J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T. Kobayashi [T. Kobayashi, Casson-Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553-573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] proved the existence of arbitrarily high distance Heegaard splittings.In this work we explicitly define an infinite sequence of 3-manifolds {M n } via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of M n is at least n.Moreover, we show that π 1 (M n ) surjects onto π 1 (M n−1 ). Hence, if we assume that M 0 has nontrivial boundary then it follows that the first Betti number β 1 (M n ) > 0 for all n 1. Therefore, the sequence {M n } consists of Haken 3-manifolds for n 1 and hyperbolizable 3-manifolds for n 3.

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The original version of this article unfortunately contained an incorrect text. Monash University and the University of Melbourne should be "two Australian universities."Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Lecture capture (LC), the process of recording face-to-face lectures for future viewing, has become a common technology in Western universities in the twenty-first century, yet research on its effectiveness has lagged behind its implementation. Despite the rapid, widespread implementation, research regarding the impact LC has on pedagogy and student attainment is limited and not conclusive in its findings. It is still unclear if there is a causal or a correlated relationship between attainment and usage of LC. This systematic review sought to collate and compare the current literature on the efficacy of LC in tertiary mathematics education and provide practical advice for institutions that use or plan to use LC. The literature is consistent in the opinion that students and administrators positively view LC for its utility and flexibility despite the moderately strong evidence that most institutions face attendance drops. However, most students do tend to see attending lectures/watching recordings as an “either-or.” The literature predominantly reports a negative association between attainment and the use of LC as a substitute to live lectures. The proportion of students who choose to skip live lectures has steadily increased over the last decade as the student campus culture adjusts to LC. Within this group, LC is used imperfectly, providing false benefits and promoting surface learning strategies. There is evidence that regular use of LC by this large group of students may diminish the quality of their learning. We offer research-informed, evidence-based recommendations to mitigate the unplanned and counterproductive impact of LC implementation.

Our study addresses a systemic issue facing higher education – a lack of rigorous educational research alongside new technology-assisted ways of teaching and learning. The issue highlights the disciplinary disconnect as many academics do not research outside their discipline, yet are tasked with educational modernisation through trying out new educational technology. Addressing this issue, we present our conceptual framework, the course transaction space (CT-space), and use it to analyse the impact of an intervention we designed that involved the use of regular online pre-lecture quizzes in a university mathematics course. The aim of the intervention was to optimise the effect of distributed (spaced) practice on long-term retention. Our findings suggest that a relatively small change in course instruction can improve the efficiency and effectiveness of educational exchange. Our analyses of data from multiple sources provide evidence that our intervention resulted in a sustained increase in the frequency of students’ engagement with the content, increased attendance of lectures, and improved grades. Additionally, we discuss the impact of our intervention on the quality of student engagement with reference to competence related beliefs and self-efficacy. Finally, we discuss how our intervention can be used in other contexts for supporting an evidence-based approach to teaching and learning. Implications for practice or policy  For teachers designing an intervention with the aim of improving students’ learning engagement during a course of tertiary study, we advise incorporating a series of frequent low stakes online quizzes with low level of difficulty. For students, these will act as an incentive, enabling improvement in the frequency of their learning engagement and its quality. The course transaction space (CT-space) model can be used to explore and analyse the impact of a variety of interventions introduced in tertiary courses through the lens of engagement.