Research Article

Year 2019,
Volume: 8 Issue: 4, 237 - 257, 31.10.2019
### Abstract

### Keywords

### Thanks

### References

This study investigated Turkish middle school students’ proportional

reasoning and provided a diagnostic assessment of their strengths and

weaknesses on the ratio and proportion concepts. A proportional reasoning test

with 22 multiple-choice items was developed from the context of the log-linear

cognitive diagnosis model. The test was developed around four core cognitive

skills (attributes) that required in solving middle school ratio and proportion

problems. These skills included understanding ratios, directly, inversely, and

nonproportional relationships. The test was applied to 282 seventh grade

students, and the collected data were analyzed using the Mplus software. The

analysis showed that approximately 62% of the students were able to recognize

directly proportional relationships. Whereas, roughly 48% of them were able to

recognize inversely proportional relationships. Moreover, while 25% of the

students did not master any of the four cognitive skills, 39.1% mastered all

four of these skills. In addition, many students had difficulty distinguishing

proportional relationships from nonproportional relationships. Diagnostic

feedbacks on the students’ strengths and weaknesses were provided based on the

findings.

Diagnostic assessment Diagnostic classification models Middle school students Ratios and proportions Proportional reasoning

I would like to thank Dr. Sedat Şen and Dr. Ragıp Terzi for their valuable feedback.

- Arican, M. (2018). Preservice middle and high school mathematics teachers’ strategies when solving proportion problems. International Journal of Science and Mathematics Education, 16(2), 315–335. DOI: 10.1007/s10763-016-9775-1
- Arican, M. (2019). Preservice mathematics teachers’ understanding of and abilities to differentiate proportional relationships from nonproportional relationships. International Journal of Science and Mathematics Education, 17(7), 1423–1443. DOI: 10.1007/s10763-018-9931-x
- Arican, M., & Kuzu, O. (2019). Diagnosing preservice teachers’ understanding of statistics and probability: Developing a test for cognitive assessment. International Journal of Science and Mathematics Education, 1–20. DOI: 10.1007/s10763-019-09985-0
- Atabas, S., & Oner, D. (2017). An examination of Turkish middle school students’ proportional reasoning. Boğaziçi University Journal of Education, 33(1), 63–85.
- Ayan, R., & Isiksal-Bostan, M. (2018). 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, 1–17. DOI: 10.1080/0020739X.2018.1468042
- Beckmann, S. (2011). Mathematics for elementary teachers (3rd. ed.). Boston, MA: Pearson.
- Bradshaw, L., & Cohen, A. (2010). Accuracy of multidimensional item response model parameters estimated under small sample sizes. In A. Izsák (Chair), Using cognitive attributes to develop mathematics assessments, opportunities, and challenges. Symposium conducted at the annual American Educational Research Association conference in Denver, CO.
- Bradshaw, L., Izsak, A., Templin, J., & Jacobson, E. (2014). Diagnosing teachers’ understandings of rational numbers: Building a multidimensional test within the diagnostic classification framework. Educational Measurement: Issues and Practice, 33(1), 2–14. DOI: 10.1111/emip.12020
- Choi, K. M., Lee, Y. S., & Park, Y. S. (2015). What CDM can tell about what students have learned: An analysis of TIMSS eighth grade mathematics. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1563–1577. DOI: 10.12973/eurasia.2015.1421a
- Common Core State Standards Initiative. (2010). The common core state standards for mathematics. Washington, D.C.: Author.
- Cramer, K., & Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60(6), 342–346.
- Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 159–178). New York, NY: Macmillan.
- De Bock, D., Verschaffel, L., & Janssens, D. (1998). The predominance of the linear model in secondary school students’ solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics, 35(1), 65–83. DOI: 10.1023/A:1003151011999
- Degrande, T., Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2017). Open word problems: Taking the additive or the multiplicative road? ZDM, 50(1), 1–12. DOI: 10.1007/s11858-017-0900-6
- de la Torre, J. (2008). An empirically based method of Q‐matrix validation for the DINA model: Development and applications. Journal of Educational Measurement, 45(4), 343–362. DOI: 10.1111/j.1745-3984.2008.00069.x
- de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76(2), 179–199. DOI: 10.1007/s11336-011-9207-7
- DiBello, L. V., Stout, W. F., & Roussos, L. A. (1995). Unified cognitive/psychometric diagnostic assessment likelihood–based classification techniques. In P. Nichols, S. Chipman, & R. Brennan (Eds.), Cognitively diagnostic assessment (pp. 361–390). Hillsdale, NJ: Lawrence Erlbaum.
- Dogan, E., & Tatsuoka, K. (2008). An international comparison using a diagnostic testing model: Turkish students’ profile of mathematical skills on TIMSS–R. Educational Studies in Mathematics, 68(3), 263–272. DOI: 10.1007/s10649-007-9099-8
- Fisher, L. C. (1988). Strategies used by secondary mathematics teachers to solve proportion problems. Journal for Research in Mathematics Education, 19(2), 157–168.
- Hartz, S. (2002). A Bayesian framework for the Unified Model for assessing cognitive abilities: Blending theory with practice (Unpublished doctoral dissertation). University of Illinois at Urbana–Champaign.
- Henson, R., & Douglas, J. (2005). Test construction for cognitive diagnostics. Applied Psychological Measurement, 29(4), 262–277. DOI: 10.1177/0146621604272623
- Henson, R., Roussos, L., Douglas, J., & He, X. (2008). Cognitive diagnostic attribute–level discrimination indices. Applied Psychological Measurement, 32(4), 275–288. DOI: 10.1177/0146621607302478
- Henson, R., Templin, J., & Willse, J. (2009). Defining a family of cognitive diagnosis models using log–linear models with latent variables. Psychometrika, 74(2), 191–210. DOI: 10.1007/s11336-008-9089-5
- Huebner, A. (2010). An overview of recent developments in cognitive diagnostic computer adaptive assessments. Practical Assessment, Research & Evaluation, 15(3), 1–7.
- Im, S., & Park, H. J. (2010). A comparison of US and Korean students’ mathematics skills using a cognitive diagnostic testing method: Linkage to instruction. Educational Research and Evaluation, 16(3), 287–301. DOI: 10.1080/13803611.2010.523294
- Izsák, A., & Jacobson, E. (2017). Preservice teachers’ reasoning about relationships that are and are not proportional: A knowledge-in-pieces account. Journal for Research in Mathematics Education, 48(3), 300–339. DOI: 10.5951/jresematheduc.48.3.0300
- Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25(3), 258–272. DOI: 10.1177/01466210122032064
- Jurich, D. P., & Bradshaw, L. P. (2014). An illustration of diagnostic classification modeling in student learning outcomes assessment. International Journal of Testing, 14(1), 49–72. DOI: 10.1080/15305058.2013.835728
- Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
- Kuzu, O. (2017). Matematik ve fen bilgisi öğretmen adaylarının integral konusundaki kazanımlarının incelenmesi. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi, 18(3), 948–970. DOI: 10.29299/kefad.2017.18.3.049
- Lamon, S. (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 (Vol 1, pp. 629–667). Charlotte, NC: Information Age Publishing.
- Lee, Y. S., Park, Y. S., & Taylan, D. (2011). A cognitive diagnostic modeling of attribute mastery in Massachusetts, Minnesota, and the US national sample using the TIMSS 2007. International Journal of Testing, 11(2), 144–177. DOI: 10.1080/15305058.2010.534571
- Lei, P. W., & Li, H. (2016). Fit indices’ performance in choosing cognitive diagnostic models and Q-matrices. Paper presented at the annual meeting of the National Council on Measurement in Education (NCME), Philadelphia, PA.
- 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). Reston, VA: National Council of Teachers of Mathematics.
- Lim, K. (2009). Burning the candle at just one end: Using nonproportional examples helps students determine when proportional strategies apply. Mathematics Teaching in the Middle School, 14(8), 492–500.
- Milli Eğitim Bakanlığı (2018). Matematik dersi öğretim programı (1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar) [Mathematics curriculum (1, 2, 3, 4, 5, 6, 7, and 8. Grades]. Ankara: Talim ve Terbiye Kurulu Başkanlığı.
- Misailadou, C., & Williams, J. (2003). Measuring children’s proportional reasoning, the “tendency” for an additive strategy and the effect of models. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 293–300). Honolulu, HI: University of Hawaii.
- Modestou, M., & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity”. Educational Psychology, 27(1), 75–92. DOI: 10.1080/01443410601061462
- Muthen, L. K., & Muthen, B. O. (2011). Mplus user’s guide (6th ed.). Los Angeles, CA: Muthen & Muthen.
- National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.
- Ranjbaran, F., & Alavi, S. M. (2017). Developing a reading comprehension test for cognitive diagnostic assessment: A RUM analysis. Studies in Educational Evaluation, 55, 167–179. DOI: 10.1016/j.stueduc.2017.10.007
- Ravand, H., & Robitzsch, A. (2015). Cognitive diagnostic modeling using R. Practical Assessment, Research & Evaluation, 20(11), 1–12.
- Ravand, H., & Robitzsch, A. (2018). Cognitive diagnostic model of best choice: A study of reading comprehension. Educational Psychology, 38(10), 1255–1277. DOI: 10.1080/01443410.2018.1489524
- R Core Team. (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from http://www.R–project.org/
- Rupp, A., & Templin, J. (2008). Effects of Q–matrix misspecification on parameter estimates and misclassification rates in the DINA model. Educational and Psychological Measurement, 68(1), 78–98. DOI: 10.1177/0013164407301545
- Rupp, A., Templin, J., & Henson, R. A. (2010). Diagnostic measurement: Theory, methods, and applications. Guilford Press.
- Satorra, A., & Bentler, P. M. (2010). Ensuring positiveness of the scaled difference chi–square test statistic. Psychometrika, 75(2), 243–248. DOI: 10.1007/s11336-009-9135-y
- Sen, S., & Arican, M. (2015). A diagnostic comparison of Turkish and Korean students’ mathematics performances on the TIMSS 2011 assessment. Journal of Measurement and Evaluation in Education and Psychology, 6(2), 238–253. DOI: 10.21031/epod.65266
- Stemn, B. S. (2008). Building middle school students’ understanding of proportional reasoning through mathematical investigation. Education 3–13, 36(4), 383–392. DOI: 10.1080/03004270801959734
- Tatsuoka, K. (1985). A probabilistic model for diagnosing misconceptions by the pattern classification approach. Journal of Educational Statistics, 10(1), 55–73. DOI: 10.3102/10769986010001055
- Templin, J. (2008). Test construction item discrimination. Lecture presented at the Diagnostic Modelling Seminar at the University of Georgia, Athens. Retrieved from https://jonathantemplin.com/files/dcm/ersh9800f08/ersh9800f08_lecture11.pdf
- Templin, J., & Bradshaw, L. (2013). Measuring the reliability of diagnostic classification model examinee estimates. Journal of Classification, 30(2), 251–275. DOI: 10.1007/s00357-013-9129-4
- Templin, J., & Henson, R. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11(3), 287–305. DOI: 10.1037/1082-989X.11.3.287
- Terzi, R., & Sen, S. (2019). A nondiagnostic assessment for diagnostic purposes: Q-matrix validation and item-based model fit evaluation for the TIMSS 2011 assessment. SAGE Open, 1–11. DOI: 10.1177/2158244019832684
- Toker, T., & Green, K. (2012). An application of cognitive diagnostic assessment on TIMMS–2007 8th grade mathematics items. Paper presented at the annual meeting of the American Educational Research Association, Vancouver, British Columbia, Canada.
- Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23(1), 57–86. DOI: 10.1207/s1532690xci2301_3
- Van Dooren, W., De Bock, D., Janssens, D., & Verschaffel, L. (2007). Pupils’ overreliance on linearity: A scholastic effect? British Journal of Educational Psychology, 77(2), 307–321. DOI: 10.1348/000709906X115967
- von Davier, M. (2005). A general diagnostic model applied to language testing data. ETS Research Report. Princeton, NJ: Educational Testing Service.
- Werner, C., & Schermelleh-Engel, K. (2010). Deciding between competing models: Chi–square difference tests. In Introduction to Structural Equation Modeling with LISREL (pp. 1–3). Frankfurt, Germany: Goethe University.

Year 2019,
Volume: 8 Issue: 4, 237 - 257, 31.10.2019
### Abstract

### Keywords

### References

Bu çalışmada ortaokul

öğrencilerinin orantısal akıl yürütmeleri araştırılmış ve oran ve orantı

konuları için güçlü ve zayıf yönlerinin bilişsel bir tanısal değerlendirmesi

sağlanmıştır. Yirmi iki çoktan seçmeli madde içeren bir orantısal akıl yürütme

testi log-linear bilişsel tanı modeli perspektifinden faydalanılarak

geliştirilmiştir. Test, ortaokul öğrencilerinin oran ve orantı problemlerini

çözmeleri için gerekli olan dört temel bilişsel beceri etrafında tasarlanmıştır.

Bu beceriler sırasıyla oran, doğru orantılı ilişki, ters orantılı ilişki ve

orantısal olmayan ilişki kavramlarını anlamayı içermektedir. Test 282 yedinci

sınıf öğrencisine uygulanmış ve toplanan veriler Mplus yazılımı kullanılarak

analiz edilmiştir. Yapılan analizler neticesinde öğrencilerin en çok (yaklaşık

62%) doğru orantılı ilişkileri tanıma becerisine ve en az (yaklaşık 48%) ters

orantılı ilişkileri tanıma becerisine sahip oldukları görülmüştür. Ayrıca,

öğrencilerin 25%’inin dört temel becerinin hiçbirisine sahip olmadıkları,

39,1%’inin ise bütün becerilere sahip oldukları görülmüştür. Bunlara ek olarak,

pek çok öğrencinin orantısal ilişkileri orantısal olmayanlardan ayırt etmede

zorlandıkları görülmüştür. Elde edilen bulgular yorumlanarak öğrencilerin güçlü

ve zayıf yönleri ile ilgili tanısal geri bildirimler verilmiştir.

Bilişsel tanı modelleri Ortaokul öğrencileri Oran ve orantı Orantısal akıl yürütme Tanısal değerlendirme

- Arican, M. (2018). Preservice middle and high school mathematics teachers’ strategies when solving proportion problems. International Journal of Science and Mathematics Education, 16(2), 315–335. DOI: 10.1007/s10763-016-9775-1
- Arican, M. (2019). Preservice mathematics teachers’ understanding of and abilities to differentiate proportional relationships from nonproportional relationships. International Journal of Science and Mathematics Education, 17(7), 1423–1443. DOI: 10.1007/s10763-018-9931-x
- Arican, M., & Kuzu, O. (2019). Diagnosing preservice teachers’ understanding of statistics and probability: Developing a test for cognitive assessment. International Journal of Science and Mathematics Education, 1–20. DOI: 10.1007/s10763-019-09985-0
- Atabas, S., & Oner, D. (2017). An examination of Turkish middle school students’ proportional reasoning. Boğaziçi University Journal of Education, 33(1), 63–85.
- Ayan, R., & Isiksal-Bostan, M. (2018). 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, 1–17. DOI: 10.1080/0020739X.2018.1468042
- Beckmann, S. (2011). Mathematics for elementary teachers (3rd. ed.). Boston, MA: Pearson.
- Bradshaw, L., & Cohen, A. (2010). Accuracy of multidimensional item response model parameters estimated under small sample sizes. In A. Izsák (Chair), Using cognitive attributes to develop mathematics assessments, opportunities, and challenges. Symposium conducted at the annual American Educational Research Association conference in Denver, CO.
- Bradshaw, L., Izsak, A., Templin, J., & Jacobson, E. (2014). Diagnosing teachers’ understandings of rational numbers: Building a multidimensional test within the diagnostic classification framework. Educational Measurement: Issues and Practice, 33(1), 2–14. DOI: 10.1111/emip.12020
- Choi, K. M., Lee, Y. S., & Park, Y. S. (2015). What CDM can tell about what students have learned: An analysis of TIMSS eighth grade mathematics. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1563–1577. DOI: 10.12973/eurasia.2015.1421a
- Common Core State Standards Initiative. (2010). The common core state standards for mathematics. Washington, D.C.: Author.
- Cramer, K., & Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60(6), 342–346.
- Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. Owens (Ed.), Research ideas for the classroom: Middle grades mathematics (pp. 159–178). New York, NY: Macmillan.
- De Bock, D., Verschaffel, L., & Janssens, D. (1998). The predominance of the linear model in secondary school students’ solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics, 35(1), 65–83. DOI: 10.1023/A:1003151011999
- Degrande, T., Van Hoof, J., Verschaffel, L., & Van Dooren, W. (2017). Open word problems: Taking the additive or the multiplicative road? ZDM, 50(1), 1–12. DOI: 10.1007/s11858-017-0900-6
- de la Torre, J. (2008). An empirically based method of Q‐matrix validation for the DINA model: Development and applications. Journal of Educational Measurement, 45(4), 343–362. DOI: 10.1111/j.1745-3984.2008.00069.x
- de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76(2), 179–199. DOI: 10.1007/s11336-011-9207-7
- DiBello, L. V., Stout, W. F., & Roussos, L. A. (1995). Unified cognitive/psychometric diagnostic assessment likelihood–based classification techniques. In P. Nichols, S. Chipman, & R. Brennan (Eds.), Cognitively diagnostic assessment (pp. 361–390). Hillsdale, NJ: Lawrence Erlbaum.
- Dogan, E., & Tatsuoka, K. (2008). An international comparison using a diagnostic testing model: Turkish students’ profile of mathematical skills on TIMSS–R. Educational Studies in Mathematics, 68(3), 263–272. DOI: 10.1007/s10649-007-9099-8
- Fisher, L. C. (1988). Strategies used by secondary mathematics teachers to solve proportion problems. Journal for Research in Mathematics Education, 19(2), 157–168.
- Hartz, S. (2002). A Bayesian framework for the Unified Model for assessing cognitive abilities: Blending theory with practice (Unpublished doctoral dissertation). University of Illinois at Urbana–Champaign.
- Henson, R., & Douglas, J. (2005). Test construction for cognitive diagnostics. Applied Psychological Measurement, 29(4), 262–277. DOI: 10.1177/0146621604272623
- Henson, R., Roussos, L., Douglas, J., & He, X. (2008). Cognitive diagnostic attribute–level discrimination indices. Applied Psychological Measurement, 32(4), 275–288. DOI: 10.1177/0146621607302478
- Henson, R., Templin, J., & Willse, J. (2009). Defining a family of cognitive diagnosis models using log–linear models with latent variables. Psychometrika, 74(2), 191–210. DOI: 10.1007/s11336-008-9089-5
- Huebner, A. (2010). An overview of recent developments in cognitive diagnostic computer adaptive assessments. Practical Assessment, Research & Evaluation, 15(3), 1–7.
- Im, S., & Park, H. J. (2010). A comparison of US and Korean students’ mathematics skills using a cognitive diagnostic testing method: Linkage to instruction. Educational Research and Evaluation, 16(3), 287–301. DOI: 10.1080/13803611.2010.523294
- Izsák, A., & Jacobson, E. (2017). Preservice teachers’ reasoning about relationships that are and are not proportional: A knowledge-in-pieces account. Journal for Research in Mathematics Education, 48(3), 300–339. DOI: 10.5951/jresematheduc.48.3.0300
- Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25(3), 258–272. DOI: 10.1177/01466210122032064
- Jurich, D. P., & Bradshaw, L. P. (2014). An illustration of diagnostic classification modeling in student learning outcomes assessment. International Journal of Testing, 14(1), 49–72. DOI: 10.1080/15305058.2013.835728
- Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
- Kuzu, O. (2017). Matematik ve fen bilgisi öğretmen adaylarının integral konusundaki kazanımlarının incelenmesi. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi, 18(3), 948–970. DOI: 10.29299/kefad.2017.18.3.049
- Lamon, S. (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 (Vol 1, pp. 629–667). Charlotte, NC: Information Age Publishing.
- Lee, Y. S., Park, Y. S., & Taylan, D. (2011). A cognitive diagnostic modeling of attribute mastery in Massachusetts, Minnesota, and the US national sample using the TIMSS 2007. International Journal of Testing, 11(2), 144–177. DOI: 10.1080/15305058.2010.534571
- Lei, P. W., & Li, H. (2016). Fit indices’ performance in choosing cognitive diagnostic models and Q-matrices. Paper presented at the annual meeting of the National Council on Measurement in Education (NCME), Philadelphia, PA.
- 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). Reston, VA: National Council of Teachers of Mathematics.
- Lim, K. (2009). Burning the candle at just one end: Using nonproportional examples helps students determine when proportional strategies apply. Mathematics Teaching in the Middle School, 14(8), 492–500.
- Milli Eğitim Bakanlığı (2018). Matematik dersi öğretim programı (1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar) [Mathematics curriculum (1, 2, 3, 4, 5, 6, 7, and 8. Grades]. Ankara: Talim ve Terbiye Kurulu Başkanlığı.
- Misailadou, C., & Williams, J. (2003). Measuring children’s proportional reasoning, the “tendency” for an additive strategy and the effect of models. In N. A. Pateman, B. J. Dougherty, & J. T. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 293–300). Honolulu, HI: University of Hawaii.
- Modestou, M., & Gagatsis, A. (2007). Students’ improper proportional reasoning: A result of the epistemological obstacle of “linearity”. Educational Psychology, 27(1), 75–92. DOI: 10.1080/01443410601061462
- Muthen, L. K., & Muthen, B. O. (2011). Mplus user’s guide (6th ed.). Los Angeles, CA: Muthen & Muthen.
- National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.
- Ranjbaran, F., & Alavi, S. M. (2017). Developing a reading comprehension test for cognitive diagnostic assessment: A RUM analysis. Studies in Educational Evaluation, 55, 167–179. DOI: 10.1016/j.stueduc.2017.10.007
- Ravand, H., & Robitzsch, A. (2015). Cognitive diagnostic modeling using R. Practical Assessment, Research & Evaluation, 20(11), 1–12.
- Ravand, H., & Robitzsch, A. (2018). Cognitive diagnostic model of best choice: A study of reading comprehension. Educational Psychology, 38(10), 1255–1277. DOI: 10.1080/01443410.2018.1489524
- R Core Team. (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from http://www.R–project.org/
- Rupp, A., & Templin, J. (2008). Effects of Q–matrix misspecification on parameter estimates and misclassification rates in the DINA model. Educational and Psychological Measurement, 68(1), 78–98. DOI: 10.1177/0013164407301545
- Rupp, A., Templin, J., & Henson, R. A. (2010). Diagnostic measurement: Theory, methods, and applications. Guilford Press.
- Satorra, A., & Bentler, P. M. (2010). Ensuring positiveness of the scaled difference chi–square test statistic. Psychometrika, 75(2), 243–248. DOI: 10.1007/s11336-009-9135-y
- Sen, S., & Arican, M. (2015). A diagnostic comparison of Turkish and Korean students’ mathematics performances on the TIMSS 2011 assessment. Journal of Measurement and Evaluation in Education and Psychology, 6(2), 238–253. DOI: 10.21031/epod.65266
- Stemn, B. S. (2008). Building middle school students’ understanding of proportional reasoning through mathematical investigation. Education 3–13, 36(4), 383–392. DOI: 10.1080/03004270801959734
- Tatsuoka, K. (1985). A probabilistic model for diagnosing misconceptions by the pattern classification approach. Journal of Educational Statistics, 10(1), 55–73. DOI: 10.3102/10769986010001055
- Templin, J. (2008). Test construction item discrimination. Lecture presented at the Diagnostic Modelling Seminar at the University of Georgia, Athens. Retrieved from https://jonathantemplin.com/files/dcm/ersh9800f08/ersh9800f08_lecture11.pdf
- Templin, J., & Bradshaw, L. (2013). Measuring the reliability of diagnostic classification model examinee estimates. Journal of Classification, 30(2), 251–275. DOI: 10.1007/s00357-013-9129-4
- Templin, J., & Henson, R. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11(3), 287–305. DOI: 10.1037/1082-989X.11.3.287
- Terzi, R., & Sen, S. (2019). A nondiagnostic assessment for diagnostic purposes: Q-matrix validation and item-based model fit evaluation for the TIMSS 2011 assessment. SAGE Open, 1–11. DOI: 10.1177/2158244019832684
- Toker, T., & Green, K. (2012). An application of cognitive diagnostic assessment on TIMMS–2007 8th grade mathematics items. Paper presented at the annual meeting of the American Educational Research Association, Vancouver, British Columbia, Canada.
- Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23(1), 57–86. DOI: 10.1207/s1532690xci2301_3
- Van Dooren, W., De Bock, D., Janssens, D., & Verschaffel, L. (2007). Pupils’ overreliance on linearity: A scholastic effect? British Journal of Educational Psychology, 77(2), 307–321. DOI: 10.1348/000709906X115967
- von Davier, M. (2005). A general diagnostic model applied to language testing data. ETS Research Report. Princeton, NJ: Educational Testing Service.
- Werner, C., & Schermelleh-Engel, K. (2010). Deciding between competing models: Chi–square difference tests. In Introduction to Structural Equation Modeling with LISREL (pp. 1–3). Frankfurt, Germany: Goethe University.

There are 59 citations in total.

Primary Language | English |
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Subjects | Studies on Education |

Journal Section | Research Articles |

Authors | |

Publication Date | October 31, 2019 |

Acceptance Date | October 12, 2019 |

Published in Issue | Year 2019 Volume: 8 Issue: 4 |

**Turkish Journal of Education** is licensed under CC BY-NC 4.0