The midterm seemed to accomplish its main purposes. In terms of level of difficulty, it did a good job of differentiating student understandings. Particularly as the two students that scored poorly on the exam appeared to have weaker understandings in the course from a variety of assessments. That said, Problem 6 was too hard for most of the students, and as a consequence, my grading on the question had to be modified. John was the only student in the class that was able to give a completely perfect response on the problem, and most other students had a great deal of difficulty.
In terms of testing for desired outcomes, most problems linked directly to some outcome desired in the class. These outcomes were largely from the content group, and of course, many of them were only being tested at the procedural level. Problem 6 required a very deep understanding of the material for the students to be able to solve it, since the procedure necessary had not been taught in the class, and the students would necessarily have to figure it out as they went. The last question on the exam asked for the students to summarize the argument for why the cube root of two cannot be constructed with a straightedge and compass given a unit length. To answer this question well, a student needs to have an understanding of which steps are important, and then concentrate on those.
Thus, on the exam itself, I would probably change problem 6 and have most of the problem make them use a standard basis. Then, I could move problem 6 to the final exam, which is a take home exam and thus allows for more reflection and requires less speed. I would also then make the first question (asking for two characterizations of rational numbers) a little clearer, and the question requiring the construction of the square root of 3 a little more complicated.
The last thing I judge a midterm on, however, is whether it became a learning experience for the students. On that count, I think it succeeded in that most of the students did work hard in trying to synthesize the material from the class for the midterm. Moreover, the homework following the midterm required the students to write up the midterm as a take-home exam. Thus, the students that did not have a good grasp on how to summarize and cover the high points on showing that you cannot construct the cube root of 2 using straightedge and compass were forced to go through this material again. Similarly, rewriting the exam also forced the students to attain a better understanding of the role of the basis in using matrices to represent elements of extension rings over the rational numbers.