Presenting mathematics in a first‐year engineering course at university

The approach used to teach mathematics to first‐year engineering students at the University of Warwick is described. The instruction is programmed, with weekly assignments and tutorials, so that the learning is self‐paced within each week. Our experiences with this method of teaching are discussed.     

A three-year perspective on conceptions of and orientations to learning mathematics of prospective teachers and first year university students

The paper reports a compilation of results from three studies conducted over three years to determine students' conceptions of mathematics, and orientations they follow in learning the subject. Respondents were 459 first year mathematics students from four universities and one teacher college. Results indicated that more than half the sample reported mathematics to be a subject made of numbers and formulae that could be memorized. This suggests a shallow emphasis when learning the subject, with no intention to understand. However, most students passed their examinations. It was concluded that there was no statistically significant relationship between examinations results and students' learning orientations. It is recommended that lecturers should foster students' meta-learning capabilities and an awareness of different learning strategies.     

The MACSI summer school: a case study in outreach in mathematics

To encourage the study of mathematics in Ireland, the Mathematics Applications Consortium for Science and Industry (MACSI) organizes a summer school once a year. The different aspects of this summer school are presented. Students are selected depending on their motivation, academic abilities, gender and geographical origins. Instruction and supervision is provided by academics, post-doctoral fellows and post-graduate students. The teaching programme evolves every year and reflects the interests of the people involved. Feedback from participants has been almost uniformly positive. Students favour interactive sessions and enjoy the residential aspect of the summer school. Food and accommodation are however the most costly aspects of this summer school. In this respect the support of Science Foundation Ireland has been invaluable.     

A report on the introduction of a multiple choice examination for a first year university engineering mathematics course

The paper is an account of the experience of introducing a multiple choice mathematics examination paper for the first year engineering mathematics course at Nottingham University. The form of the paper is described, as is a system for providing the candidates with feedback on their performance without the publication of the examination paper.     

How students structure their investigations and learn mathematics: insights from a long-term study

This paper reports on the mathematical thinking of participants of a long-term study, now in its 17th year, who did mathematics together through their public school and early university years. In particular, it describes how fundamental ideas and images of a cohort group of students are elaborated and presented in symbolic expressions of generalized mathematical ideas while exploring problems in grades 10 and 11. From high school and university interview data, we learn from participants how they viewed their mathematical activity in structuring their investigations and justifying their solutions.     

Informed aspirations in science and engineering with upper elementary students after 1 year of a STEM intensive university-school district partnership

This article reports findings from a study of an integrated science, technology, engineering, and mathematics (STEM) education program on student interest and awareness in science and engineering. The analysis features grade 3–5 students from a high-poverty, urban school system in the Mid-Atlantic region. Through the quantitative analysis of closed ended survey responses and the qualitative analysis of an open-ended query, we describe how the adoption of an intensive STEM-focused partnership could influence students’ early interest in and awareness of science and engineering as disciplines and careers. The analysis of the student responses revealed that after 1 year of the project, the students enrolled in the program demonstrated developing interest in science and engineering and were better able to articulate a greater understanding of engineering as a discipline. These findings have implications for the effectiveness of an integrated STEM approach for upper elementary students participating and succeeding in the STEM fields.     

A multidimensional approach to training mathematics students at a university: improving the efficiency through the unity of social,psychological and pedagogical aspects

The article deals with social, psychological and pedagogical aspects of teaching mathematics students at universities. The sociological portrait and the factors influencing a career choice of a mathematician have been investigated through the survey results of 198 first-year students of applied mathematics major at 27 state universities (Russia). Then, psychological characteristics of mathematics students have been examined based on scientific publications. The obtained results have allowed us to reveal pedagogical conditions and specific ways of training mathematics students in the process of their education at university. The article also contains the analysis of approaches to the development of mathematics education both in Russia and in other countries. The results may be useful for teaching students whose training requires in-depth knowledge of mathematics.     

Mathematical tasks,study approaches,and course grades in undergraduate mathematics: a year-by-year analysis

Students approach learning in different ways, depending on the experienced learning situation. A deep approach is geared toward long-term retention and conceptual change while a surface approach focuses on quickly acquiring knowledge for immediate use. These approaches ultimately affect the students’ academic outcomes. This study takes a cross-sectional look at the approaches to learning used by students from courses across all four years of undergraduate mathematics and analyses how these relate to the students’ grades. We find that deep learning correlates with grade in the first year and not in the upper years. Surficial learning has no correlation with grades in the first year and a strong negative correlation with grades in the upper years. Using Bloom's taxonomy, we argue that the nature of the tasks given to students is fundamentally different in lower and upper year courses. We find that first-year courses emphasize tasks that require only low-level cognitive processes. Upper year courses require higher level processes but, surprisingly, have a simultaneous greater emphasis on recall and understanding. These observations explain the differences in correlations between approaches to learning and course grades. We conclude with some concerns about the disconnect between first year and upper year mathematics courses and the effect this may have on students.     

Active learning: a case study of student engagement in college Calculus

This paper presents a case study for strategic engagement of students in a Calculus course in order to produce increased learning in the classroom. Since it has been shown that active learning can promote greater comprehension for students in science, technology, engineering, and mathematics (STEM) courses, the researcher utilized many types of active learning techniques to enhance classroom instruction. The key components implemented are presented as a model of enhanced learning through developed classroom engagement. This course redesign model entitled, Strategic Engagement for Increased Learning (SEIL), has the potential to (1) contribute to the body of knowledge on ways to improve mathematics skills for college students, (2) identify successful teaching strategies and technologies that will promote the retention of STEM students, (3) increase the success rate of students taking Calculus, and (4) help produce more STEM graduates needed for the STEM workforce in the United States of America.     

Generating scenarios of addition and subtraction: A study of Japanese university students

Students are presented with problems involving three scenario types of addition and subtraction in elementary mathematics: one dynamic (Change) and two static (Combine, Compare). Previous studies have indicated that the dynamic type is easier for school children, whereas the static types are more difficult and comprehended only gradually throughout elementary grades. However, these differences in scenario difficulty have not be adequately investigated in adults to ascertain whether they persist in adults who have used the operations for a long period of time. This study examined this with Japanese university students (ages 18–23 years). In Study 1, 99 participants were provided with one numerical equation and one picture representing one of the three scenario types, and were instructed to generate a single scenario. In Study 2, 97 participants were provided with one numerical equation and were instructed to recall the various scenario types that they had encountered previously and to generate multiple scenarios of diverse types. Results indicated that participants found the dynamic scenario types easier than static ones, and they tended to understand the given information of static types by interpreting them in dynamic forms (Study 1). In addition, they strongly preferred generating scenarios of dynamic type over static types (Study 2). Implications for mathematics learning are discussed.     

Live lectures or online videos: students’ resource choices in a first-year university mathematics module

In Maths for Business, a mathematics module for non-mathematics specialists, students are given the choice of completing the module content via short online videos, live lectures or a combination of both. In this study, we identify students’ specific usage patterns with both of these resources and discuss their reasons for the preferences they exhibit. In 2015–2016, we collected quantitative data on each student's resource usage (attendance at live lectures and access of online videos) for the entire class of 522 students and employed model-based clustering which identified four distinct resource usage patterns with lectures and/or videos. We also collected qualitative data on students’ perceptions of resource usage through a survey administered at the end of the semester, to which 161 students responded. The 161 survey responses were linked to each cluster and analysed using thematic analysis. Perceived benefits of videos include flexibility of scheduling and pace, and avoidance of large, long lectures. In contrast, the main perceived advantages of lectures are the ability to engage in group tasks, to ask questions, and to learn ‘gradually’. Students in the two clusters with high lecture attendance achieved, on average, higher marks in the module.     

Implementing a flipped classroom approach in a university numerical methods mathematics course

This paper describes and analyses the implementation of a ‘flipped classroom’ approach, in an undergraduate mathematics course on numerical methods. The approach replaced all the lecture contents by instructor-made videos and was implemented in the consecutive years 2014 and 2015. The sequential case study presented here begins with an examination of the attitudes of the 2014 cohort to the approach in general as well as analysing their use of the videos. Based on these responses, the instructor makes a number of changes (for example, the use of ‘cloze’ summary notes and the introduction of an extra, optional tutorial class) before repeating the ‘flipped classroom’ approach the following year. The attitudes to the approach and the video usage of the 2015 cohort are then compared with the 2014 cohort and further changes that could be implemented for the next cohort are suggested.     

Driving student-centred calculus: results of a comprehensive case study for Kaizen learning in the Sultanate of Oman

The art of teaching freshmen students is undergoing a rapid paradigm change. Classical forms of teaching are not applicable any more and an unmanageable offer of new multimedia tools and concepts is glutting the market. Moreover, compared to previous courses, the class size triples. In view of these challenges, we implemented a new teaching concept best described as Kaizen learning. By Kaizen learning, we define a teaching philosophy that is based on a concise mix of short learning units (with feedback loops and tests) and of carefully chosen repetitions (also with feedback loops and tests) to calibrate a course for the students. Here, this intensive blended, student-centred learning paradigm is analysed together with its direct impact on the students’ performance. This case study leads to easy-to-implement key drivers for successfully teaching science in Oman, such as (1) human–human interaction, (2) clearly communicated expectations, (3) avoidance of a short-term learning attitude, (4) a no-calculator policy, (5) continuous Kaizen learning, and (6) balanced combination of traditional teaching and e-learning.     

Discontinuous enrichment in finite elements with a partition of unity method

A technique is presented to model arbitrary discontinuities in the finite element framework by locally enriching a displacement-based approximation through a partition of unity method. This technique allows discontinuities to be represented independently of element boundaries. The method is applied to fracture mechanics, in which crack discontinuities are represented using both a jump function and the asymptotic near-tip fields. As specific examples, we consider cracks and crack growth in two-dimensional elasticity and Mindlin–Reissner plates. A domain form of the J-integral is also derived to extract the moment intensity factors. The accuracy and utility of the method is also discussed.     

Estimating Iraqi deaths: a case study with implications for mathematics education

In this paper, I present an account of attempts to quantify deaths of Iraqis during the occupation by US and other forces since the invasion of March 2003, and of the reactions to these attempts. This story illuminates many aspects of current socio-political reality, particularly, but by no means exclusively, in the United States. Here, these aspects are selectively discussed in relation to the overarching themes of what the story illuminates about the uses of statistical information in society and about shortcomings in mathematics education.     

What students outside Asia value in effective mathematics lessons: a scoping study

This article reports on a scoping study conducted in Australia and Sweden to facilitate the understanding of the values that are associated with effective learning of mathematics in Asian classrooms. Nineteen ??Western?? values have been identified, of which three (explanation, sharing and fun) were commonly embraced in the two countries. It is also evident that traits that are valued can be and are demonstrated in different forms. The implications for improving mathematics education through the harnessing of values are discussed. The distinction made between what is valued and the forms the value can take should empower classroom teachers to incorporate values flexibly in different classroom contexts.     

Student performance and attitudes in a collaborative and flipped linear algebra course

Flipped learning is gaining traction in K-12 for enhancing students’ problem-solving skills at an early age; however, there is relatively little large-scale research showing its effectiveness in promoting better learning outcomes in higher education, especially in mathematics classes. In this study, we examined the data compiled from both quantitative and qualitative measures such as item scores on a common final and attitude survey results between a flipped and a traditional Introductory Linear Algebra class taught by two individual instructors at a state university in California in Fall 2013. Students in the flipped class were asked to watch short video lectures made by the instructor and complete a short online quiz prior to each class attendance. The class time was completely devoted to problem solving in group settings where students were prompted to communicate their reasoning with proper mathematical terms and structured sentences verbally and in writing. Examination of the quality and depth of student responses from the common final exam showed that students in the flipped class produced more comprehensive and well-explained responses to the questions that required reasoning, creating examples, and more complex use of mathematical objects. Furthermore, students in the flipped class performed superiorly in the overall comprehension of the content with a 21% increase in the median final exam score. Overall, students felt more confident about their ability to learn mathematics independently, showed better retention of materials over time, and enjoyed the flipped experience.     

Gender, technology and attitude towards mathematics: a comparative longitudinal study with Mexican students

In this paper the results of a comparative longitudinal study investigating changes in girls’ and boys’ attitudes towards mathematics, and self-confidence in mathematics are presented. A 5-point Likert scale, AMMEC, was used to measure attitudes towards mathematics (AM), computer-based mathematics (AMC), and self-confidence in mathematics (CM). A total of 430 students using technology for mathematics and 109 students not using it were monitored for 3 years. At the beginning of the study, the participants were aged about 13 years. The statistical analyses of the data showed few gender differences in the way students’ attitudes and self-confidence changed over the 3 years. Significant gender differences favouring boys were found in attitudes towards mathematics in grades 8 and 9 for the group using technology. For the group using technology, significantly more boys than girls got high scores in attitudes towards computer-based mathematics in grade 7. Significantly, more girls using technology than girls not using it got high scores in grade 8. The use of technology did not have a positive impact on students’ self-confidence. Regardless of whether they used computers or not, from grades 7 to 9, there was a decrease in the self-confidence in mathematics of both boys and girls. To enrich these results and detect possible gender differences in the way attitudes were constructed, 12 girls and 13 boys were interviewed at the end of the study. The analysis of the arguments they presented to explain and justify their attitudes towards mathematics, computer-based mathematics, and their self-confidence in working in mathematics provided evidence of important gender differences in the ways in which boys and girls construct their attitude, indicating how their constructions reflect the gender stereotypes within Mexican society.     

Math clubs and their potentials: making mathematics fun and exciting. A case study of a math club

The purpose of this study is to describe the results obtained from a survey whose goal was to examine the combination of variables that have contributed to the success of a middle school math club. This is a case of a middle school in which the students are extremely successful in mathematics, and where the majority of the students voluntarily attend its math club. The results of the study show that the students have positive attitudes about mathematics and the club, and that some of the reasons that influenced them to attend the club were those of being with friends and eating donuts at the club. The results were similar for students of both genders and all grades. In addition, since positive attitudes are associated with higher levels of math achievement, such clubs have the potential to encourage students to enrol in additional mathematics classes while in high school, as well as pursuing mathematics related careers.     

An evaluation of the Supplemental Instruction programme in a first year calculus course

Supplemental Instruction (SI) incorporates collaborative learning in small, peer-led, group settings in order to integrate instruction in learning and reasoning skills with course content. Several meta-analyses speak to the efficacy of SI but fail to address selection bias due to ability/motivation and gender. In this study, SI was paired with a first year calculus for non-majors course. An ANCOVA indicated that: ability/motivation, as measured by prior grade point average, was a useful predictor of course letter grade; gender differences were statistically significant but trivial; and, SI participation was statistically and practically significant, a 1.8 letter grade improvement after correction for selection bias. For the pass/fail analysis, a sequential binary logistic regression indicated there was a sizable statistically significant improvement with SI participation after accounting for gender and ability/motivation selection biases. The odds of success were 2.7 times greater for the SI participants. No gender differences of any significance were found.