This study investigated the understanding which University of Zambia (UNZA) first year students of mathematics had of specific concepts in selected mathematics topics. Procedural and conceptual understanding underpinned the investigation. It was also the intention of the study to determine whether there exists any relationship between the students confidence levels and their procedural and conceptual understanding of particular concepts. A quantitative approach was followed and specifically a case study design was employed. Three hundred and seventy eight (378) randomly sampled first year students of mathematics wrote a test which was based on sets, functions, polynomials, trigonometry, and complex numbers as taught in the first year at UNZA. To enhance content and face validity, the administered test was preceded by document analysis, piloting, and expert judgement by UNZA lecturers of mathematics. Subsequently, a follow up questionnaire was administered to elicit the participating students experiences as they solved the test items. The test data was analysed using standard indices while the data derived through questionnaires was analysed using multivariate techniques. The study revealed that the majority of the students had procedural, and not conceptual understanding of the concepts assessed.