This article defines "higher level mathematical skills" and details an important class: that of "constructing instances" of mathematical objects satisfying certain properties. Comment is made on the frequency of higher level tasks in undergraduate work. We explain how such questions may be assessed "in practice" without the imposition on staff of an onerous marking load. Included are examples which have been implemented on a "free" computer aided assessment system. Lastly we report an investigation of students' reactions to these questions and discuss their design and impact. (Contains 2 tables.)