DIAGNOSING KNOWLEDGE STATES IN ALGEBRA USING THE RULE-SPACE MODEL

This paper illustrates the use of rule space as a tool to support cogn itive analyses of students' mathematical behavior. The rule-space appr oach is explained and is then used to classify students into one of tw o methods for solving linear algebraic equations in one unknown and to diagnose their knowledge states in this topic. A 32-item test with op en-ended questions was administered to 231 eighth and ninth graders. T wo outcomes of the rule-space model are presented: (a) a classificatio n of examinees into knowledge states resulting from the two solution a pproaches at the group level along with individual examples and (b) tr ee diagrams of the transitional relationships among the states for eac h strategy. Implications for using the feedback provided by the rule-s pace model in the context of instruction and assessment are discussed.