The study reported in this paper represents the first stage of a five-year project aimed at improving the quality of teaching and learning in undergraduate mathematics courses for science and engineering students. Students were surveyed about their confidence in their mathematics background, their confidence with current mathematics topics, and the reasons for any lack of confidence with current mathematics topics. Students also took a diagnostic test before the start of the academic year which helped them enroll in the course most suited to their mathematics background. Students with weak backgrounds were directed to remedial materials to help them overcome their deficiencies. The study highlighted a number of important issues concerning students' mathematics backgrounds, the appropriateness of course material and the method of delivery, the amount of material covered, the speed at which it is covered, the choice of lectures as the principle instructional technique, and the need to re-examine weekly tutorials and assessment structures. Some gender differences in confidence were found for confidence in mathematics background, but less so for confidence in current studies. Contains 26 references. (Author/PVD)