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**Problem statement**

**Problem statement**

My interest in student enrichment grew from my role as a lecturer and co-course coordinator teaching first year mathematics courses at the University of Pretoria. More and more students are being drawn to higher education, resulting in an increase in the numbers in first year undergraduate classes. The increase in the size of first year mathematics classes puts a strain on the teaching team; the teaching load increases; the tutoring load increases; the marking load increases and it becomes increasingly difficult to address the needs of specific subgroups. Lecturers have a defined syllabus that they strictly stick to and thus rarely get the opportunity to explore an interesting topic or an idea that links up with a concept that is being taught in class. Furthermore time constraints to teach a certain topic within a specified number of lesson periods also put pressure on the lecturer for teaching sound problem solving skills instead of algorithmic type of solution methods.

** First year mainstream mathematics**

The mainstream first year mathematics major course has a minimum entry level of a 7 (translating to a final mark of at least 80%) in the National Senior Certificate (NSC) and an Admission Point Score (APS) of 32 (out of a possible 42). The NSC is the main school-leaving certificate in South Africa. The calculation of the APS is based on a student’s achievement in six recognised subjects by using a seven-point rating scale. In the first semester students enrol for WTW114 (Calculus 1A). In the second semester they enrol for WTW128 (Calculus 1B) and WTW126 (Linear Algebra 1). Together these three courses are intended for students wishing to become professional mathematicians or secondary school mathematics teachers or for students who need to complete these courses as a co-requisite for their degree in actuarial science, physics, computer science, statistics, chemistry, etcetera.

**Structure of the thesis**

**Structure of the thesis**

This thesis is designed to explore the stated research questions mentioned in Section 1.4. The background of the study is sketched in Section 1.1, followed by the problem statement in Section 1.2. The aim of this study is outlined in Section 1.3. In Section 1.1 I indicated that my research focus was to expand on the theory of sibling curves and then to proceed to use this knowledge to implement a student enrichment project. This thesis consists of two parts: Part A (Chapters 2 to 5) documents the mathematical research on which I base the student enrichment project in part B (Chapters 6 to 9). In Part A I took a problem stemming from the teaching of the topic of complex numbers in mathematics, leading to expanding the mathematics involved. The knowledge is then ploughed back for enrichment of students as a project for first year mathematics students, which is documented in part B.

**Fehr’s idea**

**Fehr’s idea**

In [51] Harding and Engelbrecht explore an idea that appeared in an American secondary school textbook [30]. The author, Howard Fehr, was a past president of the National Council for Teachers in Mathematics in the USA and a professor at Columbia University. His idea was quite simple. Fehr considered functions f that map complex numbers onto complex numbers. Fehr then restricted the domain to those complex numbers that map the function onto real values. On this new domain the function has a range consisting of real values and can be represented in three dimensions by taking the domain as the horizontal plane and the range as the vertical axis. So the horizontal plane acts as the Argand plane, whereas the vertical axis is the new range. This allows one to give a three dimensional picture of the four dimensional function that can be derived from the function that maps complex numbers onto complex numbers. Furthermore this three dimensional cut of a four dimensional space produces all the roots of the function f, because it contains all the values for which f(z) = 0. Let us reconsider Fehr’s example [30]. Fehr noted that if you draw y = x 2 + 3x + 4 in the Cartesian plane, you obtain a graphical representation of this function. Since this graph does not intersect the x-axis, we see that there are no real roots. See Figure 2.4.5.1 taken from [30]. Thus the roots are complex and this method does not visualize the complex roots.

**The general case**

With the help of Lemma 4.3.3, we are ready to prove the main result of this chapter. Theorem 4.4.1. If f(z) is a complex polynomial of degree n, then f has n sibling curves. Proof. We will show for each real value of w that there are always n portions of sibling curves containing the solutions of f(z) = w. If these subcurves are glued together, we get the desired n sibling curves. For some fixed w ∈ R, we know f(z) = w has n solutions by the fundamental theorem of algebra. Some may have multiplicity higher than 1. Take solution z1 with multiplicity m. Then f(z)−w = q(z −z1) where q(z) = cmz m +. . .+cnz n . Hence, if q(z) = 0 then f(z +z1) = w. So we only need to show that we get m sibling subcurves containing the solutions q(z) = 0. Note there are at most n − 1 real values w such that g(z) = f(z) − w has a root with multiplicity higher than one. Noting q(z) = f(z1) − w = 0, the solution now lies in using Lemma 4.3.3. There we proved that it is possible to define m sibling subcurves of q around 0. By [19] they contain all the solutions in that neighbourhood. Furthermore each of them has the same non-zero radius of convergence. Thus each value of w produces n sub-parametrizations. Now for any real value w, we have n sub-parametrizations. Suppose R is the smallest radius of convergence for these sub-parametrizations. That is each sub-parametrization is valid on the interval (w −R, w +R). Now consider the real values w − R 2 and w + R 2 . They each have n sub-parametrizations. By glueing two sub-parametrizations that overlap together, we form n piece-wise functions on the interval [w − R 2 , w + R 2 ]. Continuing in this manner we produce the n sibling curves.

**Hyperbolic paraboloid**

In summary we noticed in the case of quadratic polynomials that one of two things can happen. One scenario is that the sibling curves meet. In this case we get two parabolas which are planar, implying that each sibling curve lies in its own plane. The second scenario is that the sibling curves never meet. Here we end up with two curves which are not parabolas. In either situation the sibling curves are congruent to each other. From the previous section it follows that we really only need to consider the quadratic polynomial f(z) = z 2 + C for some complex number C. If z = x + iy, then f(z) = (x + iy) 2 + C = x 2−y 2+Re(C)+i(2xy+Im(C)). Hence the sibling curves are on the surface z = x 2−y 2+Re(C) or on the hyperbolic paraboloid (taco-shaped surface) z = x 2−y 2 after an appropriate rotation, scaling and translation. The two typical possibilities are depicted in Figure 5.5.1 and Figure 5.5.2.

**Contents :**

- 1 Introduction
- 1.1 Background
- 1.2 Problem statement
- 1.3 Aim and purpose of the study
- 1.4 Research questions
- 1.5 Context of this study
- 1.5.1 University of Pretoria
- 1.5.2 Department of Mathematics and Applied Mathematics
- 1.5.3 First year mainstream mathematics
- 1.6 Structure of the thesis

- 2 Representing the zeroes of polynomials
- 2.1 Introduction
- 2.2 Number systems
- 2.3 History
- 2.4 Representation of zeroes
- 2.4.1 Rotating the parabola
- 2.4.2 Superimposing the Argand plane onto the Cartesian plane
- 2.4.3 Moving between the Cartesian and Argand planes
- 2.4.4 Modulus surface
- 2.4.5 Fehr’s idea

- 3 Library of sibling curves
- 3.1 Introduction
- 3.2 Polynomials
- 3.3 Rational functions
- 3.4 Exponential and trigonometric functions
- 3.5 Multi-valued functions

- 4 Sibling curves of polynomials
- 4.1 Introduction
- 4.2 An example of a sub-parametrization around a point
- 4.3 A few lemmas
- 4.4 The general case

- 5 Sibling curves of quadratic polynomials
- 5.1 Introduction
- 5.2 Real quadratics
- 5.3 Complex quadratics
- 5.4 Congruence
- 5.5 Hyperbolic paraboloid
- 5.6 θ-sibling curves
- 5.7 Further research

- 6 Literature review
- 6.1 Introduction
- 6.2 Student enrichment
- 6.3 Enrichment, acceleration and gifted education
- 6.4 Enrichment for whom?
- 6.5 Student enrichment and inquiry-based learning
- 6.6 Student enrichment in mathematics
- 6.6.1 Stimulating interest in mathematics
- 6.6.2 Problem solving
- 6.7 Positioning this study: What, who, how

- 7 Research design and methodology
- 7.1 Introduction
- 7.2 Research design
- 7.3 Phase 1: Planning phase
- 7.3.1 Selection and recruitment
- 7.3.2 Mathematical activities
- 7.3.3 Facebook
- 7.3.4 Journal
- 7.4 Phase 2: Evaluation phase
- 7.4.1 Survey questionnaire
- 7.4.2 Tutorial test marks comparison
- 7.5 Interviews
- 7.5.1 Format of the interviews
- 7.5.2 Interview questions
- 7.6 Ethics

- 8 Research findings
- 8.1 Introduction
- 8.2 My perspectives and experiences
- 8.2.1 Phase 1 experiences
- 8.2.2 Phase 2 experiences
- 8.2.3 Post Phase 2 experiences
- 8.3 Facebook group
- 8.4 Survey questionnaire
- 8.4.1 Question 1 results: Enjoyment of the project
- 8.4.2 Question 2 results: Reading on their own
- 8.4.3 Question 3 results: Enjoyment of solving new problems
- 8.4.4 Question 4 results: Coping with project and university work
- 8.4.5 Question 5 results: Working alone
- 8.4.6 Question 6 results: Enjoyment of technology
- 8.4.7 Question 7 results: Usage of technology
- 8.4.8 Question 8 results: Recommendation of technology
- 8.4.9 Question 9 results: Collaboration
- 8.4.10 Question 10 results: Prior knowledge
- 8.4.11 Question 11 and 12 results: Interest in future enrichment projects
- 8.4.12 Question 13: View of mathematics
- 8.4.13 Question 14 results: Enjoyment of the project
- 8.4.14 Question 15 results: What was learnt from the project
- 8.4.15 Question 16 results: Collaboration
- 8.4.16 Question 17 results: Ideas for improvement
- 8.5 Tutorial test
- 8.6 Interviews
- 8.6.1 Interview 1 – Peter (Student O)
- 8.6.2 Interview 2 – Sara (Student T)
- 8.6.3 Interview 3 – Jane (Student P)
- 8.6.4 Interview 4 – Tim (Student L)
- 8.6.5 Thematic analysis

- 9 Discussion and conclusions
- 9.1 Introduction
- 9.2 Research question
- 9.3 Research question
- 9.4 Research question
- 9.5 Limitations of the study
- 9.6 Recommendations
- 9.7 Future research
- 9.7.1 Mathematical research
- 9.7.2 Educational research
- 9.8 Value of the study
- 9.9 Concluding remarks

- References

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