Post-16 Further Mathematics blended learning: learner self-regulation, mathematical resilience and technology

This paper reports on a study set in Wales where the Further Mathematics Support Programme Wales supports the provision of an advanced qualification in mathematics for 16- to 18-year-old students with courses delivered in reduced teaching time. The study aimed to understand how the students experienced the Further Mathematics (FM) courses which are delivered either face-to-face or online and, more generally, to negotiate a place of alternative forms of delivery in post-16 mathematics curriculum. Sixteen students, eight of whom studied through the online course, were interviewed; overall, although they found the course challenging both in terms of the content and relatively limited teaching time, they enjoyed it and appeared to perceive benefits from taking the course. Most volunteered ‘tips’ about coping with the challenges of the course and the tips can be seen as strategies of self-regulation. Self-regulation strategies were reported more strongly by the students taking the course online than those attending face-to-face classes. In view of the evidence of technology creating new learning environments perceived as advantageous by students, it is hypothesized that introducing blended learning as part of post-16 mathematics curriculum could be beneficial. Improving learner self-regulation is discussed as means of improving access to FM. Other findings included the importance of support from peers, parents and schools and gender differences

Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\R$ and $C>0$. We provide a complete characterization of the set of $(p,\sigma)\in\R^2$ such that the equation has no positive (super-)solutions, depending on the values of $A,B$ and the principle Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmen--Lindel\"of type comparison arguments and an improved version of Hardy's inequality in cone--like domains.Comment: 30 pages, 1 figur

Further Mathematics, student choice and transition to university: part 1 - Mathematics degrees

The transition from studying mathematics at school to university is known to be challenging for students. Given the desire to increase participation in science, technology, engineering and mathematics subjects at degree level, it is important to ensure that the school mathematics curriculum is providing suitable preparation for the challenges ahead, and yet remains both accessible and popular. This two-part study investigates student choices of studying the post-16 A-level Mathematics and Further Mathematics qualifications in the UK and their impact on the transition from school to university mathematics. Student opinions were accessed via a survey of undergraduate students and also individual interviews. This first part of the study considers the responses of mathematics undergraduate students and finds that both those who studied Further Mathematics and those who did not perceive studying Further Mathematics as advantageous for their degree courses. However, the advantages identified mostly relate to the familiarity with topics, while students still feel unprepared for studying more abstract and proof-based mathematics. The study found that some factors which may be beneficial for transition currently lie outside the mainstream school mathematics syllabus and include studying through blended learning provided by the Further Mathematics Support Programme, practicing more advanced extension exam papers and attending university outreach events. The choice of Further Mathematics is found to be influenced by the attitudes of the students, their teachers and their parents, to both mathematics as a subject and to Further Mathematics as a qualification as well as student perceptions of Further Mathematics and their plans in terms of degree and university choice

Further Mathematics, student choice and transition to university: part 2—non-mathematics STEM degrees

This is the second paper reporting the results of a study investigating student choices of optional post-16 advanced-level (A-level) Mathematics and Further Mathematics qualifications in the UK and their impact on the transition from school to university mathematics. Here, the opinions of non-mathematics Science, Technology, Engineering and Mathematics (STEM) undergraduate students (all of whom had previously studied A-level Mathematics) were accessed via a survey and individual interviews. The study found that Further Mathematics qualifications are perceived as advantageous for non-mathematics STEM degrees by students once they are at university but not when making A-level choices. While the students often perceived mathematics positively, this appears to influence the choice of A-level Mathematics but not Further Mathematics. The lack of support from teachers and parents, the lack of perceived utility of Further Mathematics qualifications and a perception that Further Mathematics is only useful for studying a mathematics degree could all be factors affecting the uptake of Further Mathematics. The identified perceived impact of Further Mathematics on the university transition is linked to studying more pure mathematics which may give students a better understanding of how to apply mathematics in the context of their degree. Some comparisons between the findings in Parts 1 and 2 of the study are included which suggests that the Further Mathematics qualification is better serving students intending to study a non-mathematics STEM degree rather than mathematics itself

Engagement and online mathematics enrichment for secondary students

Online may not be the ideal format for a mathematics enrichment event, but in some circumstances, it may be the only option available. This article considers a mathematics enrichment programme consisting of a series of masterclasses which were held live online for secondary students in the UK during the Covid-19 pandemic. The series of masterclasses were part of the Royal Institution of Great Britain’s Mathematics Masterclass Programme which runs annually across the UK. In this study, we investigate how and to what extent students were able to engage with this series of online masterclasses. Learner engagement is researched through in-session observations, student work, attendance data, participant feedback and interviews. While the online masterclass series lost some of its traditional in person features, such as hands-on live social interaction and a university environment, it appeared that the participants perceived the online sessions as interactive enabling them to both enjoy the sessions and enjoy learning mathematics in the sessions. The evidence found suggests that the participating students could engage behaviourally, emotionally and cognitively in online mathematics enrichment. However, constructing mathematical knowledge in online sessions can be difficult for some students and social interaction may need to rely on existing social groups among school friends

Choosing Further Mathematics

'Education in the UK is failing to provide the increases in the numbers of school-leavers with science and mathematics qualifications required by industry, business and the research community to assure the UK's future economic competitiveness' (The Royal Society, 2008: 17). Furthermore, the proportion of students in Wales following mathematics courses post 16 is lower than in England (GSR, 2014). In particular, although the situation has improved, fewer students in Wales choose to study further mathematics (FM). This paper explores the reasons for student choices in mathematics and FM in order to make recommendations about how to increase participation. Phase one of the study used a questionnaire to access the opinions of students studying mathematically based courses in sixth forms and colleges to explore the reasons behind their choices and the factors influencing their progression or otherwise in mathematics. In phase two, small focus groups of students in selected schools and colleges were interviewed to enrich the questionnaire data and provide further insight into their decisions. The study identified a lack of information from peers, siblings, parents and teachers about FM as a factor restricting choice. Current models of delivery contribute to the false perception that FM is harder than mathematics and only suitable for the most talented mathematicians. We suggest: developing teachers' knowledge and skills so that whenever possible students can be offered FM as a fully timetabled subject; promoting FM to parents; and establishing student champions to encourage participation

Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-\Delta_p u-\frac{\mu}{|x|^p}u^{p-1}=\frac{C}{|x|^{\sigma}}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $\mu\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,\sigma)\in\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the $p$-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Pr\"ufer-Transformation.Comment: 34 pages, 1 figur