Attribute Hierarchy Method - An Example of A Cognitive Model

An Example of A Cognitive Model

The following hierarchy is an example of a cognitive model task performance for the knowledge and skills in the areas of ratio, factoring, function, and substitution (called the Ratios and Algebra hierarchy). This hierarchy is divergent and composed of nine attributes which are described below. If the cognitive model is assumed to be true, then an examinee who has mastered attribute A3 is assumed to have mastered the attributes below it, namely attributes A1 and A2. Conversely, if an examinee has mastered attribute A2, then it is expected that the examinee has mastered attribute A1 but not A3.

The hierarchy contains two independent branches which share a common prerequisite – attribute A1. Aside from attribute A1, the first branch includes two additional attributes, A2 and A3, and the second branch includes a self-contained sub-hierarchy which includes attributes A4 through A9. Three independent branches compose the sub-hierarchy: attributes A4, A5, A6; attributes A4, A7, A8; and attributes A4, A9. As a prerequisite attribute, attribute A1 includes the most basic arithmetic operation skills, such as addition, subtraction, multiplication, and division of numbers. Attributes A2 and A3 both deal with factors. In attribute A2, the examinee needs to have knowledge about the property of factors. In attribute A3, the examinee not only requires knowledge of factoring (i.e., attribute A2), but also the skills of applying the rules of factoring. Therefore, attribute A3 is considered a more advanced attribute than A2.

The self-contained sub-hierarchy contains six attributes. Among these attributes, attribute A4 is the prerequisite for all other attributes in the sub-hierarchy. Attribute A4 has attribute A1 as a prerequisite because A4 not only represents basic skills in arithmetic operations (i.e., attribute A1), but it also involves the substitution of values into algebraic expressions which is more abstract and, therefore, more difficult than attribute A1. The first branch in the sub-hierarchy deals, mainly, with functional graph reading. For attribute A5, the examinee must be able to map the graph of a familiar function with its corresponding function. In an item that requires attribute A5 (e.g., item 4), attribute A4 is typically required because the examinee must find random points in the graph and substitute the points into the equation of the function to find a match between the graph and the function. Attribute A6, on the other hand, deals with the abstract properties of functions, such as recognizing the graphical representation of the relationship between independent and dependent variables. The graphs for less familiar functions, such as a function of higher-power polynomials, may be involved. Therefore, attribute A6 is considered to be more difficult than attribute A5 and placed below attribute A5 in the sub-hierarchy.

The second branch in the sub-hierarchy considers the skills associated with advanced substitution. Attribute A7 requires the examinee to substitute numbers into algebraic expressions. The complexity of attribute A7 relative to attribute A4 lies in the concurrent management of multiple pairs of numbers and multiple equations. Attribute A8 also represents the skills of advanced substitution. However, what makes attribute A8 more difficult than attribute A7 is that algebraic expressions, rather than numbers, need to be substituted into another algebraic expression. The last branch in the sub-hierarchy contains only one additional attribute, A9, related to skills associated with rule understanding and application. It is the rule, rather than the numeric value or the algebraic expression that needs to be substituted in the item to reach a solution.