Incorrect Strategies Developed by Seventh- Grade Students to Solve Proportional Reasoning Problems

Abstract

This study was conducted to determine the incorrect strategies developed by seventh-grade students to solve problems that require proportional reasoning and evaluate their solutions according to SOLO Taxonomy. This two-stage case study was conducted in a public school in the Central Black Sea region. The Proportional Reasoning Skill Test was administered to 33 seventh-grade students in the first stage. In the second stage, semi-structured interviews were held with 10 students by determining the 5 problems that students made mistakes in most. The students developed 5 different incorrect strategies when solving problems requiring proportional reasoning: additive relationships, data neglect, using numbers and no content, giving an emotional response, and failure to identify non-proportional situations. Students had difficulty identifying non-proportional situations and established additive relationships in problems requiring multiplicative relationships. Also, the levels of the participants were examined with the SOLO Taxonomy Rubric developed for proportional reasoning skills. Accordingly, the students with a high level of proportional reasoning were at abstract and relational structure levels, whereas those with moderate scores were at uni-structural and multi-structural levels. The students with low scores had low-level skills according to SOLO Taxonomy criteria.

Keywords

• proportional reasoning
• SOLO taxonomy
• ratio
• proportion
• solution strategy

References

1. Akkas, E. N. (2009). 6-8. sınıf ogrencilerinin istatistiksel dusuncelerinin incelenmesi [Evaluation of vocational school students' learning about "limit-continuity" in computer aided environment according to SOLO Taxonomy]. [Yayımlanmamıs yuksek lisans tezi]. Abant İzzet Baysal Universitesi.
2. Akkus, O., & Duatepe, P. A. (2006). Orantısal akıl yurutme becerisi testi ve teste yonelik dereceli puanlama anahtArinın gelistirilmesi [Proportional reasoning skills test and development of a rubric for the test]. EurasianJournal of Educational Research, 25(6), 1-10. https://ejer.com.tr/wp-content/uploads/2021/01/ejer_2006_issue_25.pdf.
3. Aladag, A. (2009). İlkogretim ogrencilerinin orantısal akıl yurutmeye dayalı sozel problemler ile gercekci cevap gerektiren problemleri cozme becerilerinin incelenmesi [Investigation of primary school students' ability to solve verbal problems based on proportional reasoning and problems requiring realistic answers]. [Yayımlanmamıs yuksek lisans tezi]. Cukurova Universitesi.
4. Aladag, A., & Artut, P. D. (2012). Ogrencilerin orantısal akıl yurutme ve gercekci problem cozme becerilerinin incelenmesi [Investigation of students' proportional reasoning and realistic problem solving skills]. İlkogretim Online, 11(4), 995-1009. http://www.ilkogretim-online.org/fulltext/218-1596968152.pdf?1633023246.
5. Alkan, H., & Guzel, E.B. (2005). Ogretmen adaylArinda matematiksel dusunmenin gelisimi [The development of mathematical thinking in prospective teachers]. Gazi Universitesi Gazi Egitim Fakultesi Dergisi, 25(3), 221-236. http://www.gefad.gazi.edu.tr/tr/download/article-file/77238.
6. Altayli, D. (2012). Gercekci matematik egitiminin oran orantı konusunun ogretimi ve orantısal akıl yurutme becerilerinin gelistirilmesine etkisi [The effect of realistic mathematics education on the teaching of ratio and proportion and the development of proportional reasoning skills]. [Yayımlanmamıs yuksek lisans tezi]. Ataturk Universitesi.
7. Ardic, E. O., Yılmaz, B., & Demir, E. (2012). İlkogretim 8. sınıf ogrencilerinin merkezi egilim ve yayılım olculeri hakkındaki istatistiksel okuryazarlık duzeylerinin SOLO taksonomisine gore incelenmesi [Examination of primary school 8th grade students' statistical literacy levels about central tendency and diffusion measures according to SOLO taxonomy]. X. Fen Bilimleri ve Matematik Egitimi Kongresinde sunulan bildiri. Nigde Universitesi, Nigde.
8. Ari, A. (2013). Bilissel alan sınıflamasında yenilenmis Bloom, SOLO, Fink, Dettmer Taksonomileri ve uluslararası alanda tanınma durumlAri [Revised Bloom, SOLO, Fink, Dettmer Taxonomies in cognitive domain classification and their international recognition]. Usak Universitesi Sosyal Bilimler Dergisi, 6(2), 259-290. https://doi.org/10.12780/UUSBD164.

9. Arslan, C., & İlkorucu, S. (2017). İlkogretim matematik ve fen bilgisi ogretmen adaylArinın matematiksel dusunme duzeyleri [Mathematical thinking levels of primary school mathematics and science teacher candidates]. Erzincan Universitesi Egitim Fakultesi Dergisi, 20(1), 156-166. https://doi.org/10.17556/erziefd.310384.
10. Atabas, S. (2014). An examination of fifth and sixth-grade students’ proportional reasoning. [Unpublished Master’s Thesis]. Bogazici University.
11. Avcu, R. (2010). İlkogretim 7. sınıf ogrencilerinin oran ve orantı problemlerindeki cozum stratejileri uzerine bir arastırma [A research on solution strategies of 7th grade primary school students in ratio and proportion problems]. [Yayımlanmamıs yuksek lisans tezi]. Selcuk Universitesi.
12. Bagdat, O. (2013). İlkogretim 8. sınıf ogrencilerinin cebirsel dusunme becerilerinin SOLO Taksonomisi ile incelenmesi [Investigation of primary school 8th-grade students' algebraic thinking skills with SOLO Taxonomy]. [Yayımlanmamıs yuksek lisans tezi]. Eskisehir Osmangazi Universitesi.
13. Bagdat, O., & Anapa-Saban, P. (2014). İlkogretim 8. sınıf ogrencilerinin cebirsel dusunme becerilerinin SOLO taksonomisi ile incelenmesi [Examination of primary school 8th-grade students' algebraic thinking skills with SOLO Taxonomy]. The Journal of Academic Social Science Studies, 26(2), 473-496. http://dx.doi.org/10.9761/JASSS2364.
14. Bart, W., Post, T., Behr, M., & Lesh, R. (1994). A diagnostic analysis of a proportional reasoning test item: An introduction to the properties of a semi-dense item. Focus on Learning Problems in Mathematics, 16(3), 1-11. https://eric.ed.gov/?id=EJ364133.
15. Bayazit, İ., & Kırnap-Donmez, S. M. (2017). Ogretmen adaylArinın problem kurma becerilerinin orantısal akıl yurutme gerektiren durumlar baglamında incelenmesi [Examination of prospective teachers' problem posing skills in the context of situations that require proportional reasoning]. TurkishJournal of Computer and Mathematics Education, 8(1), 130-160. https://doi.org/10.16949/turkbilmat.303759.
16. Baykul, Y. (2014). Ortaokulda matematik ogretimi (5-8. sınıflar) [Teaching mathematics in secondary schools (grades 5-8)]. Gelistirilmis İkinci Baskı. Pegem Akademi.
17. Ben-Chaim, D., Fey, J., Fitzgerald, W., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7th-grade students with different curricular experiences. Educational Studies in Mathematics, 36(3), 247-273. https://link.springer.com/article/10.1023/A:1003235712092.
18. Biggs, J., & Collis, K. (1991). Multimodal learning and the quality of intelligent behavior. Rowe, H. A. H. (Editor). Intelligence, reconceptualization and measurementicinde 57-76, New Jersey: LaurenceErlbaumAssociates.
19. Biggs, J. B. (1995). Assumptions underlying new approaches to educational assessment: Implications for Hong Kong. Curriculum Forum, 4(2), 1-22.
20. Biggs, J. B., & Collis, K. F. (1982). Evaluating the quality of learning: the SOLO Taxonomy (structure of the observed learning outcome). New York, NY: Academic Press.
21. Bilgen, O. B., & Dogan, N. (2017). Puanlayıcılar arası guvenirlik belirleme tekniklerinin karsılastırılması [Comparison of inter-rater reliability determination techniques]. Egitimde ve Psikolojide Olcme ve Degerlendirme Dergisi, 8(1), 63-78. https://doi.org/10.21031/epod.294847.
22. Bukova, E. (2006). Ogrencilerin limit kavramını algılamasında ve diger kavramlarla iliskilendirilmesinde karsılastıklAri guclukleri ortadan kaldıracak yeni bir program gelistirme [Developing a new program that will eliminate the difficulties students face in perceiving the concept of limit and associating it with other concepts]. [Yayımlanmamıs doktora tezi]. Dokuz Eylul Universitesi.
23. Bukova-Guzel, E. (2008). Yapılandırmacılık ve matematiksel dusunme surecleri [Constructivism and mathematical thinking processes]. Education Sciences, 3(4), 678-688. https://dergipark.org.tr/tr/download/article-file/185983.
24. Buyukozturk, S., Cakmak, E., Kılıc, A., Ozcan, E., Karadeniz, S., & Demirel, F. (2010). Bilimsel arastırma yontemleri [Scientific research methods]. Pegem Akademi.
25. Cajori, F. (2015). Matematik tarihi [History of mathematics] (cev. D. İlalan). ODTU Yayıncılık No: 25744. (Eserin orijinali 2014’de yayımlandı).
26. Cramer, K., & Post T. (1993). Connecting research to teaching proportional reasoning. Mathematics Teacher, 86(5), 404–407.
27. Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: research implications. Owens, D. (Editor). Research ideas for the classroom icinde 159-178, Macmillan Publishing Company.
28. Creswell, J. W. (2007). Qualitative inquiry & research design: Choosing among five approaches ( 2. Baskı). SAGE Publications.
29. Celik, A. (2010). İlkogretim ogrencilerinin orantısal akıl yurutme beceriler ile problem kurma becerileri arasındaki iliski [The relationship between proportional reasoning skills and problem posing skills of primary school students]. [Yayımlanmamıs yuksek lisans tezi]. Hacettepe Universitesi.
30. Celik, A., & Ozdemir, E. Y. (2011). İlkogretim ogrencilerinin orantısal akıl yurutme becerileri ile oran-orantı problemi kurma becerileri arasındaki iliski [The relationship between proportional reasoning skills and problem posing skills of primary school students]. Pamukkale Universitesi Egitim Fakultesi Dergisi, 30(1), 1-11. https://dergipark.org.tr/tr/download/article-file/114573.
31. Celik, D. (2007). Ogretmen adaylArinın cebirsel dusunme becerilerinin analitik incelenmesi [Analytical examination of prospective teachers' algebraic thinking skills]. [Yayımlanmamıs doktora tezi]. Karadeniz Teknik Universitesi.
32. Cetin, B., & İlhan, M. (2016). SOLO taksonomisi [SOLO taxonomy]. Bingolbali, E., Arslan, S. ve Zembat, İ. O. (Editorler). Matematik egitiminde teoriler icinde 861-879. Pegem Akademi.
33. Cetin, H. (2009). İlkogretim ikinci kademe ogrencilerinin orantısal akıl yurutme becerileri ile denklem cozme basArilAri arasındaki iliski uzerine bir calısma [A study on the relationship between proportional reasoning skills and equation solving success of primary school secondary school students]. [Yayımlanmamıs yuksek lisans tezi]. Selcuk Universitesi.
34. Cetin, İ. (2009). 7. ve 9. sınıf ogrencilerinin oran ve orantı konusundaki kavram yanılgılAri [Misconceptions of 7th and 9th grade students about ratio and proportion]. [Yayımlanmamıs yuksek lisans tezi]. Selcuk Universitesi.
35. Debreli, E. (2011). The effect of creative drama based instruction on seventh grade students’ achievement in ratio and proportion concepts and attitudes toward mathematics. [Unpublished master’s thesis]. Middle East Technical University.
36. Duatepe A., & Akkus-Cıkla O. (2002). İlkogretim matematik ogretmen adaylArinın orantısal akıl yurutme becerileri uzerine niteliksel bir calısma [A qualitative study on proportional reasoning skills of prospective elementary mathematics teachers]. Hacettepe Universitesi Egitim Fakultesi Dergisi, 23, 32-40. http://www.efdergi.hacettepe.edu.tr/shw_artcl- 947.html.
37. Duatepe A., Akkus-Cıkla O., & Kayhan M. (2005). Orantısal akıl yurutme gerektiren sorularda ogrencilerin kullandıklAri cozum stratejilerinin soru turlerine gore degisiminin incelenmesi [Investigation of the change of solution strategies used by students in questions requiring proportional reasoning according to question types]. Hacettepe Universitesi Egitim Fakultesi Dergisi, 28, 73-81. http://www.efdergi.hacettepe.edu.tr/shw_artcl-768.html.
38. Ersoy, E., & Baser, N. (2013). Matematiksel dusunme olceginin gelistirilmesi [Development of mathematical thinking scale]. Kastamonu Egitim Dergisi, 21(4), 1471-1486. https://dergipark.org.tr/tr/download/article-file/209980.
39. Fielding-Wells, J., Dole, S., & Makar, K. (2014). Inquiry pedagogy to promote emerging proportional reasoning in primary students. Mathematics Education Research Journal, 26(1), 47- 77. https://doi.org/10.1007/s13394-013-0111-6.
40. Goktepe, S., & Ozdemir, A. S. (2013). İlkogretim matematik ogretmen adaylArinın uzamsal gorsellestirme becerilerinin SOLO modeli ile incelenmesi [Examination of prospective primary school mathematics teachers’ spatial visualization skills with SOLO model]. Kalem Egitim ve İnsan Bilimleri Dergisi, 3(2),91-146. https://doi.org/10.23863/kalem.2017.26
41. Gozkaya, S. (2015). Gercekci matematik egitimi destekli ogretim yonteminin 7. sınıf oran-orantı konulArinın ogretiminde ogrenci basArisına ve ogrenmenin kalıcılıgına etkisi [The effect of realistic mathematics education supported teaching method on student success and permanence of learning in teaching 7th grade ratio-proportion topics]. [Yayımlanmamıs yuksek lisans tezi]. Erciyes Universitesi.
42. Groth, R. E., & Bergner, J.A. (2006) Pre-service elementary teachers' conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8(1), 37-63. https://doi.org/10.1207/s15327833mtl0801_3.
43. Kaplan, A., & Ozturk, M. (2012). The effect of computer based instruction method on instruction of ratio- proportion and development of proportional reasoning. Energy Education Science and Technology Part B-social And Educational Studies, 4(3), 1663- 1672. https://www.researchgate.net/publication/273136193_The_effect_of_computer_based_instruction_method_on_instruction_of_ratio- proportion_and_development_of_proportional_reasoning.
44. Kaplan, A., İsleyen, T., & Ozturk, M. (2011). 6. sınıf oran orantı konusundaki kavram yanılgılAri [Misconceptions about 6th-grade ratio and proportion]. Kastamonu Egitim Dergisi, 19(3), 953-968. https://dergipark.org.tr/tr/download/article-file/817410.
45. Karakoca, A. (2011). Altıncı sınıf ogrencilerinin problem cozmede matematiksel dusunmeyi kullanma durumlAri [Sixth grade students' use of mathematical thinking in problem solving]. [Yayımlanmamıs yuksek lisans tezi]. Eskisehir Osmangazi Universitesi.
46. Kayhan, M. (2005). 6. ve 7. sınıf ogrencilerinin oran-orantı konusuna yonelik cozum stratejilerinin; sınıf duzeyine, cinsiyete ve soru tipine gore degisiminin incelenmesi [Examination of the change in solution strategies of the 6th and 7th-grade students for ratio-proportion according to grade level, gender, and question type]. [Yayımlanmamıs yuksek lisans tezi]. Hacettepe Universitesi.
47. Kocyigit S., & Moralı H. (2020). Matematik ogretmen adaylArinın soyut matematik dersindeki bilgilerinin MATH taksonomi cercevesinde analizi [Analysis of prospective mathematics teachers' knowledge in abstract mathematics course within the framework of MATH taxonomy]. PESA Uluslararası Sosyal Arastırmalar Dergisi, 6(2), 141-161. https://doi.org/10.25272/j.2149-8385.2020.6.2.05.
48. Kocyigit Gurbuz, M. (2018). Yedinci sınıf ogrencilerinin etkinlik temelli ogrenme yaklasımı altında oran-orantı kavramlArinı olusturma sureclerinin incelenmesi: APOS Teorisi [Examination of the seventh grade students' process of forming ratio-proportion concepts under the activity-based learning approach: APOS Theory]. [Yayımlanmamıs yuksek lisans tezi]. Eskisehir Osmangazi Universitesi.
49. Kanyalihatipoglu, M. E. (2016). Ortaokul 7. sınıf ogrencilerinin analitik ve butuncul dusunme stillerinin SOLO Taksonomisi ile incelenmesi [Investigation of secondary school 7th-grade students' analytical and holistic thinking styles with SOLO Taxonomy]. [Yayımlanmamıs yuksek lisans tezi]. Recep Tayyip Erdogan Universitesi.
50. Kurdal, C. (2016). Dinamik ve etkilesimli matematik ogrenme ortamlArinda ogrencilerin kesirler ve oran orantı konusunda yaptıgı hatalar ve cozum onerileri [Mistakes and solution suggestions made by students about fractions and ratios in dynamic and interactive mathematics learning environments]. [Yayımlanmamıs yuksek lisans tezi]. Bayburt Universitesi.
51. Kupcu, A. R. (2008). Etkinlik temelli ogretim yaklasımının orantısal akil yurutmeye dayalı problem cozme basArisına etkisi [The effect of activity-based teaching approach on success at proportional reasoning-based problem solving]. [Yayımlanmamıs doktora tezi]. Marmara Universitesi Egitim Bilimleri Enstitusu, İstanbul.
52. Kupcu, A. R., & Ozdemir, A. S. (2012). İlkogretim ogrencilerinin bilissel stil, cinsiyet ve orantısal dusunme seviyelerine gore orantı iliskili problem cozme basArilAri [Proportional problem solving success of primary school students according to cognitive style, gender and proportional thinking levels]. Kastamonu Egitim Dergisi, 20(2), 451-472. https://dergipark.org.tr/tr/download/article-file/806955.
53. Langrall, C. W., & Swafford, J. (2000). Three balloons for two dollars: Developing proportional reasoning. Mathematics Teaching in the Middle School, 6(4), 254-261. Retrieved from http://www.jstor.org/stable/41180939.
54. Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. Hiebert, J. ve Behr M. (Editorler). Number concepts and operations in the middle grades icerisinde 93-118. Reston, Lawrence Erlbaum &National Council of Teachers of Mathematics.
55. MartínezOrtiz, A. (2015). examining students' proportional reasoning strategy levels as evidence of the impact of an integrated LEGO robotics and mathematics learning experience. Journal of Technology Education, 26(2), 46-69. http://doi.org/10.21061/jte.v26i2.a.3.
56. Merriam S. B. (2015). Nitel arastırma desen ve uygulama icin bir rehber [A guide to qualitative research design and practice] (cev. edt. S. Turan). Nobel YayınlAri No 349. (Eserin orijinali 2013’de yayımlandı).
57. Miles, M., & Huberman, A. (1994). Qualitative data analysis: an expanded source book. (2nd ed). Thousand Oaks, Sage.
58. 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.
59. Musan, M. S. (2012). Dinamik matematik yazılımı destekli ortamda 8. sınıf ogrencilerinin denklem ve esitsizlikleri anlama seviyelerinin SOLO Taksonomisine gore incelenmesi [Examination of 8th- grade students' understanding of equations and inequalities in a dynamic mathematics software-supported environment according to SOLO Taxonomy]. [Yayımlanmamıs yuksek lisans tezi]. Pamukkale Universitesi.
60. National Council of Teachers of Mathematics (NCTM). 2020. Principles and standards for school mathematics. Reston, Va.: NCTM.
61. Ozdemir, A. S., & Goktepe-Yıldız, S. (2015). The analysis of elementary mathematics pre-service teachers’ spatial orientation skills with SOLO model. Eurasian Journal of Educational Research (EJER), 61, 217–236. https://doi.org/10.14689/ejer.2015.61.12.
62. Ozturk, M. (2011). Bilgisayar destekli ogretim yonteminin oran orantı konusunun ogretiminde akademik basAriya etkisi [The effect of computer-assisted teaching method on academic achievement in the teaching of ratio and proportion]. [Yayımlanmamıs yuksek lisans tezi]. Ataturk Universitesi.
63. Pakmak, S. G. (2014). 6. sınıf ogrencilerinin niceliksel ve niteliksel orantısal akıl yurutme problemlerinin cozumundeki anlayıslArinın incelenmesi [Examination of 6th-grade students' understanding of solving quantitative and qualitative proportional reasoning problems]. [Yayımlanmamıs yuksek lisans tezi]. Pamukkale Universitesi.
64. Patton, M. Q. (2014). Nitel arastırma ve degerlendirme yontemleri [Qualitative research and evaluation methods]. (cevedt. M. Butun ve S. B. Demir). Pegem Akademi No: 14749. (Eserin orijinali 2014’de yayımlandı).
65. Pelen, M. S. (2014). 6. sınıf ogrencilerinin orantısal akıl yurutme becerilerinin problemlerin sınıflanması ve sayısal yapılArina gore incelenmesi [Examination of the proportional reasoning skills of 6th-grade students according to the classification of problems and their numerical structures.]. [Yayımlanmamıs yuksek lisans tezi]. Cukurova Universitesi.
66. Pittalis, M., Christou, C., & Papageorgiou, E. (2003). Students’ ability in solving proportional problems. Proceedings of the 3rd European Research Conference in Mathematics Education: Bellaria: Italy, 3.
67. Rider, R. L. (2004). The effect of multi-representational methods on students' knowledge of function concepts in developmental college mathematics. [Unpublished doctoral dissertation]. North Carolina State University.
68. Subasi, M., & Okumus, K. (2017). Bir arastırma yontemi olarak durum calısması [Case study as a research method]. Ataturk Universitesi Sosyal Bilimler Enstitusu Dergisi, 21(2), 419-426. https://dergipark.org.tr/tr/download/article-file/474049.
69. Sen, C., & Guler, G. (2017). Effect of strategy teaching for the solution of ratio problems on students' proportional reasoning skills. Malaysian Online Journal of Educational Sciences, 5(2), 1-15. https://files.eric.ed.gov/fulltext/EJ1142503.pdf.
70. Tanriogen, A. (Editor) (2012). Bilimsel arastırma yontemleri [Scientific research methods]. İkinci Baskı. Anı Yayıncılık.
71. Toluk Ucar, Z., & Bozkus, F. (2016). İlkokul ve ortaokul ogrencilerinin orantısal durumlAri orantısal olmayan durumlardan ayırt edebilme becerileri [Primary and secondary school students' ability to distinguish proportional from non-proportional situations]. Ahi Evran Universitesi Kırsehir Egitim Fakultesi Dergisi, 17(3), 281-299. https://dergipark.org.tr/tr/download/article-file/1487457.
72. Tuna, A. (2011). Trigonometri ogretiminde 5E ogrenme dongusu modelinin ogrencilerin matematiksel dusunme ve akademik basArilArina etkisi [The effect of the 5E learning cycle model on students' mathematical thinking and academic achievement in trigonometry teaching]. [Yayımlanmamıs doktora tezi]. Gazi Universitesi.
73. Umay, A. (2003). Matematiksel muhakeme yetenegi [Mathematical reasoning skills]. Hacettepe Universitesi Egitim Fakultesi Dergisi, 24, 234-243. https://dergipark.org.tr/tr/pub/hunefd/issue/7812/102550.
74. Umay, A., & Kaf, Y. (2005). Matematikte kusurlu akıl yurutme uzerine bir calısma [A study on flawed reasoning in mathematics]. Hacettepe Universitesi Egitim Fakultesi Dergisi, 28, 188- 195. https://dergipark.org.tr/tr/pub/hunefd/issue/7808/102434.
75. Unsal, A. (2009). İlkogretim 7. sınıf ogrencilerinin orantısal akıl yurutme becerilerinin basAri, tutum ve cinsiyet degiskenleri acısından incelenmesi: Bolu ili ornegi [Examination of proportional reasoning skills of primary school 7th-grade students in terms of achievement, attitudes, and gender variables: The case of Bolu province]. [Yayımlanmamıs yuksek lisans tezi]. Abant İzzet Baysal Universitesi.
76. Van de Walle, J.A., Karp, K. S., & Bay-Williams, J. M. (2013). İlkokul ve ortaokul matematigi gelisimsel yaklasımla ogretim [Teaching primary and secondary school mathematics with a developmental approach] (cev. edt. S. Durmus). Nobel YayınlAri No: 521. (Eserin orijinali 2012’de yayımlandı).
77. Wadhwa, S. (2008). A handbook of teaching and learning. New Delhi, Sarup and SonsPublishers.
78. Yesildere, S. (2006). Farklı matematiksel guce sahip ilkogretim 6, 7 ve 8. sınıf ogrencilerinin matematiksel dusunme ve bilgiyi olusturma sureclerinin incelenmesi [Examination of mathematical thinking and knowledge formation processes of primary school 6th, 7th and 8th-grade students with different mathematical strengths]. [Yayımlanmamıs doktora tezi]. Dokuz Eylul Universitesi.
79. Yildirim, A., & Simsek, H. (2008). Sosyal bilimlerde nitel arastırma yontemleri [Qualitative research methods in social sciences]. Seckin Yayıncılık.
80. Yildirim, C. (2015). Matematiksel dusunme [Mathematical thinking]. Remzi Kitabevi.