Yıl 2023,
Cilt: 24 Sayı: 2, 1222 - 1242, 15.09.2023
Hafize Gamze Kırmızıgül
,
Aziz İlhan
,
Bilal Altay
Kaynakça
- Akkuş, O., & Duatepe-Paksu, A. (2006). Construction of a proportional reasoning test and its rubrics. Eurasian Journal of Educational Research, 25(1), 1-10.
- Alfieri, L., Higashi, R., Shoop, R., & Schunn, C. D. (2015). Case studies of a robot-based game to shape interests and hone proportional reasoning skills. International Journal of STEM Education, 2(4), 1-13. https://doi.org/10.1186/s40594-015-0017-9
- Allain, A. (2000). Development of an instrument to measure proportional reasoning among fast-track middle school students. [Master’s thesis]. University of North Carolina State.
- Alpar, R. (2003). Uygulamalı çok değişkenli istatistiksel yöntemlere giriş (2. Baskı). Nobel Yayın Dağıtım.
- Arıcan, M. (2019). A diagnostic assessment to middle school students’ proportional reasoning. Turkish Journal of Education, 8(4), 237-257. https://doi.org/10.19128/turje.522839
- Ayan, R., & Işıksal-Bostan, M. (2019). Middle school students’ proportional reasoning in real life contexts in the domain of geometry and measurement. International Journal of Mathematical Education in Science and Technology, 50(1), 65-81. https://doi.org/10.1080/0020739X.2018.1468042
- Behr, M., Harel, G., Post, T.,& Lesh, R. (1992). Rational number, ratio and proportion. In D.Grouws (Eds.), Handbook of research on mathematics teaching and learning (pp. 296-333). McMillan Publishing.
- Ben-Chaim, D., Fey, J. T., Fitzgerald, W. M., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7th grade students with different curricular experiences. Educational Studies in Mathematics, 36(3), 247-273. https://doi.org/10.1023/A:1003235712092
- Büyüköztürk, Ş. (2014). Sosyal bilimler için veri analizi el kitabı (19. baskı). Pegem Akademi.
Büyüköztürk, Ş., Kılıç Çakmak, E., Akgün, Ö., Karadeniz Ş., & Demirel, F. (2014). Bilimsel araştırma yöntemleri. Pegem Akademi.
- Can, A. (2018). SPSS ile bilimsel araştırma sürecinde nicel veri analizi (6. Baskı). Pegem Akademi.
- Christensen, L.B., Johnson, R.B. & Turner, L.A. (2015). Research methods design and analysis. Cambridge: Pearson [Çeviri: Sever, M. (2015)]. Nitel ve karma yöntem araştırmaları.(Çev. Ed. A. Aypay, Araştırma yöntemleri desen ve analiz). Anı Yayıncılık.
- Cramer, K., & Post, T. (1993). Proportional reasoning. The Mathematics Teacher, 86(5), 404-407. https://www.jstor.org/stable/27968390
- Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: research implications. In D. Owens (Eds.), Research ideas for the classroom (pp. 159-178). Macmillan Publishing Company.
- Creswell, J. W. (2014). A concise introduction to mixed methods research. Sage Publications.
- Çetin, H., & Ertekin, E. (2011). The relationship between eighth grade primary school students’ proportional reasoning skills and success in solving equations. International Journal of Instruction, 4(1), 48-62. https://dergipark.org.tr/en/download/article-file/59778
- Dinç-Artut, P., & Pelen, M. S. (2015). 6th grade students’ solution strategies on proportional reasoning problems. Procedia-Social and Behavioral Sciences, 197(1), 113-119. https://doi.org/10.1016/j.sbspro.2015.07.066
Duatepe, A., Akkuş-Cıkla, O., & Kayhan, M. (2005). An investigation on students’ solution strategies for different proportional reasoning items. Hacettepe Journal of Education Faculty, 28(1), 73-81.
- Fernández, C., Llinares, S., & Valls, J. (2013). Primary school teacher's noticing of students' mathematical thinking in problem solving. The Mathematics Enthusiast, 10(1/2), 441. https://doi.org/10.54870/1551-3440.1274
- Fleiss, J. L. (1971). Measuring nominal scale agreement among many raters. Psychological Bulletin, 76(5), 378-382. https://psycnet.apa.org/doi/10.1037/h0031619
- Haja, S., & Clarke, D. (2011). Middle school students’ responses to two-tier tasks. Mathematics Education Research Journal, 23(1), 67-76. https://doi.org/10.1007/s13394-011-0004-5
- Hart, K. M. (1984). Ratio: Children's strategies and errors. A report of the strategies and errors in secondary mathematics project. NFER-Nelson.
- Hilton, A., Hilton, G., Dole, S., & Goos, M. (2013). Development and application of a two-tier diagnostic instrument to assess middle - years students’ proportional reasoning. Mathematics Education Research Journal, 25(4), 523-545. https://doi.org/10.1007/s13394-013-0083-6
- Karlı, M. G., &Yıldız, E. (2022). Incorrect strategies developed by seventh-grade students to solve proportional reasoning problems. Journal of Qualitative Research in Education, 29(1),111-148. https://doi.org/10.14689/enad.29.5
- Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents' proportional reasoning on ‘rate’ problems. Educational Studies in Mathematics, 14(3), 219-233. https://doi.org/10.1007/BF00410539
- Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–668). Information Age Publishing.
- Lamon, S. (2012). Teaching fractions and ratios for understanding: Ratio and proportion (3rd ed.). Routledge.
- Landis, J. R., & Koch, G. G. (1977). An application of hierarchical kappa-type statistics in the assessment of majority agreement among multiple observers. Biometrics, 33(2), 363-374. https://doi.org/10.2307/2529786
- Langrall, C. W., & Swafford, J. (2000). Three balloons for two dollars: Developing proportional reasoning. Mathematics Teaching in The Middle School, 6(4), 254–261. https://doi.org/10.5951/MTMS.6.4.0254
- Lawton, C. A. (1993). Contextual factors affecting errors in proportional reasoning. Journal for Research in Mathematics Education, 24(5), 460-466. https://doi.org/10.2307/749154
- Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93-118). Lawrence Erlbaum & NCTM.
- Lobato, J., & Ellis, A. B. (2010). Developing essential understandings of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. National Council of Teachers of Mathematics Raport.
- Mardika, F., & Mahmudi, A.(2021). An analysis of proportional reasoning ability of junior high school students. Jurnal Riset Pendidikan Matematika, 8(1), 22–32. https://doi.org/10.21831/jrpm.v8i1.14995
- Mersin, N. (2018). An evaluation of proportional reasoning of middle school 5th, 6th and 7th grade students according to a two-tier diagnostic test. Cumhuriyet International Journal of Education, 7(4), 319-348. https://doi.org/10.30703/cije.4266271
- Milli Eğitim Bakanlığı [MEB], (2018). Matematik dersi öğretim programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar). Milli Eğitim Bakanlığı Talim ve Terbiye Kurulu Başkanlığı.
- Misailidou, C., & Williams, J. (2003). Diagnostic assessment of children’s proportional reasoning. Journal of Mathematical Behavior, 22(3), 335-368. https://doi.org/10.1016/S0732-3123(03)00025-7
- Modestou, M., & Gagatsis, A. (2008). Proportional reasoning in elementary and secondary education: Moving beyond the percentages. In A. Gagatsis (Ed.), Research in mathematics education (pp. 147-162). University of Cyprus.
- Modestou, M., & Gagatsis, A. (2010). Cognitive and metacognitive aspects of proportional reasoning. Mathematical Thinking and Learning, 12(1), 36-53. https://doi.org/10.1080/10986060903465822
- Muijs, D. (2004). Doing quantitative research in education with SPSS. London: Sage Publications.
National Council of Teachers of Mathematics (NCTM) (2000). Principles and Standards for School Mathematics. National Council of Teachers of Mathematics.
- Noelting, G. (1980). The development of proportional reasoning and the ratio concept: Part 1-Differentiation of stages. Educational Studies in Mathematics, 11(1), 217-253. https://doi.org/10.1007/BF00304357
- Özgün-Koca, S. A., & Altay, M. K. (2009). An investigation of proportional reasoning skills of middle school students. Investigations in Mathematics Learning, 2(1), 26-48. https://doi.org/10.1080/24727466.2009.11790289
Pakmak, G. S. (2014). 6. sınıf öğrencilerinin niceliksel ve niteliksel orantısal akıl yürütme problemlerinin çözümündeki anlayışlarının incelenmesi [Yüksek Lisans Tezi]. Pamukkale Universitesi, Eğitim Bilimleri Enstitüsü, Denizli.
- Pelen, M. S. (2014). 6. sınıf ögrencilerinin orantısal akıl yürütme becerilerinin problemlerin sınıflanması ve sayısal yapılarına göre incelenmesi [Yüksek Lisans Tezi]. Çukurova Universitesi, Eğitim Bilimleri Enstitüsü, Adana.
- Pelen, M. S., & Dinç-Artut, P. (2015). 7th grade students’problem solving success rates on proportional reasoning problems. The Eurasia Proceedings of Educational and Social Sciences, 2(1), 96-100. https://doi.org/10.21890/ijres.71245
Ramful, A., & Narod, F. B. (2014). Proportional reasoning in the learning of chemistry: levels of complexity. Mathematics Education Research Journal, 26(1), 25-46. https://doi.org/10.1007/s13394-013-0110-7
- Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 43(1), 271-292. https://doi.org/10.1023/A:1011976904850
- Teddlie, C. & Tashakkori, A. (2015). Karma yöntem araştırmalarının temelleri. Anı Yayıncılık.
Tavşancıl E. (2002). Tutumların ölçülmesi ve SPSS ile veri analizi. Nobel Yayıncılık.
- Tekin, H. (2010). Eğitimde ölçme ve değerlendirme (20. Baskı). Yargı Yayınevi.
- Toluk-Uçar, Z. & Bozkuş, F. (2016). İlkokul ve ortaokul öğrencilerinin orantısal durumları orantısal olmayan durumlardan ayırt edebilme becerileri. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi, 17(3) , 281-299. Retrieved from https://dergipark.org.tr/tr/pub/kefad/issue/59425/853495
- Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204. https://doi.org/10.1007/BF02400937
- Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: teaching developmentally. Pearson Education, Inc.
- Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360-381. https://doi.org/10.1080/07370008.2010.488306
- Vanluydt, E., Verschaffel, L., & Van Dooren, W. (2018). Emergent proportional reasoning: Searching for early traces in four-to five-year olds. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd conference of the International Group for the Psychology of Mathematics Education (pp. 247–254). Sweden: Umeå.
- Vanluydt, E., Wijns, N., Torbeyns, J., & Van Dooren, W. (2021). Early childhood mathematical development: the association between patterning and proportional reasoning. Educational Studies in Mathematics, 107(1), 93–110. https://doi.org/10.1007/s10649-020-10017-w
- Weinberg, S. L. (2002). Proportional reasoning: One problem, many solutions! In B. Litwiller (Eds.), Making sense of fractions, ratios, and proportions (pp. 138-144). National Council of Teachers of Mathematics.
7. Sınıf Öğrencilerine Yönelik Bir Orantısal Akıl Yürütme Beceri Testi (OAYBT) Geliştirilmesi
Yıl 2023,
Cilt: 24 Sayı: 2, 1222 - 1242, 15.09.2023
Hafize Gamze Kırmızıgül
,
Aziz İlhan
,
Bilal Altay
Öz
Araştırmanın amacı 7. sınıf öğrencilerinde orantısal akıl yürütme becerilerini teşhis etmek adına geçerli ve güvenilirliği kanıtlanmış bir beceri testi geliştirmektir. Bu bağlamda testin geliştirilme sürecinde karma yöntemin desenlerinden keşfedici sıralı desene göre araştırma süreci yürütülmüştür. Araştırmanın nitel kısmında doküman incelemesi yapılmış, uzman akademisyenlerden ve öğretmenlerden görüş alma işlemi gerçekleştirilmiştir. Nitel kısım ile kapssamlı bir alanyazın taraması gerçekleştirilmiş ve testin genel çerçevesi çizilmiştir. Nicel kısımda ise nitel kısımda toplanan veriler istatistiksel yöntemlerle analiz edilmiş ve ölçme aracına son şekli verilmiştir. Araştırmanın katılımcıları Türkiye’nin Doğu Anadolu bölgesinde bulunan orta büyüklükteki bir ilinde beş farklı devlet ortokulunda öğrenimlerine devam etmekte olan 307 ortaokul 7. sınıf öğrencisinden oluşmaktadır. 21 çoktan seçmeli sorudan oluşan test için kapsam geçerliği madde indeks analizi ve yapı geçerliği çalışmaları sırasıyla yapılmıştır. Bu doğrultuda Microsoft Excel ve Factor programları kullanılmıştır. Güvenirlik çalışmaları kapsamında ise KR-20 iç tutarlılık katsayısı 0,76 olarak hesaplanmıştır. Geçerlik ve güvenirlik analizleri sonucunda elde edilen teste yönelik ileride yapılacak araştırmalara önerilerde bulunulmuştur.
Kaynakça
- Akkuş, O., & Duatepe-Paksu, A. (2006). Construction of a proportional reasoning test and its rubrics. Eurasian Journal of Educational Research, 25(1), 1-10.
- Alfieri, L., Higashi, R., Shoop, R., & Schunn, C. D. (2015). Case studies of a robot-based game to shape interests and hone proportional reasoning skills. International Journal of STEM Education, 2(4), 1-13. https://doi.org/10.1186/s40594-015-0017-9
- Allain, A. (2000). Development of an instrument to measure proportional reasoning among fast-track middle school students. [Master’s thesis]. University of North Carolina State.
- Alpar, R. (2003). Uygulamalı çok değişkenli istatistiksel yöntemlere giriş (2. Baskı). Nobel Yayın Dağıtım.
- Arıcan, M. (2019). A diagnostic assessment to middle school students’ proportional reasoning. Turkish Journal of Education, 8(4), 237-257. https://doi.org/10.19128/turje.522839
- Ayan, R., & Işıksal-Bostan, M. (2019). Middle school students’ proportional reasoning in real life contexts in the domain of geometry and measurement. International Journal of Mathematical Education in Science and Technology, 50(1), 65-81. https://doi.org/10.1080/0020739X.2018.1468042
- Behr, M., Harel, G., Post, T.,& Lesh, R. (1992). Rational number, ratio and proportion. In D.Grouws (Eds.), Handbook of research on mathematics teaching and learning (pp. 296-333). McMillan Publishing.
- Ben-Chaim, D., Fey, J. T., Fitzgerald, W. M., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7th grade students with different curricular experiences. Educational Studies in Mathematics, 36(3), 247-273. https://doi.org/10.1023/A:1003235712092
- Büyüköztürk, Ş. (2014). Sosyal bilimler için veri analizi el kitabı (19. baskı). Pegem Akademi.
Büyüköztürk, Ş., Kılıç Çakmak, E., Akgün, Ö., Karadeniz Ş., & Demirel, F. (2014). Bilimsel araştırma yöntemleri. Pegem Akademi.
- Can, A. (2018). SPSS ile bilimsel araştırma sürecinde nicel veri analizi (6. Baskı). Pegem Akademi.
- Christensen, L.B., Johnson, R.B. & Turner, L.A. (2015). Research methods design and analysis. Cambridge: Pearson [Çeviri: Sever, M. (2015)]. Nitel ve karma yöntem araştırmaları.(Çev. Ed. A. Aypay, Araştırma yöntemleri desen ve analiz). Anı Yayıncılık.
- Cramer, K., & Post, T. (1993). Proportional reasoning. The Mathematics Teacher, 86(5), 404-407. https://www.jstor.org/stable/27968390
- Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: research implications. In D. Owens (Eds.), Research ideas for the classroom (pp. 159-178). Macmillan Publishing Company.
- Creswell, J. W. (2014). A concise introduction to mixed methods research. Sage Publications.
- Çetin, H., & Ertekin, E. (2011). The relationship between eighth grade primary school students’ proportional reasoning skills and success in solving equations. International Journal of Instruction, 4(1), 48-62. https://dergipark.org.tr/en/download/article-file/59778
- Dinç-Artut, P., & Pelen, M. S. (2015). 6th grade students’ solution strategies on proportional reasoning problems. Procedia-Social and Behavioral Sciences, 197(1), 113-119. https://doi.org/10.1016/j.sbspro.2015.07.066
Duatepe, A., Akkuş-Cıkla, O., & Kayhan, M. (2005). An investigation on students’ solution strategies for different proportional reasoning items. Hacettepe Journal of Education Faculty, 28(1), 73-81.
- Fernández, C., Llinares, S., & Valls, J. (2013). Primary school teacher's noticing of students' mathematical thinking in problem solving. The Mathematics Enthusiast, 10(1/2), 441. https://doi.org/10.54870/1551-3440.1274
- Fleiss, J. L. (1971). Measuring nominal scale agreement among many raters. Psychological Bulletin, 76(5), 378-382. https://psycnet.apa.org/doi/10.1037/h0031619
- Haja, S., & Clarke, D. (2011). Middle school students’ responses to two-tier tasks. Mathematics Education Research Journal, 23(1), 67-76. https://doi.org/10.1007/s13394-011-0004-5
- Hart, K. M. (1984). Ratio: Children's strategies and errors. A report of the strategies and errors in secondary mathematics project. NFER-Nelson.
- Hilton, A., Hilton, G., Dole, S., & Goos, M. (2013). Development and application of a two-tier diagnostic instrument to assess middle - years students’ proportional reasoning. Mathematics Education Research Journal, 25(4), 523-545. https://doi.org/10.1007/s13394-013-0083-6
- Karlı, M. G., &Yıldız, E. (2022). Incorrect strategies developed by seventh-grade students to solve proportional reasoning problems. Journal of Qualitative Research in Education, 29(1),111-148. https://doi.org/10.14689/enad.29.5
- Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents' proportional reasoning on ‘rate’ problems. Educational Studies in Mathematics, 14(3), 219-233. https://doi.org/10.1007/BF00410539
- Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–668). Information Age Publishing.
- Lamon, S. (2012). Teaching fractions and ratios for understanding: Ratio and proportion (3rd ed.). Routledge.
- Landis, J. R., & Koch, G. G. (1977). An application of hierarchical kappa-type statistics in the assessment of majority agreement among multiple observers. Biometrics, 33(2), 363-374. https://doi.org/10.2307/2529786
- Langrall, C. W., & Swafford, J. (2000). Three balloons for two dollars: Developing proportional reasoning. Mathematics Teaching in The Middle School, 6(4), 254–261. https://doi.org/10.5951/MTMS.6.4.0254
- Lawton, C. A. (1993). Contextual factors affecting errors in proportional reasoning. Journal for Research in Mathematics Education, 24(5), 460-466. https://doi.org/10.2307/749154
- Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93-118). Lawrence Erlbaum & NCTM.
- Lobato, J., & Ellis, A. B. (2010). Developing essential understandings of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. National Council of Teachers of Mathematics Raport.
- Mardika, F., & Mahmudi, A.(2021). An analysis of proportional reasoning ability of junior high school students. Jurnal Riset Pendidikan Matematika, 8(1), 22–32. https://doi.org/10.21831/jrpm.v8i1.14995
- Mersin, N. (2018). An evaluation of proportional reasoning of middle school 5th, 6th and 7th grade students according to a two-tier diagnostic test. Cumhuriyet International Journal of Education, 7(4), 319-348. https://doi.org/10.30703/cije.4266271
- Milli Eğitim Bakanlığı [MEB], (2018). Matematik dersi öğretim programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar). Milli Eğitim Bakanlığı Talim ve Terbiye Kurulu Başkanlığı.
- Misailidou, C., & Williams, J. (2003). Diagnostic assessment of children’s proportional reasoning. Journal of Mathematical Behavior, 22(3), 335-368. https://doi.org/10.1016/S0732-3123(03)00025-7
- Modestou, M., & Gagatsis, A. (2008). Proportional reasoning in elementary and secondary education: Moving beyond the percentages. In A. Gagatsis (Ed.), Research in mathematics education (pp. 147-162). University of Cyprus.
- Modestou, M., & Gagatsis, A. (2010). Cognitive and metacognitive aspects of proportional reasoning. Mathematical Thinking and Learning, 12(1), 36-53. https://doi.org/10.1080/10986060903465822
- Muijs, D. (2004). Doing quantitative research in education with SPSS. London: Sage Publications.
National Council of Teachers of Mathematics (NCTM) (2000). Principles and Standards for School Mathematics. National Council of Teachers of Mathematics.
- Noelting, G. (1980). The development of proportional reasoning and the ratio concept: Part 1-Differentiation of stages. Educational Studies in Mathematics, 11(1), 217-253. https://doi.org/10.1007/BF00304357
- Özgün-Koca, S. A., & Altay, M. K. (2009). An investigation of proportional reasoning skills of middle school students. Investigations in Mathematics Learning, 2(1), 26-48. https://doi.org/10.1080/24727466.2009.11790289
Pakmak, G. S. (2014). 6. sınıf öğrencilerinin niceliksel ve niteliksel orantısal akıl yürütme problemlerinin çözümündeki anlayışlarının incelenmesi [Yüksek Lisans Tezi]. Pamukkale Universitesi, Eğitim Bilimleri Enstitüsü, Denizli.
- Pelen, M. S. (2014). 6. sınıf ögrencilerinin orantısal akıl yürütme becerilerinin problemlerin sınıflanması ve sayısal yapılarına göre incelenmesi [Yüksek Lisans Tezi]. Çukurova Universitesi, Eğitim Bilimleri Enstitüsü, Adana.
- Pelen, M. S., & Dinç-Artut, P. (2015). 7th grade students’problem solving success rates on proportional reasoning problems. The Eurasia Proceedings of Educational and Social Sciences, 2(1), 96-100. https://doi.org/10.21890/ijres.71245
Ramful, A., & Narod, F. B. (2014). Proportional reasoning in the learning of chemistry: levels of complexity. Mathematics Education Research Journal, 26(1), 25-46. https://doi.org/10.1007/s13394-013-0110-7
- Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 43(1), 271-292. https://doi.org/10.1023/A:1011976904850
- Teddlie, C. & Tashakkori, A. (2015). Karma yöntem araştırmalarının temelleri. Anı Yayıncılık.
Tavşancıl E. (2002). Tutumların ölçülmesi ve SPSS ile veri analizi. Nobel Yayıncılık.
- Tekin, H. (2010). Eğitimde ölçme ve değerlendirme (20. Baskı). Yargı Yayınevi.
- Toluk-Uçar, Z. & Bozkuş, F. (2016). İlkokul ve ortaokul öğrencilerinin orantısal durumları orantısal olmayan durumlardan ayırt edebilme becerileri. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi, 17(3) , 281-299. Retrieved from https://dergipark.org.tr/tr/pub/kefad/issue/59425/853495
- Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204. https://doi.org/10.1007/BF02400937
- Van De Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: teaching developmentally. Pearson Education, Inc.
- Van Dooren, W., De Bock, D., & Verschaffel, L. (2010). From addition to multiplication and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360-381. https://doi.org/10.1080/07370008.2010.488306
- Vanluydt, E., Verschaffel, L., & Van Dooren, W. (2018). Emergent proportional reasoning: Searching for early traces in four-to five-year olds. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd conference of the International Group for the Psychology of Mathematics Education (pp. 247–254). Sweden: Umeå.
- Vanluydt, E., Wijns, N., Torbeyns, J., & Van Dooren, W. (2021). Early childhood mathematical development: the association between patterning and proportional reasoning. Educational Studies in Mathematics, 107(1), 93–110. https://doi.org/10.1007/s10649-020-10017-w
- Weinberg, S. L. (2002). Proportional reasoning: One problem, many solutions! In B. Litwiller (Eds.), Making sense of fractions, ratios, and proportions (pp. 138-144). National Council of Teachers of Mathematics.