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Diagnosing Knowledge States in Algebra Using the Rule Space Model

Birenbaum, Menucha; Kelly, Anthony E.; Tatsuoka, Kikumi K.
Publication Year:
Report Number:
ETS Research Report
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Subject/Key Words:
Algebra, Educational Diagnosis, Error Patterns, Israel, Mathematics, Rule Space Analysis


This paper illustrates the use of rule space as a tool to support cognitive analyses of students' mathematical behavior. The rule space approach is explained and is then used to classify students into one of two methods for solving linear algebraic equations in one unknown and to diagnose their knowledge states in this topic. A 32-item test with open-ended questions was administered to 231 eighth and ninth graders. The following outcomes of the rule space model were presented: (a) a classification of examinees into knowledge states resulting from the two solution approaches at the group level along with individual examples; (b) tree-diagrams of the transitional relationships among the states for each strategy. Implications for using a feedback provided by the rule space model in the context of instruction and assessment are discussed.

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