Abstract
The cognitive structure of an algebra test was defined and validated using the linear logistic test model (LLTM) and quadratic assignment (QA), respectively. The LLTM is an extension of the Rasch model with a linear constraint that describes the difficulty of a test item in terms of the cognitive operations required to solve it. The cognitive structure of a test is specified using the weight matrix W. The cognitive structure defined here was based on a set of eight production rules that represented the mathematical procedures employed in solving linear equations with one variable. A 29-item test was constructed and administered to 235 ninth-graders. Item response data were analyzed using Fischer & Formann's (1972) LLTM computer program. A QA confirmatory approach was used to validate the cognitive structure of the test. The structure was validated—examinees solved the items using the set of rules specified in the W matrix. The parameters estimated using the LLTM are quantitative indexes of the difficulties of each of the cognitive rules included in the W matrix.
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