Given the need to read computer output, students should see analyses on exams that present computer output and be expected to read and interpret it. Calculators are a distraction on exams. Students tend to rely on them blindly hoping that by hitting the right keys they'll get the right answer. This encourages student to think that the right answer is a number. Without this crutch, students are encouraged to think more about the data and the analysis of it, and to recognize that statistics is not about getting the right numeric result.
Tables serve quite well for normal, t, and chi-square critical values.
The argument that if students learn with calculators they should be able to use them on exams is specious. It is quite possible to design good exams that do not require calculators. I've done it for many years. With only a very slight tweak there would have been no use for a calculator at all. The two we have seen thus far made only slight use of calculators, and then mostly just for arithmetic.