## The transition from secondary to tertiary mathematics : exploring means to assist students and lecturers

##### Abstract

Early in 2009 it became apparent from articles in the newspapers that first year mathematics students were not performing as well as the students of previous years. There was great concern regarding the insufficient transition from secondary to tertiary mathematics, as well as the preparedness of first year students for university studies. This research focuses on the different factors that are potential causes of the underachievement of first year mathematics students. Students‟ and lecturers‟ beliefs are shaped by their experiences, the impact of continuous perceptions from the world around them, the present dominant paradigm, as well as the beliefs of their teachers. The different views of the nature of school mathematics show how a worldview has an effect on these views and the implications of this on the teaching of mathematics in secondary, as well as tertiary institutions. The paradigm shift from the modern era to the post-modern era caused an awareness of and interest in the construction of meaningful mathematical understanding. The gap between first year students‟ and lecturers‟ beliefs regarding the nature of mathematics and how mathematics is learned became apparent. The changes in the thoughts about the structure of mathematics were investigated and a better understanding of the processes through which mathematical understanding develops emerged. This brought insight into the gap between the reasoning abilities of incoming students from secondary schools and the reasoning needed to succeed in university mathematics. The theoretical study of the global theories of Piaget and Van Hiele gave insight into conceptual development through different stages and that a person should be on an appropriate conceptual level to make sense of what they learn. If not, then rote learning is likely to occur. The local theory of Tall implies that to facilitate understanding of a concept in mathematics, one should go through three worlds of mathematics: the embodied world, symbolic world and the formal world. The embodied view helps someone to give deep meaning to a page iv concept, otherwise one can be trapped in the symbolic world and not be able to move on to the formal world of mathematical thinking. The theoretical investigations led to an empirical study in three phases. Phase 1 was an investigation into the views of mathematics held by the students and the lecturers. In phase 2 an investigation was done to establish the students‟ preferences on how they learn mathematics and how mathematics should be taught, using the Index of Learning Styles (ILS) questionnaire of Felder and Silverman. The results were compared with the way lecturers want their students to learn and how they themselves prefer to teach. Phase 3 included a classification of the questions in the first mathematics test written at tertiary level and subsequent analysis of the answers of students to obtain information on the type of reasoning required from students at tertiary level, as well as the reasoning abilities of the students. The empirical study assisted in understanding the problematic transition from secondary to tertiary mathematics with regard to the nature of mathematics, the beliefs on teaching and learning of mathematics, as well as the reasoning skills that the students possess when entering university