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In this study, we redesigned paper-and-pencil activities and adapted them to an introductory multivariate calculus course supported by 3D dynamic geometry software. We used semi-structured interviews and students’ written productions during the semester to analyse the use of technology in supporting students’ learning. Considering the work of two students, we find that the software has transformative potential but that it did not promote students’ learning as expected. We discuss possible reasons for this. [ABSTRACT FROM AUTHOR]

‘Group work’ is a vague description of an instructional activity, because many factors shape its character and effect on students. One important factor is group formation, that is, how groups are formed by the instructor. In this paper, we sought to better understand the variation of group work with respect to group formation by addressing: How do university mathematics instructors form groups over the course of a semester? To this end, we examined eight instructors' methods of forming groups in one multi-section introduction to proof course. Our findings include a classification of group formation methods and descriptions of how instructors varied their formation methods across the semester. Further, we sought to understand: How do students experience different group formation methods? We analysed interviews with 29 students from the eight instructors' classes and identified central themes among students' experiences of different group formation methods. Finally, we discuss the sometimes conflicting research on (when) which group formation method is most appropriate and offer our thoughts on how the differences between typical undergraduate and K-12 mathematics classrooms may contribute to different recommendations. [ABSTRACT FROM AUTHOR]

In this paper we studied the personal mathematical work of 14 students in initial teacher training, in their first year of studies at a public university in Chile, based on two tasks on complex numbers. These tasks were set in a Computer Aided Assessment System (CAA) based on Moodle and Wiris and it was proposed to solve them using different Computer Algebra Systems (CAS), such as GeoGebra, Symbolab, Photomath and Wolfram Alpha among others. Difficulties and potentialities were observed in the mathematical work of the students. In the difficulties, it was observed that the students had problems interpreting the information from the CAS and the CAA feedback because their previous mathematical knowledge was not solid enough to do so. In the potentialities, it was observed that different characteristics of the task, together with the articulation of two or more artifacts allowed students to give meaning to the mathematical objects involved. [ABSTRACT FROM AUTHOR]

Creativity and problem solving are considered to be twenty-first-century competencies, therefore promoting mathematical creativity should be an important part of school mathematics. The study presented in this paper is inspired by the notion of mathematical creativity and the utilization of multiple solution tasks (MSTs) to investigate students' creativity. Multiple solution tasks are mathematical tasks with an explicit requirement to be solved with different methods. For the purpose of the study, we used a textbook question which allows multiple solution pathways as our research instrument. This MST was administered to first-year mathematics students (18–20 years) along with additional questions related to their experience in school mathematics. The results of the study showed that students had difficulties complying with the demands of the task, but generally have a positive disposition toward MSTs. Moreover, the results underline the necessity of incorporating MSTs in mathematics classroom teaching to create a more coherent corpus of mathematical knowledge. The study also provides an example of how a regular textbook task can be used as an MST. [ABSTRACT FROM AUTHOR]

The increase in online education has implications for mathematics teacher educators who are tasked with teaching mathematics content and pedagogy to pre-service teachers (PSTs) who do not attend on-campus. Concerns about PSTs' content knowledge continues to be an ongoing issue, and ways to address this, especially in an online environment, require further investigation. While online courses offer accessibility and flexibility, it can be challenging to translate synchronous on campus teaching and experiences into the online environment. The importance of productive mathematical discourse has been identified as an appropriate pedagogy, yet the opportunity to engage in such discourse can be limited in the online environment. This paper provides the results from a case study of an online forum, which highlights how pre-service teachers can be engaged in productive online mathematical discussions, particularly when facilitated by the instructor's and other learners' interactions and presence. The findings indicated that social, cognitive, and teaching presence influenced the nature and frequency of online mathematical discourse. The study contributes to existing research through conceptualizing how productive mathematical conversations can occur outside of the classroom setting, with implications for mathematics teacher educators who are tasked with teaching content, pedagogy and practices to PSTs. [ABSTRACT FROM AUTHOR]

Numerous studies affirm the importance of students' attitudes for mathematical education. This study uses mathematical autobiographical essays to examine the mathematical experiences and attitudes of undergraduate students enrolled in mathematics classes at a Midwestern University in the United States. The essays of 41 students (20 men, 21 women) out of 170 enrolled in College Algebra, Precalculus, and Calculus were analyzed using the categories of (1) confidence in mathematics ability, (2) value of mathematics, and (3) liking of mathematics. This paper provides context and theoretical depth to student reports of their attitudes in these categories. Statements at the intersections of both confidence and liking, and value and liking provide additional insights. First, students who perceive mathematics as valuable are more likely to like it. Second, the vast majority of students talk about confidence and liking together, and with a positive correlation. Further, these students tend to talk about liking or not liking mathematics because of their perceived ability in mathematics. Our study suggests the importance of proactive teaching strategies to improve students' perceptions of the value, liking of, and their confidence in, mathematics, and also suggests self-efficacy as a potential theoretical basis for this work. [ABSTRACT FROM AUTHOR]

This paper set out to contribute to mathematics education research by elaborating on a methodology developed during a study, trying to understand, view and follow shifts in prospective teachers' discursive patterns. The methodology aims to illustrate and describe how prospective teachers adapt to the context of teaching through a flexible process. This flexible process is then described in the result as a narrative. It is argued that the methodology can be used in relation to different theoretical directions, such as research about beliefs, knowledge, or identity. Another contribution is that the methodology presented gives insights into bridging the gap between different analytical levels, micro and macro. With a theoretical foundation in 'Cultural Worlds' [Holland, D., Skinner, D., Lachicotte, W., & Cain, C. (1998). Identity and agency in cultural worlds. Harvard University Press.] the Social Semiotic approach of Systemic Functional Linguistics, SFL [Halliday, M., & Hasan, R. (1989). Language, context, and text: Aspects of language in a social-semiotic perspective (2nd ed.). Oxford University Press] is used as a methodological tool. SFL offers a toolkit that allows the analysis of meaning at the clause level to uncover how and why a speaker produces a particular wording rather than any other in a specific social practice. The paper aims to illustrate and describe how to go beyond findings in the micro-analysis and then present the result as a narrative case. [ABSTRACT FROM AUTHOR]

This paper aims to examine the extent to which pre-service teachers' mathematical connection skills were promoted through the course designed within the scope of pedagogies of practice (Grossman et al. [2009]. Teaching practice: A cross-professional perspective. Teachers College Record, 111(9), 2055–2100. https://doi.org/10.1177/016146810911100905) in the teacher education programme. Data were collected in a course that was designed to develop pre-service mathematics teachers' (PSTs) mathematical connections skills through a classroom teaching experiment with individual contributions to the whole class learning process. Thirty-six PSTs who were enrolled in the course participated in the present study. Initial and revised lesson plans were used as primary data collection instruments. In the most general sense, the findings demonstrated that most of the PSTs improved their mathematical connection skills during the course. The results of the current study provide insight into how the courses in undergraduate education should be implemented to give PSTs opportunities to develop their mathematical connection skills. [ABSTRACT FROM AUTHOR]

This paper reports on the development of the Teaching Mathematics for Social Justice beliefs (TMfSJ-B) scale and its psychometric properties. Preservice elementary education teachers enrolled in teacher education programmes at seven universities completed multiple surveys designed to understand the evolution of attitudes and beliefs about teaching mathematics. To our knowledge, the TMfSJ-B scale is the only survey specifically designed to measure pre- and in-service teachers' beliefs about teaching mathematics for social justice. Confirmatory factor analyses (n = 655) provided here indicate that the TMfSJ-B scale is a valid and reliable assessment of beliefs about social justice in mathematics teacher education, with three factors: (1) value of mathematics for all students; (2) integration of social justice into mathematics instruction; and (3) mathematics and its relation to inclusiveness. The purpose of this study was to validate measures produced by the TMfSJ-beliefs scale and discuss its potential use in teacher education programmes and research. [ABSTRACT FROM AUTHOR]

Teachers' and students' values influenced the teaching and learning process in the mathematics classroom. Few studies were conducted to compare teachers' and students' value in mathematics learning. This paper explores Malaysian mathematics teachers' and students' values in mathematics learning. Fifty-three mathematics teachers and 354 secondary school students participated in the online survey. This study reports ten semantic differential items and an open-ended question related to mathematics learning. An independent t-test was conducted on the semantic differential items. Thematic coding analysis was performed to code the open-ended questions based on the framework of three intersecting values: mathematical values (the discipline), mathematics educational values (the pedagogy), and general educational values (the ethical and moral principles). The findings showed significant mean differences in mathematics educational values between teachers and students. No significant difference was found in mathematical values between teachers and students. Moreover, the qualitative data confirmed the similarities in mathematical values and differences in mathematics educational values between teachers and students. The current study also discusses possible reasons for the differences in teachers' and students' values. The findings imply that teachers' and students' values are crucial in enhancing teaching and learning in the mathematics classroom. [ABSTRACT FROM AUTHOR]

International literature is increasingly disclosing the relevance of cultural aspects within the processes of teaching and learning mathematics. Knowledge is inextricably linked to the activities in which the subjects engage, and must be considered in relation to the socio-cultural context wherein the activity takes place. Literature reveals the relationship between the background culture (e.g. language, nationality, etc.) of prospective elementary teachers and their beliefs about mathematics and its teaching. In this paper, we define culture with reference to the wider discussion from cross-cultural psychology literature about cultural values, and we investigate if and how differences in individuals' cultural values are related to prospective teachers' beliefs about being successful in mathematics and in its teaching. We adopt a questionnaire from the studies by Schwartz to measure participants' values. We assign each prospective teacher of our sample to a cluster of beliefs and we analyse how the beliefs of prospective teachers are related to their values. Results show that cultural values and beliefs about mathematics are related, while this is not the case for beliefs about the teaching of this subject. [ABSTRACT FROM AUTHOR]

Teacher practice is rife with tensions. Tensions around what and how best to teach, how to manage situations with students, and how to manage situations with colleagues, administrators, and parents. These tensions are often seen as pairs of opposing internal and external forces: this assessment is better, but it takes a lot of time. These forces are inescapable, and teachers have to learn to either manage them or resolve them. In this paper we look closely at tensions through the lens of opposing forces and, more interestingly, how teachers either learn to live with them or work to resolve them. And, in particular, we look at how this work differs if the tension exists between internal forces, external forces, or a tension between an internal and external force. Drawing on case studies of seven different participants we dive deep into the murky world of the lived experiences of teachers to understand better the way that tensions contribute to their beliefs, decisions, and actions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the learning experiences, beliefs and motivations of students in classes where the mathematics teachers have received support for using inquiry-based learning activities. Data were collected from 248 students in the age-range 11–16 using electronic questionnaires. Our results show that key features of inquiry-based mathematics were only moderately reflected in these students' beliefs about the subject, their dispositions towards mathematics were less positive across the transition from primary to secondary school, and with respect to motivation this decline was stronger for girls than for boys. Furthermore, medium to strong correlations between belief- and motivation subdomains were found, for instance, students who view mathematics as a creative subject and/or have a growth mindset of mathematics also tend to find this subject enjoyable and perceive it as useful. Finally, our results indicate that inquiry-based teaching has a potential for fostering positive dispositions towards mathematics, as students who often experience inquiry-related activities in class also tend to see mathematics as a creative and interesting subject that will be useful for them in the future. [ABSTRACT FROM AUTHOR]

This paper targets patterns of expressed emotional directions towards mathematics. By using TIMSS data and recoding two of the instruments, we compared responses from the same student cohort in Grade 4 (11 years old) and Grade 8 (15 years old) in Sweden. Two hypotheses were tested: negative emotions including negative motivations increase with age, and there will be gender differences. By using statistical analyses, including Cramer's V, the two hypotheses were confirmed. The results also illuminated patterns within the responses, such that boys and girls in Grade 4 differ in expressed intrinsic motivation but not extrinsic motivation. There are also indications that girls might be more likely to form negative emotional directions between grades 4 and 8 than boys, and some implications of this is discussed. [ABSTRACT FROM AUTHOR]

Although research in mathematics education has yielded a great number of results on mathematics teachers' beliefs in recent decades, it is still an open question as to whether these teachers' beliefs are stable or not. However, since mathematics teachers' beliefs are perceived to be the default of their classroom practices, it is important to gain insight into these beliefs and into their development. In this paper, we contribute to this field of research by providing an approach designed to analyse a change in teachers' belief systems. We study a sample of 20 prospective teachers during their first practical experience. We analyse the structure of teachers' belief systems with interviews and a questionnaire set twice within one year. To analyse development of these teachers' belief systems, we assigned the young teachers of the first and second survey to clusters that were developed in a former study and re-analysed for our purposes here. Our results show considerable changes in mathematics teachers' central beliefs during their first practical experience. However, a general pattern of belief changes is not found. [ABSTRACT FROM AUTHOR]

Improving the mathematical skills of undergraduate Engineering and Information Technology (EIT) students has been an enduring challenge for educators. Lecturers continually implement new strategies that try to overcome the challenges encountered in teaching university students with varying levels of success being encountered. This paper provides insight into how educators have handled the challenges students face in their mathematics learning journey. In addition, the paper highlights some of the methods used to improve the performance of these students as problem solvers, as well as the development and implementation of creative techniques that may have made the situation more successful and less challenging. The literature was searched through using a structured literature review approach based on the primary databases: Scopus; the Education Research Complete (ERC); and the Education Resources Information Centre (ERIC). This study followed the three main steps of a systematic literature review: retrieval, selection, and analysis of the literature. Analysis of the literature showed that there is still a need to further develop and deliver innovative pedagogical strategies for teaching mathematics to EIT students. It is recommended that an interdisciplinary approach, inclusive of technology, will lead to increased understanding, improve performance and, ultimately, enhance student satisfaction. [ABSTRACT FROM AUTHOR]

Mathematical memes are an Internet phenomenon with an epistemic potential noteworthy for the teaching and learning of mathematics. The aim of this paper is to conceptualize this phenomenon on an empirical base, elucidating its educational potential. To pursue this goal, a two-year ethnographic research study has been conducted in a selection of social media communities that exchange mathematical memes. In the paper, a twofold theoretical lens is used to investigate the phenomenon from a memetic perspective focused on the general characteristics of Internet memes, and from a cultural perspective considering the Web 2.0 environment. Analysis tools derived from the memetic and cultural theoretical background are applied to an instrumental case study, to reconstruct how mathematical memes activate the interaction among members of an online community. The investigation elicits key characteristics of the phenomenon, provides grounds to conceptualize mathematical memes as hybrid representations of mathematical statements, and sheds light on the traits of the epistemic culture they establish in the community. [ABSTRACT FROM AUTHOR]

Despite research demonstrating positive effects of writing on student achievement, writing-to-learn remains under-utilised in mathematics and mathematics teacher education. In this paper, we explore how writing-to-learn tasks in a mathematics methods course influenced elementary preservice teachers’ beliefs about teaching in two domains: mathematics and writing. Preservice teachers engaged in weekly writing-to-learn tasks designed to support their understanding of mathematics content and mathematics pedagogy over the course of a semester. Using survey data, we found no significant difference in their beliefs from pre to post. Such null findings led us to ask new qualitative questions, resulting in a concurrent mixed methods design. Through qualitative analyses of student writing, we found several meaningful and non-meaningful practices for engaging in writing-to-learn tasks. This research supports teacher educators, teachers, and researchers to understand how to implement writing-to-learn tasks and to support students towards meaningful engagement in the tasks. [ABSTRACT FROM AUTHOR]

Even prior to COVID-19, there had been an increase in the use of technology in undergraduate mathematics education to support the development of mathematical understanding and skill, and to enhance the student learning environment. Technology-enhanced resources can be used to provide formative assessment which can help students identify gaps in their knowledge and point towards appropriate action. In this paper, we report on the evaluations of a number of technology-enhanced resources that used formative assessment techniques and were provided for students attending non-specialist first-year undergraduate mathematics modules in Ireland. Analysis of the evaluations and consultation with literature identified 12 decisive factors that encourage students’ successful engagement with the resources, covering aspects such as the educational context, affordances and types of technology, technological communication and formative assessment. These factors constitute important knowledge for practitioners as they can use them in the planning, design, and development of technology-enhanced resources for use in their modules. The measures observed during the evaluations indicated both positive and negative engagement providing a link between observations and resultant student engagement, which was found lacking in previous studies on student engagement. [ABSTRACT FROM AUTHOR]

In order to learn the concept of statistical techniques one needs to run real experiments that generate reliable data. In practice, the data from some well-defined process or system is very costly and time consuming. It is difficult to run real experiments during the teaching period in the university. To overcome these difficulties, statisticians developed simple and very economical experiments, which can be performed in the class by the students. Stone studied the variation, bias, stability and statistical quality control through the Blind Paper Cutting (BPC) experiment. In this article, the Blind Paper Cutting experiment is demonstrated on the basis of basic principles of experiment design. A BPC experiment is performed considering different factors, and important factors to optimise the response are identified through complete factorial design. The appropriate response model using important factors has been constructed. [ABSTRACT FROM AUTHOR]

The paper is the result of extensive research carried out among Czech students and focuses on a conception of infinity. The questionnaire survey was taken by 861 students ranging from grades 7 to 13. The aim of the research was to describe the development of students' conceptions of infinity. These conceptions are built on the intuitive phenomenon of the horizon. We monitor the proportional representation of these conceptions in four combinations of views (into the distance and in depth) and contexts (arithmetical and geometrical). It can safely be maintained that the development of the proportional representation of the earliest conception comprehension of the concept of infinity, the so-called natural infinity, is not concurrent with the students' age. The development of the proportional representation of the conception of actual infinity is non-decreasing, at least in the view into the distance, in both contexts with respect to age. In general, it can be stated that the proportional representation of individual conceptions of infinity is strongly dependent on both context and view. [ABSTRACT FROM AUTHOR]

In this paper, we will show a characterization for pairs of commensurable diagonals of regular n-gons. This result gives a positive answer to a recent question. [ABSTRACT FROM AUTHOR]

Introducing the distributivity property of multiplication over addition is a well-known challenge in mathematics education, especially in primary school. As a contribution, this paper presents the results of a cycle of design research that focuses on the design, implementation and evaluation of a modelling activity in which 2nd-grade students are introduced to the key concept of distributivity of multiplication over addition. The results show that modelling activities designed through the heuristics of didactical phenomenology, guided reinvention and emergent modelling may support the accessibility of distributivity, as these students were able to reinvent this concept solving an experientially significant problem. This result can be attributed to a combination of several factors: the choice of a realistic and rich problem, that stimulated students to elaborate formal mathematical concepts mathematizing their informal solving strategies, rooting new understandings in experientially real phenomena; the use of a suitable artifact, that presented mathematics as a means of interpreting and understanding reality and increasing the opportunities for observing mathematics outside of the school context; the role of the teacher, who guided students in reinventing mathematics in an active way. [ABSTRACT FROM AUTHOR]

Capitalizing on cognitive load theory and learning by analogy, we propose two instructional methods to learn a complex linear equation (e.g. two-step equation) by building on prior knowledge of a simpler linear equation (e.g. one-step equation). We will examine the proposal theoretically in this paper. In line with the design principles of cognitive load theory, we propose to strengthen students' prior knowledge of simpler linear equations before they learn complex linear equations with the aid of worked examples. Because a subset of the complex linear equation shares the same schema as the simpler linear equation, students can draw on their schema for the simpler linear equation to understand the complex linear equation, thus alleviating the limitation on working memory load. Based on the principles of learning by analogy, we place a simpler linear equation and a complex linear equation side-by-side and label the solution procedure of both linear equations to encourage active analogical comparison between these two equations. Making both the simpler linear equation and the complex linear equation visible to learners may help to reduce cognitive load demands in retrieving the simpler linear equation in order to facilitate the learning of the complex linear equation. [ABSTRACT FROM AUTHOR]

In this paper a class of regular surfaces, generated by rotation and twist of a rigid line segment, is considered. Firstly a complete description of those surfaces by means of local charts is obtained and then its orientability is characterized in terms of the number of half-twists, taking into account their direction (clockwise or counterclockwise) since changes in the sense of twist are allowed. Indeed a procedure to construct nonorientable surfaces is provided. Illustrative examples and computer-generated plots using the free software Maxima are included. [ABSTRACT FROM AUTHOR]

LEGO® stop-motion animations were developed to engage first-year university students in the topics of matrices, vectors, linear geometry and linear transformations. LEGO® provided a versatile medium through which a wide range of concrete and abstract concepts could be physically demonstrated in both two and three dimensions. It was also familiar to most students, making the mathematics videos accessible and fun. When designed according to key principles around reducing cognitive load in multimedia learning, animations in education have been shown to increase learning compared to static images for concepts involving dynamic systems and processes. Multimodal approaches including video and narration have also been shown to be beneficial to students in terms of demonstrating mathematical problem solving. The videos created in this project were a mix of conceptual explanations and problem-solving examples, where the word problems were physically modelled before being converted to formal mathematical notation and then solved. Some animations were short clips that were integrated into online lecture recordings, others were stand-alone videos designed either as short introductions to topics or examples of common questions. This paper describes the theory underpinning the design of the animations. [ABSTRACT FROM AUTHOR]

This article is a follow-up to an earlier paper on the mathematics support learning tutorial programme (SLT), an intervention programme at The University of Queensland that targets students considered to be at risk of failing Calculus and Linear Algebra I, the first tertiary level mathematics subject at The University of Queensland. The first paper (Hillock, P., Jennings, M., Roberts, A., & Scharaschkin, V. (2013). Amathematics support programme for first-year engineering students. International Journal of Mathematical Education in Science and Technology, 44(7), 1030–1044) reported on the inaugural programme implemented in 2012. This article provides an update of the progress of the SLT since 2012. We provide statistics for the subsequent 12 semesters to Semester 2, 2018 and describe the evolution of the SLT since its implementation. Statistical analysis of the additional data and student feedback indicate that the SLT continues to have a positive impact on student learning, with weak students making significant gains from attending the programme. [ABSTRACT FROM AUTHOR]

This paper focuses on the students who are registered in the University of Pretoria's academic development programme, named the Four-year Programme (FYP). The programme was introduced as a gateway for students who are underprepared but have the potential to succeed and then continue their studies into the mainstream science programmes. Our research focuses on measuring the change in the academic maturity of these students. In the theoretical framework that we developed, academic maturity is subdivided into two components namely non-subject based maturity and subject based maturity (mathematical maturity). This paper focuses on measuring non-subject based academic maturity. The survey used for this purpose is called the Student Academic Readiness Survey (STARS), taken at the beginning of the year and after the first semester respectively. The results of the surveys are compared to measure the change in students' views. Results show that in all constructs there is a surprising decline in students' perceptions regarding their own abilities over the first semester at university. We use the Dunning–Kruger effect to explain this unexpected decline, in that students seem to develop a more realistic view of their own maturity, which in itself can be seen as a growth in academic maturity. [ABSTRACT FROM AUTHOR]

In Australian universities, mathematics is often an essential core subject for disciplines such as engineering, information technology and science. These first-year mathematics subjects may have large enrolments of which a significant number fail the subject and repeat. While interventions to retain students are common, those who have failed before and are repeating the subject are often overlooked. In this paper, we undertook a statistical analysis based on demographic and performance data of six cohorts of students undertaking a first-year university-level mathematics subject. We found that pass rates for students on their second attempt were significantly lower than pass rates for students enrolled on their first attempt, and that age, degree enrolled in and pathway into university were predictors for who was more likely to repeat. Students in their second attempt were less likely to complete all assessment items and were more than twice as likely not to attend the final exam. We argue for an increased focus on and efforts targeting repeating students; that demographic data are used to predict students likely to repeat before they have failed a subject, and that this group of students, as well as those repeating, are supported to increase their chances to pass. [ABSTRACT FROM AUTHOR]

This paper uses a framework established by one of the author's in a previous study and refines it to develop a rubric for mathematics tutor training in third-level education. The paper further opens the discussion on the necessity of, and practical methods for, providing high quality, efficient tutor training at third-level. Models of mathematics teacher knowledge are unanimous in stating that solely possessing strong subject matter knowledge (SMK) is not sufficient for one to be deemed to be a good teacher. Despite the necessity of high quality tutoring at third-level, past research indicates that though tutors' SMK is usually of high quality, their ability to transform their knowledge so that students will understand the content, as well as their general teaching skills (e.g. board work, speech), often has room to improve. This paper discusses the development of the rubric and its use in a mathematics tutor training programme at third-level. The tutor training programme was found to lead to improvements in multiple elements of the tutors' teaching. [ABSTRACT FROM AUTHOR]

In this paper we discuss the important Abel's summation formula, which is a very powerful tool for analysing series of real or complex numbers. We derive from it an integral test which may be useful in cases where the classical integral test may not be applied. We also discuss how this new integral test may be used when one is dealing with alternating series of real or complex numbers. The main goal of this note is to provide a supplementary material for courses dealing with infinite series of real or complex numbers. [ABSTRACT FROM AUTHOR]

This paper reports on themes that arose in an investigation of university lecturers' views on the teaching of linear algebra. This focus on themes was the initial part of a study concentrating on four areas: What is important to teach in a first course in linear algebra? Are there teaching methods which are particularly suited for such a course? Are there tools that should/should not be used; and do the answers to these questions vary according to the degree (Engineering or Mathematics) the students follow? Questionnaire data was coded using thematic analysis which generated 11 themes related to the four questions above. The Results section presents the themes. The Discussion section considers the themes as a whole; splits – dualities – in teaching linear algebra; students' challenges with abstraction; aspects of doing mathematics; and pedagogical issues. [ABSTRACT FROM AUTHOR]

In this paper, we pursue a materialist approach to tool use to account for the material dimensions of learning mathematics with technology while decentralizing the human body from activity. We root this vision in theoretical advances on embodied mathematics learning and tool use that have been recently offered as alternatives to traditional theories of learning that draw attention only to the individual learner or the mind. Focus is on the ways that learners, tools and mathematics are entangled in provisional reconfigurations of the world and on how the tool partakes in this movement. Starting from our theoretical commitment, we discuss some episodes from a research study in which grade 9 students were introduced to function through a graphical approach. The study took advantage of graphing motion technology, which allows working with couples of position over time graphs, capturing spatio-temporal relationships. The episodes provide insight into the complex nature of learning in this context, valuing the mobile way that mathematics emerges from activity with the tool. [ABSTRACT FROM AUTHOR]

This paper examines the effect of the use of dynamic geometry environments on young children's (ages 5–6, kindergarten/grade 1) thinking about angle. It provides a detailed description of introductory sessions of a geometry unit about angle, during which children worked in a whole classroom setting in which they could interact directly with Sketchpad through an interactive whiteboard. Using Sfard's [2008, Thinking as communicating: Human development, the growth of discourses and mathematizing. Cambridge: Cambridge University Press] communicational approach, an attempt is made to show how children developed a reified discourse on angles and were able to develop an understanding of angle as 'turn', that is, of angle as describing an amount of turn. This discourse focused on the behaviour of a pre-constructed sketch which instantiated both dynamic and static representations of angle. Extending prior research on children's difficulties in unifying static and dynamic conceptions of angles, this study provides one way of establishing a relationship between angle-as-turn and angle-as-shape conceptions. Gestures and motion played an important role in children's developing conceptions of angles. It presents implications of considering young children's embodied forms for communications along with their verbal communication for understanding their mathematical thinking. [ABSTRACT FROM AUTHOR]

Partial Latin squares constitute an interesting approach to improve the teaching of different subjects in the mathematics classroom. This paper delves into this topic by introducing the use of a Dynamic Geometry System to deal with these combinatorial structures as a source of loci of points in the Euclidean plane. It is assumed to this end that the distance matrix among all these points contains a zero-diagonal symmetric partial Latin square with at least one non-zero filled cell per row. [ABSTRACT FROM AUTHOR]

The initial training of mathematics teachers in Spain can start in the final year of the mathematics degree when the students, and future teachers, select courses related to teaching the discipline. At that time, since over the next two years they will probably be certified to enter the profession, it is worth asking oneself what mathematical knowledge they should have and what is needed to teach at the secondary education level. This paper describes one case study in which a group of final-year students solves a task containing aspects of mathematical understanding that are present in solving problems using GeoGebra. This was done using a solution protocol intended to emphasize the different possibilities offered by GeoGebra, presenting three approaches for the solution: dynamic, numeric and algebraic. We noticed that the use of technology is key to bringing out some of aspect and to demonstrating an understanding of mathematics for teaching. [ABSTRACT FROM AUTHOR]

Nomographs (or nomograms, or alignment charts) are graphical representations of mathematical relationships (extending to empirical relationships of data) which are used by simply applying a straightedge across the plot through points on scales representing independent variables, which then crosses the corresponding datum point for the dependent variable; the choice among independent and dependent variable is arbitrary so that each variable may be determined in terms of the others. Examples of nomographs in common current use compute the lift available for a hot-air balloon, the boiling points of solvents under reduced pressure in the chemistry laboratory, and the relative forces in a centrifuge in a biochemical laboratory. Sundials represent another ancient yet widely familiar example. With the advent and ready accessibility of the computer, printed mathematical tables, slide rules and nomographs became generally redundant. However, nomographs provide insight into mathematical relationships, are useful for rapid and repeated application, even in the absence of calculational facilities, and can reliably be used in the field. Many nomographs for various purposes may be found online. This paper describes the origins and development of nomographs, illustrating their use with some relevant examples. A supplementary interactive Excel file demonstrates their application for some simple mathematical operations. [ABSTRACT FROM AUTHOR]

To date, the way that academic misconduct is manifested in undergraduate mathematics coursework has been unexamined in the literature, with the consequence that policy and preventive education can fail to address it appropriately. This paper describes the particular features of the responses expected in mathematical tasks and provides concrete examples of what lapses of integrity look like in this context. The intent is to lay groundwork for discussion and a response to this issue from within our discipline. [ABSTRACT FROM AUTHOR]

This paper explores analysis of feedback data collected from student consultations at two mathematics support centres at universities in Australia and Ireland. Unstructured text data was collected over six years and includes qualitative data on student queries collected during the consultations from mathematics and statistics related subjects. Topic modelling and clustering algorithms are used to uncover key themes in the data across stages. Common areas of difficulty experienced by undergraduate students at both universities are investigated and a comparison between them is shown. The results suggest that, despite institutional differences, there is considerable overlap in the types of mathematical and statistical difficulties experienced by students in their first and second year of university at these institutions. We discuss how the ability to uncover such common mathematical and statistical themes with the aid of text mining techniques can be used to improve the support provided by mathematics support centres in terms of providing an efficient and effective service. The code for analyses at both institutions is provided in a GitHub repository so other academic support centres may use it. Outcomes of this analysis have implications for mainstream mathematics and statistics instructors who wish to gain further insights into their students' learning. [ABSTRACT FROM AUTHOR]

Undergraduate students usually study Laurent series in a standard course of Complex Analysis. One of the major applications of Laurent series is the classification of isolated singular points of complex functions. Although students are able to find series representations of functions, they may struggle to understand the meaning of the behaviour of the function near isolated singularities. In this paper, I briefly describe the method of domain colouring to create enhanced phase portraits to visualize and study isolated singularities of complex functions. Ultimately this method for plotting complex functions might help to enhance students' insight, in the spirit of learning by experimentation. By analysing the representations of singularities and the behaviour of the functions near their singularities, students can make conjectures and test them mathematically, which can help to create significant connections between visual representations, algebraic calculations and abstract mathematical concepts. [ABSTRACT FROM AUTHOR]

This paper presents apps designed to assist students in understanding and developing proofs in geometric theorems. These technologies focus on triangle congruence, triangle similarity and properties of parallelograms. Focus group discussions and initial testing of the apps revealed that the apps offered a more engaging medium for learning proving and were capable of facilitating proof-writing skills in geometry. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper investigates the teaching practices used by university mathematics teachers when lecturing, a topic within university mathematics education research which is gaining an increasing interest. In the study, a view of mathematics teaching as a discursive practice is taken, and Sfard's commognitive framework is used to investigate the teaching practices of seven Swedish university mathematics teachers on the topic of functions. The present paper looks at the discourse of mathematics teaching, presenting a categorization of the didactical routines into three categories –explanation,motivationandquestion posingroutines. All of these are present in the discourses of all seven teachers, but within these general categories, a number of different sub-categories of routines are found, used in different ways and to different extent by the various teachers. The explanation routines includeknown mathematical facts,summary and repetition, different representations,everyday language, andconcretization and metaphor; the motivation routines include reference toutility,the nature of mathematics,humourandresult focus; and the question posing routines includecontrol questions, asking forfacts,enquiriesandrhetorical questions. This categorization of question posing routines, for instance, complements those already found in the literature. In addition to providing a valuable insight into the teaching of functions at the university level, the categorizations presented in the study can also be useful for investigating the teaching of other mathematical topics. [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]

This paper addresses a topic within university mathematics education which has been somewhat underexplored: the teaching practices actually used by university mathematics teachers when giving lectures. The study investigates the teaching practices of seven Swedish university teachers on the topic of functions using a discursive approach, the commognitive framework of Sfard. In the paper a categorization of the construction and substantiation routines used by the teachers is presented, for instance various routines for constructing definitions and examples, and for verifying whether an example satisfies a given definition. The findings show that although the overall form of the lectures is similar, with teachers using ‘chalk talk’, and overt student participation limited to asking and answering questions, there are in fact significant differences in the way the teachers present and do mathematics in their lectures. These differences present themselves both on the level of discursive routines and on a more general level in how the process of doing mathematics is made visible in the teachers’ teaching practices. Moreover, I believe that many of the results of the study could be relevant for investigating the teaching of other mathematical topics. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra and geometry, like other branches of mathematics, are interrelated. [ABSTRACT FROM AUTHOR]

The paper discusses the interaction between intuitive biases of probabilistic thinking and mathematical knowledge. It would appear that students may answer numerical problems correctly but falter on simple descriptive solutions. Students appear to relinquish formal knowledge for simpler heuristics when attempting to describe the outcome of an event. This paper explores some possible theoretical causes of the dissociation between procedural and estimative abilities. [ABSTRACT FROM AUTHOR]

This paper, the second of a two part article, expands on an idea that appeared in literature in the 1950s to show that by resticting the domain to those complex numbers that map onto real numbers, representations of functions other than the ones in the real plane are obtained. In other words, the well-known curves in the real plane only depict part of a bigger whole. This expanded representation brings new insight into visualising complex roots. The suggestion is that this new approach be introduced to students firstly through relating the path in history and secondly by imparting the visual presentation as exposed in the paper to offer a richer teaching and learning approach to the topic. Furthermore this approach provides a new way of employing technology to visualise concepts and curves that were previously not noticed. [ABSTRACT FROM AUTHOR]

With Simson's 1753 paper as a starting point, the current paper reports investigations of Simson's identity (also known as Cassini's) for the Fibonacci sequence as a means to explore some fundamental ideas about recursion. Simple algebraic operations allow one to reduce the standard linear Fibonacci recursion to the nonlinear Simon's recursion that is equivalent to Simson's identity and then further to a nonlinear recursion dependent only on a single preceding term. This leads to a striking nonrecursive characterization of Fibonacci numbers that is much less well-known than it should be. It is then discovered that Simson's recursion itself implies a family of linear recursions and characterizes a class of generalized Fibonacci sequences. [ABSTRACT FROM AUTHOR]

This paper is concerned with identifying and assessing the impacts of various technologically enriched approaches to mathematics learning. Its purpose is to address the usefulness of emerging knowledge to enhance practice, and to contribute to theorizing about technology-based learning. Hence the main drivers are intentions to identify and elaborate on obstacles, errors, and unresolved problems on the one hand, and positives and insights on the other. The paper tells a cautionary tale about expectations of technology-enhanced learning, while simultaneously uncovering a rich base from which to theorize and test new appreciations of what is involved when students, technology, and mathematics connect in learning settings. A summary assessment is that this area still very much represents work in progress, but there is now a heightened realization, at least among those not transfixed by technological blindness, that the search for ultimate answers will require much greater exploration of machine–mathematics–learner relationships. [ABSTRACT FROM AUTHOR]