Just recently experienced another area with an assignment for some teachers where a hierarchical classification of quadrilaterals is also important. For example, asking them which are necessary & sufficient conditions by which to define certain quadrilaterals.
Many got right answers for some questions, but did so by incorrect reasoning and providing incorrect counter-examples! For example, all of them correctly said that 'equal diagonals' is a necessary, but not sufficient condition for a quadrilateral to be rectangle. However, all of them gave as 'counter-example' (to show it is false) a square, but did not realize this was an INVALID counter-example as a square IS a rectangle! They didn't realize they had to produce an example of a quadrilateral that has equal diagonals, but is NOT a rectangle, for example, a general isosceles trapezoid to show that the condition is insufficient.