South Africa has opened up access to higher education over the past 20 years. The massive increase in enrolments (with almost 70% first-generation students) substantially affects progress and graduation rates in Science programmes in higher education. First-year students in Science realise that university mathematics requires knowledge and skills that are not part of their academic repertoires. Science students at the University of Johannesburg register for a two-week, credit-bearing First-year Seminar (FYS). The research question that this paper addresses is: What is the relationship between the First-year Seminar and the mathematics performance of first-year students in Science? The specific purpose is to determine the relationship between: (1) students' school mathematics background; (2) the problem-solving skills sessions of the FYS; and (3) their first-year performance in mathematics. It was found that the FYS enhances students' ability to make a successful transition to university, with problem-solving ability acting as a fair predictor of performance in first-year mathematics. The empirical data was collected through a before-and-after test performed by the 2014 cohort with regards to students' attendance of the FYS. Notably, the data indicate that the value added by the problem-solving test can be applied to identify and engage students who have high probability of becoming students at risk (STARs).