Teaching and Testing Mathematics Reading

When we discuss teaching undergraduate mathematics with our colleagues, we frequently distinguish two sorts of goals: first, we want students to be able to do computations related to the course and, second, we want students to know the theory from the course. In a calculus course, we want students to be able to integrate polynomials, so in class we tell them how, and on the exam; we give them polynomials to integrate. In linear algebra class, we want the students to know the theorems concerning determinants, so we prove the theorems from the book, and on the exam, we ask them to prove that if A is an invertible matrix, then det(A-1) = 1/det(A). Our hallway conversations elicit general agreement that most of our students do learn to do the calculations we put on the exam and most don't master the theory to the point that they can write a coherent proof of any but the most trivial theorems. I want to argue in this note that there is another goal, between these in accessibility, that should be a conscious part of our teaching effort: