All of the psychometric models discussed in this chapter reflect exactly this kind of reasoning, and all of them can in fact be expressed as particular implementations of Bayes nets. The models described above each evolved in their own special niches, with researchers in each gaining experience, writing computer programs, and developing a catalog of exemplars. Issues of substance and theory have accumulated in each case. As statistical methodology continues to advance (see, e.g., Gelman, Carlin, Stern, and Rubin, 1995), the common perspective and general tools of Bayesian modeling may become a dominant approach to the technical aspects of modeling in assessment. Yet researchers will continue to draw on the knowledge gained from both theoretical explorations and practical applications based on models such as those described above and will use those models as building blocks in Bayes nets for more complex assessments. The applications of Bayes nets to assessment described below have this character, reflecting their heritage in a psychometric history even as they attack problems that lie beyond the span of standard models. After presenting the rationale for Bayes nets in assessment, we provide an example and offer some speculations on the role of Bayes nets in assessment in the coming years.

Bayes nets bring together insights about the structuring of complex arguments (Wigmore, 1937) and the machinery of probability to synthesize the information contained by various nuggets of evidence. The three elements of the assessment triangle lend themselves to being expressed in this kind of framework.

Example: Mixed-Number Subtraction

The form of the data in this example is familiar—right/wrong responses to open-ended mixed-number subtraction problems—but inferences are to be drawn in terms of a more complex student model suggested by cognitive analyses. The model aims to provide short-term instructional guidance for students and teachers. It is designed to investigate which of two strategies students apply to problems and whether they can carry out the procedures necessary to solve the problems using those strategies. While competence in domains such as this can be modeled at a much finer grain size (see, e.g., VanLehn’s [1990] analysis of whole-number subtraction), the model in this example does incorporate how the difficulty of an item depends on the strategy a student employs. Rather than treating this interaction as error, as would be done under CTT or IRM, the model leverages this interaction as a source of evidence about a student’s strategy usage.

The example is based on studies of middle school students conducted by Tatsuoka (1987, 1990). The students she studied characteristically solved mixed-number subtraction problems using one of two strategies:

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