Teachers' Encounters with Horizon Content Knowledge Investigating Knowledge Sensibilities for Teaching Mathematics

Abstract

Mathematics education researchers have sought to understand the knowledge that teachers need to teach mathematics effectively. Teachers need to know more than merely knowing how to "do the math" at a particular grade level. However, the research community differs on the nature of that knowledge. The construct of "horizon content knowledge" has emerged in the literature as a promising way to characterize advanced mathematical knowledge (AMK) as it relates specifically to teaching practice. Ball, Thames, and Phelps (2008) and Ball and Bass (2009) propose a kind of knowledge that is neither common nor specialized, that is not about curriculum progression, but is more about having a sense of the broader mathematical environment of the discipline. They call this horizon content knowledge (HCK) and argue that knowledge of the mathematical horizon can support teachers in hearing students' mathematical insights, orienting instruction to the discipline, and making judgments about what is mathematically important. However, operationalizing HCK in practice is still under development. Jakobson, Thames, and Ribeiro (2013) offer an overarching definition of HCK, which foregrounds some inherent characteristics of this knowledge. This dissertation examined cases of teaching and teachers for the purpose of collecting and analyzing examples of HCK in practice, understanding the interaction between teachers’ management of what I call "encounters with mathematics at the horizon" and students’ learning experiences in the classrooms, and to characterize the knowledge resources that teachers draw upon to make sense of the mathematics at the horizon. I identified and articulated a new domain of knowledge resources that the teachers draw upon, called professional practice knowledge (PPK). I define PPK as a form of mathematical knowledge derived from practice and experience. As PPK is knowledge that is shaped by experience, the culture in school, role of leadership, and kind of students’ and parents’ involvement impacts PPK. If PPK is the only resource available to the teacher, then teachers’ explanations of mathematical deductions are often pseudo-mathematical. Pseudo-mathematical descriptions are generated by the teachers in such ways that they do not explain the concept, term, or formula but instead focus on memorization. These center on the visual patterns or syntactic patterns, use colloquial meanings of the mathematical terms, and often have a cue to remember the term, concept, or formula. These explanations can block mathematical access for the students to investigate or build further. However, if PPK remains rooted in other domains of HCK elaborated in the dissertation, teachers are able to manage encounters with HCK in more meaningful ways.