Prior decisions and experiences about mathematics of students in bridging courses

We report on the survey responses of 51 students attending mathematics bridging courses at a major Australian university, investigating what mathematics, if any, these students had studied in the senior years of schooling and what factors affected their decisions about the level of mathematics chosen. Quantitative findings are augmented by qualitative responses to open-ended questions in the survey as well as excerpts from follow-up emails. The findings show that the major reasons for students taking lower levels of mathematics in senior year(s), or dropping mathematics, include finding enough time for non-mathematics subjects, confidence in mathematical capability, advice and maximizing potential ranking for university admission.     

A hybrid model of mathematics support for science students emphasizing basic skills and discipline relevance

The problem of students entering university lacking basic mathematical skills is a critical issue in the Australian higher-education sector and relevant globally. The Maths Skills programme at La Trobe University has been developed to address under preparation in the first-year science cohort in the absence of an institutional mathematics support centre. The programme was delivered through first-year science and statistics subjects with large enrolments and focused on basic mathematical skills relevant to each science discipline. The programme offered a new approach to the traditional mathematical support centre or class. It was designed through close collaboration between science subject coordinators and the project leader, a mathematician, and includes resources relevant to science and mathematics questions written in context. Evaluation of the programme showed it improved the confidence of the participating students who found it helpful and relevant. The programme was delivered through three learning modes to allow students to select activities most suitable for them, which was appreciated by students. Mathematics skills appeared to increase following completion of the programme and student participation in the programme correlated positively and highly with academic grades in their relevant science subjects. This programme offers an alternative model for mathematics support tailored to science disciplines.     

Trends in basic mathematical competencies of beginning undergraduates in Ireland, 2003–2013

Deficiencies in beginning undergraduate students’ basic mathematical skills has been an issue of concern in higher education, particularly in the past 15 years. This issue has been tracked and analysed in a number of universities in Ireland and internationally through student scores recorded in mathematics diagnostic tests. Students beginning their science-based and technology-based undergraduate courses in the University of Limerick have had their basic mathematics skills tested without any prior warning through a 40 question diagnostic test during their initial service mathematics lecture since 1998. Data gathered through this diagnostic test have been recorded in a database kept at the university and explored to track trends in mathematical competency of these beginning undergraduates. This paper details findings surrounding an analysis of the database between 2003 and 2013, outlining changes in mathematical competencies of these beginning undergraduates in an attempt to determine reasons for such changes. The analysis found that the proportion of students tested through this diagnostic test that are predicted to be at risk of failing their service mathematics end-of-semester examinations has increased significantly between 2003 and 2013. Furthermore, when students' performance in secondary level mathematics was controlled, it was determined that the performance of beginning undergraduates in 2013 was statistically significantly below that of the performance of the beginning undergraduates recorded 10 years previously.     

Subject design and factors affecting achievement in mathematics for biomedical science

Reports such as Bio2010 emphasize the importance of integrating mathematical modelling skills into undergraduate biology and life science programmes, to ensure students have the skills and knowledge needed for biological research in the twenty-first century. One way to do this is by developing a dedicated mathematics subject to teach modelling and mathematical concepts in biological contexts. We describe such a subject at a research-intensive Australian university, and discuss the considerations informing its design. We also present an investigation into the effect of mathematical and biological background, prior mathematical achievement, and gender, on student achievement in the subject. The investigation shows that several factors known to predict performance in standard calculus subjects apply also to specialized discipline-specific mathematics subjects, and give some insight into the relative importance of mathematical versus biological background for a biology-focused mathematics subject.     

Investigating students' levels of engagement with mathematics: critical events,motivations, and influences on behaviour

Universities invest significant resources in the provision of mathematics tuition to first year students, through both traditional and non-traditional means. Research has shown that a significant minority of students do not engage with these resources appropriately. This paper presents findings from a study of two groups of students at Maynooth University. Both groups had similar mathematical backgrounds on entry to university. The first group consisted of seven students who had failed first year mathematics and had very low levels of engagement with available supports. The second group consisted of nine students who had passed first year mathematics and had engaged with the supports to a significant extent. It emerged that while both groups initially displayed similar tactics and encountered similar difficulties, their levels of reaction to a number of critical events in their mathematical education were key to their engagement levels and their subsequent progression. Further analysis revealed aspects of the students' behaviour which caused them to approach or avoid difficulties. The reasons behind the different student behaviours were investigated, and the main categories of influence on student behaviour which emerged from the interview data were fear, social factors, and motivation.     

Are beginning calculus and engineering students adequately prepared for higher education? An assessment of students’ basic mathematical knowledge

Are students transitioning from the secondary level to university studies in mathematics and engineering adequately prepared for education at the tertiary level? In this study, we discuss the prior mathematical knowledge and skills demonstrated by Norwegian engineering (N?=?1537) and calculus (N?=?626) university students by using data from a mathematics assessment administered by the Norwegian Mathematical Council. The assessment examines students’ conceptual understanding, computation skills and problem solving skills on the basis of the mathematics curriculum of lower secondary education. We found that calculus students significantly outperformed engineering students, but both student groups struggled to solve the test, with the calculus and engineering groups scoring an average of 60% and 46%, respectively. Beginning students who fail to master basic skills, such as solving arithmetic and algebra problems, will most likely face difficulties in their further courses. Although few female students enrol in calculus and engineering programmes compared with male ones and are thus underrepresented, male and female students at the same ability level achieved comparable test scores. Furthermore, students reported high levels of intrinsic and extrinsic motivation, and a positive relationship was observed between intrinsic motivation and achievement.     

Mathematics learning support and engagement in first year engineering

This paper considers the effects of both free optional mathematics learning support and engagement on the mathematics performance in a foundation mathematics subject of a cohort of engineering students entering university with poor mathematical skills. New engineering students were directed to either a foundation or standard mathematics subject based on the results of a placement test. For students in the foundation subject, it was found that high levels of learning support were associated with greater improvement over the semester. Some form of learning support was used by 57.9% of the students, a reasonably high proportion of the cohort. Some factors for this high level of use of learning support are considered. One possible factor, the placement test, appears to have had a positive effect. Engagement in the subject activities as measured by tutorial attendance and learning management system use was found to have a positive association with final mark. Students who utilized a high level of learning support were more likely to be more engaged with the subject, making it impossible to draw conclusions about improvements being solely due to the use of learning support.     

The student experience of mathematical proof at university level

While proofs are central to university level mathematics courses, research indicates that some students may complete their degrees with an incomplete picture of what constitutes a proof and how proofs are developed. The paper sets out to review what is known of the student experience of mathematical proof at university level. In particular, some evidence is presented of the conceptions of mathematical proof that recent mathematics graduates bring to their postgraduate course to teach high school mathematics. Such evidence suggests that while the least well-qualified graduates may have the poorest grasp of mathematical proof, the most highly qualified may not necessarily have the richest form of subject matter knowledge needed for the most effective teaching. Some indication of the likely causes of this incomplete student perspective on proof are presented.     

Mathematical Readiness of Entering College Freshmen: An Exploration of Perceptions of Mathematics Faculty

The National Council of Teachers of Mathematics has set ambitious goals for the teaching and learning of mathematics that include preparing students for both the workplace and higher education. While this suggests that it is important for students to develop strong mathematical competencies by the end of high school, there is evidence to indicate that overall this is not the case. Both national and international studies corroborate the concern that, on the whole, US 12th grade students do not demonstrate mathematical proficiency, suggesting that students making the transition from high school to college mathematics may not be ready for its rigors. In order to investigate mathematical readiness of entering college students, this study surveyed mathematics faculty. Specifically, faculty members were asked their perceptions of average entering students' readiness related to relevant mathematical skills and concepts, and the importance of the same skills and concepts as foundations for college mathematics. Results demonstrated that the faculty perceived that average freshman students are generally not mathematically prepared; further, the skills and concepts rated as highly important — namely, algebraic skills and reasoning and generalization — were among those rated the lowest in terms of student competencies.     

An index to measure the ability of first year science students to transfer mathematics

This project aimed to quantitatively measure the ability of first year science students to transfer their mathematical skills and knowledge to a physical context. An instrument used in earlier studies to measure transfer of mathematical knowledge was modified for this project, and tested with 49 student volunteers. This paper presents the Transfer Index, a novel way of measuring the degree of transfer of the students' mathematics skills and knowledge. It is demonstrated that the Transfer Index is a useful way of quantifying transfer and that it has predictive validity in a natural educational setting.     

Post-primary students’ images of mathematics: findings from a survey of Irish ordinary level mathematics students

A questionnaire survey was carried out as part of a PhD research study to investigate the image of mathematics held by post-primary students in Ireland. The study focused on students in fifth year of post-primary education studying ordinary level mathematics for the Irish Leaving Certificate examination – the final examination for students in second-level or post-primary education. At the time this study was conducted, ordinary level mathematics students constituted approximately 72% of Leaving Certificate students. Students were aged between 15 and 18 years. A definition for ‘image of mathematics’ was adapted from Lim and Wilson, with image of mathematics hypothesized as comprising attitudes, beliefs, self-concept, motivation, emotions and past experiences of mathematics. A questionnaire was composed incorporating 84 fixed-response items chosen from eight pre-established scales by Aiken, Fennema and Sherman, Gourgey and Schoenfeld. This paper focuses on the findings from the questionnaire survey. Students’ images of mathematics are compared with regard to gender, type of post-primary school attended and prior mathematical achievement.     

Interdisciplinary education – a predator–prey model for developing a skill set in mathematics,biology and technology

The science of biology has been transforming dramatically and so the need for a stronger mathematical background for biology students has increased. Biological students reaching the senior or post-graduate level often come to realize that their mathematical background is insufficient. Similarly, students in a mathematics programme, interested in biological phenomena, find it difficult to master the complex systems encountered in biology. In short, the biologists do not have enough mathematics and the mathematicians are not being taught enough biology. The need for interdisciplinary curricula that includes disciplines such as biology, physical science, and mathematics is widely recognized, but has not been widely implemented. In this paper, it is suggested that students develop a skill set of ecology, mathematics and technology to encourage working across disciplinary boundaries. To illustrate such a skill set, a predator–prey model that contains self-limiting factors for both predator and prey is suggested. The general idea of dynamics, is introduced and students are encouraged to discover the applicability of this approach to more complex biological systems. The level of mathematics and technology required is not advanced; therefore, it is ideal for inclusion in a senior-level or introductory graduate-level course for students interested in mathematical biology.     

‘I’m worried about the correctness’: undergraduate students as producers of screencasts of mathematical explanations for their peers – lecturer and student perceptions

Undergraduate mathematics is traditionally designed and taught by content experts with little contribution from students. Indeed, there are signs that there is resistance from mathematics lecturers to involve students in the creation of material to support their peers – notwithstanding the fact that students have been successfully engaged as co-creators of material in other disciplines. There appears to be little research into what issues may lead to reservations to using student-created content in mathematics learning. This paper takes a case study approach to investigate the reasons for lecturers’ resistance to undergraduate student contributions to learning material, in particular with a view to the production of screencasts of mathematical explanations. It also investigates the views of students producing mathematical screencasts. This study is part of a larger research project investigating undergraduate involvement in mathematics module design. Four second-year students, who were producing mathematics screencasts as part of an internship, and five academics, were interviewed to gain an understanding of their views of the value of student screencasts. The interviews focused on the particular contributions students make to screencasts, outcomes for the students and level of lecturer acceptance of these resources. We argue that students benefit from creating screencasts for their peers by gaining deeper mathematical understanding, improved technological skills and developing other generic skills required of today's graduates. In contrast, we confirm lecturer resistance to using student-generated screencasts in their teaching materials. Lecturer reservations pertain to students’ lack of mathematical maturity and concerns over the mathematical integrity of the content that students produce. We conclude that close collaboration between students and lecturers during the design and production phases of screencasts may help lecturers overcome reservations, whilst preserving the benefits for students. In addition, we provide evidence that the process is a valuable professional development opportunity for the lecturers themselves.     

The development of reasoning skills during compulsory 16 to 18 mathematics education

The belief that studying mathematics improves reasoning skills, known as the Theory of Formal Discipline (TFD), has been held since the time of Plato. Research evidence supports this idea, at least in the context of students who had chosen to study post-compulsory mathematics. Here we examined the development of reasoning skills in 16- to 18-year-old Cypriot students, who are required to study mathematics until age 18. One hundred and eighty-eight students, studying high- or low-intensity mathematics, completed the abstract Conditional Inference Task and the contextual Belief Bias Syllogisms task at ages 16, 17 and 18. While the high-intensity group improved on the conditional inference task and showed a reduction in belief bias, the low-intensity group did not change on either measure. This is promising for the TFD, but suggests that a certain level of mathematical study may be necessary for students' general reasoning skills to develop.     

Applying an alternative mathematics pedagogy for students with weak mathematics: meta-analysis of alternative pedagogies

Student mathematics performance and the need for work-ready graduates to be mathematics-competent is a core issue for many universities. While both student and teacher are responsible for learning outcomes, there is a need to explicitly acknowledge the weak mathematics foundation of many university students. A systematic literature review was undertaken of identified innovations and/or interventions that may lead to improvement in student outcomes for university mathematics-based units of study. The review revealed the importance of understanding the foundations of student performance in higher education mathematics learning, especially in first year. Pre-university mathematics skills were identified as significant in student retention and mathematics success at university, and a specific focus on student pre-university mathematics skill level was found to be more effective in providing help, rather than simply focusing on a particular at-risk group. Diagnostics tools were found to be important in identifying (1) student background and (2) appropriate intervention. The studies highlighted the importance of appropriate and validated interventions in mathematics teaching and learning, and the need to improve the learning model for mathematics-based subjects, communication and technology innovations.     

Can't do maths--understanding students' maths anxiety

The number of students continuing with their mathematics educationpost GCSE level has declined in recent years and hence studentsentering Engineering degrees are reducing. The University ofBirmingham recognized this problem and introduced the Suiteof Technology programme (STP) which no longer requires studentsto have A-level mathematics. Therefore lecturers at universityare now faced with teaching A-level mathematics in order togive the students the mathematical skills for their technologydegree. With little experience of teaching at this level, lecturersfrequently face the challenge of choosing the most appropriatelevel for a lecture that encourages students to engage withand learn a subject that they are novices in. It turned outthat some students have a mathematics anxiety and hence thebiggest challenge for a lecturer is supporting the studentsovercoming this fear of mathematics. Choosing the appropriatestarting level for any lecture and the fear of mathematics wasacknowledged during a peer learning group meeting as part ofthe Post Graduate Certificate in Learning and Teaching in HigherEducation (PGCLTHE) at the University of Birmingham. In orderto explore this further, a session of mathematics was taughtby a Civil Engineering lecturer to fellow peers who do not havean engineering background. This article describes the mathematicsteaching session, reflections from the lecturer and the learners,and the impact that this had on teaching mathematics to undergraduates.Further, the article explores the difficulties and challengesexperienced by lecturers when teaching mathematics as a servicesubject.     

Enhancing student engagement to positively impact mathematics anxiety,confidence and achievement for interdisciplinary science subjects

Contemporary science educators must equip their students with the knowledge and practical know-how to connect multiple disciplines like mathematics, computing and the natural sciences to gain a richer and deeper understanding of a scientific problem. However, many biology and earth science students are prejudiced against mathematics due to negative emotions like high mathematical anxiety and low mathematical confidence. Here, we present a theoretical framework that investigates linkages between student engagement, mathematical anxiety, mathematical confidence, student achievement and subject mastery. We implement this framework in a large, first-year interdisciplinary science subject and monitor its impact over several years from 2010 to 2015. The implementation of the framework coincided with an easing of anxiety and enhanced confidence, as well as higher student satisfaction, retention and achievement. The framework offers interdisciplinary science educators greater flexibility and confidence in their approach to designing and delivering subjects that rely on mathematical concepts and practices.     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

How to make mathematics relevant to first-year engineering students: perceptions of students on student-produced resources

Many approaches to make mathematics relevant to first-year engineering students have been described. These include teaching practical engineering applications, or a close collaboration between engineering and mathematics teaching staff on unit design and teaching. In this paper, we report on a novel approach where we gave higher year engineering and multimedia students the task to ‘make maths relevant’ for first-year students. This approach is novel as we moved away from the traditional thinking that staff should produce these resources to students producing the same. These students have more recently undertaken first-year mathematical study themselves and can also provide a more mature student perspective to the task than first-year students. Two final-year engineering students and three final-year multimedia students worked on this project over the Australian summer term and produced two animated videos showing where concepts taught in first-year mathematics are applied by professional engineers. It is this student perspective on how to make mathematics relevant to first-year students that we investigate in this paper. We analyse interviews with higher year students as well as focus groups with first-year students who had been shown the videos in class, with a focus on answering the following three research questions: (1) How would students demonstrate the relevance of mathematics in engineering? (2) What are first-year students' views on the resources produced for them? (3) Who should produce resources to demonstrate the relevance of mathematics? There seemed to be some disagreement between first- and final-year students as to how the importance of mathematics should be demonstrated in a video. We therefore argue that it should ideally be a collaboration between higher year students and first-year students, with advice from lecturers, to produce such resources.     

Distinguishing between mathematics classrooms in Australia, China, Japan, Korea and the USA through the lens of the distribution of responsibility for knowledge generation: public oral interactivity and mathematical orality

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study documented the voicing of mathematical ideas in public discussion and in teacher–student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher–student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons studied in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation and to student learning ourcomes are central to the development of any theory of mathematics instruction and learning.