ABSTRACT. This talk will cover two related sequences of research in undergraduate mathematics education: a first sequence on proof comprehension, and a second on conditional inference.  The first sequence takes a global view of how people read mathematical arguments, using experimental and eye-tracking studies to compare student and expert reading and to study mathematical self-explanation training.  The second takes a local view of how people evaluate specific inferences, drawing on extensive work in cognitive psychology to design and test the first mathematical conditional inference task.  Both sequences provide insight into the complexity that students must navigate when interpreting everyday language in mathematical arguments.  Both, however, have positive outcomes: mathematics students reason about mathematical content in a way that conforms quite closely to normative validity, and can improve their comprehension with light-touch generic training.

ABSTRACT. The paper considers different representations of numbers and investigates how these representations may influence the way mathematical courses are taught at university level. The theory of procepts provides a theoretical framework for our consideration. Being analysed as procepts numbers can be seen as a flexible didactical tool. Several examples with focus on mathematical constants are discussed in detail to show how mathematical knowledge and understanding of the Calculus content can be enhanced within the educational theoretical framework.

ABSTRACT. The recent pandemic caused a brutal upheaval in approaches to learning and teaching and our ability and capacity, in adverse circumstances, to engage with students and help them develop important skills and understanding. Necessity is the mother of invention and there may be silver linings emanating from lessons learnt by us, as educators, when forced to make abrupt changes or rethink, with little notice, our teaching approaches and preferences, which may be long-held and ingrained. Assessment and the protection of academic integrity have always been thorny and controversial issues, but even more so during a pandemic. This talk focuses on contrasting roles taken by the extensive use of multiple-choice questions and aims to tease out important underlying principles in using them for formative and summative assessment tasks, separately and in combination. Functionality may change, depending on whether the learning environment is synchronous (in-person, face-to-face) or asynchronous (online, typically remote, using a learning management system). A variety of examples will be presented. If there is time, connections may be made to models of learning, such as constructive alignment and the role of SOLO (Biggs, 2003), the theory of threshold concepts (Meyer and Land, 2005) and navigation of liminal space (Cousins 2006), and ways and means of engaging students and moving them through the passive-active interface (Easdown, 2007).

Biggs, J. (2003). Teaching for Quality Learning at University (2nd ed.). Berkshire: Open University Press.

Cousin, G. (2006). An introduction to threshold concepts. Planet, 4-5.

Easdown, D. (2007) The importance of true-false statements in mathematics teaching and learning. UniServe Science Teaching and Learning Research Proceedings, 164-167

Meyer, J. H. F. & Land, R. (2005). Threshold concepts and troublesome knowledge (2): Epistemological considerations and a conceptual framework for teaching and learning. Higher Education 49(3), 373-388.

ABSTRACT. In a move to prioritise Science, Technology, Engineering and Mathematics (STEM) education, the Department of Education developed the Mathematics Teaching and Learning Framework which prescribes that Mathematics be taught for Conceptual Understanding. Along with the development of the Mathematics Teaching and Learning Framework, and in response to the skills required for the Fourth Industrial Revolution (4IR), the Department has also introduced new subjects. These include Technical Mathematics and Science, Coding and Robotics, Aerospace, Biomedical, Ocean and Marine Engineering. One topic that is indispensable to all of these subjects is the topic of Complex Numbers. Now, given that teachers tend to teach how they were taught, in cases where they have to teach a topic which they were never taught begs the question: how can teachers with little or no prior encounters with complex numbers be supported to teach the topic of complex numbers for conceptual understanding? My hypothesis is that understanding complex numbers conceptually requires viewing complex numbers relationally. That is, both algebraically and geometrically. Also, that Transformations Geometry provides us with, not only a way of visually interpreting complex numbers, but also with a language that can support learners’/students’ conceptual understanding of complex numbers. I argue, firstly, that the South African Mathematics Curriculum was designed so that the topic of transformations developed up to Grade 9 can serve as a language that can mediate this conceptual understanding in the Further Education and Training (FET) phase. Secondly, that for learners/students to develop a conceptual understanding of complex numbers, pre-service and in-service teachers must be enabled to support learners/students towards coordinating geometric and algebraic interpretations of complex numbers and the interactions between them through the language of transformations.

ABSTRACT. In the year 2019, the world was struck with a global Covid-19 pandemic which led to many universities, including University of Cape Town (UCT), in South Africa to adapt an online way of teaching and learning for both tutorials and lectures. Online learning can take the form of asynchronous and synchronous tutorials and lectures. In this study both categories were used where MS Teams was used for synchronous tutorials and the Learning Management System (LMS) Vula was used asynchronously. There are benefits and challenges linked to online tutorials but these differ depending on the contextual circumstances of implementation (Motaung et al, 2021) .online tutorials can create a platform where students can engage with tutors and content outside of the classroom, these include and are not limited to engaging with materials in their own time, being able to access the content anywhere engaging with peers remotely which in turn makes learning more flexible and relaxed. This study investigated the technological and pedagogical experiences of an online tutorial system in an undergraduate Quantitative Literacy course at UCT. The research methodology employed in this study is design-based research (DBR) (Doig & Groves, 2011; Hunter & Back, 2011). In this study the tutors implemented and improved the way they tutored online by assessing the tutorial session and implementing new ways to improve the tutorial session. A qualitative approach was used to make sense of the experiences from the tutors by using a questionnaire and recorded online tutorial sessions. The concerns raised by tutors contained both technical and pedagogical concerns with online learning. These concerns were poor attendance, low participation, network/connectivity, and low preparation from students. As a result, Synchronous tutorial sessions were found to be more engaging than non-asynchronous tutorials but still lacked the benefits of observing struggles among students that face-to-face tutorials offer.

ABSTRACT. This study investigates aspects of undergraduate mathematics students’ learning in their first encounter with Group Theory. Research in the learning of Group Theory proves significant, since various studies have reported that novice students consider it as one of the most demanding subjects in their syllabus. In particular, this qualitative study investigates undergraduate mathematics students’ responses to two mathematical tasks on equivalence relations. For the purposes of this study there has been used the Commognitive Theoretical Framework. Analysis suggests that these students’ first encounter with equivalence relations is challenging. There have emerged three categories of errors and inaccuracies in students’ solutions. The first category of errors is related to the proof of the size of equivalence classes, which is predominantly due to incomplete object-level learning of the form and structure of equivalence classes as well as the notion of bijection. The second category includes several errors regarding the proof of symmetry and transitivity. The third category is related to the distinction between the elements of the set X and the elements of the group Sym(X), when these coexist in the same context.

ABSTRACT. Mathematics and Statistics are disciplines that many would say are complicated formulae or funny symbols and it is all too confusing for many students. Educators have tried to reduce this confusion by allowing students to have their own notes during tests and examinations. According to Charles P. Corcoran, in 2020, the use of notes had no significant effect on the learning outcomes, and he noted there was very little literature available on this topic. At Curtin College, which is pathway college to university, students hope to obtain entry to the university at either first or second year. Most classes during the trimester are interactive, with small groups of students collaborating and creating solutions on whiteboards, facilitated by the educator. Last trimester, a reverse in lecture style was trialled with a cohort of engineering students at Curtin College. The minor twist is students will create notes suitable for the upcoming assessment on whiteboards, with only minor prompts by educator, then use the notes to answer a question in class and finally revise and complete notes. There are many benefits for both student and educator, not only how time-consuming writing notes can be for the student but for the educator you can see the misconceptions and failure to recognise key points. It is hoped that this experience may help students provide better notes in all disciplines. The paper will elaborate on the experience in the classroom setting.

ABSTRACT. Mathematics is traditionally considered necessary for engineering courses. Over the last three decades, the mathematics requirements for entry into engineering programmes has steadily weakened in Australia. Further, the mathematics component of engineering programmes has progressively decreased. This research aims to investigate the following two questions. First, is mathematics a barrier for students to complete an engineering programme? And second, is performance in mathematics associated with performance in engineering? We identified the significant factors associated with weighted average mark and the completion status of engineering studies at both an undergraduate level and a Masters level. Of particular interest was the students' mathematical background. Furthermore, a survey of students enrolled in engineering at the University of Western Australia was conducted to obtain more in depth views of student attitudes and perceptions towards how mathematics has affected their engineering studies. Binary logistic models were fitted to the survey data. Additionally, focus group interviews were conducted to gain insight on student perspectives regarding the effectiveness of mathematics teaching in engineering. The results are discussed in relation to the importance of mathematics and statistics for the engineering curriculum.

IJMEST special issue: https://doi.org/10.1080/0020739X.2023.2256319

ABSTRACT. The accelerated adoption of technology in mathematics education has been a focal point in recent educational discourse. As educators increasingly recognise technology's transformative potential, its integration offers a promising avenue to enrich the academic journey. Yet, seamlessly weaving technology into teaching presents multifaceted challenges. This reflective study delves into technology's pivotal role, especially under the unique conditions precipitated by the Covid-19 pandemic. Engaging in thoughtful introspection, the authors traverse the experiences of participants navigating emerging technologies and pedagogical innovations. The Technological Pedagogical Content Knowledge (TPACK) framework underpins this exploration. It emphasises the vital expertise educators must cultivate to holistically integrate technology into their instructional strategies. Drawn from a diverse pool, the study encapsulates perspectives from 20 first-year mathematics educators affiliated with Universities of Technology, Comprehensive Universities and Traditional Universities in South Africa. The study employed a qualitative research method to gather and analyse data. The target population consisted of lecturers responsible for instructing first-year mathematics courses across 26 universities in South Africa. To collect data, the researchers utilised semi-structured interviews, a technique that allows for open-ended responses and provides the flexibility to explore emerging themes. Following the data collection phase, the information was systematically analysed using ATLAS.ti software, a renowned tool for managing and examining qualitative data. Preliminary findings accentuate technology's indispensable role in amplifying student engagement and fortifying learning outcomes. Moreover, the study sheds light on the tangible benefits and potential of hybrid or online teaching models. The insights gleaned serve as a beacon for educators striving to seamlessly fuse technology into their teaching matrix. To culminate, the onus is on educational institutions to ensure equitable technological access. This entails prioritising financial incentives, offering intensive training, and fostering a robust infrastructure to buttress hybrid pedagogical models, ensuring academic continuity amidst disruptions.

ABSTRACT. A literature review is one of the major pillars of academic work, as it is the basis upon which scholars establish context for their research, identify gaps in the literature, formulate their own research problems and objectives, and develop the research design and data collection tools (Benbellal et al., 2021). ATLAS.ti is a powerful computer-assisted qualitative data analysis software (CAQDAS) that facilitates analysis of textual and media data in any discipline and for diverse research topics. In addition to assisting with analysis of data, the tools of ATLAS.ti can also be applied to the literature review process particularly when access to library and university facilities is limited due to the global challenge of COVID-19. The workshop consists of both instruction and hands-on exercises in ATLAS.ti. By the end of the workshop, it is hoped that participants will have the conceptual and practical tools necessary to use ATLAS.ti to assist organise, manage and analyse literature related to their current or future research projects.

ABSTRACT. Engineering programs in South African universities struggle with large diverse classes; diverse in student abilities, prior knowledge and learning styles. This results in some students failing courses, feelings of alienation and low throughput rates. Academic development initiatives, like extended degree programs, aim to redress but face hurdles due to students' hidden abilities and backgrounds. To enhance teaching and learning, understanding the current environment and designing effective interventions is crucial.

This study explores what pedagogical views educators hold of teaching for promoting learning in the first year of the degree programme at university, how students experience learning events designed to promote learning in an extended engineering degree programme and what the structures or mechanisms are that influence student learning in this program?

Grounded Theory methodology guided this research, which encompassed two cycles involving educators (including distinguished teacher award recipients and extended degree educators) and two cycles involving students (2nd year mechanical engineering students and 1st year mathematics students, both in the extended degree program). The theory is presented using qualitative systems dynamics modelling, shedding light on potential structures that align with empirical findings.

Two primary findings emerged: Firstly, educators can enhance student-focused teaching through responses to diversity, employing care and effort in teaching, fostering quality engagement opportunities, and creating meaningful learning experiences. Secondly, students experiencing alienation tend to either disengage or proactively seek or establish learning communities, ultimately improving engagement quality. The identified structure highlights the significance of care in teaching and the role of learning communities as key drivers and intervention focus points.

ABSTRACT. This report is about an initiative to provide university bridging students who are mainly looking towards a non-Science pathway, with a mathematics course. Every student on the bridging programme must pass at least one mathematics course, so prior to 2019, most students took the standard mathematics course, Maths 91F, which is pitched at upper secondary school levels. The increase in student numbers on the programme has resulted in a much broader range of entry abilities, with several now being pre-assessed as not (yet) being at the 91F level. With the 91F course no longer meeting the needs of candidates at the lower end of the entry assessment, a new course, Maths 89F has been offered. Maths 89F or Mathematics for Arts, was created so these students could meet the mathematics requirements, but also to provide them with contexts that will connect them to some mathematics. Themes have included Problem solving, Probability and false positives and negatives, local political systems (including MMP), environmental issues, and loan sharks. Students have been assessed with group tasks, assignments, a video assignment, and a Final Test. What is evident over the five years 89F has run is the need to also provide the cohort with several calculated proportional thinking interventions, within and without contexts, with an eye to future academic and adult numeracy needs.

ABSTRACT. Languages and literacies are central to meaning making of knowledges, cultures, identities, social relations, and values. In undergraduate mathematics, we traditionally value formal mathematics register, written and symbolic modes, and discourses of defining, proving, etc., in the medium of instruction of the institution. The University of Cape Town (UCT) is an elite, historically ‘white’, English-medium university, in a diverse, multilingual context. UCT mathematics students report difficulties following lectures in English, asking questions, understanding assessment items, and linking school and university mathematics, with implications for their performance, and sense of belonging and being (le Roux, 2017; Shay et al., 2020). In this postcolonial context, mathematics educators are challenged to answer questions of access and transformation: How to use languages and literacies towards equity of student access to and success in a locally relevant curriculum in a globally connected world?

I offer conceptual tools that educators can use for thinking about languages and literacies for mathematics curriculum design and pedagogy towards tackling this challenge. This was developed using scholarship and policy at UCT and elsewhere (e.g., le Roux et al., 2022; Prediger & Hein, 2017; DHET, 2020). Languages are conceptualised broadly as registers, modes, genres, discourses, language codes, dialects, and accents. Literacies are actions with languages in context: writing, reading, talking, listening, drawing, and using information and technologies. A student uses these: to access mathematics (in a lecture, textbook); to learn mathematics in formal and informal spaces; to communicate understanding in assessments; to demonstrate as an outcome their use of formal written mathematics. I elucidate these tools with examples from a first-year mathematics course at UCT. However, they can be adapted for other contexts characterised by asymmetries in what knowledges, languages and literacies are valued, and in which mathematics lectures and students bring diverse language and literacy experiences to the classroom.

ABSTRACT. This paper is the result of a reflection on students’ inverse function understanding, in pursuit of a teaching sequence to broaden and challenge their limited understanding of inverse functions. Although the commonly used procedure of switching-and-solving is effective for determining and sketching a one-to-one function and its inverse function on the same set of axes, the deeper conceptual inverse function understanding needed for contextualized problems is problematic. The contextualized problem described in this paper is a simple linear economics model of the markets. These students are first-year economics students, registered for the service module MTHS112 (Mathematical Techniques for BCom) at the North-West University. This paper reflects on students’ prior knowledge of inverse functions and unpacks the influence of their limited understanding of a contextual problem. This is augmented by examples from a teaching sequence. The paper concludes with directions for future research.

ABSTRACT. Mathematics and physics are two disciplines separated by a common language. Since the 1990s, we have been developing curricular materials that attempt to bridge this gap, as part of the Paradigms in Physics and Vector Calculus Bridge projects (Paradigms Team, 2019–2023), leading also to the development of an online textbook (Dray and Manogue, 2009–2023). Most of our work focuses on the transition from multivariable calculus to upper-division physics courses such as electromagnetism.

Vector line integrals are an important mathematical object occurring in multiple subdisciplines throughout upper-division physics. Our research question is to compare standard textbook treatments of vector line integrals in lower-division mathematics and physics so as to identify and compare the disciplinary approaches using representational transformation diagrams (Bajracharya et al., 2019). We identify two principal approaches that are loosely correlated with these two disciplines, and which differ primarily in how they treat the dot product. These textbook approaches are compared to existing characterizations for (single-variable) integration in the theory literature (Ely & Jones, 2023), including our characterization of chop, multiply, add. Finally, we present a hypothetical learning trajectory for vector line integrals (Dray & Manogue, 2023) designed to scaffold student acquisition of rich concept images that bridge the approaches of these two disciplines. We conclude with a discussion of possible implications for the teaching of single-variable calculus.

ABSTRACT. Designing and implementing purposefully developed tasks have been linked to students’ understanding of mathematical topics. We report on design principles and intentions that guided the development of inquiry-oriented tasks. Through a qualitative analysis of the tasks, we identified seven themes that encapsulated these design intentions: Provide Review; Introduce New Content; Provide Guidance; Prompt Graphing and Visualisation; Prompt Generalisation; Foster Mathematical Creativity; and Prompt Reflection. In the discussion, we align these themes with existing literature and recognise an intersection between designing inquiry-oriented tasks, creativity-fostering tasks and reflective assignments. We additionally provide some recommendations for practitioners and report on students’ feedback on these tasks.

IJMEST special issue: https://doi.org/10.1080/0020739X.2023.2257699

ABSTRACT. This article explores the conceptual challenges that engineering students encounter with double integrals in a vector calculus course. Drawing on previous literature and utilising APOS (Activity- Process- Objects- Schema) as a theoretical framework, this study investigates the difficulties that students face in understanding and applying double integrals. By analysing responses to a written test that assesses students’ conceptual understanding of double integrals and student responses in interviews, this research sheds light on specific areas where students struggle, such as curve sketching, identifying regions of integration and changing the order of integration. Swapping the order of integration in double integrals is the focus of this research study. The findings of this study have the potential to contribute to the enhancement of teaching and learning strategies for double integrals to better support engineering students’ conceptual comprehension and proficiency in this important mathematical concept.

IJMEST Special issue: https://doi.org/10.1080/0020739X.2023.2265948

ABSTRACT. This paper discusses findings from an ongoing study investigating mental mechanisms involved in the conceptualisation of linear transformations from the perspective of Action (A), Process (P), Object (O), and Schema (S) (APOS) theory. Data reported in this paper came from 44 first-year linear algebra students’ responses on a task regarding the range of a linear transformation. Our analysis revealed parallels between Levels/Stages of the range concept and the use of representations of matrix multiplications. More importantly, these representations appeared to have been the enablers of transitions from lower to higher APOS Stages for the range. Conversely, mental mechanisms employing other means showed little to no progressions, some, furthermore, revealed faulty knowledge structures.

IJMEST special issue: https://doi.org/10.1080/0020739X.2023.2260790

ABSTRACT. The concept of limit is a common topic in mathematics education papers and books (see for example Juter (2005), Kidron & Zehavi (2002), Szydlik (2000), among others). It is not so common to find works about one-sided limits, however, this fact does not make this topic less important (Fernández-Plaza et al., 2015). Many students - and not a few teachers - see limits as a theoretical tool, very useful for the definition of other concepts such as derivatives, integrals, and series, but not as a useful tool for mathematical modeling and applications. After analyzing some elements of the theoretical framework of modeling and applications, we will see some examples that show how limits and one-sided limits are also important due to their applications to other disciplines. Particularly, we will analyze examples related to chemistry, pharmacokinetics, physical chemistry, and reactor design, among others (Martinez-Luaces, 2017). Finally, examples of non-convergent series will be analyzed, and also how through the concept of limit it is possible to assign a value to those series, with applications in quantum mechanics and optics, among other disciplines (Herman, in press). Based on the above, some suggestions are proposed, mainly oriented to mathematics courses for other sciences and engineering.