Expert-Based Attribute Identification and Validation on Fraction Subtraction: A Cognitively Diagnostic Assessment Application

Lokman Akbay, Ragip Terzi, Mehmet Kaplan, Katibe Gizem Karaaslan


In this study we describe the methodology used to identify and validate a set of expert-defined fraction subtraction related attributes. These attributes are expected to be mastered by 6th grade students toward proficiency. This research argues and demonstrates that state standards guiding subject instruction plays an important role in identification of the domain related fundamental attributes. This study also illustrates throughout implementation of cognitive diagnosis model framework, which is used to extract diagnostic information about students' specific strengths and weaknesses.


Behr, M. J., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grows (Ed.), Handbook of research on mathematics teaching and learning (pp. 296-333). New York, NY: Macmillan Publishing Company

Birgin, O., & Gürbüz, R. (2009). İlköğretim II. kademe öğrencilerinin rasyonel sayılar konusundaki işlemsel ve kavramsal bilgi düzeylerinin incelenmesi. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 22, 529-550.

Chi, M. T. (1997). Quantifying qualitative analyses of verbal data: A practical guide. The Journal of the Learning Sciences, 6, 271-315.

de la Torre, J. (2009). DINA model and parameter estimation: A didactic. Journal of Educational and Behavioral Statistics, 34, 115-130.

de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179-199.

de la Torre, J., Hong, Y., & Deng, W. (2010). Factors affecting the item parameter estimation and classification accuracy of the DINA model. Journal of Educational Measurement, 47, 227-249.

de la Torre, J., & Karelitz, T. M. (2009). Impact of diagnosticity on the adequacy of models for cognitive diagnosis under a linear attribute structure: A simulation study. Journal of Educational Measurement, 46, 450-469.

de la Torre, J., & Lee, Y. S. (2010). A note on the invariance of the DINA model parameters. Journal of Educational Measurement, 47, 115-127.

de la Torre, J., & Minchen, N. (2014). Cognitively diagnostic assessments and the cognitive diagnosis model framework. Psicologia Educative, 20, 89-97.

DiBello, L. V., & Stout, W. (2007). Guest editors' introduction and overview: IRT-based cognitive diagnostic models and related methods. Journal of Educational Measurement, 44, 285-291.

Embretson, S. E. (1994). Applications of cognitive design systems to test development. In C. R. Reynolds (Ed.), Cognitive assessment: A multidisciplinary perspective (pp. 107-135). New York: Plenum Press.

Embretson, S. E. (1998). A cognitive design system approach to generating valid tests: Application to abstract reasoning. Psychological Methods, 3, 380-396.

Ericsson, K. A. (2006). Protocol analysis and expert thought: Concurrent verbalizations of thinking during experts' performance on representative tasks. In K. A. Ericsson, N. Charness, R. R. Hoffman, & P. J. Feltovich (Eds.). The Cambridge handbook of expertise and expert performance (p.p. 223-241). New York, NY: Cambridge University Press.

Haser, Ç., & Ubuz, B. (2002). Kesirlerde kavramsal ve işlemsel performans [Conceptual and Procedural Performance in Fractions]. Eğitim ve Bilim, 27(126), 53-61.

Hiebert, J. & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsade, NJ: Lawrence Erlbaum Associates.

Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25, 258-272.

Leighton, J. P. (2004). Avoiding misconception, misuse, and missed opportunities: The collection of verbal reports in educational achievement testing. Educational Measurement: Issues and Practice, 23 (4), 6-15.

Leighton, J. P., Cui, Y., & Cor, M. K. (2009). Testing expert-based and student-based cognitive models: an application of the attribute hierarchy method and hierarchy consistency index. Applied Measurement in Education, 22, 229-254.

Leighton, J. P., & Gierl, M. J. (2007). Defining and evaluating models of cognition used in educational measurement to make inferences about examinees' thinking processes. Educational Measurement: Issues and Practice, 26 (2), 3-16.

Leighton, J. P., Gierl, M. J., & Hunka, S. M. (2004). The attribute hierarchy method for cognitive assessment: A variation on Tatsuoka's rule-space approach. Journal of Educational Measurement, 41, 205-237.

Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal for Research in Mathematics Education, 26, 422- 441.

Maris, E. (1995). Psychometric latent response models. Psychometrika, 60, 523-547.

Maris, E. (1999). Estimating multiple classification latent class models. Psychometrika, 64, 187-212.

Misquitta, R. (2011). A review of the literature: Fraction instruction for struggling learners in mathematics. Learning Disabilities Research and Practice, 26, 109-119.

Ni, Y. J. (1999). The understanding of the meaning and nature of fraction of Grade fıfth and sixth. Psychological Development and Education, 11, 26-30.

No Child Left Behind (NCLB) Act of 2001, 20 U.S.C. § 6301 et seq. (2002).

Pellegrino, J. W., Baxter, G. P., & Glaser, R. (1999). Addressing the “two discipline” problem: Linking theories of cognition and learning with assessment and instructional practices. In A.Iran-Nejad & P. D.Pearson (Eds.), Review of Research in Education (pp. 307-353). Washington, DC: American Educational Research Association.

R Core Team (2014). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL

Revelle, W. (2015). “Psych: Procedures for psychological, psychometric, and personality research”. In R package Version 1.5.6. Retrieved from

Rupp, A. A., & Templin, J. L. (2008). Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art. Measurement, 6, 219-262.

Son, J. W. (2012). A cross-national comparison of reform curricula in Korea and the US in terms of cognitive complexity: The case of fraction addition and subtraction. ZDM The International Journal on Mathematics Education, 44, 161-174.

Son, J., & Crespo, S. (2009). Prospective teachers' reasoning about students' nontraditional strategies when dividing fractions. Journal of Mathematics Teacher Education, 12, 236-261.

Tatsuoka, K. K. (1984). Analysis of errors in fraction addition and subtraction problems. Report No. NIE-G-81-0002). Urbana, IL: University of Illinois, CERL.

Tatsuoka, K. K. (1990). Toward an integration of item-response theory and cognitive error diagnosis. In N. Frederiksen, R. Glaser, A. Lesgold, & M. Shafto (Eds.), Diagnostic monitoring of skill and knowledge acquisition (pp.453-488). Hillsdale, NJ:Erlbaum.

Tirosh, D. (2000). Enhancing prospective teachers' knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31, 5-25.

Van de Walle, J., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics methods: Teaching developmentally. New York, NY: Allyn and Bacon.

Vinner, S., Hershkowitz, R., & Bruckheimer, M. (1981). Some cognitive factors as causes of mistakes in the addition of fractions. Journal for Research in Mathematics Education, 12, 70-76.


  • There are currently no refbacks.

Journal on Mathematics Education
Program S3 Pendidikan Matematika FKIP Universitas Sriwijaya
Kampus FKIP Bukit Besar
Jl. Srijaya Negara, Bukit Besar
Palembang - 30139

p-ISSN: 2087-8885 | e-ISSN: 2407-0610
Journal on Mathematics Education is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License

View My Stats

Indexed in: