# The development of mathematics-for-teaching: The case of fraction multiplication

#### Abstract

The parallel research traditions of explicit-objective and tacit-emergent vary greatly in how they define, assess, and enable development of teacher mathematical knowledge. Despite these diversities, widespread agreement exists in mathematics education research that a teacher's mathematical knowledge is a key competency of an effective teacher. This research report investigates the nature and development of teacher mathematical knowledge of fraction multiplication defined from a tacit-emergent perspective. Questions about the nature and development of teacher mathematical knowledge for fraction multiplication were investigated in this report at the individual and collective levels. In addition, this research report also investigated the developmental links between these levels. The concept study design and the framework for teacher knowledge used in this report derived from the work of Davis and colleagues (Davis & Simmt, 2006; Davis & Renert, 2014). The results from this report were multifaceted for both the individual and collective levels of mathematical knowledge. Teachers' individual mathematics-for-teaching (M4T) knowledge of fraction multiplication developed throughout their participation in the mathematical environments of the concept study. Furthermore, two types of collective action emerged as proposed links between the collective and individual development of teachers' M4T knowledge of fraction multiplication. These proposed links, titled synergistic realizations and recursive elaborations emerged in this report as patterns of mathematical action existent in moments of coaction. Recursive elaboration defines the decision-making mechanism where the collective expands the realm of what is possible for a single mathematical realization. Synergistic realization defines the collective decision action in which all previous realizations are abandoned for one innovation in the mathematical realization of a mathematical concept. A discussion of the implications for defining teachers' mathematical knowledge of fraction multiplication as nested systems of individual and collective knowledge is included in the conclusion of this report.

#### Degree

Ph.D.

#### Advisors

Kastberg, Purdue University.

#### Subject Area

Mathematics education|Mathematics|Cognitive psychology

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