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 reasoningand-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-andproving 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. digitalcommons.unl.edu

Explicit reasoning-and-proving opportunities in the United States are often relegated to a single secondary geometry course. This study analyzed the reasoningand-proving opportunities in six U.S. geometry textbooks, giving particular attention to the chapter that introduced proof. Analysis focused on the types of reasoningand-proving activities expected of students and the type of mathematical statement around which the reasoning-and-proving took place, be it general or particular. Results include the fact that reasoning-and-proving opportunities in student exercises were predominantly of the particular type, whereas textbook exposition most commonly had general statements. Within the chapters introducing proof, opportunities for students to develop proofs were less common than exercises involving conjectures and statements or exercises about the reasoning-and-proving process. Opportunities to reflect on the reasoning-and-proving process were prevalent in the introduction chapters, though rare in the remainder of the books.

For many American students, high school geometry provides their only focused experience in writing proofs (Herbst 2002), and proof is often viewed as the application of recently learned theorems rather than a means of establishing and understanding the truth of general results (Soucy McCrone and Martin 2009).

Most mathematics curricula include contextual problems, but not all these problems have the same potential as modeling tasks. The authors describe how to select tasks that give students opportunities to use modeling.

Consider these seven techniques to extend students’ learning.