One of the comments on my blog on Types of question for exams
A True/False math question where they have to prove their answer. A student who picks the wrong answer can figure that out during the proof and then correct their answer. A student who picks the wrong answer and proves it has proven they really don't have a clueActually I once did an experiment about this! It's only one so I don't know what to read into it, but I will describe it and speculate.
CMSC 250 is the Sophomore Discrete Math course, required for all majors. CS 3 is a co-req. It's a course on how to prove simple things. We DO go over how a FOR ALL statement can be true vacuously (E.g.,all of the students over 10 feet tall will get an A+). Around 150 students take the course. In the spring there is an honors section of about 20. I PLANNED the following:
- In Spring of 2008 one of the questions on the final was a set of FIVE statements where the students had to, for each statement, say if its TRUE or FALSE and NO JUSTIFICATION NECC. One of the statements was If A is a set of natural numbers such that the powerset of A has 5 elements then A is infinite.
- In Spring of 2010 one of the questions on the final was a set of FIVE statement where the students had to, for each statement, say if it's TRUE or FALSE and IF TRUE THEN GIVE A SHORT JUSTIFICATION, IF FALSE THEN GIVE A COUNTEREXAMPLE.
Note that the statement is TRUE since there are NO such sets A.
So, how did they do?
- When NOT needing to justify or give a counterexample, of the 150 students in the class, 14 got it right. There was no correlation (or perhaps a very weak one) between those who got it right and those who did well in the course or those that were in the honors section.
- When the DID need to justify or give counterexample, of the 152 students in the class, 19 got it right. Slightly stronger correlation to those who got it right and those who did well in the course and to those in the honors section.
tricky question which some non-theory faculty members had trouble with when I explained this story to them. Exam Pressure was likely NOT a factor as my exams have very little time pressure- by the end of the exam
there were only about 30 students left taking it.
Here are the answers I got:
- FALSE- clearly A is finite.
- FALSE- too obvious to say why.
- FALSE- there is no such A
- Variants of the above.
- Incoherent things that may be similar to the above.