Prospective Teachers’ Beliefs About Mathematical Mistakes: An Exploratory Case Study

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Duncan, Matthew David
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Middle Tennessee State University
Mathematics education reform documents consider mathematical mistakes to be vital for the teaching and learning of mathematics. Additionally, mathematical mistakes are viewed as opportunities for learning and catalysts for discussion. However, research has demonstrated that some teachers do not see mathematical mistakes as productive for learning and often avoid students’ mathematical mistakes. Seminal educational research demonstrated that teachers’ beliefs influence the ways in which teachers teach, and thus, teachers’ beliefs concerning mathematical mistakes influence the ways in which they use mathematical mistakes. Researchers have found that in-service teachers’ (ISTs) beliefs are not aligned with reform documents’ ways of treating mistakes; however, there is a dearth of research concerning the beliefs of prospective teachers (PTs). The purpose of this study was to investigate the beliefs of PTs concerning mathematical mistakes and answer two research questions: 1. What beliefs do PTs hold about mathematical mistakes? 2. What are the noticeable changes in PTs’ beliefs systems concerning mathematical mistakes, if any, given the classroom context? An exploratory multiple-case study design was implemented to examine the beliefs of two PTs while enrolled in a content course for teachers designed specifically to use mathematical mistakes in ways aligned with mathematics reform documents. The study produced findings that were significant in at least five ways. First, the study produced results that demonstrated a change in PTs’ beliefs while enrolled in a content course for teachers. This adds to a larger body of seminal research concerning PTs’ beliefs and how they can change. Additionally, a content course being the context in which this study took place adds to the significance that content courses have in teacher preparation. Second, the findings of this study contribute to a growing body of research on beliefs concerning mathematical mistakes. There is a scarcity of research on PTs’ beliefs concerning mathematical mistakes, but the findings of this study complement research on ISTs’ beliefs concerning mathematical mistakes. Third, the study revealed that the PTs’ beliefs changed during a course with a positive error climate. This finding has potential applications for teacher educators and teacher preparation course designers including a need to create positive error climates to reduce the influence of affective qualities on when PTs share their mathematical mistakes. Fourth, the findings of the study suggested that mechanisms that promote changes in PTs’ beliefs should include the error climate of the classroom. This has potential applications for teacher educators and teacher preparation course designers as well including a need to create positive error climates to support alignment of beliefs concerning mathematical mistakes with mathematics education reform documents. Finally, the findings of this study revealed the importance of PTs’ teacher identities in how they interpreted their experiences with mathematical mistakes and, subsequently, their beliefs concerning mathematical mistakes. Future research with PTs’ beliefs concerning mathematical mistakes should account for PTs’ teacher identities as they played a substantial role in the findings of this study.