Some years ago, Gila Hanna offered the very insightful and useful distinction between mathematical proofs that prove and those that also explain. Proofs that explain not only state the deductive logic that justifies their mathematical claims, they lay bare the mechanisms and structures that underlie that logic, making content and logic more properly align. In so doing, they more effectively teach mathematics than do proofs that only prove. In this article, the author wishes to argue for a related perspective on the nature and importance of reviews of submissions to "JRME". He states that as reviewers, they can choose between preparing reviews that primarily "evaluate" and reviews that also "teach", and they should try--whenever submissions call for it--to submit reviews that teach. He sees this proposal as being quite consistent with Ed Silver's recent distinction between the "evaluative" and "educative" functions of reviews. Writing reviews that teach may be more demanding standard than preparing educative reviews. (Contains 2 footnotes.)