Understanding logical necessity is an important component of proof and reasoning for teachers of grades K-8. The ability to determine exactly where young students' arguments are faulty offers teachers the chance to give youngsters feedback as they progress toward writing mathematically valid deductive proofs. As defined, logical necessity is the…

Online conversations help teachers engage in constructive criticism and attend more carefully to aligning lesson plans with problem solving. This article describes a strategy for using an online discussion board to help teachers evaluate one another's lesson plans. The online peer-review activity suggests that fostering critical pedagogical…