Reviews That Teach

Abstract

Some years ago, Gila Hanna offered the very insightful and useful distinction between mathematical proofs that prove and those that also explain (Hanna, 1989). 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. I wish to argue for a related perspective on the nature and importance of reviews of submissions to JRME.' As reviewers, we can choose between preparing reviews that primarily evaluate and reviews that also teach, and we should try-whenever submissions call for it-to submit reviews that teach. I see this proposal as being quite consistent with Ed Silver's recent distinction between the evaluative and functions of reviews (Silver, 2003). But writing reviews that teach may be a more demanding standard than preparing educative reviews. At least, choosing the right teaching stance can provide a useful context for preparing reviews. My thesis is that all who review JRME submissions should be willing-when needed-to conceptualize their task as one of mentoring authors. Taking this perspective shifts the focus in reviewing from evaluation to teaching. Evaluation is certainly a necessary dimension, as summary recommendations (e.g., accept, reject, reject but invite revision) are required for the editor to make and support final judgments about publication. But as proofs that explain do not stop at flawless logic, reviews that teach do not stop at stating and justifying a reviewer's recommendation. Instead, reviews that teach-and do so sensitively-clearly articulate the main reasons why authors' current arguments are inadequate and also present strategies they might pursue in correcting those deficiencies. Before I develop this proposal a bit further, I want to be clear. I do not think that JRME's review process is deficient; it is just the opposite. Because there is