At a time of transition, when the increasing availability and affordability of Computer Algebra Systems (CAS) presents mathematics educators with new challenges, this thesis explores two facets of students’ abilities and understanding that impact on the use of CAS in teaching and learning mathematics. In this thesis, these are called ‘Algebraic Insight’ and ‘Effective Use of CAS’. A framework is presented and described for each construct and then the frameworks are explored within the context of a course in introductory calculus, taught by the researcher to a class of 21 undergraduate tertiary students. Algebraic Insight is the subset of Symbol Sense required when using CAS for the mathematical solution phase of problem solving. The framework breaks Algebraic Insight into two aspects: ability to Link Representations (symbolic, numeric, graphical); and Algebraic Expectation, the cognitive skill required to monitor symbolic work (comparable to arithmetic estimation for monitoring numeric work). The framework of Effective Use of CAS is also divided into two aspects: Technical, using syntax and program features; and Personal, the willingness to use CAS in a judicious manner.