ALBERT

Description: In this paper we explore the attitudes of under-privileged secondary school pupils in Ireland towards mathematics and investigate the impact of attending a 4-week engagement programme on these attitudes. The pupils involved in this research attended schools recognized by the Department of Education & Skills as socio-economically deprived. Pupils attending these schools, known as Delivering Equality of Opportunity in Schools (DEIS), are 40% less likely than their counterparts in non-DEIS schools to pursue mathematics at a higher level in state examinations (Smyth, E., Mccoy, S. & Kingston, G., 2015, Learning From the Evaluation of DEIS. Dublin: Economic and Social Research Institute). However, little research has reported on these pupils’ experiences of and attitudes towards mathematics at senior secondary level. An engagement programme entitled ‘Maths Sparks’ was purposefully designed for secondary pupils from DEIS schools, with the aim of positively influencing their attitudes towards and confidence in mathematics. The programme consisted of weekly out-of-school workshops exploring extra-curricular mathematics topics, designed and delivered by undergraduate mathematics students. Questionnaires were utilized to evaluate pupils’ attitudes towards mathematics before and after their participation in the programme. Despite its relatively short time frame, qualitative and quantitative analysis suggests an increase in participating pupils’ attitudes towards, enjoyment of and self-confidence in mathematics due to their participation in the programme. Findings also suggest that while these pupils liked the subject of mathematics, their experience of learning the subject in school was not always positive and was sometimes hindered by the absence of higher-level mathematics as an option in school. The high-stakes examination content and teachers’ beliefs in the ability of their students also sometimes negatively impacted learners’ intentions to pursue mathematics at a higher level. Findings suggest that longitudinal mathematics engagement programmes, which focus on problem solving, involve extra-curricular mathematical concepts and are presented by undergraduate mathematics students, may provide a valuable way of positively impacting pupils’ intentions to pursue the subject.

Description: This study focuses on the concept of pairability as a fundamental metaphorical concept in the Cantorian set theory regarding comparison of infinite sets. In this theory, pairability appears in a hierarchical manner of generating and developing as a one-to-one correspondence by arrows in finite, and then in infinite states, a bijection map through an explicit formula and the concept of cardinality. Adapting these hierarchical components of pairability and the APOS theory, about description of constructing and understanding a mathematical concept in a hierarchical manner, this study examines the construction of the concept of pairability in the individuals’ mind. In this way, their imaginations of pairability and practical performances in different situations will also be surveyed. In so doing, a total of 20 mathematics teachers and university lecturers holding at least an M.Sc. degree in mathematics were chosen. To collect the data, interviews were conducted in which the participants not only answered questions about the concept of the various types of pairability but also compared infinite countable sets in different situations. Our findings revealed that a bijection map via an explicit formula was the individuals’ dominant conception of pairability and most of the incorrect answers were related to unsuccessful attempts to recall a formula or method as the only possible way, and the encapsulated concept of cardinality was used less frequently in practice. Therefore, there was not a total schema of actions, processes and objects of the concept of pairability in the individuals’ mind.

Description: Mathematics learning support centres (MLSC) are widely established and evaluated in English-speaking countries (such as the UK, Ireland and Australia). In most of these countries, several national surveys on MLSCs exist. They give an overview of the number of MLSCs as well as their characteristics in these countries. In Germany, there is a lack of studies on MLSCs and the landscape of MLSCs has not been described yet. This article presents basic information concerning counts of MLSCs and their characteristics at universities. Based on a three-step approach of analysing university homepages and additional personal contact via email or phone calls, we gathered typical MLSC features (e.g. staff quantities and qualification, opening and support hours, supported study programmes). We analysed 190 universities from a web-based register on study programmes. In total, we found 61 MLSCs located at 51 German universities. Another 16 support centres were specialized on mathematics didactics, which means they focussed on didactical and methodological support for preservice teacher students and often provided different teaching materials. Thirty-eight centres were located at universities (62.3%) and 23 MLSCs at universities of applied sciences and comparable universities (37.7%). The MLSCs were different in their sizes of staff and opening hours, and both the numbers of staff and the service hours differed greatly. The student groups MLSCs at German universities focus on differ concerning characteristics like study programme or semester. We will provide the main categories describing these groups. We seek to answer research questions concerning the characteristics of MLSCs in Germany and discuss the results compared to international findings. This information is useful for further international collaborative research, for example a standardized international survey. From a national perspective, these findings support networking and collaborations between the MLSCs as well. As some German MLSCs are facing financial cuts, these results might help in gaining additional funding.

Description: In England, a relatively new set of post-16 qualifications has been developed under the umbrella term ‘Core Maths’, with a focus on the application of mathematics in context, including the kinds of mathematics needed to support other subjects, to provide a sound basis for the mathematical demands of higher education and employment and to develop problem-solving skills for use in life. The UK government has an ambition that all post-16 students should be studying some mathematics, and Core Maths was designed in part with this aspiration in mind. In this paper, longitudinal questionnaire data from over 100 Core Maths students in 13 case study institutions are analysed to measure students’ views of teaching as transmissionist, their mathematical dispositions and self-identification and how these change over a year of studying Core Maths. We find some evidence that pedagogy in lessons is perceived as being less transmissionist than it was in school mathematics pre-16. There is also some evidence of a negative change in students’ mathematical dispositions over a year of Core Maths. We conclude that supporting teachers in embedding new pedagogical approaches remains a challenge and that this issue could inhibit the growth of new qualifications like Core Maths.

Description: School-university transition in mathematics is of global concern, with multiple cognitive, social and affective disjunctures evidenced. Access to, successful participation in and retention on, competitive mathematically intensive degree courses remain particular challenges in England, especially for disadvantaged young people with high mathematical aptitude. One response has been to establish mathematics specialist schools aimed at such students aged 16–18. Early cohorts have achieved encouraging progression to and through such university courses, but more qualitative and longitudinal outcomes have been less well evidenced. The reported study harnessed a student lens and documentary scrutiny to analyse the contribution to building for successful transition of the particular approaches used. Data suggest that the model adopted has initially supported transition to target degree courses well. I relate the findings to known transition challenges in the global issue of successful passage into and through university mathematics education. I argue many of those are in principle transferable to other post-16 contexts. The study therefore offers evidence suggesting broadly applicable specific strategies that can begin to address widely problematic disjunctures in transition.

Description: This paper derives from a study which main purpose was to investigate how a group of adults with low schooling can have access to powerful mathematical ideas when working with activities that involve the use of technology resources and that take into account the adults’ previous experience with mathematics. Specifically, adults’ previous experience with area calculation was considered. Principles of the Theory of Didactical Situations (TDS) formulated by Brousseau guided the study design, and Pick’s theorem was recreated in a dynamic digital setting, with which it is possible to calculate the area of regular and irregular polygons. In this approach, intuitive notions of area and perimeter are resorted to, seeking to promote the experience with powerful ideas such as ‘the generality of a method’, ‘realizing the existence of different methods used for one and the same end’ and ‘realizing that each method possesses advantages and limitations’. Analysis of interview protocols from three noteworthy cases (which include both adults’ work in the digital setting and their discussions with the researcher) suggests the presence of powerful underlying mathematical ideas, such as the idea of generality and the power of a method and the features of the constituent elements of a geometric figure that are involved in calculating its attributes, attributes such as area.

Description: Undergraduate research increasingly features in university mathematics degrees. Despite this, research papers are used infrequently in mathematics teaching, and this is especially the case for first-year undergraduates. Mathematical subjects are more likely than other STEM disciplines to pinpoint cognitive difficulty as the principal reason for not exposing undergraduate students to research papers. In this paper, we test whether first-year students can engage effectively with research papers. We describe an intervention that exposes first-year, first term undergraduate students to current research in probability and statistics by asking them to read a research paper and summarize it for a general readership following an interview with the paper’s author. Our findings show that the activity introduced students to new fields of knowledge and helped to develop a clearer understanding of scientific process, leading to a heightened sense of personal satisfaction at engaging closely with current research. We argue that structured reading of research papers can lead to productive and rewarding engagement with difficult content, recent and current research and with research processes and that this should make us reconsider the role of research papers in the undergraduate mathematics curriculum.

Description: Mathematics undergraduates often encounter a variety of digital representations which are more idiosyncratic than the ones they have experienced in school and which often require the use of more sophisticated digital tools. This article analyses a collection of digital representations common to undergraduate dynamical systems courses, considers the significant ways in which the representations are interconnected and examines how they are similar or differ from those students are likely to have experienced at school. A key approach in the analysis is the identification of mathematical objects corresponding to manipulative elements of the representations that are most essential for typical exploratory tasks. As a result of the analysis, augmentations of familiar representations are proposed that address the gap between local and global perspectives, and a case is made for greater use of isoperiodic diagrams. In particular, these diagrams are proposed as a new stimulus for students to generate their own explorations of fundamental properties of the Mandelbrot set. The ideas presented are expected to inform the practice of teachers seeking to develop visually rich exploratory tasks which pre-empt some of the issues of instrumentation that mathematics undergraduates experience when introduced to new digital tools. The overarching aim is to address significant questions concerning visualization and inscriptions in mathematics education.

Description: This paper presents ‘expert opinions’ on what should be taught in a first-year linear algebra course at university; the aim is to gain a generic picture and general guiding principles for such a course. Drawing on a Delphi method, 14 university professors—called ‘experts’ in this study—addressed the following questions: What should be on a first-year linear algebra undergraduate course for engineering and/or mathematics students? How could such courses be taught? What tools (if any) are essential to these two groups of students? The results of the investigation, these experts’ opinions, mainly concern what should be in a linear algebra course (e.g. problem-solving and applications) and what students should be able to do. The experts also emphasized that certain theoretical aspects (e.g. proofs, abstract structures, definitions and relationships) were more important to mathematics students. There was no real consensus among the experts on teaching methods or the use of digital tools, but this lack of consensus is interesting in itself. The results are discussed in relation to extant research.

Description: The MATH taxonomy classifies questions according to the mathematical skills required to answer them. It was created to aid the development of more balanced assessments in undergraduate mathematics and has since been used to compare different assessment regimes across school and university. To date, there has been no systematic investigation of the reliability of the taxonomy when applied by multiple coders, and it has only been applied in a limited range of contexts. In this paper, we outline a calibration process which enabled four novice coders to attain a high level of inter-rater reliability. In addition, we report on the results of applying the taxonomy to different secondary school exams and to all assessment questions in a first-year university mathematics module. The results confirm previous findings that there is a difference between the mix of skills assessed in school and university mathematics exams, although we find a notably different assessment profile in the university module than in previous work. The calibration process we describe has the potential to be used more widely, enabling reliable use of the MATH taxonomy to give insight into assessment practices.