In this research I explored how mathematics teachers can inform their teaching practice through a meta-reflective inquiry into methods of facilitating Whole Brain® learning in mathematics. Herrmann’s Whole Brain® theory was used as a lens through which to explore leading theories in the fields of constructivism, mathematics education and cognitive psychology by means of a participatory action research innovation, stretching over approximately 3 years. An analysis of these theories validated Herrmann’s Whole Brain® theory as the foundation for a synthesised integrated theory of practice, which also formed the epicentre of the conceptual framework for the research. The conceptual framework was also at the core of the participatory action research. The Herrmann Brain Dominance Instrument® (HBDI®) was administered to 8 teacher participants in a school mathematics department. Learners of each of the teacher participants also completed a questionnaire on how they perceived their teachers to facilitate learning and assessment of mathematics. These results were compared to the teacher participants’ Herrmann’s Brain Dominance Instrument®. Findings indicate that the Herrmann Brain Dominance Instrument® initiated scholarly reflection with teacher participants involved in facilitating and assessing the learning of mathematics. The collective reflexive practice was both part of the action research process and an outcome of the research itself. Findings also indicate that the thinking preferences of teacher participants, as tested by the Herrmann Brain Dominance Instrument®, are not necessarily indicative of their teaching style and teachers involved in post-graduate studies indicated an ability to access their non-dominant thinking mode situationally.

Challenges experienced by first-year students transitioning from secondary to tertiary mathematics education are examined through the lens of the didactical contract. The didactical contract describes the expectations of both lecturer and students about their mutual obligations towards teaching and learning. First-year students’ beliefs about the nature of mathematics and mathematics teaching/learning need to be challenged to renegotiate the didactical contract at tertiary level. The study focuses on how to elicit and confront transitioning students’ beliefs in order to support their learning and influence a shift in the didactical contract. A Likert scale questionnaire was deployed at the beginning of students’ first year to gauge their beliefs about mathematics and mathematics teaching/learning and redeployed near the end of the first semester (or term) to observe possible changes in their beliefs and hence the didactical contract. The intervention consisted of personal response system (PRS) sessions regularly incorporated into the traditional transmission mode lecture to flip the classroom and create a student-centred learning environment, aimed at influencing students’ beliefs in order to make them aware of their own learning and their responsibility for learning. Questionnaire data were quantified and compared for the before and after surveys. There is evidence of a shift towards students taking ownership of their learning and a renegotiation of the didactical contract. Qualitative data generated by focus group interviews confirm the role of the PRS sessions in influencing student beliefs and the didactical contract.

This article focuses on the unique contribution of the QT-clicker regarding formative and summative assessment in a large flipped first year statistics module. In this module, the flipped classroom as pedagogical model first substituted the traditional teaching model. QT-clickers were subsequently introduced to enable active and cooperative learning for face-to-face engagement inside the classroom. The various input capabilities of the QT-clicker, providing for the possibility of partial grade crediting, offer a distinguishing advantage. These clickers were initially only used for formative assessment, but soon extended to summative assessment. Two cohorts—2014 (no QT-clickers) and 2017 (with QT-clickers)—were compared. The intervention of using QT-clickers was evaluated along two lines: the pedagogical influence of the QT-clicker and the effect of partial grade crediting. Several general linear models (GLMs) were fitted to the data to investigate how QT-clicker use is related to the students’ examination performance. The outcome of the GLM models indicates that the association of higher examination marks with QT-clicker use holds for the 2017 cohort with and without partial credit. A qualitative component of the study reports on the student voice testifying to positive experience.

This paper originated from the desire to develop elementary calculus based tools to empower students, not necessarily with a strong mathematical background, to test predator-prey related models for boundedness of solutions and for the existence of limit cycles. There are several well-known methods available to prove, or disprove, the existence of bounded solutions to systems of differential equations. These methods rely on LiГ©nard's theorem or using Dulac or Lyaponov functions. The level of mathematics required in the study of differential equations is not addressed in the courses presented on the first year level, and students in biology, ecology, economics and other fields are often not suitably equipped to perform these advanced techniques.The conditions under which a unique limit cycle exists in predator-prey systems is considered a primary problem in mathematical ecology. A great deal of mathematical effort has gone into trying to establish simple, yet general, theorems which will allow one to decide whether a given set of nonlinear equations has a limit cycle or not. We introduce a method to first determine the boundedness of solution trajectories in such a way that the transformation to a LiГ©nard system or the use of a Dulac function can be avoided. Once boundedness of trajectories has been established, the nature of the equilibrium points reduces to simple eigenvalue analysis. The Elemental Limit Cycle method (ELC) provides elementary criteria to evaluate the nature of the pivotal functions of a system which will indicate boundedness and may be applicable to more general models.

This paper focuses on the students who are registered in the University of Pretoria's academic development programme, named the Four-year Programme (FYP). The programme was introduced as a gateway for students who are underprepared but have the potential to succeed and then continue their studies into the mainstream science programmes. Our research focuses on measuring the change in the academic maturity of these students. In the theoretical framework that we developed, academic maturity is subdivided into two components namely non-subject based maturity and subject based maturity (mathematical maturity). This paper focuses on measuring non-subject based academic maturity. The survey used for this purpose is called the Student Academic Readiness Survey (STARS), taken at the beginning of the year and after the first semester respectively. The results of the surveys are compared to measure the change in students' views. Results show that in all constructs there is a surprising decline in students' perceptions regarding their own abilities over the first semester at university. We use the Dunning-Kruger effect to explain this unexpected decline, in that students seem to develop a more realistic view of their own maturity, which in itself can be seen as a growth in academic maturity.

It is well-known that the Lotka-Volterra predator-prey model has a family of periodic orbits, but does not possess limit cycles and therefore the model is said to be structurally unstable. The Lotka-Volterra model is a special case of a much larger group namely the quadratic population models and it can be shown that none of them can produce limit cycles. The surprising finding is that by combining two quadratic models a quadratic population model with two limit cycles is uncovered. Although the model looks simple at first glance it provides a rich source of dynamics and deserves attention. In this paper, we revisit a model that has its origin in the work of Dubois and Closset. A set of two quadratic population models interact as piecewise defined differential equations. The model has been discussed by Ren Yongtai and Han Li, cryptically written and showing some linguistic and typographical errors, but providing an excellent vehicle for developing skills in mathematical modelling, differential equations and technology for the young researcher. We explore the model in clearer detail and supplement the theory with rich graphical illustration. The paper has the purpose of providing an example of how a young researcher, such as a postgraduate student in biomathematics, can expand on an existing model by making use of current technology.

One of the problems encountered when teaching complex numbers arises from an inability to visualise the complex roots, the so-called "imaginary" roots of a polynomial. Being four dimensional, it is problematic to visualize graphs and roots of polynomials with complex coefficients in spite of many attempts through centuries. An innovative way is described to visualize the graphs and roots of functions, by restricting the domain of the complex function to those complex numbers that map onto real values, leading to the concept of three dimensional sibling curves. Using this approach we see that a parabola is but a singular case of a complex quadratic. We see that sibling curves of a complex quadratic lie on a three-dimensional hyperbolic paraboloid. Finally, we show that the restriction to a real range causes no loss of generality.

ABSTRACT It is well-known that the Lotka–Volterra predator-prey model has a family of periodic orbits, but does not possess limit cycles and therefore the model is said to be structurally unstable. The Lotka–Volterra model is a special case of a much larger group namely the quadratic population models and it can be shown that none of them can produce limit cycles. The surprising finding is that by combining two quadratic models a quadratic population model with two limit cycles is uncovered. Although the model looks simple at first glance it provides a rich source of dynamics and deserves attention. In this paper, we revisit a model that has its origin in the work of Dubois and Closset. A set of two quadratic population models interact as piecewise defined differential equations. The model has been discussed by Ren Yongtai and Han Li, cryptically written and showing some linguistic and typographical errors, but providing an excellent vehicle for developing skills in mathematical modelling, differential equations and technology for the young researcher. We explore the model in clearer detail and supplement the theory with rich graphical illustration. The paper has the purpose of providing an example of how a young researcher, such as a postgraduate student in biomathematics, can expand on an existing model by making use of current technology.

The study in this paper reports on whether an online homework system in combination with a flipped classroom has a positive effect on success rates of first year statistics students. Departing from a traditional teaching model (Baseline), the technology-based intervention strategies implemented were firstly the Aplia interactive online homework system (Phase 1) and then, in addition, the flipped classroom (Phase 2). In the flipped classroom model, the transfer of information took place by pre-reading and pre-class Aplia homework assignments. This approach intended to allow more quality time to be spent on revisiting difficult concepts and engaging in problems inside the lecture hall, thus enhancing the learning process. The average final marks of the three cohorts were compared using an ANOVA test. Chi-square tests were used to evaluate the effects of the changing teaching models from Baseline through Phase 1 to Phase 2. The success rates of three different cohorts (n = 1485, 1343 and 1466 respectively), consisting of only first-time statistics students with a Grade 12 mathematics mark of at least 60%, were assessed. For Phase 1 compared with the Baseline the increase in the success rate was insignificant, but the success rates for Phase 2 compared with both Phase 1 and Baseline improved significantly with a small to medium effect size, because of the very large samples. The practical importance of this is that the number of successful students increased by more than 100 from Phase 1 to Phase 2 and more than 130 from Baseline to Phase 2. A significant relationship was also found between the different cohorts and their mark distributions, confirming that the online homework system in combination with the flipped classroom in particular is effective in increasing the success rate.

This paper offers a narrative of ideas, events and opinions addressing the underexposed area of storytelling in tertiary mathematics. A short discussion on storytelling is followed by a brief account of the history of storytelling. Features of stories are discussed as well as options for when a story should be told and the requirements of a good story. The main thrust of the paper is a personal account of experiences of storytelling in a tertiary mathematics classroom. The study involves a large group of engineering students doing a calculus module. The storytelling discussed in this paper takes the form of a structured activity in a specific timeslot. Student feedback presents an unexpected angle, deviating from the intended purpose of entertain, inspire and educate, namely, giving a perception of caring from the teacher's side.

The purpose of this case study is to explore the integration of technology into teaching at a mathematics department at a large South African University. Both quantitative and qualitative data were collected from staff teaching undergraduate mathematics. The study shows that many staff members feel that chalkboards are still more suitable than technology for teaching mathematics. This finding supports the idea of a strong subject culture. Age does not emerge as a determinant for preference of either technology or the chalkboard, although gender and academic qualifications do. Subject culture is strongly rooted under the male members of staff, while female staff members feel more positive towards the use of technology in teaching. Use of chalkboards has decreased significantly over the past 10years, while the use of modern technologies has increased accordingly. Teaching of large groups has necessitated the use of technology in the classroom. Despite the strong subject culture, a shift in attitude towards technology use in teaching is noticed and there is a definite trend of moving towards using new technologies.

This paper presents an enrichment case study to showcase a possible avenue for attending to the needs of academically strong mathematics students. We report on a group of university students who were presented with the opportunity of exploring a specific first year mathematics topic deeper, using an inquiry-based learning approach as part of an enrichment programme. Following the intervention, students completed a questionnaire and a few were interviewed to establish their experiences of the enrichment programme. We discuss the successes and pitfalls of the intervention and report on the impact it had on the participants.

Sibling curves were demonstrated in [1, 2] as a novel way to visualize the zeroes of real valued functions. In [3] it was shown that a polynomial of degree n has n sibling curves. This paper focuses on the algebraic and geometric properites of the sibling curves of real and complex quadratic polynomials.

Medical students spend over 85% of their clinical learning time on hospital placements, but there has been comparatively little detailed analytical investigation. This work therefore seeks to further the understanding of clinical learning in hospitals. The study adopted a focussed ethnographic approach using quasi-participant observation of third-year medical students, on one hospital placement over a period of two years. Observations revealed repeating types of learning episodes, which are presented as vignettes. These vignettes are analysed using Actor-Network-Theory (ANT), a branch of material semiotics. ANT seeks to account for both the social and material aspects of learning relevant to complex socio-technical environments such as hospitals. Although theoretically attractive, socio-material approaches such as ANT have been difficult to operationalise for empirical use. I have developed a number of bespoke methodological and analytic approaches that are clearly articulated to enable critique and future use. Analysis suggests that clinical learning can usefully be conceptualised by learning networks that produce varying opportunities for learning. The networks comprise human and material participants (or actors), interacting in complex but definable ways. The material actors figure prominently, and often inhibit network formation. Within learning networks, differing actor combinations generate a range of learning processes that produce a corresponding variety of learning opportunities. The networks are time consuming to initiate, fragile and short-lived. When operational, networks can contribute to learning technical proficiency, but opportunities to learn clinical skills are rare. The analysis contributes towards the understanding of medical education by identifying new material and human actors. The analytic process also introduces a systematic way of describing how the actors interact to produce learning. Identification of new actors and relationships has led to opportunities to improve clinical learning at the observations site and generated several opportunities for further research.

This study reports on student experiences in an access programme in the science faculty at the University of Pretoria, South Africa, named the BSc Four Year Programme (BFYP). The programme has a preparatory 18 months phase after which students join the mainstream programme. The aim of the paper is to give a voice to students who enrolled in 2008 and are either still in the system or have successfully completed the degree programme. We identify three performance bands – good, moderate and poor performers, and focus on their experiences in the preparatory phase of the BFYP, reporting on personal perceptions of the structure of the programme, on challenges faced and on preparedness upon transition to the mainstream programme. The distinguishing feature of the paper is that experiences are reported on through the lens of the different performance bands, adding shades of intensity to the different perspectives. Whilst the structure and features of the BFYP were experienced positively by most students across the performance bands the voice of the poor performing students emerged in this study, expressing their sense of frustration, their inability to cope and their failure to identify their challenges and to seek assistance. The need to equip under-performing students more effectively in terms of academic and life skills is the key finding of this work and should be of interest to the wider audience of educators and counsellors involved in access programmes.

This study focuses on the mathematics department at a South African university and in particular on teaching of calculus to first year engineering students. The paper reports on a cause-effect analysis, often used for business improvement. The cause-effect analysis indicates that there are many factors that impact on secondary school teaching of mathematics, factors that the tertiary sector has no control over. The analysis also indicates the undesirable issues that are at the root of impeding success in the calculus module. Most important is that students are not encouraged to become independent thinkers from an early age. This triggers problems in follow-up courses where students are expected to have learned to deal with the work load and understanding of certain concepts. A new model was designed to lessen the impact of these undesirable issues.

Co-chairs: Ansie Harding (South Africa), Juha Oikkonen (Finland); Team Members: Christopher Sangwin (UK), Sepideh Stewart (New Zealand), Miroslav Lovric (Canada), Sung-Ock Kim (Korea); Liaison IPC Member: Johann Engelbrecht (South Africa).