AN INVESTIGATION INTO FOUNDATIONAL CONCEPTS RELATED TO SLOPE: AN APPLICATION OF THE ATTRIBUTE HIERARCHY METHOD
Broaddus, Angela Elizabeth
University of Kansas
Curriculum and Teaching
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The mathematics education community has been working for more than two decades to reform K - 12 school mathematics programs. The literature consistently emphasizes the importance of students understanding and making sense of mathematics, which is characterized by complex networks of knowledge reflecting the conscious organization of related facts and processes. Understanding of slope is an essential milestone in a person's mathematical development, yet assessments often measure student knowledge of slope in terms of their procedural fluency rather than their conceptual understanding. The purpose of this study was to explore the concepts students should possess to demonstrate understanding of selected foundational concepts related to slope, to determine a cognitive model hypothesizing how this understanding develops, and to design an instrument to assess understanding of selected foundational concepts related to understanding slope as described in the model. The instrument was administered to a sample of Kansas students in middle and high school mathematics courses. This study provided an example of one way to implement components of Evidence-Centered Design. The study was conducted in two phases. The first phase included a domain analysis and yielded a theoretical cognitive model of how selected foundational concepts related to slope are acquired. The second phase included a task analysis and the development of an assessment containing items that targeted the knowledge described in the cognitive model. Test responses were analyzed using Item Response Theory, and students were classified into knowledge states based on their test response data using the Attribute Hierarchy Method (AHM). Students demonstrated varying levels of knowledge with regard to the selected foundational concepts of slope. The AHM revealed that students who participated in this study demonstrated three main levels of understanding of the selected foundational concepts of slope. First, students demonstrate the ability to identify quantities that are related as covariates. Second, students demonstrate the ability to identify the direction of covariation in a problem setting. Third, students demonstrate the ability to interpret a slope ratio in terms of a problem's context variables.
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