An investigation into how the undergraduate mathematics topic of sibling curves of functions can be developed and used for student enrichment

Abstract

The aim of this research study is to investigate how undergraduate mathematics can be en- hanced through research and how this enriched content can be exploited for the enrichment of academically stronger students. The study o ers a unique blend of mathematical and educa- tional research. In the rst part of the study a problem stemming from teaching the undergraduate topic of complex numbers, namely on how to represent the zeroes of functions, particularly polynomi- als, is researched. The notion of sibling curves ([51], [52]) o ers an elegant and natural way to represent the zeroes of a polynomial, which is explored and expanded in this thesis. A library of sibling curves for well-known functions is developed and presented. Signi cant research results are that every polynomial of degree n has n sibling curves. This result gives a more geometric interpretation of the roots of a polynomial than the Fundamental Theorem of Algebra. I then focus on quadratic polynomials with complex coe cients and prove that, although the siblings are not always parabolas, the two sibling curves are always congruent and that they lie on a hyperbolic paraboloid determined by the coe cients of the polynomial. Some of these results are reported on in [115]. The second part of the study centres on utilising the researched knowledge on sibling curves for student enrichment. A group of rst year students were guided through a number of designed activities using an inquiry-based learning approach to explore polynomials, complex numbers and ultimately sibling curves. Implementation of the programme as well as experiences are reported on, following a research approach of evaluation research. Student as well as facilitator experiences, discoveries and learning curves were captured in order to analyse perspectives. Re- sults show that there is a need to stimulate and challenge academically strong undergraduate students. The study further shows that all the participants of this enrichment programme ben- e ted from this experience. The students were engaged with the work and had the opportunity to delve deeper into the mathematical topic while sharpening their problem solving skills. I, as facilitator, had the opportunity to interact closely with academically strong students and experience their needs rst hand, which added a new dimension to my teaching. This research also demonstrates how enrichment programmes can be a vehicle to expose enriched content to academically strong students. The dual value of the study is that it adds not only to the knowledge base of complex number theory, but also to the body of reported experiences on student enrichment in undergraduate mathematics teaching. It is envisaged that research ndings reported on in this study will lead to an increased focus on student enrichment at tertiary level. This study exposes this element of teaching academically strong students and o ers possible avenues of challenging these students.