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The overall objective of this study was to examine the effect of blended learning on improving students’ mathematical cognitive demand and their extraneous and germane cognitive load in learning calculus of several variables. Guided by constructivist and cognitive load theories, a quasi-experimental design was conducted in three public universities. A total of 67 second-year undergraduate mathematics students participated for 4 months. The study used a mixed research method with social constructivism augmented by pragmatism. Data were analysed using analysis of co-variance procedure and thematic analysis and discussed using both quantitative and qualitative methods. The findings revealed that the treatment group students’ mathematical cognitive demand (treatment (TG1) (M = 70.9, SD = 10.8) and treatment (TG2) (M = 69.3, SD = 20.5) were significantly higher than the posttest scores of students in the comparison group in CG (M = 58.8, SD = 13.5), (p = 0.00)) with effect size of 0.25. Moreover, their germane cognitive load had improved and extraneous cognitive load reduced significantly compared to the comparison group. To conclude, mathematics cognitive demand can be improved with blended learning. These findings imply that blending mathematics instruction in higher education institutions is useful to policymakers. The finding also revealed that cognitive load was found a negative and significant predictor of mathematics cognitive demand. [ABSTRACT FROM AUTHOR]

Combinatorial proofs of binomial identities involve establishing an identity by arguing that each side enumerates a certain set of outcomes. In this paper, we share results from interviews with experienced provers (mathematicians and upper-division undergraduate mathematics students) and examine one particular aspect of combinatorial proof, namely the kinds of contexts that experienced provers used to establish combinatorial proofs of binomial identities. Our findings show that overall, our participants used a variety of contexts in their work; we also demonstrate ways in which previous experiences influenced the contexts they chose, and we offer some instances in which features of a context supported their combinatorial proof production. We offer some theoretical implications of our work, and we conclude with a discussion of limitations and avenues for future work. [ABSTRACT FROM AUTHOR]

Many students who take a mathematics course as part of their post-secondary education are not enrolled in a mathematically-intensive degree program. This poses a challenge to mathematics departments that, for example, value and privilege more traditional mathematics curriculum: should students in such courses be taught differently than those in more traditional mathematics courses? What should constitute the curricula of these courses and who should teach them? How might the local institutional context influence the courses that are ultimately offered? This paper sketches a theory intended to help frame responses to these questions. We then present an example of this theory in action, in the form of a general education mathematics course for first-year, underprepared students. The ongoing design, teaching, and evaluation of this course might inspire further revisions to general education in mathematics at the undergraduate level. We present the motivation for the design of the course, a summary of what was ultimately enacted, and our reflections of this ongoing event. Our intention here is not to present our course and an evaluation of its 'success' – however that might be conceptualised – but rather to display a rigorous vision for post-secondary general education mathematics. [ABSTRACT FROM AUTHOR]

Technical skills related to basic mathematics in higher education have been a key foundation for further allied courses across primary programmes. While the traditional learning strategies appear to be inadequate to enhance the basic mathematics skills of university students, digital mathematical games pose the potential to improve such a set of skills by means of mobile learning. This work focuses on investigating the influence of digital mathematical games on students' basic mathematics skills under four various academic programmes. To illustrate the proposed scheme, a case study is conducted in a state university in the Philippines across undergraduate programmes in information technology, hospitality management, industrial engineering, and education. It was found that the use of digital games can improve the skills of students in basic mathematics. Thus, the educational system may be redesigned along with the positive effects of digital games on traditional skill enhancement initiatives. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic has posed one of the most challenging situations for the academicians as well as the students all around the world. Students being the biggest stake holders of the education process, their challenges during this pandemic become even more pivotal. These challenges vary in intensity and type depending on multiple items. In this paper, we analyse and document the challenges particular to the courses of mathematics and statistics in higher education. A survey was carried out for students, who have taken mathematics/statistics courses at universities mainly in UAE, to analyse various items affecting the learning process during the COVID-19 pandemic. The results of the survey showed that a significant portion of students are satisfied with distance education. [ABSTRACT FROM AUTHOR]

It is generally accepted that teachers' beliefs impact their instructional choices, but characterizations of that relationship are limited in college settings. Furthermore, examinations of instructor beliefs, instruction, and student beliefs together in one setting are rarely described. Based on interviews with two Abstract Algebra instructors, classroom video from three units of instruction, and survey and interview data from students in the classes, this paper examines instructors' stated beliefs, ways these beliefs manifested in their teaching, and students' beliefs across the course. Both instructors made curricular choices clearly aligned with their stated views of the nature of mathematics, learning, and teaching. Day-to-day instructional choices reflected these stated beliefs as well, but the difficulty of material and tensions with other beliefs like the importance of interactivity manifested. Characterizations of the interactivity of classes and placement of the mathematical authority in class are provided through descriptive and quantitative measures. These characterizations of instruction provide nuanced portrayals of classroom norms and changes in those norms throughout the semester. Furthermore, subtle shifts in student beliefs about teaching and learning are noticeable, suggesting students' beliefs about teaching and learning mathematics can be influenced even by modest changes in instructional practice. [ABSTRACT FROM AUTHOR]

This study examines how collective argumentation in the integration of the ACODESA method (collaborative learning, scientific debate and self-reflection) and GeoGebra can help students understand parametric equations. The participants of the study consist of 24 university students enrolled in a mathematics education programme at a state university in Turkey. Data were collected through students' written productions, screen recorder software, and transcriptions of the students' argumentations for selected groups. The collected data were analysed based on Toulmin's model. Results of the study indicate the use of GeoGebra in ACODESA method triggered a sense of understanding of parametric equations. It became apparent that the participants constructed parametric equations from existing knowledge of trigonometric identities, definition of function, and general form of the equation of a circle by making connections between algebraic and geometric representations. The integration of GeoGebra into ACODESA method also fostered some process aspects of mathematical reasoning (generalizing, conjecturing, and justifying) on parametric equations. [ABSTRACT FROM AUTHOR]

In this study, performance data from students on a mathematics graduate programme from Spring 2019 to Spring 2020 were collected and analysed. The results showed that, if the right course delivery method was implemented, the impact of the coronavirus pandemic (COVID-19) on students' performance could be minimal. Based on the study, some recommendations are made on how to improve students' performances in graduate-level mathematics courses. [ABSTRACT FROM AUTHOR]

We present action research of a problem posed as part of a multi-participant national (Israeli) test checking the mathematical knowledge of high school students at the ages of 16–17, where some of those who solved this problem made an error by using the converse to a well-known theorem, where the converse is not true. In order to examine the danger of using a wrong converse, the problem was posed as field research to a group of pre-service teachers and training teachers in the 'methods (of teaching mathematics)' course, where similar failures were discovered. The conclusion is that a large part of the students was not aware that the converse is not always true, and its correctness is to be tested before it can be used. The investigation of the problem was accompanied by a dynamic geometry environment software that allowed us to examine the problem in the general case and so to make conclusions for particular cases. [ABSTRACT FROM AUTHOR]

A partially-automated method of assessment is proposed, in which automated question setting is used to generate individualized versions of a coursework assignment, which is completed by students and marked by hand. This is designed to be (a) comparable to a traditional written coursework assignment in validity, in that complex and open-ended tasks can be set with diverse submission formats that would not be suitable for written examination or automated marking; and, (b) comparable to e-assessment in terms of reduction of academic misconduct, with individualization acting as a barrier to copying and collusion. This method of assessment is implemented in practice. Evaluation focuses on expert second-marking, student feedback and analysis of marks, and aims to establish that the partially-automated method can be useful in practice. The partially-automated method proposed appears to be capable of adapting a coursework assignment to make it less sensitive to copying and collusion (and therefore more reliable) while maintaining its validity, though leading to reduced efficiency for the marker. This paper therefore contributes the introduction of a novel approach to assessment which offers a way to bring automated individualization to the assessment of higher order skills in higher education mathematics. [ABSTRACT FROM AUTHOR]

The systematic design and analysis of tasks which can be implemented in first-year university courses and point to advanced inner- and extra-mathematically rich issues and their rationales is an open problem in the didactics of mathematics in higher education. Potentials of such tasks can be seen with regard to learning processes in the first year of study, which are less compartmentalized and allow for an extended acquisition of rationales of advanced mathematical practices and concepts. Against this background, subject-specific potentials of advanced mathematics are examined in this contribution. The Anthropological Theory of the Didactic (ATD) serves as the theoretical framework and in the analyses notions from its 4T-model are applied. Structural observations in praxeological terms are illustrated by examples chosen from a presentation of a classical result in Nonlinear Approximation. At the specific focus of the praxeological analyses are aspects for bridging and extending concepts within and across Analysis. On this basis, tasks designed for first-year university Analysis courses are analysed in detail. Moreover, general characteristics of the methodological approach realized in this contribution are discussed. [ABSTRACT FROM AUTHOR]

This study examines undergraduate student attitudes to acting as peer-leaders in Applied Mathematics computing workshops. Through the use of data from end-of-trimester surveys and an on-line reflection worksheet, it is shown that students had positive attitudes to being leaders and were engaged in the experience. Benefits mentioned by students included an increased understanding of the content, increased confidence in communicating with and approaching others, and an improved connection with peers. The worksheet also gave students the option of suggesting improvements for the workshops. It was found that those who took up this option provided valuable feedback to the instructor from a student perspective, and that this improved the workshops for the next iteration of the course. Finally, it was concluded that the students acted as partners, not only by collaborating in the re-design of the workshop materials, but also because they were engaged in and contributed to learning and teaching in the course. [ABSTRACT FROM AUTHOR]

First-year engineering students often struggle to see the relevance of theoretical mathematical concepts for their future studies and professional careers. This is an issue, as students who do not see relevance in fundamental parts of their studies may disengage from these parts and focus their efforts on other subjects they think will be more useful to them. In this study, we surveyed engineering students enrolled in a first-year mathematics subject on their perceptions of the relevance of the individual mathematical topics taught. Surveys were administered at the start of semester when some of these topics were unknown to them, and again at the end of semester when students had not only studied all these topics but also watched a set of animated videos. These videos had been produced by higher-year students to explain where they had seen applications of the mathematical concepts presented in the first year. We notice differences between the perceived relevance of topics for future study and for professional careers, with relevance to study rated higher than relevance to careers. We also find that the animations are seen as helpful in understanding the relevance of first-year mathematics. The majority of students indicated that lecturers with students as partners should work collaboratively to produce future videos. [ABSTRACT FROM PUBLISHER]

The mathematics education literature shows that encouraging students to develop multiple solutions for given problems has a positive effect on students’ understanding and creativity. In this paper, we present an example of multiple-solution problems in statistics involving a set of non-traditional dice. In particular, we consider the exact probability mass distribution for the sum of face values. Four different ways of solving the problem are discussed. The solutions span various basic concepts in different mathematical disciplines (sample space in probability theory, the probability generating function in statistics, integer partition in basic combinatorics and individual risk model in actuarial science) and thus promotes upper undergraduate students’ awareness of knowledge connections between their courses. All solutions of the example are implemented using the R statistical software package. [ABSTRACT FROM PUBLISHER]

Previous studies have produced several typologies of teacher questions in mathematics. Probing questions that ask students to explain are often included in the types of questions. However, only rare studies have created subtypes for probing questions or investigated how questioning differs depending on whether technology is used or not. The aims of this study are to elaborate on different ways of asking students to give explanations in inquiry-based mathematics teaching and to investigate whether questioning in GeoGebra lessons differs from questioning in other lessons. Data was collected by video recording 29 Finnish mathematics student teachers’ lessons in secondary and upper secondary schools. The lesson videos were coded for the student teachers’ probing questions. After this, categories for the types of probing questions were created, which is elaborated in this paper. It was found that the student teachers who used GeoGebra emphasized conceptual probing questions during the explore phase of a lesson slightly more than the other student teachers. [ABSTRACT FROM AUTHOR]

By helping create a shared, supportive, learning community, the creative use of custom-designed spaces outside the classroom has a major impact on student engagement. The intention is to create spaces that promote peer interaction within and across year groups, encourage closer working relationships between staff and students and support specific coursework activities – particularly group work. Such spaces make better use of time since students are motivated to stay and work during long gaps in their timetable, can provide a sense of ‘home’ within the institution and can lead to a cohesive community of practice. In this paper, we describe how this has been achieved and currently delivered in Mathematics at Sheffield Hallam University and provide some detailed analysis of the student usage of the space. [ABSTRACT FROM PUBLISHER]

Counting problems offer rich opportunities for students to engage in mathematical thinking, but they can be difficult for students to solve. In this paper, we present a study that examines student thinking about one concept within counting, factorials, which are a key aspect of many combinatorial ideas. In an effort to better understand students’ conceptions of factorials, we conducted interviews with 20 undergraduate students. We present a key distinction between computational versus combinatorial conceptions, and we explore three aspects of data that shed light on students’ conceptions (their initial characterizations, their definitions of 0!, and their responses to Likert-response questions). We present implications this may have for mathematics educators both within and separate from combinatorics. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

The goal of this paper is to promote computational thinking among mathematics, engineering, science and technology students, through hands-on computer experiments. These activities have the potential to empower students to learn, create and invent with technology, and they engage computational thinking through simulations, visualizations and data analysis. We present nine computer experiments and suggest a few more, with applications to calculus, probability and data analysis, which engage computational thinking through simulations, visualizations and data analysis. We are using the free (open-source) statistical programming language R. Our goal is to give a taste of what R offers rather than to present a comprehensive tutorial on the R language. In our experience, these kinds of interactive computer activities can be easily integrated into a smart classroom. Furthermore, these activities do tend to keep students motivated and actively engaged in the process of learning, problem solving and developing a better intuition for understanding complex mathematical concepts. [ABSTRACT FROM AUTHOR]

A study conducted with 25 Year 6 primary school students investigated the potential for a short classroom intervention to begin the development of aModellingconception of mathematics on the way to developing a sense of mathematics as a way of thinking about life. The study documents the developmental roots of the cognitive activity, actions and conceptions of both modelling and mathematics that these beginners to modelling displayed. Understanding the conceptions of mathematics that students might hold or be developing and how these can be influenced in early schooling are essential ingredients in any plans for introducing modelling seriously into primary school classrooms. The majority of the students (22/25) were identified as displaying a developing conception of modelling asa way of problem handling. The three other students displayed the developmental roots ofa way of understanding the worldconception of modelling. These three students also displayed aModellingconception of mathematics with one showing indications of developing towards aLifeconception of mathematics. [ABSTRACT FROM AUTHOR]

In this article, we revisit the concept of strong differentiability of real functions of one variable, underlying the concept of differentiability. Our discussion is guided by the Straddle Lemma, which plays a key role in this context. The proofs of the results presented are designed to meet a young audience in mathematics, typical of students in a first course of Real Analysis or an honors-level Calculus course. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Dr RL Moore was undoubtedly one of the finest mathematics teachers ever. He developed a unique teaching method designed to teach his students to think like mathematicians. His method was not designed to convey any particular mathematical knowledge. Instead, it was designed to teach his students to think. Today, his method has been modified to focus on using student participation toward the goal of the conveyance of mathematical knowledge rather than on Dr Moore's goal of teaching students to think. This article proposes that undergraduates would be better served if they took at least one course using Dr Moore's original method and his original goal. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The article addresses the problématique of where mathematization is taught in the educational system, and who teaches it. Mathematization is usually not a part of mathematics programs at the upper secondary level, but we argue that physics teaching has something to offer in this respect, if it focuses on solving so-called unformalized problems, where a major challenge is to formalize the problems in mathematics and physics terms. We analyse four concrete examples of unformalized problems for which the formalization involves different order of mathematization and applying physics to the problem, but all require mathematization. The analysis leads to the formulation of a model by which we attempt to capture the important steps of the process of solving unformalized problems by means of mathematization and physicalization. [ABSTRACT FROM AUTHOR]

This study focuses on the mathematics department at a South African university and in particular on teaching of calculus to first year engineering students. The paper reports on a cause–effect analysis, often used for business improvement. The cause–effect analysis indicates that there are many factors that impact on secondary school teaching of mathematics, factors that the tertiary sector has no control over. The analysis also indicates the undesirable issues that are at the root of impeding success in the calculus module. Most important is that students are not encouraged to become independent thinkers from an early age. This triggers problems in follow-up courses where students are expected to have learned to deal with the work load and understanding of certain concepts. A new model was designed to lessen the impact of these undesirable issues. [ABSTRACT FROM AUTHOR]

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs are given under the headings: standard proof, elegant proof, and the proof without words. The solutions were obtained through a combination of mathematical tools and by dynamic investigation of the geometrical properties. [ABSTRACT FROM PUBLISHER]

In recent years, changes in the Brazilian economic and social scenario have generated a growing demand for higher education in the country. In response to this new context, the federal government launched in 2007 a programme aiming at expansion of the enrolment to public higher education. In such environment of changes, several proposals have emerged to adapt the Brazilian federal universities to a new reality. Taking this context into account, the focus of our study is on proposals from the Mathematics Department of the Federal University of Minas Gerais. Their aim is to create a new model of teaching practices for the freshman lessons of the Exact Sciences area, which at first were being experimented in special classrooms of students attending their first course on differential and integral calculus. The data were collected from interviews with students and professors from the mathematics department. They were analyzed and systematized using an activity theory approach. We became instigated by a model developed by Engeström in his study on the changes in the Finnish public health system, considering it as a test (testbench) for activity theory in its application to a particular case. Following Engeström's footsteps when developing his research, we arrived at our own model showing the internal tensions in the activity of reformulation of the courses offered by the Department of Mathematics and in the conceptions of teachers that promote – and constraint – the proposals for change. [ABSTRACT FROM PUBLISHER]

Two separate studies, Jonsson et al. (J. Math Behav. 2014;36: 20–32) and Karlsson Wirebring et al. (Trends Neurosci Educ. 2015;4(1–2):6–14), showed that learning mathematics using creative mathematical reasoning and constructing their own solution methods can be more efficient than if students use algorithmic reasoning and are given the solution procedures. It was argued that effortful struggle was the key that explained this difference. It was also argued that the results could not be explained by the effects of transfer-appropriate processing, although this was not empirically investigated. This study evaluated the hypotheses of transfer-appropriate processing and effortful struggle in relation to the specific characteristics associated with algorithmic reasoning task and creative mathematical reasoning task. In a between-subjects design, upper-secondary students were matched according to their working memory capacity. The main finding was that the superior performance associated with practicing creative mathematical reasoning was mainly supported by effortful struggle, however, there was also an effect of transfer-appropriate processing. It is argued that students need to struggle with important mathematics that in turn facilitates the construction of knowledge. It is further argued that the way we construct mathematical tasks have consequences for how much effort students allocate to their task-solving attempt. [ABSTRACT FROM PUBLISHER]

This study aimed to reveal the effects of teaching with concrete learning objects taken from the history of mathematics on student achievement. Being a quasi-experimental study, it was conducted with two grade 8 classes in a secondary school located in Trabzon. The experimental group consisted of 27 students and the control group consisted of 25. Data were collected by using worksheets, an achievement exam and written opinion forms. The data from the achievement exam were analysed by using the Mann-WhitneyU-test while the data from written opinion forms were analysed through content analysis. The Mann–WhitneyU-test results showed a significant difference between the mean ranks of the experimental and control groups in favour of the former. Findings from the written opinion forms suggested that the students found the activities to be instructive and fun, enjoyed using concrete models in their classes, and learned from discovering the rules. It was also found that students had previously not engaged in similar activities and had only experienced the history of mathematics through the life stories and works of mathematicians and the representation of ancient numbers at the beginning of each unit. [ABSTRACT FROM PUBLISHER]

Basic arithmetic forms the foundation of the math courses that students will face in their undergraduate careers. It is therefore crucial that students have a solid understanding of these fundamental concepts. At an open-access university offering both two-year and four-year degrees, incoming freshmen who were identified as lacking in basic arithmetic skills were engaged in an experimental technology-enhanced workshop designed to provide them with a deeper understanding of arithmetic prior to their initial remedial coursework. Customized online content was created specifically for this experiment, and the first implementation (n= 27) yielded statistically significant improvement, not only from pre-test to post-test, but also in the subsequent remedial course. This paper also analyses the accuracy of students’ self-assessment from pre-test to post-test, as well as student attitudes about this experimental approach. [ABSTRACT FROM PUBLISHER]

One goal of an undergraduate education in mathematics is to help students develop a productive disposition towards mathematics. A way of conceiving of this is as helping mathematical novices transition to more expert-like perceptions of mathematics. This conceptualization creates a need for a way to characterize students' perceptions of mathematics in authentic educational settings. This article presents a survey, the Mathematics Attitudes and Perceptions Survey (MAPS), designed to address this need. We present the development of the MAPS instrument and its validation on a large (N = 3411) set of student data. Results from various MAPS implementations corroborate results from analogous instruments in other STEM disciplines. We present these results and highlight some in particular: MAPS scores correlate with course grades; students tend to move away from expert-like orientations over a semester or year of taking a mathematics course; and interactive-engagement type lectures have less of a negative impact, but no positive impact, on students' overall orientations than traditional lecturing. We include the MAPS instrument in this article and suggest ways in which it may deepen our understanding of undergraduate mathematics education. [ABSTRACT FROM AUTHOR]

Learning mathematics in a technology-rich environment enables us to revive classical topics which have been removed from the curriculum a long time ago. Both theoretical issues and applications can be studied with an experimental process. We present how envelopes of 1-parameter families of plane curves and some of their applications can be presented early in the curriculum either for pre-service teachers or for in-service teachers. This approach may be useful for students in an engineering curriculum. Working with technology yields important effects, such as reviving classical topics, broadening perspectives on already known topics, and enhancing the learner's experimental skills, where conversion between various registers of representation is an important issue. [ABSTRACT FROM PUBLISHER]

This paper describes a case study of two pure mathematicians who flipped their lecture to teach matrix determinants in two large mathematics service courses (one at Stage I and the other at Stage II). The purpose of the study was to transform the passive lecture into an active learning opportunity and to introduce valuable mathematical skills, such as debate, argument and disagreement. The students were told in advance to use the online material to prepare, which had a short handout on matrix determinants posted, as the lesson would be interactive and would rely on them having studied this. At the beginning of the lesson, the two mathematicians worked together to model the skill of professional disagreement, one arguing for the cofactor expansion method and the other for the row reduction method. After voting for their preferred method, the students worked in small groups on examples to defend their choice. Each group elected a spokesperson and a political style debate followed as the students argued the pros and cons of each technique. Although one lecture does not establish whether the flipped lecture model is preferable for student instruction, the paper presents a case study for pursuing this approach and for further research on incorporating this style of teaching in Science, Technology, Engineering and Mathematics subjects. [ABSTRACT FROM PUBLISHER]

In this article, we give two examples of creating portable chalkboards using chalkboard paint for students to use during cooperative learning. This provides a creative method for professors to facilitate active learning in the undergraduate mathematics classroom. [ABSTRACT FROM PUBLISHER]

This article provides ready-to-use supplementary material on recursive sequences for a second-semester calculus class. It equips first-year calculus students with a basic methodical procedure based on which they can conduct a rigorous convergence or divergence analysis of many simple recursive sequences on their own without the need to invoke inductive arguments as is typically required in calculus textbooks. The sequences that are accessible to this kind of analysis are predominantly (eventually) monotonic, but also certain recursive sequences that alternate around their limit point as they converge can be considered. [ABSTRACT FROM AUTHOR]

Teaching mathematics in university levels is one of the most important fields of research in the area of mathematics education. Nevertheless, there is little information about teaching knowledge of mathematics university teachers. Pedagogical content knowledge (PCK) provides a suitable framework to study knowledge of teachers. The purpose of this paper is to make explicit the perception of mathematics university teachers about PCK. For this purpose, a phenomenological study was done. Data resources included semi-structured interviews with 10 mathematics university teachers who were in different places of the mathematics university teaching experience spectrum. Data analysis indicated a model consisting of four cognitive themes which are mathematics syntactic knowledge, knowledge about mathematics curriculum planning, knowledge about students' mathematics learning and knowledge about creating an influential mathematics teaching–learning environment. Besides, it was found out that three contextual themes influenced on PCK for teaching mathematics in university levels which were the nature of mathematics subjects, university teachers' features and terms of learning environment. [ABSTRACT FROM PUBLISHER]

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations. [ABSTRACT FROM AUTHOR]

Studies have shown that at the end of an introductory statistics course, students struggle with building block concepts, such as mean and standard deviation, and rely on procedural understandings of the concepts. This study aims to investigate the understandings entering freshman of a department of mathematics and statistics (including mathematics education), students who are presumably better prepared in terms of mathematics and statistics than the average university student, have of introductory statistics. This case study found that these students enter college with common statistical misunderstandings, lack of knowledge, and idiosyncratic collections of correct statistical knowledge. Moreover, they also have a wide range of beliefs about their knowledge with some of the students who believe that they have the strongest knowledge also having significant misconceptions. More attention to these statistical building blocks may be required in a university introduction statistics course. [ABSTRACT FROM AUTHOR]

In spite of sustained efforts tertiary institutions implement to try and improve student academic performance, the number of students succeeding in first-year mathematics courses remains disturbingly low. For most students, the gap between their mathematical capability and the competencies they are expected and need to develop to function effectively in these courses persists even after course instruction. In this study, an instrument for identifying and examining factors affecting student performance and success in a first-year Mathematics university course was developed and administered to 86 students. The overall Cronbach's Alpha coefficient for the questionnaire was found to be 0.916. Having identified variables from prior research known to affect student performance, factor analysis was used to identify variables exhibiting the greatest impact on student performance. The variables included prior academic knowledge, workload, student approaches to learning, assessment, student support teaching quality, methods and resources. From the analysis, students' perceptions of their workload emerged as the factor having the greatest impact on student's performance, followed by the matriculation examination score. The findings are discussed and strategies that can be used to improve teaching and contribute to student success in a first-year mathematics course in a South African context are presented. [ABSTRACT FROM PUBLISHER]

Many students enter the Canadian college system with insufficient mathematical ability and leave the system with little improvement. Those students who enter with poor mathematics ability typically take a developmental mathematics course as their first and possibly only mathematics course. The educational experiences that comprise a developmental mathematics course vary widely and are, too often, ineffective at improving students’ ability. This trend is concerning, since low mathematics ability is known to be related to lower rates of success in subsequent courses. To date, little attention has been paid to the selection of an instructional approach to consistently apply across developmental mathematics courses. Prior research suggests that an appropriate instructional method would involve explicit instruction and practising mathematical procedures linked to a mathematical concept. This study reports on a randomized field trial of a developmental mathematics approach at a college in Ontario, Canada. The new approach is an adaptation of the JUMP Math program, an explicit instruction method designed for primary and secondary school curriculae, to the college learning environment. In this study, a subset of courses was assigned to JUMP Math and the remainder was taught in the same style as in the previous years. We found consistent, modest improvement in the JUMP Math sections compared to the non-JUMP sections, after accounting for potential covariates. The findings from this randomized field trial, along with prior research on effective education for developmental mathematics students, suggest that JUMP Math is a promising way to improve college student outcomes. [ABSTRACT FROM PUBLISHER]

The article discusses papers within the issue adapted from the Southern Hemisphere Conferences on the Teaching and Learning of Undergraduate Mathematics and Statistics with topics such as engineering, finance and group theory.

Rarely do university departments of mathematics redesign their basic mathematics courses. Through developing an online version of our associate degree in engineering in collaboration with Open Universities Australia, we redesigned the first in a sequence of five engineering mathematics courses. The online cohort proved different to our face-to-face experience. We embarked on a process of refining the unit using experiential learning and action research. The 13 week unit is delivered up to four times a year and this paper reviews the first 10 cycles of enhancements over 3 years and unpacks the layers of hypotheses underlying development decisions. Several category themes were identified with a focus on students, teachers and learning activities. Investment in online developments for mathematics can have multiple flow-on impacts for other teaching modes. Good curriculum design, regardless of environment, will always be a cornerstone of effective course development processes. [ABSTRACT FROM PUBLISHER]

This study forms part of a collaboration project between universities in South Africa and Sweden in which we investigate whether the emphasis in undergraduate mathematics courses for engineering students would benefit from being more conceptually oriented than a traditional more procedurally oriented way of teaching. In this paper, we report in some detail from two interviews with professional engineers, selected to represent two different ‘poles’ of engineering work. The aim was to explore different kinds of arguments regarding the role of mathematics in engineering work, as well as some common across contexts. Both interviewees feel that conceptual mathematics is more important for engineering work, although the role of the procedural aspect was seen by one of the interviewees also to be important, but in a very intricate way. [ABSTRACT FROM PUBLISHER]

Open book assessment is not a new idea, but it does not seem to have gained ground in higher education. In particular, not much literature is available on open book examinations in mathematics and statistics in higher education. The objective of this paper is to investigate the appropriateness of open book assessments in a first-year business statistics course. Data over two semesters of open book assessments provided some interesting results when compared with the closed book assessment regime in the following semester. The relevance of the results is discussed and compared with findings from the literature. The implications of insights gained for further practice in the assessment of mathematics and statistics is also discussed. [ABSTRACT FROM PUBLISHER]

The flipped classroom model is being used more frequently in undergraduate mathematics courses. As with any new teaching model, in-depth investigations of both various implementation styles and how the new model improves student learning are needed. Currently, many practitioners have been sharing their implementations of this model. However, there has not yet been an investigation of the various implementations of the model to discern general trends in this movement. With this research goal in mind, we conducted a study exploring various implementations of the flipped classroom model by interviewing 19 faculty members who experienced using this model at 14 different institutes. Results indicate that participants had similar motivations for implementation; however, subsequent implementations were different. In addition, we share participants' perspectives on (a) student learning of pre-requisite, procedural and conceptual knowledge, and (b) how this particular model promotes such knowledge developments. Finally, we provide suggestions for future implementations and research regarding this particular teaching model. [ABSTRACT FROM PUBLISHER]

In keeping with the national mandate of increasing graduates in the sciences in South Africa, a concerted effort in improving the first year experience becomes imperative. First year mathematics courses commonly provide the base knowledge necessary for progression in different degree programmes at university. Success in mathematics courses influences throughputs, retention and graduation rates of various degree programmes. Due to the highly complex and integrated nature of issues pertaining to improving teaching and learning in these courses, a multi-dimensional approach was conceptualized and implemented at the University of Pretoria. This paper reports on the development of a coherent framework, and the process and strategy for improving student success through a number of teaching and learning interventions in the first year mathematics courses, addressing the different dimensions of the framework. The process embarked upon resulted in a coherent, resource-focused approach with a replicable model for similar contexts. [ABSTRACT FROM PUBLISHER]

Over the last two decades, many Australian universities have relaxed their selection requirements for mathematics-dependent degrees, shifting from hard prerequisites to assumed knowledge standards which provide students with an indication of the prior learning that is expected. This has been regarded by some as a positive move, since students who may be returning to study, or who are changing career paths but do not have particular prerequisite study, now have more flexible pathways. However, there is mounting evidence to indicate that there are also significant negative impacts associated with assumed knowledge approaches, with large numbers of students enrolling in degrees without the stated assumed knowledge. For students, there are negative impacts on pass rates and retention rates and limitations to pathways within particular degrees. For institutions, the necessity to offer additional mathematics subjects at a lower level than normal and more support services for under-prepared students impacts on workloads and resources. In this paper, we discuss early research from the First Year in Maths project, which begins to shed light on the realities of a system that may in fact be too flexible. [ABSTRACT FROM PUBLISHER]

Supporting student success in entry-level mathematics courses at the undergraduate level has and continues to be a challenge. Recently we have seen an increased reliance on technological supports including software to supplement more traditional in-class instruction. In this paper, we explore the effects on student performance of the use of a computer software program to supplement instruction in an entry-level mathematics course at the undergraduate level, specifically, a pre-calculus course. Relying on data from multiple sections of the course over various semesters, we compare student performance in those classes utilizing the software against those in which it was not used. Quantitative analysis of the data then leads us to conclusions about the effectiveness of the software as well as recommendations for future iterations of the course and others like it. [ABSTRACT FROM PUBLISHER]

Mathematics has long been known to be problematic for university engineering students and their teachers, for example, Scanlan.[1] This paper presents recent data gathered from interviews with engineering students who experienced problems with mathematics and their lecturers during their transition through the first year in different programme contexts. Our interviews with the students reveal how they understand the relation between engineering and mathematics and we draw on the concept of ‘use- and exchange-value’ to explore this relationship more fully. This paper challenges both the pedagogical practice of teaching non-contextualized mathematics and the lack of transparency regarding the significance of mathematics to engineering. We conclude that the value of mathematics in engineering remains a central problem, and argue that mathematics should be a fundamental concern in the design and practice of first-year engineering. [ABSTRACT FROM PUBLISHER]