Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

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### Shiv Karunakaran, Ben Freeburn, Nursen Konuk, and Fran Arbaugh

### Darla R. Berks and Amber N. Vlasnik

Two teachers discuss the planning and observed results of an introductory problem to help students nail a conceptual approach to solving systems of equations.

### Harold B. Reiter, John Thornton, and G. Patrick Vennebush

Through KenKen puzzles, students can explore parity, counting, subsets, and various problem-solving strategies.

### Bobson Wong and Larisa Bukalov

Parallel geometry tasks with four levels of complexity involve students in writing and understanding proof.

### David A. Yopp

Asked to “fix” a false conjecture, students combine their reasoning and observations about absolute value inequalities, signed numbers, and distance to write true mathematical statements.

### Michael K. Weiss and Deborah Moore-Russo

The moves that mathematicians use to generate new questions can also be used by teachers and students to tie content together and spur exploration.

### Dustin L. Jones and Max Coleman

Many everyday objects–paper cups, muffins, and flowerpots–are examples of conical frustums. Shouldn't the volume of such figures have a central place in the geometry curriculum?

### R. Alan Russell

In trying to find the ideal dimensions of rectangular paper for folding origami, students explore various paper sizes, encountering basic number theory, geometry, and algebra along the way.

### Ayana Touval

Through movement-a welcome change of pace-students explore the properties of the perpendicular bisector.

### Carol J. Bell

Reasoning and Proof is one of the process standards set forth in NCTM's *principles and standards for school mathematics* (2000).