Middle school students' representational understandings and justification schemes: gleanings from cognitive interviews
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This dissertation investigated several aspects of middle graders’ mathematical understanding based on representational models. Twenty (11 male, 9 female) sixth grade students were interviewed about their solution strategies and answer justifications when solving difficult mathematics problems. The interview participants represented a stratified demographic sampling of the student body of a culturally diverse middle school in a suburban school district in the southwestern United States. Data from the interviews were analyzed qualitatively. This involved “chunking” cognitive interview transcripts into sections. Major themes were identified and manuscripts were developed around those themes. One theme examined the interviewers’ ethic of care behaviors. Carol Gilligan noted differences in male and female ethic of care behaviors, but it was Nel Noddings who discussed the importance of such behaviors in the educational community. So what impact could the gender of the interviewer have on cognitive interviews? After considering ethic of care behaviors explicated by Hayes, Ryan and Zseller’s (1994) study with middle grades students, the interview transcripts were examined for specific positive and negative ethic of care behaviors. The theme of students’ justifications of mathematical solutions was also selected. The major undertaking involved developing a justification scheme applicable across mathematical strands and grade levels. The justification scheme that emerged was based on the work of Guershon Harel and Larry Sowder. The first-level schemes of Language, Mechanistic, Authoritarian, and Visual were used to classify and define the justifications. Several second-level schemes were also defined. The justification scheme framework was applied to students’ cognitive interview responses on four difficult mathematics problems. The third theme investigated the symbiosis of justification schemes with mathematical representations. This study examined possible links between representational formats and justification scheme categories. The premise of this study was that representations “trigger” students’ choices of justification schemes. Student responses were analyzed as to which aspect of the mathematical representation received the students’ initial attention. The students’ understanding of the representation was pivotal to their solution, as well as the students’ reasoning, or justification, of the answer. Students focused on key aspects of the problem and developed solutions based on that information.
Matteson, Shirley Marie (2007). Middle school students' representational understandings and justification schemes: gleanings from cognitive interviews. Doctoral dissertation, Texas A&M University. Available electronically from