Current methods for detecting growth of students’ problem-solving skills in math focus mainly on analyzing changes in test scores. Score-level analysis, however, may fail to reflect subtle changes that might be evident at the item level. This article demonstrates a method for studying item-level changes using data from a multiwave experiment with a teaching method called enhanced anchored instruction (EAI). The analysis combines a mixture Rasch model for detecting individual differences within latent groups with a latent transition analysis model for tracking changes in latent group membership over the course of EAI. The analysis clearly indicates the effects of EAI and how they differ for members of each latent class. Comparisons are provided with a standard analysis of changes in test scores. Implications of the new approach are discussed for detecting subtle transformations in the math performance of students across a range of ability levels.
American Educational Research Association, American Psychological Association, & National Council on Measurement in Education. ( 1999). Standards for educational and psychological. Washington, DC: Authors.
Barrows, H.L. ( 1996). Problem-based learning in medicine and beyond: A brief overview. New Directions for Teaching and Learning, 68, 3-12.
4.
Bolt, D.M., Cohen, A.S., & Wollack, J.A. ( 2001). A mixture item response for multiple-choice data. Journal of Educational and Behavioral Statistics, 26, 381-409.
5.
Bottge, B.A., Heinrichs, M., Mehta, Z., & Hung, Y. ( 2002). Weighing the benefits of anchored math instruction for students with disabilities in general education classes. Journal of Special Education, 35, 186-200.
6.
Bottge, B.A., Heinrichs, M., Mehta, Z.D., Rueda, E., Hung, Y., & Danneker, J. ( 2004). Teaching mathematical problem solving to middle school students in math, technology education, and special education classrooms. Research in Middle Level Education Online, 27(1). Retrieved from http://www.nmsa.org/Publications/RMLEOnline/Articles/Vol27No1Article1/tabid/529/Default.aspx .
7.
Bottge, B.A., Rueda, E., Serlin, R.C., Hung, Y.H., & Kwon, J.M. ( 2007). Shrinking achievement differences with anchored math problems: Challenges and possibilities. Journal of Special Education, 41, 31-49.
8.
Bransford, J.D., Brown, A.L., Cocking, R.R., Donovan, M.S., & Pellegrino, J.W. (2000). How people learn: Brain, mind and school. Washington, DC: National Academies Press.
9.
Brown, J.S., Collins, A., & Duguid, P. ( 1989). Situated cognition and the culture of learning. Educational Researcher, 17(1), 32-41.
10.
Cho, S.J., Cohen, A.S., Kim, S.H., & Bottge, B. ( 2010). Latent transition analysis with a mixture item response theory measurement model. Applied Psychological Measurement , 34, 483-504.
11.
Cognition and Technology Group at Vanderbilt. ( 1997). The Jasper Project: Lessons in curriculum, instruction, assessment, and professional development. Mahwah, NJ: Lawrence Erlbaum.
12.
Cognition and Technology Group at Vanderbilt University. ( 1990). Anchored instruction and its relationship to situated cognition. Educational Researcher, 19(3), 2-10.
13.
Collins, L.M., & Wugalter, S.E. (1992). Latent class models for stage-sequential dynamic latent variables. Multivariate Behavioral Research, 27, 131-157.
Embretson, S.E. ( 1991). A multidimensional latent trait model for measuring learning and change. Psychometrika, 56, 495-515.
16.
Embretson, S.E., & Reise, S.P. ( 2000). Item response theory for psychologists. Mahwah, NJ: Lawrence Erlbaum.
17.
Fischer, G.H. ( 1976). Some probabilistic models for measuring change. In D. N. M. de Gruijter & L. J. Th. van der Kamp (Eds.), Advances in psychological and educational measurement (pp. 97-110). New York: John Wiley.
18.
Fischer, G.H. ( 1989). An IRT-based model for dichotomous longitudinal data . Psychometrika, 54, 599-624.
19.
Fischer, G.H. ( 1995). Linear logistic models for change. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 157-180). New York: Springer-Verlag.
20.
Fuchs, L.S. ( 2003). Assessing intervention responsiveness: Conceptual and technical issues. Learning Disabilities Research & Practice , 18, 172-186.
21.
Fuchs, D., & Deshler, D. ( 2007). What we need to know about responsiveness to intervention (and shouldn’t be afraid to ask). Learning Disabilities Research & Practice, 22, 129-136.
22.
Fuchs, D., & Fuchs, L.S. ( 2006). Introduction to response to intervention: What, why, and how valid is it?Reading Research Quarterly, 41, 93-99.
23.
Gijbels, D., Dochy, F., Van den Bossche, P., & Segers, M. ( 2005). Effects of problem-based learning: A meta-analysis from the angle of assessment. Review of Educational Research, 75, 27-61.
24.
Greeno, J.G., & the Middle School Mathematics Through Applications Project Group. (1998). The situativity of knowing, learning, and research. American Psychologist, 53, 5-26.
25.
Hoover, H.D., Dunbar, S.N., & Frisbie, D.A. ( 2001). The Iowa Tests of Basic Skills (ITBS, Form A). Itasca, IL: Riverside Publishing.
26.
Li, F., Cohen, A.S., Kim, S.-H., & Cho, S.-J. (2009). Model selection methods for dichotomous mixture IRT models. Applied Psychological Measurement, 33, 53-373.
27.
Maij-de Meij, A.M., Kelderman, H., & van der Flier, H. (2008). Fitting a mixture item response theory model to personality questionnaire data: Characterizing latent classes and investigating possibilities for improving prediction. Applied Psychological Measurement, 32, 611-631.
28.
Mislevy, R.J., & Verhelst, N. ( 1990). Modeling item responses when different subjects employ different solution strategies. Psychometrika, 55, 195-215.
National Council of Teachers of Mathematics. ( 2000). Principles and standards for school mathematics. Reston, VA: Author.
31.
National Mathematics Advisory Panel. (2008 ). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
32.
Pellegrino, J., Chudowsky, N., & Glaser, R. (Eds.), for the National Research Council Committee on the Foundations of Assessment. (2001). Knowing what students know: The science and design of educational assessment. Washington, DC: National Academies Press.
33.
Rost, J. ( 1990). Rasch models in latent classes: An integration of two approaches to item analysis. Applied Psychological Measurement , 14, 271-282.
34.
Rost, J. ( 1997). Logistic mixture models. In W. J. van der Linden & R. K. Hambleton (Eds.), Handbook of modern item response theory (pp. 449-463). New York: Springer.
35.
Sireci, S.G., Scarpati, S.E., & Li, S. ( 2005). Test accommodations for students with disabilities: An analysis of the interaction hypothesis. Review of Educational Research, 75, 457-490.
36.
Sulzer-Azaroff, B., & Mayer, G.R. ( 1977). Applying behavior analysis procedures with children and youth. New York: Holt, Rinehart & Winston.
37.
von Davier, M. ( 2001). WINMIRA [Computer software]. St. Paul, MN: Assessment Systems Corporation.
38.
Willett, J.B., & Sayer, A.G. ( 1994). Using covariance structure analysis to detect correlates and predictors of individual change over time. Psychological Bulletin, 116, 363-381.
39.
Wisconsin Department of Public Instruction. ( 2006). Specific learning disabilities eligibility criteria resources . Retrieved October 23, 2006, from http://dpi.wi.gov/sped/eligild.html
