraic Notation: 11–15. Educational Studies in Mathematics, 33, 1-19. DOI:10.1023/A:1002970913563
- Mevarech, Z.R., (1999). Effects of Metacognitive Training Embedded in Cooperative Settings on Mathematical Problem
- Solving. The Journal of Educational Research, 92(4), 195-205. DOI:1080/00220679909597597
- Miles, M.B. and Huberman, M.A., (1994). An Expanded Sourcebook Qualitative Data Analysis. London: Sage Publication.
- Miller, S.P. and Hudson, P.J., (2007). Using Evidence-Based Practices to Build Mathematics Competence Related to Conceptual, Procedural, and Declarative Knowledge. Learning Disabilities Research & Practice, 22(1), 47–57. DOI:10.1111/j.1540582007.00230.x
- Milli Eğitim Bakanlığı (MEB), (2009). İlköğretim Matematik Dersi 6- Sınıflar Öğretim Programı ve Kılavuzu. Ankara: MEB.
- Moyer, P.S. and Jones, M.G., (2004). Controlling Choice: Teachers, Students, and Manipulatives in Mathematics Classrooms. School Science and Mathematics, 104(1), 16-31. DOI:1111/j.1949-8594.2004.tb17978.x
- Nosegbe, I.C., (2001). Middle School Students’ Sense Making of Their Solutions to Mathematical Word Problems. USA: Indiana University.
- Ormond, C., Luszcz, M.A., Mann, L., and Beswick, G., (1991). A Metacognitive Analysis of Decision Making in Adolescence. Journal of Adolescence, 14(3), 275-291. DOI:10.1016/01401971(91)90021-I
- Orton, A. and Orton, J., (1999). Pattern and the Approach to Algebra. In A. Orton (Eds.), Pattern in the Teaching and Learning of Mathematics (pp.104-120). London: Cassel.
- Panaoura, A. and Philippou, G. (2003). The Construct Validity of an Inventory for the Measurement of Young Pupils’ Metacognitive Abilities in Mathematics. In N.A. Pateman, B.J. Doherty and J. Zilliox (Eds.), In the Proceedings of the 27 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 437-444). Honolulu, USA: Pme.
- Panaoura, A. and Philippou, G. (2007). The Developmental Change of Young Pupils’ Metacognitive Ability in Mathematics in Relation to Their Cognitive Abilities. Cognitive Development, 22(2), 149-164. DOI:10.1016/j.cogdev.2006.08.004
- Panaoura, A., Philippou, G., and Christou, C., (2003). Young Pupils’ Metacognitive Ability in Mathematics. http://fibonacci.dm.unipi.it/clusterpages/didattica/CERME3/proce edings/Groups/TG3/TG3_Panaoura_cerme3.pdf, Erişim tarihi: 15 Haziran 2008.
- Perso, T.F., (1992). Using Diagnostic Teaching to Overcome Misconceptions in Algebra. The Mathematical Association of Western Australia.
- Pintrich, P.R., (2002) . The Role of Metacognitive Knowledge in Learning, Teaching and Assessing. Theory into Practice, 41(4), 219-2 DOI:10.1207/s15430421tip4104_3
- Pope, S. and Sharma, R., (2001). Symbol Sense: Teacher’s and Student’s Understanding. In the Proceedings of the British Society for Research into Learning Mathematics, 21(3). http://www.bsrlm.org.uk/IPs/ip21-3/BSRLM-IP-21-3-12.pdf, Erişim tarihi:17 Mayıs 2009.
- Pugalee, D.K., (2001). Writing, Mathematics, and Meta-Cognition: Looking for Connections through Students’ Work in Mathematical Problem Solving. School Science and Mathematics, 101(5), 2362 DOI:10.1111/j.1949-8594.2001.tb18026.x
- Pugalee, D.K., (2004). A Comparison of Verbal and Written Description of Students’ Problem Solving Processes. Educational Studies in Mathematics, 55(1-3), 27-47. DOI:1023/B:EDUC.0000017666.11367.c7
- Sakonidis, H. and Bliss, J., (1990). Children’s Writing about the Idea of Variable in the Context of Formula. In the Proceedings of The Annual Conference of The International Group for The Psychology of Mathematics Education with The North American Chapter 12 th PMEANA Conference (Vol. 2, pp.133-140). Mexico: PME.
- Schoenfeld, A.H., (1985). Metacognitive and Epistemological Issues in Mathematical Understanding. In E.A. Silver (Eds.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives (pp. 361-379). Hillsdale, Nj: Erlbaum.
- Schraw, G., (1998). Promoting General Metacognitive Awareness. Instructional Science, 26(1-2), 113-125. DOI:1023/A:1003044231033
- Schraw, G., Crippen, K.J., and Hartley, K., (2006). Promoting Self-Regulation in Science Education: Metacognition as Part of a Broader Perspective on Learning. Research in Science Education, 36, 111-139. DOI: 10.1007/s11165-005-3917-8
- Sheffield, L.J. and Cruikshank, D.E., (2005). Teaching and Learning Mathematics. Pre-kindergarten Though Middle School(5th Ed.) New York: J. Wiley.
- Smilkstein, R., (1993). Acquiring Knowledge and Using İt. http://www.eric.ed.gov/ERICWebPortal/search/detailmini.jsp?_nfpb =true&_&ERICExtSearch_SearchValue_0=ED382238, Erişim tarihi:12 Haziran 2009.
- Smith, P.L. and Ragan, T.J., (1993). Instructional Design. New york: Macmillan.
- Sperling, R.A., Howard, B.C., Miller, L.A. and Murphy, C., (2002). Measures of Children’s Knowledge and Regulation of Cognition. Contemporary Educational Psychology, 27, 51-79. DOI:1006/ceps.2001.1091
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İLKÖĞRETİM ÖĞRENCİLERİNİN CEBİRSEL PROBLEMLERİ ÇÖZME BAŞARILARININ ÜSTBİLİŞSEL BİLGİ BAĞLAMINDA İNCELENMESİ
Yıl 2013,
Cilt: 8 Sayı: 4, 411 - 427, 01.05.2013
Sare Şengül
Fatma Erdoğan
Öz
Bu araştırmada, ilköğretim öğrencilerinin cebirsel problemleri çözme başarılarını üstbilişsel bilgi türlerine göre değerlendirerek, başarı düzeylerini sınıf seviyelerine göre karşılaştırmak amaçlanmıştır. Araştırma tarama türü betimsel bir araştırmadır. Araştırmanın örneklemini, 2011-2012 öğretim yılında İstanbul ili Avrupa yakasındaki bir ilköğretim okulunda öğrenim görmekte olan öğrenciler arasından rastlantısal olarak seçilen 235 öğrenci oluşturmaktadır. Araştırma verileri "Üstbilişsel Bilgiye Yönelik Matematik Başarı Testi(ÜBYMBT)" ile elde edilmiştir. Toplanan verilerin analizinde, frekans tabloları ve varyans analizinden yararlanılmıştır. Araştırma bulgularına göre, ÜBYMBT`nin tanıtıcı bilgi boyutuna yönelik öğrenci başarıları arasında sınıf seviyelerine göre anlamlı bir farklılık bulunmadığı ortaya çıkarken, işlemsel ve koşullu bilgi boyutlarına yönelik öğrenci başarılar arasında sınıf seviyelerine göre anlamlı farklılık olduğu belirlenmiştir. Elde edilen sonuçlara dayalı olarak ilköğretim matematik öğrenimi ve öğretimine yönelik öneriler geliştirilmiştir.
Kaynakça
- Akkan, Y., Baki, A., ve Çakıroğlu, Ü., (2012). 5-8. Sınıf Öğrencilerinin Aritmetikten Cebire Geçiş Süreçlerinin Problem Çözme Bağlamında İncelenmesi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 43, 1-13.
- Akkaya, R. ve Durmuş, S., (2006). İlköğretim 6-8. Sınıf Öğrencilerinin Cebir Öğrenme Alanındaki Kavram Yanılgıları. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 31, 1-12.
- Alibali, M.W., Knuth, E.J., Hattikudur, S., McNeil, N.M., and Stephens, A.C., (2007). Alongitudinal Examination of Middle School Students’ Understanding of the Equal Sign and Equivalent Equations. Mathematical Thinking and Learning, 9(3), 221-247. DOI:1080/10986060701360902
- Atılgan, H., Kan, A. ve Doğan, N., (2006). Eğitimde Ölçme ve Değerlendirme. Ankara: Anı Yayınları.
- Biryukov, P., (2004). Metacognitive Aspects of Solving Combinatorics Problems. International Journal for Mathematics Teaching and Learning, 1-19. DOI:10.1.1.178.3238
- Booth, L. and Herscovics, N., (1986). Algebra. In the Proceedings of the 5 th International Congress on Mathematical Education (pp. 150-155). Boston, USA.
- Büyüköztürk, Ş., (2012). Sosyal Bilimler İçin Veri Analizi El Kitabi. Ankara: PegemA Yayıncılık.
- Cooper, T.J., Boulton ‐Lewis, G., Athew, B., Willss, L., and Mutch, S., (1997). The Transition Arithmetic to Algebra: Initial Understandings of Equals, Operations and Variable. International Group for the Psychology of Matematics Education, 21(2), 89-96. Dede, Y. ve Peker, M., (2007). Öğrencilerin Cebire Yönelik Hata ve Yanlış Anlamaları: Matematik Öğretmen Adaylarının Bunları Tahmin Becerileri ve Çözüm Önerileri. İlköğretim Online, 6(1), 35Desoete, A., (2001). Off-Line Metacognition in Children with Mathematics Learning Disabilities. Unpublished doctoral dissertation. Belgium: Universiteit Gent.
- Desoete, A., Roeyers, H. and Buysse, A. (2001). Metacognition and Mathematical Problem Solving in Grade 3. Journal of Learning Disability, 34(5), 435–449. DOI: 10.1177/002221940103400505
- Erbaş, A.K., Çetinkaya, B. ve Ersoy, Y., (2009). Öğrencilerin Basit Doğrusal Denklemlerin Çözümünde Karşılaştıkları Güçlükler ve Kavram Yanılgıları. Eğitim ve Bilim, 34(152), 44-59.
- Falkner, K.P., Levi, L., and Carpenter, T.P., (1999). Children’s Understanding of Equality: A Foundation for Algebra. Teaching Children Mathematics, 6, 232-236.
- Flavell, J.H., (1979). Metacognition and Cognitive Monitoring: A New Area of Cognitive Developmental İnquiry. American Psychologist, 34(10), 906–911. DOI:10.1037/0003-066X.34.10.906 Flavell, J.H., (1992). Perspectives on Perspective Taking. In H. Beilin and P. Pufall (Eds.), Piaget’s theory: Prospects and Possibilities (pp. 107–139). Hillsdale, NJ: Erlbaum.
- Fortunato, I., Hecht, D., Title, C., and Alvarez, L., (1991). Metacognition and Problem Solving. Arithmetic Teacher, 39(4), 38Fouche, K.K., (1997). Algebra for Everyone: Start Early. Mathematics Teaching in the Middle School, 2(4), 226-229.
- Gürbüz, R. ve Akkan, Y., (2008). Farklı Öğrenim Seviyesindeki Öğrencilerin Aritmetikten Cebire Geçiş Düzeylerinin Karşılaştırılması: Denklem Örneği. Eğitim ve Bilim, 33(148), 64
- Hoch, M. and Dreyfus, T., (2004). Structure sense in high school algebra: The effect of brackets. In M.J. Hİines and A.B. Fuglestad (Eds.), In the Proceedings of the 28 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp.49-56). Bergen, Norway: PME.
- Ifenthaler, D., (2012). Determining the Effectiveness of Prompts for Self-Regulated Learning in Problem-Solving Scenarios. Educational Technology & Society, 15(1), 38–52.
- Karasar, N., (2009). Bilimsel Araştırma Yöntemi. Ankara: Nobel yayıncılık.
- Kieran, C., (1992). The Learning and Teaching of School Algebra. In D.A. Grouws (Eds.), Handbook of Research on Mathematics Teaching and Learning (pp. 390 ‐419). New York: Macmillan.
- Kieran, C., (2007). Learning and Teaching Algebra at the Middle School through College Levels. In Proceedings of the Second Handbook of Research on Teaching and Learning Mathematics (pp.707–762). Charlotte: Information Age.
- Kieran, C. and Chaloug, L., (1993). Prealgebra: The Transitions from Arithmetic to Algebra. In D. T. Owens (Eds.), Research İdeas for the Classroom: Middle Grades Mathematics (pp.179-198). New York: Macmillan.
- Kramarski, B., Mevarech, Z.R., and Liberman, A., (2001). The Effects of Multilevel Versus Unilevel-Metacognitive Training on Mathematical Reasoning. The Journal of Educational Research, 94(5), 292-300. DOI:10.1080/00220670109598765
- Kuchemann, D., (1981). Algebra. In K. Hart (Eds.), Children’s Understanding of Mathematics: 11-16 (pp. 102-119). London: Murray.
- Lane, S., (1993). The Conceptual Framework for the Development of a Mathematics Performance Assessment Instrument. Educational Measurement: Issues & Practices, 12(2), 16-23. DOI:1111/j.1745-3992.1993.tb00529.x
- Lesh, R., Post, T., and Behr, M., (1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. In C. Janvier (Eds.), Problems of Representation in the Teaching and Learning of Mathematics (pp.33-40). New Jersey: Lawrence Erlbaum Associates.
- MacGregor, M. and Stacey, K., (1997). Students’ Understanding of Algebraic Notation: 11–15. Educational Studies in Mathematics, 33, 1-19. DOI:10.1023/A:1002970913563
- Mevarech, Z.R., (1999). Effects of Metacognitive Training Embedded in Cooperative Settings on Mathematical Problem
- Solving. The Journal of Educational Research, 92(4), 195-205. DOI:1080/00220679909597597
- Miles, M.B. and Huberman, M.A., (1994). An Expanded Sourcebook Qualitative Data Analysis. London: Sage Publication.
- Miller, S.P. and Hudson, P.J., (2007). Using Evidence-Based Practices to Build Mathematics Competence Related to Conceptual, Procedural, and Declarative Knowledge. Learning Disabilities Research & Practice, 22(1), 47–57. DOI:10.1111/j.1540582007.00230.x
- Milli Eğitim Bakanlığı (MEB), (2009). İlköğretim Matematik Dersi 6- Sınıflar Öğretim Programı ve Kılavuzu. Ankara: MEB.
- Moyer, P.S. and Jones, M.G., (2004). Controlling Choice: Teachers, Students, and Manipulatives in Mathematics Classrooms. School Science and Mathematics, 104(1), 16-31. DOI:1111/j.1949-8594.2004.tb17978.x
- Nosegbe, I.C., (2001). Middle School Students’ Sense Making of Their Solutions to Mathematical Word Problems. USA: Indiana University.
- Ormond, C., Luszcz, M.A., Mann, L., and Beswick, G., (1991). A Metacognitive Analysis of Decision Making in Adolescence. Journal of Adolescence, 14(3), 275-291. DOI:10.1016/01401971(91)90021-I
- Orton, A. and Orton, J., (1999). Pattern and the Approach to Algebra. In A. Orton (Eds.), Pattern in the Teaching and Learning of Mathematics (pp.104-120). London: Cassel.
- Panaoura, A. and Philippou, G. (2003). The Construct Validity of an Inventory for the Measurement of Young Pupils’ Metacognitive Abilities in Mathematics. In N.A. Pateman, B.J. Doherty and J. Zilliox (Eds.), In the Proceedings of the 27 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 437-444). Honolulu, USA: Pme.
- Panaoura, A. and Philippou, G. (2007). The Developmental Change of Young Pupils’ Metacognitive Ability in Mathematics in Relation to Their Cognitive Abilities. Cognitive Development, 22(2), 149-164. DOI:10.1016/j.cogdev.2006.08.004
- Panaoura, A., Philippou, G., and Christou, C., (2003). Young Pupils’ Metacognitive Ability in Mathematics. http://fibonacci.dm.unipi.it/clusterpages/didattica/CERME3/proce edings/Groups/TG3/TG3_Panaoura_cerme3.pdf, Erişim tarihi: 15 Haziran 2008.
- Perso, T.F., (1992). Using Diagnostic Teaching to Overcome Misconceptions in Algebra. The Mathematical Association of Western Australia.
- Pintrich, P.R., (2002) . The Role of Metacognitive Knowledge in Learning, Teaching and Assessing. Theory into Practice, 41(4), 219-2 DOI:10.1207/s15430421tip4104_3
- Pope, S. and Sharma, R., (2001). Symbol Sense: Teacher’s and Student’s Understanding. In the Proceedings of the British Society for Research into Learning Mathematics, 21(3). http://www.bsrlm.org.uk/IPs/ip21-3/BSRLM-IP-21-3-12.pdf, Erişim tarihi:17 Mayıs 2009.
- Pugalee, D.K., (2001). Writing, Mathematics, and Meta-Cognition: Looking for Connections through Students’ Work in Mathematical Problem Solving. School Science and Mathematics, 101(5), 2362 DOI:10.1111/j.1949-8594.2001.tb18026.x
- Pugalee, D.K., (2004). A Comparison of Verbal and Written Description of Students’ Problem Solving Processes. Educational Studies in Mathematics, 55(1-3), 27-47. DOI:1023/B:EDUC.0000017666.11367.c7
- Sakonidis, H. and Bliss, J., (1990). Children’s Writing about the Idea of Variable in the Context of Formula. In the Proceedings of The Annual Conference of The International Group for The Psychology of Mathematics Education with The North American Chapter 12 th PMEANA Conference (Vol. 2, pp.133-140). Mexico: PME.
- Schoenfeld, A.H., (1985). Metacognitive and Epistemological Issues in Mathematical Understanding. In E.A. Silver (Eds.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives (pp. 361-379). Hillsdale, Nj: Erlbaum.
- Schraw, G., (1998). Promoting General Metacognitive Awareness. Instructional Science, 26(1-2), 113-125. DOI:1023/A:1003044231033
- Schraw, G., Crippen, K.J., and Hartley, K., (2006). Promoting Self-Regulation in Science Education: Metacognition as Part of a Broader Perspective on Learning. Research in Science Education, 36, 111-139. DOI: 10.1007/s11165-005-3917-8
- Sheffield, L.J. and Cruikshank, D.E., (2005). Teaching and Learning Mathematics. Pre-kindergarten Though Middle School(5th Ed.) New York: J. Wiley.
- Smilkstein, R., (1993). Acquiring Knowledge and Using İt. http://www.eric.ed.gov/ERICWebPortal/search/detailmini.jsp?_nfpb =true&_&ERICExtSearch_SearchValue_0=ED382238, Erişim tarihi:12 Haziran 2009.
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