Search

We describe an organising principle for online learning materials we term coherently organised digital exercises and expositions. Larger in scale than individual lessons but smaller than a programme of study, this innovation in instructional practice is increasingly guiding our thinking in the development of university mathematics courses. Essentially we have taken the book and put it inside automatically assessed online quizzes. In doing this we embrace the potential provided by new technology to implement evidence-based practices such as spaced retrieval practice. This paper discusses details of this innovation, and how we have implemented it. On the basis of these experiences, we believe this innovation has the potential to change the model of education for university mathematics courses in substantial and non-trivial ways.

This paper describes the collaborative development of an agenda for research on e-assessment in university mathematics. We adapted an established approach to develop the agenda from the contributions of 22 mathematics education researchers, university teachers and learning technologists interested in this topic. The resulting set of 55 questions are grouped into 5 broad themes: errors and feedback, students, design and implementation, affordances, and mathematical skills. This agenda gives the framework for a programme of research aligned with practical concerns, that will contribute to both theoretical and practical development.

We describe the design and evaluation of an innovative course for beginning undergraduate mathematics students. The course is delivered almost entirely online, making extensive use of computer-aided assessment to provide students with practice problems. We outline various ideas from education research that informed the design of the course, and describe how these are put into practice. We present quantitative evaluation of the impact on students' subsequent performance (N = 1401), as well as qualitative analysis of interviews with a sample of 14 students who took the course. We find evidence that the course has helped to reduce an attainment gap among incoming students, and suggest that the design ideas could be applied more widely to other courses.

In this paper we investigate undergraduate mathematics students' conceptions of rigour and insight. We conducted comparative judgement experiments in which students were asked to judge different proofs of the same theorem with five separate criteria: rigour, insight, understanding, simplicity and assessment marks. We predicted, and our experiment found, that rigour is a reliable construct. We predicted that insight is also a reliable construct but asking students to judge on the basis of ``which proof gives you more insight into why a theorem is true'' did not result in a reliable judging scale. Our analysis suggests two distinct dimensions: rigour and marks contribute to one factor whereas simplicity and personal understanding relate to a second factor. We suggest three reasons why insight was related almost equally to both factors. First, while comparative judgement was suitable for assessing some aesthetic criteria it may not be suited to investigating students conceptions of insight. Second, students may not have developed an aesthetic sense in which they appreciate insight in ways which are regularly discussed by mathematics educators. Lastly, insight may not be a coherent and well-defined construct after all.

This document sets out a model for focussed small-group workshops in mathematics. The purpose of the model is to support students to complete challenging (for-credit) assignments to a high standard. The model consists of a pattern of activities where students work on problems before and after a live session. The preparation is a way to maximise the benefit of live sessions. We begin by describing the standard model that has been used for small-group workshops for several years in the School of Mathematics at The Univerisity of Edinburgh before going on to describe the focussed model in detail. Due to the COVID-19 pandemic, we assume the model is being implemented entirely online. However, there is no need for the model to be implemented online; we think it would work just as well after a return to on-campus teaching.

In this paper, I investigate whether two-stage exams aid learning in undergraduate mathematics, as measured by students’ performance on subsequent exams. In a two-stage exam, students complete the exam individually then form into groups to solve it again, with grades based on a combination of the two stages. Previous research in other disciplines has found mixed results about their effect on subsequent performance, and little is known about their use in undergraduate mathematics. Here, I report on three studies which investigate the use of two-stage exams in different undergraduate mathematics contexts. The first two studies replicate observational methods from previous research, and find indications of a positive impact from group collaboration. The third study is experimental and finds that, in a delayed post-test, there is no difference in performance between students who answered related questions in a two-stage exam format and a control group which had no collaborative second stage. The findings suggest that two-stage exams may have little impact on longer-term learning of mathematics, but instructors may still wish to use them to emphasise a collaborative classroom pedagogy.

In response to Covid-19, university mathematics courses will switch to remote teaching and replace traditional exams with alternatives. The School of Mathematics at the University of Edinburgh has agreed to use take home exams for courses in years 3 and later. In the summer of 2020, each exam will be an open-book version of the original exam, taken remotely and unsupervised. This has potentially serious consequences because the exam questions may take on a different character in open- versus closed-book settings, and because of concerns about possible academic misconduct. This document outlines some ways that existing exam questions can be evaluated and modified to address these concerns. We hope this will provide colleagues with some practical suggestions for adjusting their exam papers, as well as showing students examples of how exam questions might look different.

Lectures are a commonly used teaching method in higher education, but there is significant debate about the relative merits of different classroom practices. Various classroom observation tools have been developed to try to give insight into these practices, beyond the simple dichotomy of “traditional lecturing versus active learning”. Here we review of a selection of classroom observation protocols from an ethological perspective, and describe how this informed the development of a new protocol, FILL+. We demonstrate that FILL+ can be applied reliably by undergraduate students after minimal training. We analysed a sample of 208 lecture recordings from Mathematics, Physics, and Veterinary Medicine and found a wide variety of classroom practices, e.g. on average lecturers spent 2.1% (±2.6%) of the time asking questions, and 79.3% (±19%) of the lecture talking, but individuals varied considerably. The FILL+ protocol has the potential to be widely used, both in research on effective teaching practices, and in informing discussion of pedagogical approaches within institutions and disciplines.

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.

This poster reports on a project to apply the theoretical framework of psychometric testing to evaluate the performance of an online diagnostic test in mathematics, and to design and implement consequent improvements. The steps of the evaluation methodology are described, with a view to the techniques being more routinely used in the re-design of digital tasks. A key finding is that for the diagnostic test, an exploratory factor analysis gave results consistent with an entirely separate classification according to an established taxonomy.

In brief: Based on the belief that physical education majors should have above-average fitness levels, 128 students were instructed to design and execute their own fitness development project to improve strength, endurance, and flexibility. Most relied on jogging, stair running, swimming, and cycling; others performed weight training, calisthenics, racket sports, and aerobic dancing. Generally, both men and women showed statistically significant increases in all the fitness tests. The authors believe that success was encouraged by the students' daily contact with instructors, the fact that their fitness improvement affected their grades, and the knowledge they gained from the course.

This paper is a report on the results of a study of 266 members of the Alpine Club of Canada, Calgary Section. The study attempted to gain further insight into why people climb mountains. The results indicate that an everchanging mosaic of primary and secondary motives for climbing exist. Age, ability and sex would appear to be three of the stronger determining variables that influence this mosaic. A model for characterizing motives for participation in physical activity is suggested. The categories of the model include: 1) social experience, 2) health and fitness, 3) excitement, 4) expressive, 5) relaxation, 6) competitive achievement, 7) non-competitive achievement, and 8) the love of nature.

An investigation of the amount of exchangeable potassium found in the surface and subsoils of a 33-year rotation experiment at Blacksburg, Virginia, was made. The rotation is located on Dunmore silt loam, Which is rated as a relatively fertile soil in Virginia. There are indications that sheet erosion has removed considerable amounts of the surface soil from the steeper slopes of the first nine treatments, and it is evident that an accumulation of this material has occurred on the four lower plots. It is believed by the writer, and others familiar with these plots, that.this erosion occurred prior to the beginning of the rotation in 1909. Composite soil samples were taken from each of the 8 treatments in each of the 4 series of this rotation and resulted in a total of thirty-two fertilized subplot samples, and a corresponding thirty-two unfertilized subplot samples were collected and analyzed for exchangeable potassium. Similar analyses were ma.de on the thirty-two subsoils of both the fertilized and the unfertilized treatments. The resulting data from the surface and subsoils were then statistically analyzed and the following conclusion drawn: 1. The applications of both muriate of potash and manure significantly increased the amount of exchangeable potassium found in both the surface and subsoils. 2. The surface soils are significantly higher in exchangeable potassium than the subsoils. 3. Exchangeable potassium may be leached from the surface horizon to the subsoil when applied in a complete fertilizer in excess of plant needs. 4. Manure applications which contained considerably less potassium than the potassium added to the soils in the muriate applications, resulted in larger amounts of exchangeable potassium in the surface soils, a 30 per cent increase in crop yields, and significantly less exchangeable potassium leached to the subsoils. 5. The difference in exchangeable potassium between the fertilized subplots receiving muriate of potash. and the unfertilized subplots. resulted in an increase in the amount of potassium found in the surface soils which was equivalent to 996 pounds of 50 per cent potash fertilizer, and this figure would represent an amount equivalent to 15 per cent of the total muriate added to the surface soils. 6. The difference in exchangeable potassium found in the subsoil between the fertilized and unfertilized plots represents an equivalent of approximately 9 per cent of the total potash added. 7. Significantly larger amounts of exchangeable potassium were leached to the subsoil from plots, receiving muriate of potash than soils which received manurial treatments. Greater utilization of the applied potassium is indicated by the manurial applications. 8. A chemical analysis should be made of the manure which is applied to this experiment in order to obtain a more comprehensive understanding of the yields, and physical and chemical properties of these soils. M.S.