Rethinking Mathematics Misconceptions: Using Knowledge Structures to Explain Systematic Errors within and across Content Domains

The present study examined the ability of content domain (algebra, geometry, rational number, probability) to classify mathematics misconceptions. The study was conducted with 1,133 students in 53 algebra and geometry classes taught by 17 teachers from three high schools and one middle school across three school districts in a Midwestern state. Students answered 17 multiple choice items from the National Assessment of Educational Progress. A prompt was added to each item for students to explain the reasoning for their answer. Explanations were examined qualitatively to determine whether the responses were indicative of misconceptions. The responses were then analyzed quantitatively with structural equation modeling. Several knowledge structures (variable, measurement, spatial reasoning, additive/multiplicative structures, absolute/relative comparison) were found to be foundational to understanding multiple content domains. Multiple foundational structures were found to be related to misconceptions in each content domain, and misconceptions for multiple content domains were found within each foundational structure. The quantitative analyses found that the best classification of misconceptions included correlations among the content domains and a factor independent of content domain. The qualitative analyses led to the conclusion that interactions among the foundational knowledge structures best described the factor independent of content domain.