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Traditionally, teachers evaluate students’ abilities via their total test scores. Recently, cognitive diagnostic models (CDMs) have begun to provide information about the presence or absence of students’ skills or misconceptions. Nevertheless, CDMs are typically applied to tests with multiple-choice (MC) items, which provide less diagnostic information than constructed-response (CR) items. This paper introduces new CDMs for tests with both MC and CR items, and illustrates how to use them to analyse MC and CR data, and thus, identify students’ skills and misconceptions in a mathematics domain. Analyses of real data, the responses of 497 sixth-grade students randomly selected from four Taiwanese primary schools to eight direct proportion items, were conducted to demonstrate the application of the new models. The results show that the new models can better determine students’ skills and misconceptions, in that they have higher inter-rater agreement rates than traditional CDMs. [ABSTRACT FROM AUTHOR]

Using the capabilities of the knowledge structure theory, the researcher prepared a model for the experts’ knowledge structure of the valid test items of the university Calculus topic “trigonometric substitution of indefinite integral about integrands involving quadratic square root”. After the Calculus course was taught and tested, this knowledge structure model was combined with Bayesian network analysis to find the best recognition rate of Bayesian network and to identify easily the error types that students might make in learning “trigonometric substitution” unit. Then, the statistical information was analyzed using the Bayesian probability analysis methods to establish the “trigonometric substitution” unit model as an example of a Bayesian network adaptive diagnostic test system, to determine four established Bayesian network models to compare their recognition rates, and to establish this structure could be used as the foundation for developing remedial teaching approaches. [ABSTRACT FROM AUTHOR]