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.
