Reconceptualizing a mathematical domain on the basis of student reasoning: Considering teachers' perspectives about integers

Integers have historically been approached as a system of rules. However, to teach any mathematical domain for understanding, teachers must conceptualize it as comprised of more than procedures. Using as a lens the four types of integer reasoning identified by Bishop et al. (2014a, 2014b), we interviewed 7th-grade teachers to investigate their own integer reasoning and how this corresponds to their approaches to teaching integers and to their interpretations of students' reasoning. The teachers not only correctly solved integer tasks but also most reasoned using more than rules, demonstrating a flexibility of strategies. Additionally, although they attempted to introduce integers in meaningful ways, most teachers viewed teaching integers as helping their students apply procedures, an orientation that constrained their understanding of students' integer reasoning. Results indicate that teachers possess productive conceptual resources but need a structure to leverage their understandings to teach integers as more than a set of rules.