Samuel Otten

The Mathematical Nature of Reasoning-and-Proving Opportunities in Geometry Textbooks

International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.

Improving Instructional Leadership Through the Development of Leadership Content Knowledge The Case of Principal Learning in Algebra

Instructional leadership is integral to improving mathematics teaching in secondary schools. However, administrators often lack sufficient content knowledge in mathematics to be effective in this role. This study examined the impact of professional development focused on developing leadership content knowledge in algebra. Data included written assessments, case discussions, and interviews from 10 principals. Analysis identified shifts in principals’ algebra content knowledge and their frames for interpreting algebra instruction. Principles improved their connections between mathematical representations and shifted from using frames highlighting teacher characteristics toward using frames highlighting teacher and student thinking. Implications for leadership professional development design are discussed.