A Study of Students’ Conceptual, Procedural Knowledge, Logical Thinking and Creativity During the First Year of Tertiary Mathematics

Abstract

This study focuses on students in first year environmental science degree programs, where traditionally mathematical emphasis has been much less than within the strict science or math majors. The importance now placed on applied mathematics, however, means that students need to gain more conceptual and quantitative knowledge of mathematics in not only the environmental degree programs but also in most, if not all, sciences for example, health, education, ecology and biology. In this paper, the authors attempt to gain insight into why many students fail mathematical courses even when the mathematical requirements are not as demanding. This is done by examining students' conceptual thinking patterns and strategies, as evident in students' prepared scripts. A total of 133 students were requested to prepare a focus sheet that summarizes their knowledge of topics learned to a point in time. To motivate students to prepare focus sheets, the students were also told that they could use them during exams as notes. This indeed motivated the students to complete the sheets, as they prepared weekly summaries that were later summarized for use in exams. Detailed examination of such sheets allowed researchers to examine and study students' knowledge, based in terms of procedural work, mathematical skills, strategies and depth of conceptual knowledge as evidenced in the written form. A study of linear and quadratic functions and limit sections led to interesting insights not only regarding mathematical knowledge but also, and importantly, the higher order skills such as problem solving tasks during preparation. Logical and creative competencies were assessed in terms of planning, organisation and linkage; that is, how and which aspects of the knowledge each student focused upon, transferred or linked across topics in order to facilitate use and application of knowledge. The results show average levels of procedural and conceptual competence, but low levels of logical and creative competence in terms of the categories studied. Almost 50% of students lacked competency in procedural work, while around 54% lacked conceptual competency. Given the emphasis placed on procedural skills by students, generally the levels were lower than expected. The lack of structure, planning, preparation and organisation found in the focus sheets was worrying, in that students did not demonstrate deeper levels of understanding of the topics learned. These findings have implications for the first year mathematics teaching teams at universities, especially in the non-specialist mathematical majors, in terms of where the focus of teaching may lie. KEYWORDS: Tertiary mathematics, learning algebra, conceptual, procedural, logic, higher order thinking, first year university mathematics, environmental science, mathematics education