Elephant Delta 2015: Think Big!

The Southern Hemisphere Conferences on the Teaching and Learning of Undergraduate Mathematics and Statistics have been taking place (with slightly shifting names) since 1997, located in South Africa, New Zealand, Australia and Argentina. Otherwise known as ‘Delta’ conferences, the 2015 conference, the tenth of its name, is Elephant Delta, with the theme ‘Think Big!’ The papers included in this special issue resonate with that theme in their global scope, their grain size of engagement with knowledge generation and in their disciplinary scope across mathematics, statistics and finance. These few, select, papers together with the conference Proceedings as well as the presentations themselves in Port Elizabeth, South Africa, November 2015, will form a body of work representing the researchers within this Delta community, a community which indeed Thinks Big.

In their paper, ‘Issues and trends: a review of Delta conference papers from 1997 to 2011’, Henderson and Britton [1] categorized Delta papers published in the International Journal of Mathematical Education Science and Technology (iJMEST) and the Proceedings. When making a call for reviewers for this issue, in expectation of submissions, we adapted and modified this classification as two different categories:

Category I: Calculus; Linear Algebra; Differential Equations; Advanced Analysis; Abstract Algebra; Numerical Analysis; Discrete Mathematics; Modelling and Applications; Proof; Statistics; Geometry.

Category II: Technology and Visual Learning; Teaching and Learning; General Pedagogy; Student Resources; Transition from High School to Tertiary; High School Topics; Learning Difficulties; Pre-service and In-service Teachers; Professional Development; Assessment.

The papers submitted for this special issue and accepted after a process of peer review clearly could be tagged with these categories; however, we close our editorial with a suggestion of two additional categories.

This special issue of the iJMEST comprises papers of a high standard from different scholars globally, reporting on their research. The nine papers published in this issue fall under different topics from both categories in the fields of engineering, finance, pure mathematics and statistics. With this issue, we have the opportunity to gain insight into what theoretical resources and new developments are currently of interest to the Delta community globally and the consequent opportunities for debate and further research. Each paper offers quite different insights from the others and we discuss them individually.

The literature of mathematics and statistics education within the tertiary environment is frequently concerned with the first one or two years of mathematical studies, an emphasis which is understandable due to the larger classes at the earlier levels and challenges related to the school–university transition. It is always interesting, therefore, to read the literature on mathematics education at the more advanced level, as reported in Cornock's paper on using Rubik's cube to teach group theoretic concepts. The group theory module was a pure mathematics module presented to students studying applied sciences at Sheffield Hallam University. The abstract concepts of group theory were explored through use of the cubes, as far as the constraints of that physical tool would allow, and the algebraic understanding of group theory in turn brought about understanding of the affordances of the cubes. Cornock discusses the strengths and limitations of the teaching method, concluding that it was a broadly successful initiative and was positively experienced by the students.

From a novel teaching method to a (relatively) new teaching model; the ‘flipped classroom’ model is a relatively recent teaching model across higher education, taking advantage of the wide range of teaching and learning options available through today's easy access to technology. Much of the literature on using this model in the context of undergraduate mathematics reports on individual initiatives. Naccarato and Karakok offer a timeous study of flipped classroom initiatives across 14 institutions in the United States. They find that, while motivations for implementation are similar across cases, practicalities and characteristics of implementation vary widely. Naccarato and Karakok not only report on the different implementations, but consider them in the broader context of knowledge development, what knowledge is supported by different implementations of the flipped classroom model and lecturer perspectives on successful knowledge development.

Bergsten, Engelbrecht and Kågesten continue their collaboration in a series of studies across South Africa and Sweden within which they approach conceptual and procedural knowledge from a variety of directions. In this paper, they consider whether engineering mathematics curricula should have a greater conceptual or procedural focus and, within this context, provide arguments from two professional engineers representing different types of engineering, one research and development and the other on site practice. Many engineering mathematics courses are taught in a largely procedural manner – calculus techniques, and so forth – however, both professional engineers, speaking for different engineering fields, emphasize the importance of conceptual mathematical knowledge to engineering studies, de-emphasizing the procedural content of mathematics. This dichotomy between the relative importance of procedural and conceptual knowledge and the way engineering mathematics courses is traditionally taught has implications for the readiness of engineering graduates for professional work.

Quinn, Albrecht and Webby are also concerned with engineering mathematics curriculum design, in their case of an online 13-week unit on algebra and trigonometry, the first in a series of five engineering mathematics courses. Over three years, the unit underwent ten cycles of development and refinement through a process of experiential learning and active research. Teaching and learning category themes were related to students struggling with the technological demands and being naïve with respect to university-level learning, as well as staff being underprepared and needing to focus on the teaching demands. Ten cycles of refinement over the three-year period resulted in better student retention and staff preparedness. Of further interest, the authors observe that the refinements of the online course have benefits outside the course itself, with resources and processes developed within the course being used elsewhere.

Hoadley, Kyng, Tickle and Wood share the concerns of other authors in this issue in looking at issues of knowledge and curriculum, this time not in engineering mathematics, but in finance. In looking for ways to improve curriculum design and delivery, they investigate and compare how students and academics identify threshold concepts (essential conceptual knowledge) in finance. In this paper, the authors report on the student perspective and observe that, while students frequently identify similar important disciplinary features to academics, they rarely do so by referring directly to specific concepts and do not seem to differentiate between different types of knowledge. Hoadley et al. observe that using the framing of threshold concepts to investigate a (finance or other) curriculum allows for comparing student and academic views of the discipline.

While Hoadley et al. look at teaching and learning at the large scale, the next paper narrows the grain size with a study of teaching and learning in calculus problem solving. LaRue and Infante explored students’ problem-solving skills, focusing on the classic optimization problem of finding the minimum amount of fencing required to enclose a fixed area in a calculus class. They were primarily interested in students’ evoked concept images for the mathematical concepts that play a role in the construction of the function whose minimum is needed. The paper taps on several concepts impacting on optimization.

Reflecting global concern for the challenges related to the school–university transition, King and Cattlin report on the strengths and weaknesses of a system of access requirements, while Engelbrecht and Harding discuss initiatives undertaken to encourage success after university entrance. Engelbrecht and Harding discuss the different interventions that the University of Pretoria, South Africa, engaged in in order to address issues regarding student retention as students transit from high school to university. They report on how a multi-dimensional approach was conceptualized in an attempt to improve teaching and learning of mathematics courses across several faculties. Different forms of mathematics intervention initiatives were introduced and the challenges thereof are discussed.

In their paper regarding the impact of assumed knowledge, King and Cattlin report on how many Australian universities have relaxed their entry requirements for mathematics-dependent degrees and hence increased their enrollments, where there is shift from hard prerequisites to assumed knowledge standards which provide students with an indication of the prior learning that is expected. They report on the significant negative impacts associated with assumed knowledge approaches, with large numbers of students enrolling in mathematics-dependent degrees without the stated assumed knowledge which therefore has an impact on students’ pass rates, retention rates and attrition. This paper offers different interpretations of the meaning of assumed knowledge by students, academics and university administration and the consequences of these interpretations.

In the only paper in this special issue focusing on assessment, Khan reports on an observational study to investigate assessment in terms of open and closed book examinations in a first-year business statistics course. The argument raised in this paper revolves around how well open and closed book examinations test understanding and application rather than memory as well as the advantages of each. Khan's paper, by looking at assessment and reporting data which are primarily quantitative and statistical in nature, stands out from the other papers in this special issue; however, it forms part of a thematic strand in that it is also concerned with the types of knowledge supported by these different forms of assessment.

In this special issue of iJMEST, we find examples of several of the categories recognized by Henderson and Britton, namely Calculus, Abstract Algebra, Statistics, Technology and Visual Learning, Teaching and Learning, General Pedagogy, Transition from High School to Tertiary, High School Topics, and Assessment. Had Henderson and Britton been able to include this issue (and that of 2013) in their survey of Delta publications, we consider that two more categories would have made the list, namely Service and/or Disciplinary courses, as well as Knowledge and Curriculum. Engineering mathematics courses, in particular, are often discussed at Delta meetings and this special issue contributes to the ongoing conversation; in addition, this issue features a discussion on mathematics in finance. Knowledge and curriculum forms a strong theme in this issue, with much grappling with issues of conceptual and/or procedural knowledge as well as considering the types of knowledge supported by different modes of teaching and of assessment.