# Musings on the Midterm

This week the students sat for their midterm. And sat. And sat. None of them finished completely in the allotted time. Every submitted test had at least one question only partially answered. No student left early (which is a good sign since students tend to leave early when the test is too hard or too easy—maybe this was the Goldilocks test).

I find it difficult to create good tests. In part because I’m not sure what a good test is, especially before I give it (and isn’t that disrespectful—me giving the test and not the students taking it—and what does that mean, to ‘take a test’–this reflection on teaching can, I’m afraid, become a gaze into a carnival hall-of-mirrors).  Perhaps I recognize a good test, after the fact, when I grade it? Is it good when the grade distribution is normal? Or when it is not normal? Is it good when a student reaches a correct answer but only through a torturous, almost incomprehensible chain of reasoning?  Is it good when a student easily gives the correct answer but ignores the request for justification?

Maybe I’m so unsettled today because of the lesson that followed the midterm, set theory. We started our discussion of sets with the Russell paradox that arises from imagining a set S that contains all sets that don’t contain themselves. I’ll let you explore the question, ‘Is set S an element of set S?”

What do I try to balance when writing and endlessly editing a test? (Yes, many parts of teaching seem to never end.) [Aside (I teach math after all). This week we also studied the halting problem in computer science—can we write a universal program that will be able to determine from the description of any program and a list of data to be processed by the program, whether or not the program will halt or run endlessly. The answer is related to the Russell paradox.]  Back to balance–for sure I don’t write a balanced test by measuring it using some exotic formula with multiple variables—as much as that might appeal to me as a math teacher. I think I must cite the tired analogy of riding a bicycle—some actions are done unconsciously once the skill has been learned—and how does one learn the skill? Can it only be through trial and error?  And who is the judge of attainment?  [In the manner of Socrates, when you don’t have an answer, ask another question.]

What do I try to balance?  What are some tensions that create a balance?  What is the gravity of testing that calls for balance?

• Coverage: “This test covered too much.” versus “You picked the one area I didn’t study.”
• Difficulty: “Yes, I’ve seen all this before.” versus “When did you show us that?”
• Challenge: “Look at me, I’m finished really early.” versus “Is it still possible to withdraw?”
• Grading: “I’ll have your results tomorrow.” versus “The don’t pay me enough for this job.”
• Gravity: “I need an A.” versus “What have I learned?”

That’s enough; before I further test your patience and my balance.