matrix serves an important test item development blueprint where items can be created to measure each specific combination of attributes. In this way, each component of the cognitive model can be evaluated systematically. In this example, a minimum of 9 items are required to measure all the attribute combinations specified in the Qr matrix. The expected examinee response patterns can now be generated using the Qr matrix. An expected examinee is conceptualized as a hypothetical examinee who correctly answers items that require cognitive attributes that the examinee has mastered. The expected response matrix (E) is created, using Boolean inclusion, by comparing each row of the attribute pattern matrix (which is the transpose of the Qr matrix) to the columns of the Qr matrix. The expected response matrix is of order (j,i), where j is the number of examinees and i is the reduced number of items resulting from the constraints imposed by the hierarchy. The E matrix for the Ratio and Algebra hierarchy is shown below. If the cognitive model is true, then 58 unique item response patterns should be produced by examinees who write these cognitively-based items. A row of 0s is usually added to the E matrix which represents an examinee who has not mastered any attributes. To summarize, if the attribute pattern of the examinee contains the attributes required by the item, then the examinee is expected to answer the item correctly. However, if the examinee's attribute pattern is missing one or more of the cognitive attributes required by the item, the examinee is not expected to answer the item correctly. Test development Role of the cognitive model in item development The cognitive model in the form of an attribute hierarchy has direct implications for item development. Items that measure each attribute must maintain the hierarchical ordering of the attributes as specified by the cognitive model while also measuring increasingly complex cognitive processes. These item types may be in either multiple choice or constructed response format. To date, the AHM has been used with items that are scored dichotomously where 1 corresponds to a correct answer and 0 corresponds to an incorrect answer. Therefore, a student's test performance can be summarized by a vector of correct and incorrect responses in the form of 1's and 0's. This vector then serves as the input for the psychometric analysis where the examinee's attribute mastery is estimated. Approach to item