ALBERT

Description: The article investigates the role of attention in the reflective thinking of school mathematics teachers. It analyses teachers’ ability to pay attention to detail and use their mathematical knowledge. The vast majority of teachers can be expected to have an excellent knowledge of mathematical techniques. The question examined here is whether this kind of knowledge might structure their attention in such a way that the emphasis on procedures deflects their attention from the essential details. Four groups of participant teachers from New Zealand, Hong Kong, Germany and Ukraine were given a mini-test containing seven simple mathematics questions. Most questions in the test were provocative in the sense that they looked like routine questions but in fact had some catch. The results of the test were startling—the vast majority of the participants gave incorrect answers to most questions in the test. After the test, the participants were given a short questionnaire to reflect on their performance on the test. Their responses were analysed using the theories of selective, divided and focused attention and Mason’s concept of the discipline of noticing. Implementations of the results of the study in assessment and professional development are discussed.

Description: Issues of disconnect and isolation from other disciplines seem to plague numerous subjects taught in high schools today. Although they share common roots, mathematics, natural sciences and humanities appeared to have developed rich, but separated curricula with only occasionally emphasized and seemingly weak connections between their respective contents. The absence of truly meaningful context that allows students to build connections to material they are learning often results in procedural, factual and regrettably short-term knowledge. One approach to repair the disarray within our educational structure may be by developing integrated projects that span not only across different facets of one particular subject, but across multiple disciplines as well. Most educational reforms have been strongly supporting the idea of integration of curricula, emphasizing that the noticeable lack of connections between the subjects is affecting not only mathematics, but also language arts, social studies and various other disciplines. The primary purpose of this article is to describe an integrated project that successfully merged the teaching standards of mathematics, social studies and language arts, while utilizing technology as the means to an innovative learning medium for high school students. The focus of the article is the development process of an integrated project, more specifically the obstacles that needed to be overcome for its successful completion. Through this particular example of a mathematics-based integrated project, we demonstrate how content standards of various disciplines can be effectively combined to produce a meaningful teaching and learning context. It is our hope that the presented development process can be generalized and applied in various collaborative teaching efforts.

Description: If assessment drives learning and the closed book examination dominates the pattern of assessment for undergraduate mathematics (as it does in the UK), lecturers need to ensure that examinations reflect the learning they value. This article uses a mixed method approach to explore lecturers’ views of the closed book examination in relation to other assessment methods, their preferences for different types of methods and the extent to which they discriminate between stronger and weaker students. A survey of staff views confirms the dominance of closed book examinations, but an accompanying interview study shows hidden complexity to this view. In particular a tension between the potential and the reality of examinations is uncovered and suggestions are made for resolving that tension.

Description: It is widely recognized that mathematical typesetting is more difficult than typesetting in most other disciplines due to the need for specialized mathematical notation and symbols. While most mathematicians type mathematical documents using LaTeX, with varying levels of proficiency, students often use other options or handwrite mathematics. Here, we investigate students’ perceptions of the mathematical editor available since Word 2007. This investigation is timely as there is anecdotal evidence that mathematicians do not think that Word is capable of quality mathematical typesetting, and there have not been many studies in the literature on this topic. In this case study, we ask the following questions: Is Word a suitable tool for students to typeset mathematics? And: How does Word compare to other mathematical typesetting packages students have used? We asked master’s level students to typeset mathematics in Word and then comment on their experience, relating to the ease of use, quality of output and overall experience. We also asked the students to compare Word to other tools they may have used in the past. The results show that the current versions of Word are indeed capable of producing quality mathematical typesetting, that the learning curve is not high, and that Word should not be dismissed as a tool for typesetting mathematics. While there were concerns that editing takes longer than in other tools and that typesetting in general takes time, overall students were positively surprised by Word’s mathematical features and commented favourably.

Description: Fermi problems are problems which, due to their difficulty, can be satisfactorily solved by being broken down into smaller pieces that are solved separately. In this article, we present different sequences of activities involving Fermi problems that can be carried out in Secondary School classes. The aim of these activities is to discuss problem-solving strategies and to introduce modelling processes in compulsory education.

Description: This article draws on extensive experience working with secondary and tertiary teachers and educators using an applet to display rational polynomials (up to degree 7 in numerator and denominator), as support for the challenge to deduce as much as possible about the graph from the graphs of the numerator and the denominator. Pedagogical and design aspects of this work are described.

Description: The provision of some level of Mathematics Learning Support is now commonplace in the majority of Higher Education Institutions in the UK and Ireland. Most of these supports were initially established with the aim of trying to address the problem of large numbers of first-year students with weak mathematical backgrounds. The centres provide students the opportunity to overcome mathematical issues in their transition from school to Higher Education. This article presents findings from a recent quantitative study at a university, where the range of supports available and levels of engagement have increased dramatically since they were first offered in 2007. We investigated if the Mathematics Support Centre there was still helping the students who needed it most; in particular, we considered students’ mathematical backgrounds, the number and length of their visits and their end of year module grades.

Description: It is well documented in both the British press and mathematics education literature that many consider A-levels (qualifications for secondary school leavers in England, Wales and Northern Ireland) to be insufficient preparation for undergraduate mathematics study in the UK. A-level Mathematics and Further Mathematics are not solely intended for those wishing to go on to study undergraduate mathematics, though this is often the only mathematics study and preparation available to, and done by, most students. A number of British universities now require applicants to perform well on entrance examinations or ‘extension’ papers in addition to achieving certain grades in A-level Mathematics and, sometimes, Further Mathematics. This article outlines research which used the Mathematical Assessment Task Hierarchy (Smith et al. , 1996) and builds upon the work of Darlington (2014) to describe the mathematical skills required to answer the questions in extension papers. These are then contrasted with skills required at A- and undergraduate level. Each of the three extension papers analysed (Advanced Extension Awards, Sixth Term Examination Papers and university admissions tests) were found to differ from each other, as well as from A-level and undergraduate examinations. This suggests that there are benefits for students doing such papers before university mathematics study. The advantages of doing these papers are not only for admissions tutors, who can use them as additional indicators of candidates’ aptitudes, but also for the students themselves as they gain insights into the nature of undergraduate mathematics and related further mathematical challenges.

Description: We consider here a number of ideas for the classroom or lecture theatre associated with the mensuration of solids. In particular, the volumes of various tetrahedra are obtained in an indirect manner (by way of prisms and square-based pyramids). This activity develops problem-solving skills, spatial visualization and a from-first-principles approach to integration. Several suggestions are also provided for taking things further.

Description: We aimed at investigating knowledge revealed by the tutors of a continuing education course for mathematics teachers offered at a distance. Initially, we observed the work of 32 tutors over a year, in order to typify their interventions in discussion forums with course participants. From the results, we offered training to a new group of tutors, in order to improve actions that we considered were below our expectations. Between August 2012 and July 2013, we observed six tutors who are the subjects of this research. An adaptation of the Technological Pedagogical Content Knowledge (TPACK) theoretical framework was used in their training. This qualitative research made use of the observation of the tutors’ work, which was analysed by means of discursive typologies found in tutors’ interventions in discussion forums. Our analysis indicated that affective and attitudinal components play a key role in tutoring in this context.

Description: This study addresses the embodied approach of convergence of numerical sequences using the GeoGebra software. We discuss activities that were applied in regular calculus classes, as a part of a research which used a qualitative methodology and aimed to identify contributions of the development of activities based on the embodiment of concepts, with the use of software, to the understanding of the convergence of numerical sequences and to the transition of mathematical thinking from elementary to advanced. Such activities had an exploratory nature and were constructed based on the theoretical frameworks of Advanced Mathematical Thinking and the Three Worlds of Mathematics. For the data collection were used the recordings of the computer screens, audio recordings of the discussion between students, student records in the responses of the activities and field notes of the researchers. The data were analyzed considering relations between the embodiment of the concept of convergence and proceptualization processes and axiomatization and also considering the transition from elementary mathematical thinking to advanced. The results indicate that the activities enabled the formation of the mental image that the convergence of sequences occurs when the sequence terms are approaching a certain value, and thus propitiated the embodiment of the concept of convergence and established cognitive roots for formal definition of convergence by the limit and further theoretical development. They also indicate that relations between different representations were established, which contributed to abstraction of the concept of convergence of sequences and thus for the transition between elementary and advanced mathematical thinking.

Description: Complex functions, generally feature some interesting peculiarities, seen as extensions of real functions. The visualization of complex functions properties usually requires the simultaneous visualization of two-dimensional spaces. The multiple Windows of GeoGebra, combined with its ability of algebraic computation with complex numbers, allow the study and exploration of complex functions of one complex variable not only through the traditional techniques but also by the use of Domain Colouring techniques. Here, we will show how we can use GeoGebra to explore complex functions, using several representations obtained by the creation of new tools, which complement the ones already provided by this software. Our tools were designed to be used by students attending the first academic year of engineering or science courses. They can and should be used as an educational tool in collaborative learning environments. The main advantage in their use is the promotion of deductive reasoning (conjecture/proof) in individual terms, and the promotion of mathematical discussions in collective terms. In the literature review done so far, few references were found involving the study of this topic by the use of a single software.

Description: For the past 4 years, we have been involved in a project that aims to enhance the teaching and learning of experimental analysis and statistics, of environmental and biological sciences students, through computational programming activities (using R code). In this project, through an iterative design, we have developed sequences of R-code-based activities that have been implemented in three institutions in Mexico and Portugal, in eight postgraduate and five undergraduate courses; these are hands-on sets of tasks in R script that include computer programming work and are meant to be carried out collaboratively (a sample of an Analysis of Variance (ANOVA) activity is given). Here, we focus on results that indicate that students tend to enjoy the courses, lose their fear of statistics, as well as develop some competencies for applying statistical methods and using computational tools, such as R, on their own data that may deepen their understanding of the biological phenomena they have to analyse.

Description: This study focuses on teacher’s Knowledge for Teaching Mathematics with Technology (KTMT), paying a special attention to teacher’s representational fluency. It intends to characterize how the teacher uses and integrates the different representations provided by the graphing calculator on the process of teaching and learning functions at the high school level. Specifically, it intends to understand the balance established between the use of the different representations, and the way these representations are articulated. The study adopts a qualitative approach undertaking one teacher case study. Data were collected for two school years, at 10th and 11th grades, and included class observation, semi-structured interviews and documents gathering. Data analysis was mainly descriptive and interpretative in nature, considering the problem under study. The conclusions reached reveal an active use of the graphical and algebraic representations and a scarce use of the tabular representation. The lack of balance on the use of representations also includes the work within a representation. In this case, the graphical representation is the only one that was explored. The conclusions also indicate a flexible articulation between the two representations usually used. It was possible to identify different patterns on the use of the representations and a frequent use of an interactive approach, marked by repeated alternations between representations. Globally, this study emphasizes teacher’s KTMT and raises questions about the impact of technology on teacher’s representational fluency and about the difference between a numerical and a tabular representation.

Description: A wide body of literature has highlighted how high achievement in mathematics in secondary school does not necessarily motivate students to both choose and succeed on mathematically demanding programmes at post-compulsory level. The recent Enterprising Science project (Archer et al. (2015, J. Res. Sci. Teach., 52, 922–948)) and before that, the ASPIRES project (Archer et al. (2013, London: Kings College)), have both highlighted that access to science capital is perhaps more important than prior achievement in shaping students’ aspirations and their future trajectories in Science, Technology, Engineering and Mathematics. In this article, we critically analyse the notion of science capital and its role in mediating students’ choice of and experience of studying mathematically demanding degree programmes at university. Drawing on data from the TransMaths project, we present two cases—Stacey and Elton—who are both enrolled on the same ‘Mathematics for Physics’ course at university. We show that although both discuss access to science capital in narrating their choice of degree, they do so in different ways and this invariably interplays with different forms of identification with ‘Mathematics for Physics’. We conclude that there is a need to re-conceptualize science capital so that the dialectic relationship between its exchange and use value is theorized more fully. Whilst some students may access science capital as a means to accumulate capital (e.g. qualifications) for its own sake (exchange value), others appear to recognize the ‘use value’ of science learning and knowledge and this produces different forms of engagement with science (and mathematics). We therefore argue that authoring oneself in the name of a STEM identity is crucial in mediating how one perceives science capital. Finally, we argue that mathematics should be a central part of this framework since it significantly contributes to the exchange value of science as a form of capital (especially Physics), but it also offers use value in scientific labour (e.g. in modelling scientific problems).

Description: This article builds on previous results of the Transmaths studies concerning transmissionist teaching practices—and especially adds the significance of students’ perceptions of these practices—in their association with students’ declining dispositions for studying mathematics. It addresses a gap in this work, and the literature in general, regarding the relationship between teachers’ and students’ perceptions of pedagogy. Drawing on data analyses from a recent, large survey of teaching and learning mathematics in secondary schools, the article: (a) demonstrates and validates two new measures of perceptions of transmissionist practices, as experienced from students’ and teachers’ perspectives, (b) investigates the comparability of these two measures, and (c) identifies their associations with students’ dispositions to study mathematics. Analysis draws on measures of students in Years 7 to 11 (involving 13,000+ students) and from 132 of their mathematics teachers, and shows low correlation at class level and negligible correlation at student level. Results of regression analysis confirm previous work with older students, i.e. that teachers’ self-reported transmissionism is negatively associated with learners’ dispositions, but adds that students’ perceptions of transmissionism are much more strongly negatively associated with these dispositions, and largely mediate the effect of teachers’ (self-reported) transmissionism. Further, the differences between year groups and gender show how girls and older learners suffer significantly larger negative effects. The article concludes with a brief discussion of these complexities and some implications for students’ trajectories and transitions into (and out of) mathematics.

Description: In this editorial we explain the background to the research papers reported in this special issue, and to some extent how each paper relates to this body of work. In particular we outline the research projects and research teams that worked together between 2006 and 2014 on projects that related to the theme of transition, and we provide the knowledge base on which these papers build.

Description: Drawing on large-scale survey data and interviews with students during their first year at university, and case studies in their institutions, we explore the problems faced by students taking mathematically demanding courses, e.g. physics and engineering. These students are often taught mathematics as a service subject by lecturers of mathematics. Analysis of students’ perceptions of transition suggests that ‘the lecture’ in Higher Education continues to pose problems. Thematic analysis of interview data shows that these problems relate to the way lectures involve ‘time pressure’ and ‘lack of dialogue/interaction’ which are practices that we associate with transmissionist pedagogy generally and can also create negative dispositions. A case study of one mathematics course for engineering that we argue made a difference is presented, and conclusions drawn for developing practice which are especially pertinent with the introduction of the Teaching Excellence Framework to monitor and assess teaching in universities.

Description: Students at a large urban community college enrolled in seven classes of an experimental remedial algebra programme, which integrated study skills instruction and collaborative problem solving. A control group of seven classes was taught in a traditional lecture format without study skills instruction. Student performance in the course was measured by a common 25-question multiple-choice exit exam. After controlling for cognitive and affective differences among students (N = 233), when all other variables were held constant, students in the experimental classes answered two more questions out of 25 correctly than students in the control group classes. The results suggest problem solving and study skills are potential areas for further improvement in learning.

Description: In this article we report and discuss a contextual problem solving task that was proposed to a class of 8th grade (13–14-year-old) students. These students had been developing a reasonable experience in the use of the spreadsheet to model relations within contextual problems and chose to use this tool to solve the mentioned problem, engaging in the process of translating relations between variables and combining them in chained models, while working with fractions, multiples and expressions. We intend to highlight the role of the spreadsheet in students’ processes of variable identification and translation of the problem conditions, their numerical approaches to algebraic models and their experimental forms of finding solutions to equations.

Description: This theoretical study is an attempt to explore the potential of the dynamic and interactive mathematics learning environments (DIMLE) in relation to the technological pedagogical content knowledge (TPACK) framework. DIMLE are developed with intent to support learning mathematics through free exploration in a less constrained environment. A typical DIMLE software package has interactivity and dynamism as key affordances; these are especially suitable for enhancing learning and teaching with technology of the essentially dynamic mathematics concepts. Moreover, we propose that DIMLE and their affordances should be studied under the TPACK framework because this framework is explicit in considering technology-supported mathematics learning as a qualitative add-on as contrasted to what would be a simple totalling of technological, pedagogical and mathematical knowledge. As an example, we focus in our discussion on using a DIMLE in order to support learner in development of the limit concept.

Description: The present article reports a study concerning the analysis of 19 activity plans (we call them ‘scenarios’) developed by mathematics teacher educators-in-training for the pedagogical use of digital tools. The development of these scenarios took place during their training program and was designed as an activity for increasing reflection, for expressing creative pedagogical ideas and for an active engagement in the design of curricula enriched with the use of technology. Our analysis showed that the trainee teacher educators deconstructed and reconstructed respective parts of the formal curriculum regarding the mathematical concepts they chose to embody in their scenarios.

Description: In this research we present the use of some technologies in problem solving activities (at different secondary school grades), aimed at finding a model for a geometric configuration, and representing this model in various ways: through a construction, through a Cartesian graph, etc. The task is part of a teaching experiment, in which students used paper and pencil, and technological tools: a sensor and a calculator (at a lower grade), GeoGebra and TI-Nspire (at a higher grade). We show results in terms of the passage from static to dynamic representations and back, to observe how technology may foster dynamic thinking for students solving mathematical problems. Data suggest that the dynamic features of technology support the genesis of conjectures, and their validation or refutation, along with the choice of independent and dependent variables. Results are used to prepare materials for teacher training in an e-learning Moodle platform (Comenius EdUmatics Project).

Description: Teachers of mathematics recognize the difficulty of reaching every student when the range of student abilities puts a considerable strain on the classroom discussion and time. In a response to the problem, students are grouped so that those with greater mathematical aptitude help those who have difficulties. While this approach is to be appreciated, it tends to mean that the more able students have less opportunity to explore further their own initiatives in mathematics, while those who have more difficulties find themselves on the receiving end with little opportunity to be in the role of enriching the mathematics experience for everyone, including themselves. A ‘multiple-centres’ approach is designed to overcome these problems. In this variation of differentiated instruction, all students get the chance to engage the material from a vantage point and at a level they find interesting and challenging as a consequence of their selecting extensions of the teacher’s initial focus problem. This article will present some findings of 11th year (roughly Fifth Form) average mathematics students at a US Independent School in transforming the standard quadratic equation to represent fountain parabolic trajectories, which was the teacher’s focus problem, along with some multiple-centre investigations they chose. A further set of opportunities with commentaries providing additional centres for student inquiry are included.

Description: One-third of the current A-level mathematics curriculum is determined by choice, constructed out of ‘applied mathematics’ modules in mechanics, statistics and decision mathematics. Although this choice arguably involves the most sizeable instance of choice in the current English school mathematics curriculum, and it has a significant impact on students’ post-compulsory study of mathematics, it is not well understood how this choice is navigated. This article explores how mathematics teachers perceive each of these three areas of applied mathematics, how widely each of the modules is offered and in what ways perception might be connected to provision. Data from an online questionnaire and follow-up interviews demonstrate that teachers are influenced by a wide range of factors, including strategic concerns and views on the relative worth of each strand. The results also highlight the presence of inertia in centres’ provision. In this way, this article offers some insight into current perception and provision of applied mathematics in England, and speaks to contemporary debates about curriculum content and reform. It argues that students’ exposure to the powerful utility of mathematics is often unhelpfully steered or limited at a critical point in their education.

Description: As mathematicians, we assign rigid meanings to words that may have a variety of interpretations in common language. This article considers meanings of ‘if’ and ‘or’ from everyday English that have caused students to misinterpret mathematical statements, and that are consistently overlooked by instructional materials in addressing students’ mistakes. To fill this gap, this article presents three studies for the classroom that engage students in confronting the differences between mathematical and everyday meaning in statements of implication and statements of disjunction.

Description: Some mathematical activities and investigations for the classroom or the lecture theatre can appear rather contrived. This cannot, however, be levelled at the idea given here, since it is based on a perfectly sensible question concerning distributional approximations that was posed by an undergraduate student. Out of this simple question, and subsequent investigation, a wealth of mathematical and statistical material is covered. We go on to discuss the educational benefits of carrying out activities such as the one suggested here.

Description: In this article, we report on the findings of an exploratory study into the experience of undergraduate students as they learn new mathematical models. Qualitative and quantitative data based around the students’ approaches to learning new mathematical models were collected. The data revealed that students actively adopt three approaches to understanding a new mathematical model: gathering information for the task of understanding the model, practising with and using the model, and finding interrelationships between elements of the model. We found that the students appreciate mathematical models that have a real world application and that this can be used to engage students in higher level learning approaches.

Description: Mathematical conjectures and theorems are most often of the form $$P(x)\Rightarrow Q(x)$$ , meaning $$\forall x,P(x)\Rightarrow Q(x)$$ . The hidden quantifier $$\forall x$$ is crucial in understanding the implication as a statement with a truth value. Here P ( x ) and Q ( x ) alone are only predicates, without truth values, since they contain unquantified variables. But standard textbook instruction on implication, in particular in writing negations, relies mainly on truth tables, treating P and Q as statements themselves with their own truth values. The lack of careful and thorough explanations on handling implications of the form $$P(x)\Rightarrow Q(x)$$ creates difficulties for students, in particular in proof by contradiction, where one begins with the negation of the statement to be proved. Through analysis of interesting errors involving hidden quantifiers in implication, this article offers ways to improve standard instruction to treat these quantifiers correctly and explicitly.

Description: We report here on students’ views of example generation tasks assigned to them in two first year undergraduate Calculus courses. The design and use of such tasks was undertaken as part of a project which aimed to afford students opportunities to develop their thinking skills and their conceptual understanding. In interviews with 10 students, we found that on the whole they viewed the example generation tasks as unfamiliar and sometimes difficult, but also as beneficial for promoting conceptual understanding and independent thinking. In addition, some students characterized these tasks as ‘the backwards ones’.

Description: There is growing support for making the study of mathematics to the age of 18 years compulsory for all young people in England. This article aims to inform this debate through new insights into historic A-Level Mathematics participation trends. We analyse full-year cohorts of 16-year-old students from the Department for Education's National Pupil Database from 2004 to 2010, a total of just over 4.5 million young people. Using a cohort-tracking approach, we aim to better understand the flow of young people through upper secondary mathematics education. Earlier work identified General Certificate of Secondary Education (GCSE) attainment as the strongest predictor of A-Level Mathematics participation. In this article, we show that the percentage of students progressing to A-Level by GCSE grade has not changed significantly over the period in question, with some exceptions. This implies that the increase in A-Level Mathematics numbers is largely explained by the growing proportion of higher GCSE grades. We discuss the implications for policy that this raises, e.g. the possible impact of making GCSE mathematics more demanding.

Description: We demonstrate how the DEWIS e-Assessment system may use embedded R code to facilitate the assessment of students’ ability to perform involved statistical analyses. The R code has been written to emulate SPSS output and thus the statistical results for each bespoke data set can be generated efficiently and accurately using standard R routines. This enables students’ answers, generated from their application of SPSS, to be marked and appropriate feedback supplied back to them automatically. Staging is used between different parts of the e-Assessment to replicate the natural stages of a complete statistical analysis. This allows students the freedom to work away from the e-Assessment; they may view the relevant stage an unlimited number of times prior to submitting their answers for that stage. The technical challenges of setting up the e-Assessment in this way are discussed as well as the rationale for adopting this pioneering approach.

Description: In seeking to align the everyday decision-making and lesson delivery of secondary teachers to current research in mathematics education, this piece provides an application of theory into classroom practice. By focusing on a sample of 13 quantitative and qualitative research studies of pedagogical best practice published since 2000, a set of recommendations about improving teaching and learning in the secondary mathematics classroom are offered.

Description: Research on writing in mathematics has shown that students learn more effectively in an environment that promotes this skill and that writing is most beneficial when it is directed at the learning aspect. Writing, however, necessitates proficiency on the part of the students that may not have been developed at earlier learning stages. Research has indicated though that the burden placed on teachers and learners to master this skill is compensated by the mathematical learning in such an environment. Techniques to successfully integrate writing in the mathematics classroom can be varied. This study is conducted on students in an introductory differential equations class in which a reformed approach is adopted be it in the topics discussed, the textbook used, the technology employed or the assignments/exams given. More precisely, the article explores the effect of writing on improving student understanding of particular topics in differential equations and investigates the development of the students’ writing skills.

Description: Learning in the cellular phone environment enables utilizing the multiple functions of the cellular phone, such as mobility, availability, interactivity, verbal and voice communication, taking pictures or recording audio and video, measuring time and transferring information. These functions together with mathematics-designated cellular phone applications facilitate authentic learning based on real-life phenomena, and widen the range of mathematical activities possible to carry out. This research shows that the cellular phone provides the mathematics students with rich, diverse and colourful learning environment in and out of the classroom. The participating students worked with the cellular phone to carry out activities involving formal mathematical phenomena and at the same time authentic real-life phenomena which they modelled mathematically. Doing so, they worked individually and collaboratively with diverse, specific and general mathematical concepts and at the same time practiced specific and general mathematical procedures, which expanded their mathematical knowledge and meta-knowledge. The cellular phone enabled the students to work with various representations of mathematical objects which encouraged them to investigate these objects independently or with the guidance of their teacher. The students also could tackle advanced mathematical objects intuitively based on their visual representation and actual realization in real-life situations.

Description: Based on the results of a survey of the science and mathematics students at our university, we observed that students do not consider mathematics and science to be creative endeavors, though the traditional artistic disciplines rank high in this regard. To address this problem in perception, the authors used photography as a means to encourage students to find the deep-rooted connections between science and mathematics and the arts. The photography project was used in a formal classroom setting as well as an outside activity, i.e. in a more informal setting. In this article, we describe the project and its outcomes.

Description: This article investigates the attitudes and beliefs towards studying mathematics by university level students. A total of 970 randomly chosen, first year, Estonian bachelor students participated in the study (of which 498 were science students). Data were collected using a Likert-type scale questionnaire and analysed with a respect to field of study (science and non-science). Results of this study show that science students have a more positive attitude towards mathematics.

Description: No risk, no fun—betting on sports events costs the gamblers a lot of money and brings excellent profits to those who offer the bets. Among the people who bet on a regular basis, the proportion of young adults is frighteningly high. We now suggest a concept (as part of a basic mathematics course) for acquiring the necessary mathematical knowledge to be able to look behind the screen of the gambling game, as betting via the Internet has become increasingly popular. Learners should organize a sports event (or a simulation of such an event), found in several different types of betting offices and offer odds. Other participants receive virtual money in order to bet. When the results are fixed, the learners calculate their gains (profits) and their losses. As a consequence, the participants learn to use mathematics as a necessary tool for real-life situations by considering the exploring teaching method. Results of an experimental class who had to discuss this topic are shown below. First, we start with some background information about bets and the mathematical calculation of odds, then go on with a suggestion for a self-organized learning process including a kicking simulation in the classroom to understand the mathematical and ethical field of gambling, and we offer some more ways of reflecting the problem and learning about probability.

Description: Much work is currently being undertaken to investigate the effectiveness of informal, ‘drop-in’ mathematics support services. This can be a difficult task; students are individuals with different expectations and backgrounds, and support that is effective for one student, may be less effective in relation to another. This article investigates the effectiveness of the mathematics support service at Cardiff University by statistically analysing the relationship between student confidence and ability before and after attending a session. Some covariates, such as the effect of time and the student rating of the tutor, are also explored.

Description: A new application of logarithmic differentiation is presented, which provides an alternative elegant proof of two basic rules of differentiation: the product rule and the quotient rule. The proof can intrigue students, help promote their critical thinking and rigorous reasoning and deepen their understanding of previously encountered concepts. The author also stresses the advantages of presenting the formulas for the product rule and the quotient rule in the forms recommended here over other forms which more commonly appear in the current mainstream calculus textbooks.

Description: This article uses Rasch analysis to explore the efficacy of a questionnaire designed to assist university teaching staff in identifying those Level 4 students most in need of mathematics support. The students were all taking a mathematics module as part of their first year Computing curriculum, and the questionnaire explores the students' previous mathematical experience and asks them to evaluate their basic skills using Likert scale responses. Analysis of the 124 student responses identifies those questionnaire items that do not fit the Rasch model well and indicates some areas where further improvements to the questionnaire can be made.

Description: This article considers the role of socio-cognitive conflict in addressing students’ mechanics preconceptions. A discussion problem was posed to sixth-form students taking an introductory mechanics unit. The problem involved a child choosing either to slide down a smooth slide or to jump off the top. Students were invited to predict which route would be quicker. Their experience suggested that jumping would be faster, but they struggled to use their knowledge of mechanics to explain why sliding (in the absence of friction) would be slower. The resolution of the cognitive conflict between their reasoning about the physical situation and their everyday experience led to a better understanding of resolved forces and how they affect acceleration.

Description: The quadratic mean, geometric mean and harmonic mean of two numbers a and b are obtained as weighted averages by using segments parallel to the bases a and b of a trapezoid.

Description: It is the case that some activities claiming to reside under the STEM umbrella do not, in fact, give participants the opportunity to engage in anything other than routine mathematics. With this in mind, we explore here the potential for developing and then delivering STEM activities based on the discipline of mathematical epidemiology. We argue that this field does indeed provide us with a rich source of such activities which, in addition to allowing students to encounter some interesting and demanding mathematics, encourages non-trivial forays into aspects of the biological sciences, computer technology, chemistry, medicine and statistics. Projects may be tailored to the needs of both secondary-school pupils and undergraduates. The examples presented here will, as well as providing ideas for use in the classroom or lecture theatre, be utilized in order to highlight a number of the key issues associated with the STEM agenda.

Description: This article adapts the problem-solving model developed by Pólya to investigate and give an innovative approach to discuss and solve an optimization problem in geometry: the Regiomontanus Problem and its application to football. Various mathematical tools, such as calculus, inequality and the properties of circles, are used to explore and reflect on the different aspects of the problem and its solution. In addition, other than the traditional calculus approach to solve this problem, an elegant geometric approach is introduced.

Description: Damped harmonic oscillations appear naturally in many applications involving mechanical and electrical systems as well as in biological systems. Most students are introduced to harmonic motion in an elementary ordinary differential equation (ODE) course. Solutions to ODEs that describe simple harmonic motion are usually found by investigating the roots of a corresponding characteristic polynomial and ultimately written out as a time series to be graphed. Although this approach is standard, students often fail to connect the relationship between position and velocity that one achieves using phase plane analysis. The purpose of this article is to illustrate how the phase plane analysis and transformations can be used to gain a deep understanding of damped harmonic oscillations.

Description: Trigonometry is one of the topics in mathematics that the students in both high school and pre-undergraduate levels need to learn. Generally, the topic covers trigonometric functions, trigonometric equations, trigonometric identities and solving oblique triangles using the Laws of Sines and Cosines. However, when solving the oblique triangles, Mollweide's formula is most likely to be omitted from the discussion. Mollweide's formula—which exhibits a cyclical nature—is particularly useful in checking one's result after solving an oblique triangle since all six components of the triangle are involved. It is interesting to note that proving Mollweide's formula can be performed without words. Furthermore, the Law of Tangents can be derived straightforwardly from this equation. In this article, we revisit Mollweide's formula and provide classroom examples where this equation comes into alive. In addition, we suggest that this seemingly less-known equation is to be included in the mathematics syllabus on the topic of Trigonometry.

Description: The international debate about experimental approaches to the teaching and learning mathematics is very current. While number theory lends itself naturally to such approaches, elementary geometry can also provide interesting starting points for creative work in class. This article shows how simple considerations about right triangles and the golden section can be a starting point for a mathematical didactic laboratory.

Description: This article describes an activity that can be carried out in one of the final classes of Italian secondary schools. The aim of the activity is to stimulate pupil curiosity, demonstrating that Mathematics is not a barren subject and allowing the students, according to an interdisciplinary point of view, to investigate some literary works and the History of Mathematics.

Description: Courses in mathematical modelling are always in need of simple, illustrative examples. The Michaelis–Menten reaction kinetics equations have been considered to be a basic example of scaling and singular perturbation. However, the leading order approximations do not easily show the expected behaviour, and this note proposes a different perturbation approach to the same equations. The alternative analysis contains regular as well as singular perturbation and shows a simple case where a two-scale singular perturbation solution is insufficient. Additional analyses suitable for student projects are outlined.

Description: Studies incorporating technology into the teaching of trigonometry, although sparse, have demonstrated positive effects on student achievement. The optimal sequence for integrating technology with teacher-led mathematics instruction has not been determined. Our research investigated whether technology has a greater impact on student achievement and attitudes if it is implemented before or after whole class teaching. The curriculum context of the study was a set of learning objects (CLIPS: Trig) designed to support student learning of transformations of trigonometric functions. The software includes functional features identified in prior research: it relieves students of the tedium of creating graphs by hand; sliders give students control of the simulations within program parameters; there are easy transitions between algebraic and graphic representations; the environment is dynamic; animation and visualization are included with graphing functions. Twenty Canadian classrooms ( N = 489 grade 11–12 students, aged 17–18 years) were randomly assigned to two instructional sequences: CLIPS: Trig followed by whole-class teaching (CLIPS early treatment) and whole-class teaching followed by CLIPS: Trig (CLIPS late treatment). We found that in the pre-test to post-test comparisons, students who experienced CLIPS: Trig after whole-class teaching of core concepts learned more than students who began the unit with technology-supported simulations. However, there were no statistically significant differences in the pre-test to delayed post-comparisons. Beginning the trigonometry unit with CLIPS: Trig enhanced the impact of whole-class teaching, while beginning with whole-class teaching enriched students’ technology experience. The findings suggest that a tight integration of whole-class and technology-assisted instruction is preferable.

Description: When one uses Google (and many people do this!), the result of the query is a list of sites that have something to do with the item one is looking for. The specific sites are always more or less on the top, so it is not necessary to have a look on hundreds of sites to read something relevant and informative. How can Google manage this? How does Google come to the suggested list? This article is primarily written for teachers and lecturers who want to share the idea of PageRank with students without having complications arising from ‘concepts of higher mathematics’ like eigenvectors or eigenvalues. The basis is a special limit theorem (concerning Markov chains) which can be used unproved in school in order to come to interesting and elementary applications of mathematics. This example also provides a very good chance for cross-linking several mathematical fields: stochastics (probabilities, etc.), linear algebra (vectors, matrices, etc.) and analysis (limits, etc.). Another focus of this contribution is to make more visible the use of mathematics in modern society. This seems to be necessary because mathematics disappears more and more from societal perception in spite of the fact that its role rises in importance (but in most cases hidden) in our lives, it is surely a so-called key-technology.

Description: One of the biggest problems in teaching Mathematics is the motivation of students (Kloosterman, 1997; Lambic, 2008). The core of this problem is insufficient comprehension of reasons by students, why they want to learn Mathematics and in which way they could apply acquired knowledge of Mathematics (Southwell, 1994; Musto, 2008). Since programming is closely connected to Mathematics, this try has been compiled to make Mathematics closer to students with a help of working with software C++ Builder. The students from higher grades of elementary schools, high school students and university students have been participating in this project. Without previous knowledge of programming languages, the students were given tasks to create a certain computer program. The students have been making programs by inventing algorithms and setting visual components of the program, while the teacher, on their request, has been doing programming as part of the work. At the end of the project, the students have shown much better attitude towards the Mathematics class than before, and they have also obtained certain skills and knowledge in the programming field.

Description: This study examines the approaches displayed by pre-service mathematics teachers in their experiences of constructing mathematical modelling problems and the extent to which they perform the modelling process when solving the problems they construct. This case study was carried out with 35 pre-service teachers taking the Mathematical Modelling course, offered for the first time as an elective course in a secondary mathematics teacher training department in a public university in Turkey. The data were obtained from the modelling problems prepared by the pre-service mathematics teachers in the last 3 weeks of the Mathematical Modelling course, the video recordings of these problems’ presentations and the pre-service teachers’ responses to the questions they were asked about their presentations. When analyzing the participants’ approaches to construct and solve mathematical modelling problems, the main focus of attention included the appropriateness of the problems for modelling and of the solutions to the modelling process, as well as the approaches displayed in these stages. The study’s results demonstrated that the pre-service teachers considered various criteria when constructing modelling problems such as relevance to life, the possibility of being solved by mathematical knowledge and attracting attention and were found to be successful in understanding and simplifying the modelling process, however, they had difficulties in the interpreting and validating steps.

Description: A geometric model is introduced which satisfies the Euclidean parallel postulate as well as all of Hilbert's axioms except the Side-Angle-Side axiom. This model provides several teaching opportunities in those Euclidean geometry classrooms that use the axiomatic method. In presenting these models at the same time as the more familiar R 2 , R 3 , and Poincaré models, students may be less tempted to assume that familiar constructs (lines, trianges, etc.) allow them to rely on familiar assumptions.

Description: This article presents the mathematical approach of the optimal strategy to win the ‘Release the prisoners’ game and the integration of this analysis in a math class. Outline lesson plans at three different levels are given, where simulations are suggested as well as theoretical findings about the probability distribution function and its mean value.

Description: Over the last decade the ‘mathematics problem’ (students lacking basic mathematical skills on entry into higher education), and proposed solutions of this problem have been widely debated. One method to help combat this issue has been the introduction of mathematics support centres across higher education institutions. This article describes the results of an e-mail survey distributed to 40 universities to gain information regarding their mathematics support provision, with particular focus on how universities measure the effectiveness of the support available. With ever increasing pressure for impact to be validated, this article will give a contemporary standing of the current mechanisms adopted by mathematics support centres (nationally and internationally) used to determine effectiveness.

Description: In TMA, Oldknow (2009, TEAMAT , 28, 180–195) called for ways to unlock students’ skills so that they increase learning about the world of mathematics and the objects in the world around them. This article examines one way in which we may unlock the student skills. We are currently exploring the potential for students to ‘see’ mathematics in the real world through ‘marking’ mathematical features of digital images using a dynamic geometry system (GeoGebra). In this article we present, as a partial response to Oldknow, preliminary results.

Description: Engineering Students Understanding Mathematics (ESUM) is a developmental research project at a UK university. The motivating aim is that engineering students should develop a more conceptual understanding of mathematics through their participation in an innovation in teaching. A small research team has both studied and contributed to innovation, which included small group activity, a variety of forms of questioning, an assessed group project and use of the GeoGebra medium for exploring functions. Perspectives of community of practice and inquiry, and documentational genesis underpin the research approach. Issues and findings to date from the project are presented.

Description: Teaching mathematics associated with the repaying of loans has always been an awkward task—it seems superficially easy, but it contains pitfalls both for the learner and for the unwary teacher. This short article reviews some of the mathematics, and notes an interesting result.

Description: The article discusses aspects of approaches taken to increase engagement using interactive teaching elements and includes survey results from consecutive sets of student cohorts who have trialled resources contained in an Electronic Student Toolkit for Engagement in Engineering Mathematics under development at the University of Leeds. The survey results show strong support for the inclusion of a range of interactive approaches for improving engagement. Further to this, a summary of results from a quantitative study comparing engagement with an out-of-lecture online teaching and assessment tool, when used both as a formative tool and a summative tool, is included. These results demonstrate that there can be substantial engagement by students with online formative assessment tools when students feel it is integral to their course. Furthermore, engagement can be further improved when a small summative mark is associated with each task (with over 91% of the cohort actively engaging).

Description: In this article, we show a dynamic graphic in Excel that is used to introduce an important concept in our subject, Statistics I: the probability density function. This interactive graphic seeks to facilitate conceptual understanding of the main aspects analysed by the learners.

Description: With the London 2012 Olympics coming up, now is a good time for teachers to turn to track and field athletics for an application that has the potential to be intrinsically interesting to many pupils in schools new undergraduates. In this article, curves, well-known amongst former generations of pupils, are revisited by looking at the markings on the track for competitors in a range of events. Reciprocal spirals and involutes of circles emerge. These describe staggered starts and curved start lines, respectively. Modelling and other practical tasks based on measurements laid down by the IAAF are discussed.

Description: Much research and many papers on mathematics pedagogy have discussed assessment and, in particular, the need to provide a varied diet of methods by which students are assessed for the award of their degree. In this article, we explore the mix of assessment methods provided across a range of UK university mathematics departments. We examine the relationship between the mix of assessment methods and type of institution and between the type of assessment and the mathematical topic. We found that closed book exam is still the most common assessment method in undergraduate mathematics, with some new assessment methods, such as projects and presentations, now used in most UK institutions.

Description: Reports from 13 Further Mathematics Knowledge Networks supported by the National Centre for Excellence in the Teaching of Mathematics [NCETM] are analysed. After summarizing basic characteristics of the networks regarding leadership, composition and pattern of activity, each of the following aspects is examined in greater depth: Developmental aims and mathematical/pedagogical foci; Professional development activity at network meetings; Perceived impacts of the networks on professional learning and collective organization, and on classroom practice and student response. This provides some support for the claim that involving teachers in collaborative reflection and enquiry pays dividends in producing real results in the classroom. However, these findings rest predominantly on the professional judgement of participants.

Description: This article focusses on a programme of research into the teaching and learning of proof inspired by Celia Hoyles. By revisiting the first of a series of projects into justifying and proving in school mathematics developed by Celia in the 1990s and by considering how the innovative research methods adopted as well as the results obtained impacted upon our own subsequent research in Brazil, we explore how her work in this area exemplifies various aspects of her contribution to the field of mathematics education as a whole. We highlight, in particular, how we have been influenced by her commitment to understand the social shaping of mathematical practices and to create mathematical cultures, tools, activities and teaching approaches that make mathematics an attractive and accessible option to the many rather than the few.

Description: This article traces some of the influential ideas and motivations that have shaped a large part of the research on the use of new technologies in mathematics education over the past 40 years. Particular attention is focused on Papert's legacy, Celia's Hoyles' transformation of it, and how both relate to the current research landscape that features not only dramatic changes in hardware and software, but also new philosophical perspectives on the embodied nature of the learner and of mathematics.

Description: For more than 10 years, I had the honour and pleasure to work with Celia Hoyles and Richard Noss. We share a common concern for more learnable mathematics, especially in algebra, and for the need to build new representational infrastructures taking advantage of technology. Beyond this common concern, my choice to work in the French institutional context and paradigms for the Casyopee project led to a different approach. Roughly speaking, in these context and paradigms the stake is to help students access existing representations while Hoyles and Noss privilege building new more learnable representations. The aim of the article is to investigate empirically how, in spite of this gap, Hoyles and Noss’ approach can be an inspiration for enlarging the focus on a given topic, freeing oneself of a too strict dependency to one’s own context.

Description: The integration of digital tools in mathematics education is considered both promising and problematic. To deal with this issue, notions of webbing and instrumental orchestration are developed. However, the two seemed to be disconnected, and having different cultural and theoretical roots. In this article, we investigate the distinct and joint journeys of these two theoretical perspectives. Taking some key moments in recent history as points of departure, we conclude that the two perspectives share an importance attributed to digital tools, and that initial differences, such as different views on the role of digital tools and the role of the teacher, have become more nuances. The two approaches share future challenges to the organization of teachers’ collaborative work and their use of digital resources.

Description: The purposes of an online system to auto-mark students’ responses to mathematics test items are to expedite the marking process, to enhance consistency in marking and to alleviate teacher assessment workload. We propose that a semi-automatic marking and customizable feedback system better serves pedagogical objectives than a fully automatic one. The two pedagogical objectives to be addressed are that teachers should know about the range of students’ solutions and that they should provide meaningful feedback to students through the utilization of ‘customisable feedback’. Both objectives are aligned with using assessment data for learning. Our proposed IT-based system consists of a marking component and a feedback component, and it is designed to provide close linkage between IT-based marking and these pedagogical objectives.

Description: When a Computer Algebra System (CAS) is used as a pedagogical and functional tool in class and as a functional tool in exams, its effect on student achievement can be quite profound. The timing of when students are first introduced to a CAS has an impact on gains in student achievement. In this action research project, the CAS calculator was introduced to students and used to varying degrees in Years 10, 11 and 12. The effects on students' final year results were examined in terms of the timing of the introduction of the CAS device and the extent of its use in the classroom. There is some evidence of greater improvement in learning by introducing the calculator in Year 10 rather than Year 11. This gain is evident for low, average and above average mathematics achievers.

Description: This article describes how technology has been used to motivate the learning of mathematics for students of Sports Technology at Loughborough University. Sports applications are introduced whenever appropriate and Matlab is taught to enable the students to solve realistic problems. The mathematical background of the students is varied and the required prerequisite is a General Certificate of Secondary Education grade A in mathematics. Group projects include modelling the velocity of a downhill skier, the effects of lift and drag on the length of drive of a golf ball and the size of parachute required to ensure a smooth landing. All of these require the use of Matlab. In-class engagement is enhanced by the introduction of electronic voting systems. Questions involving sports applications can be posed in-class and immediate feedback received. The effect of introducing such material, on attendance and progression rates, and student engagement is reported.

Description: In light of the changing ethnic, racial and cultural demographics in US schools, there is a growing need by scholars and practitioners for teaching practices that are informed by cultural knowledge possessed by students. This comes at a time when standardized testing of student achievement has been given a central role in efforts to improve K-12 education such as the Mathematics Common Core State Standards. This article begins with a discussion of ethnomathematics as a conceptual idea, and a practical way of rethinking ideology, content and pedagogy in a rapidly changing climate of diversity. This is followed by a 5-year study on the Ethnomathematics Institute, which provides strategies for implementing promising practices. In Pacific as well as global communities, it is key to understand the role of experiential, values and place-based learning. Through ethnomathematics, we endeavour to build capacity and networks across the USA and Pacific by creating a database of materials and strategies that are relevant, contextualized and sustainable.

Description: The ever growing gap between secondary and university level mathematics is a major concern to higher education institutions. The increase in diversity of students’ background in mathematics, with entry qualifications ranging from the more traditional A-level programmes to BTEC or international qualifications is compounded where institutions attempt to widen participation. For example, work-based learners may have been out of education for prolonged periods and, consequently, are often unprepared for the marked shift in levels, and catering for all abilities is difficult in the normal lecture, tutorial format. Lack of sufficient mathematical knowledge not only affects students’ achievement on courses but also leads to disengagement and higher drop-out rates during the first 2 years of study. Many universities now offer a maths support service in an attempt to overcome these issues, but their success is varied. This article presents a novel approach to maths support designed and adopted by the University of Lincoln, School of Engineering, to bridge this transition gap for students, offer continued support through Assessment for Learning and Individual Learning Plans, and ultimately increase student achievement, engagement and retention. The article then extends this proven approach and discusses recently implemented enhancements through the use of online diagnostic testing and a ‘student expert’ system to harness mathematical knowledge held by those gifted and talented students (often overlooked by higher education institutions) and to promote peer-to-peer mentoring. The article shows that with the proven system in place, there is a marked increase in student retention compared with national benchmark data, and an increase in student engagement and achievement measured through student feedback and assessments. Although the online enhancements are in the early stages of implementation, it is expected, based on these results, that further improvements will be shown.

Description: The present article focuses on exploring the dynamics of an interdisciplinary approach in teaching mathematics as a process that cultivates creativity in the school curriculum. We specifically describe here a research project that aims: first, to encourage students to develop and express their creativity; second, to support them towards envisioning mathematics in relation to other school subjects; and third, to assist them in realizing the social role of mathematics itself as a subject closely related to citizenship. The project entitled ‘ Sensitive pendulum or heavy earth ’ was focused on actively engaging students with the main historical arguments concerning the idea of appropriating a common measure unit for length by the French National Assembly during the French Revolution. In order to motivate students and productively organize their efforts in this pedagogic setting we exploited certain drama techniques such as role-playing and role-playing debate. As a result, a short period (i.e. 6 weeks) interdisciplinary project was organized with two classes of 22 and 23 students who attend the 11th grade of an inner city Lyceum school in Athens, Greece. The students involved in this project got responses regarding the very-often-emerging question: why mathematics is useful. At the same time, students seem to change their perception of mathematics and to move towards appreciating its multi-disciplinary and creative nature as well as the broader significance of mathematics in society.

Description: Problem-solving behaviours of students have been comprehensively considered in much research. Students’ differences in their thinking processes influence their mathematical processing. In this study, we focused on whether mathematical thinking types affect what representations they use for integrals. The participants of the study were 37 undergraduate students who were enrolled for a calculus course and were selected through a purposeful sampling technique in case study design. Data collection techniques were test, observation and interview. Tests were used to identify learners' preferred representations according to their type of thinking. Participants’ views on preferences for representations are evaluated by interviews. The data gained in the study were analysed and interpreted through a classification method and descriptive statistics. Results show that, although participants’ mathematical thinking types have some effect on their representation preferences, the participants generally preferred algebraic representation. It seems that participants’ problem-solving behaviours are more affected by teaching environment than by preferred thinking types. Other findings from the study are discussed based on the literature.

Description: Advantages and disadvantages of the use of digital technologies (DT) and especially of computer algebra systems in mathematics lessons are discussed controversially worldwide. What will be the impact of DT in the next years or even the next decade? The basis of the following considerations is the long-term empirical project M 3 (Model-project New Media in Mathematics lessons) which was started 10 years ago in 2003 to test the use of symbolic calculators (SC) in Bavarian ‘Gymnasien’ (grammar schools) in Germany. In 2013, there exists extensive experience regarding the use of SC from grade 10 to 12 in classroom activities, student and teacher documents as well as test and examination results of students. The implications of this project are going to be collected in 10 theses or hypotheses about possible, beneficial developments in the future. These theses will be explained with examples taken from the project M 3 . They are addressed to mathematics teachers and mathematics educators, to people who are interested in the on-going development of the use of DT concerning a better understanding of mathematics.

Description: Touchscreen devices are increasing and becoming familiar for students. This research identifies types of screen touching during the process of solving problems using the Geometric Constructer (GC) software. We conducted a case study with Italian High School students using tablets. Since manipulation on a tablet is different from mouse clicks, this kind of research opens a new stream about dynamic geometry in multi-touch technology. Based on the research by Yook and Park, we observe two domains of manipulation, constructive and relational, which indicate geometric thinking through modes of touchscreen.

Description: This article is centred in university mathematics education. It reports the results of an extension to a research literature study identifying the competencies that mathematics majors acquire as they learn mathematics within the context of designing, programming and using Exploratory Objects, a requirement in a core mathematics programme called Mathematics Integrated with Computers and Applications (MICA) at Brock University. Some of the identified competencies are used to ascertain parallels between Exploratory Objects and Microworlds. The MICA programme was instituted in 2001 and it remains a core programme for mathematics majors. This study suggests that the research on Exploratory Objects can benefit from the rich knowledge about Microworlds, and reversely the Microworld community could benefit from research on the sustained, systemic implementation of MICA.

Description: This article describes an approach currently being carried out at the University of Manchester where engineering students are encouraged to explore maths as a complement to traditional lectures. First- and second-year electrical and electronic engineering students use a set of notebooks, developed using MuPAD, in order to explore relevant mathematical topics, working faster than can be done when every calculation is performed by hand. This allows learners to analyse more examples and hence gain an appreciation of the wider concepts surrounding the topics and how they fit into the electrical and electronic engineering syllabus. The notebooks were designed to be used both to motivate discussion in lectures and/or individual study.

Description: Traditional classroom teaching environments have used board technologies to incorporate handwritten elements within multimodal pedagogical approaches to developing mathematical thinking. In the tertiary sector, increasing use of computing technologies based on keyboard and mouse interfaces and learning environments that emphasize digital displays constrain the use of handwritten elements. This article discusses how the use of pen-enabled Tablet PCs can provide support for handwritten elements, to build on the benefits of traditional pedagogical approaches and facilitate the development of new approaches.

Description: Learning the connection between symbolic and visual representations is a key to conceptual understanding of three-dimensional (3D) vector equations. For learning such a connection, it is valuable for students to manipulate and transform the graphic objects directly while observing the simultaneous change of related symbolic equations. The interactive change of graphic and symbolic objects provides the students with opportunities to recognize these relations experimentally. This article describes how such interactivity is designed as digital worksheets, introduced into our lessons, and what reflections the worksheets received from students and teachers.

Description: This article describes part of a study which investigated the role of questions in students’ approaches to learning mathematics at the secondary–tertiary interface, focussing on the enculturation of students at the University of Oxford. Use of the Mathematical Assessment Task Hierarchy taxonomy revealed A-level Mathematics and Further Mathematics questions in England and Wales to focus on requiring students to demonstrate a routine use of procedures, whereas those in first-year undergraduate mathematics primarily required students to be able to draw implications, conclusions and to justify their answers and make conjectures. While these findings confirm the need for reforms of examinations at this level, questions must also be raised over the nature of undergraduate mathematics assessment, since it is sometimes possible for students to be awarded a first-class examination mark solely through stating known facts or reproducing something verbatim from lecture notes.

Description: An investigation into prospective mathematics/statistics teachers’ (n = 134) conceptual understanding of statistics and attitudes to statistics carried out at the University of Limerick revealed an overall positive attitude to statistics but a perception that it can be a difficult subject, in particular that it requires a great deal of discipline and time to learn. This latter result motivated a follow-up qualitative study (n = 9) to uncover the factors contributing to positive attitude and the concomitant perceptions of difficulty. What emerged was evidence of the lasting impact secondary teachers and their perceptions of statistics had on these students. The study also reveals the potential of teaching practice to act as a vehicle to challenge the perceived difficulty of learning statistics and provide prospective teachers with the opportunity to develop confidence in statistics as both learners and teachers.

Description: We consider the famous two envelope conundrum and show, in a very simple way, that there is no conundrum. Our discussion is supported by numerical simulations. We use the problem to raise the issue of disputes in mathematics, rarely touched upon in mathematics education. We suggest that it is worthwhile to expose students of mathematics to such controversies and that this problem can be introduced in the classroom via an experimental demonstration.

Description: The next edition of the European Society for Engineering Education (SEFI) Mathematics Working Group’s curriculum document will be based on the notion of mathematical competence. This approach aims particularly to capture higher-level learning goals that go beyond the traditional content-related and often small-scale description of learning outcomes. In this contribution, ‘classical’ and more recent learning arrangements are investigated regarding their potential for competence acquisition. The learning arrangements include lectures, assignments, tutorials, laboratories, projects and technology enhanced learning arrangements. We relate these to the eight competencies identified in the current SEFI curriculum discussion document and explore the contribution each of the different learning arrangements may provide for competency acquisition.

Description: Dublin Institute of Technology offers students a number of different routes into engineering, allowing many non-standard entrants the opportunity to study the discipline provided they fulfil certain criteria. The final aim of many of these students is to achieve an Honours Degree in Engineering, which takes a minimum of 4 years. Apart from the first year of the course, the other main entry point is at the start of the third year, at which stage students who have performed well in a 3-year Ordinary Degree can begin. However, these students have a wide range of mathematical abilities and prior knowledge, and many are missing the basic skills required for completion of a mathematics module at this level. It is common practice for students to be diagnostically tested upon entry to third level; however, anecdotally, it appeared that many of the mathematical issues uncovered at that point had not been rectified during the students’ subsequent studies. In an attempt to quantify the problem, it was decided to pilot an Advanced Mathematics Diagnostic Test that covered many of the key concepts from the early years of Engineering Mathematics. A pass-mark of 90% was set in this assessment. About 167 third-year students studying for an Honours Engineering degree were tested during the pilot study, only two of whom achieved the pass mark on the first sitting. To encourage the other students to revise this crucial material, multiple re-sit opportunities were provided, and a weighting of 10% of the continuous assessment mark for the mathematics module was given to the diagnostic test. Online resources and special classes covering the relevant material were also provided, with the result that 131 of the 167 students reached the necessary threshold by the end of the semester.

Description: A study was conducted at the Technical University Berlin involving students who twice failed the written examination in the first semester course Linear Algebra for Engineers in order to better understand the reasons behind their failure. The study considered student understanding in terms of Bloom’s taxonomy and the ways in which students approached problem solving. The results indicate that students rely on lower-order thinking processes and these processes are linked to solution approaches. Thus, by investigating solution strategies in homework sets and in classwork, an instructor can easily identify students at risk of not understanding at the appropriate level. In this contribution, the study is related to the framework set forth by the European Society for Engineering Education (SEFI) Mathematics Working Group.

Description: Computer-aided assessment (CAA) has been used at a university with one of the largest mathematics and engineering undergraduate cohorts in the UK for more than ten years. Lecturers teaching mathematics to first year students were asked about their current use of CAA in a questionnaire and in interviews. This article presents the issues that these lecturers faced as they made use of this assessment tool. Lecturers explained how they attempted to overcome these issues. The findings show that while the lecturers were happy to use the CAA system because it is efficient and timesaving, there were concerns that it might not always be beneficial for students. The bases for lecturers’ concerns were that some students developed tendencies to depend on the feedback to complete assessments and to develop procedural, context-dependent strategies for solving problems.

Description: Competency has been the buzzword of higher education for at least a decade. The reasonable approach of describing what students should be able to do after completing coursework, however, falls short of the fact that mathematics, like any subject matter, contains inherent difficulties for students. Students usually need assistance in overcoming such difficulties. Competency-driven approaches to the curriculum tend to ignore this issue. In this article, this problematic issue will be exemplified by investigating students’ difficulties with the concept of the empty set on one hand and the framework of the European Society for Engineering Education (SEFI) for a mathematics curriculum on the other hand.

Description: Large numbers of students entering higher education take some level of mathematics as part of their degrees, and it is widely reported that a considerable minority of these students demonstrate a lack of the basic mathematical skills that they require to succeed. A common response has been the establishment of mathematics learning supports to give students the opportunity to reach the levels required. Research has shown that in general, although the supports appear to impact positively on students who avail of them, a significant number of students do not engage appropriately. This article presents preliminary findings from a national survey carried out at nine Higher Education Institutions in Ireland, focusing on the reasons given by students for their lack of engagement with the extra supports. It looks at the students’ mathematical backgrounds; the type of institution they attend, and discusses what these students reported would encourage them to avail of the supports.

Description: The purpose of this study was to observe students’ retention, success and attitudes towards mathematics in a community college quantitative literacy course, taught in a lab-based format. The redesigned course implemented the daily use of Microsoft Excel in the classroom demonstrations, group activities and individual assignments, and utilized data from many fields of study. Results showed statistically significant growth in attitudes towards real-world application problems, the use of computers in mathematics, and the consideration of taking additional mathematics courses. There was also marginally significant growth in students’ attitudes towards the relevance and utility of mathematics. Higher retention and success rates in the redesigned course were also observed, although those rates were not found to be statistically significant.

Description: Mathematics Support Centres (MSCs) have been established at universities in the UK and a number of other countries, of which colleagues from Australia and Ireland have been the most prolific in publishing about their work. Their main functions are to address issues surrounding the transition to university mathematics and to support students’ learning of mathematics and statistics across the wide variety of undergraduate courses. There is a growing body of research examining the operation and impact of MSCs. This article will review and synthesize available published research evidence so that informed decisions can be made about the value of mathematics support activity and the targeting of future funding. Evidence will be shown of the evaluation of MSCs in each of the following areas: the collection of data and the challenges that are presented in both quantitative and qualitative studies; analysis demonstrating MSC usage and activity; analysis showing the impact of MSCs on students, staff and the institution. The article will conclude by identifying areas where further research would be helpful.

Description: Many higher education institutions have introduced some kind of mathematics learning support provision in response to the well-documented ‘mathematics problem’. In 2001 and 2004 two independent studies were undertaken to assess the number of universities offering mathematics learning support to students in addition to that provided through lectures, tutorials and the personal tutorial system. The results of these surveys showed a growth in the number of institutions providing support from 46 to 66. In this article we report on a survey carried out in 2012 to establish the current position regarding the provision of mathematics learning support in UK universities. In addition to determining the number of institutions providing mathematics learning support—there has been a further rise to 88—the article analyses the distribution of mathematics learning support by university mission group and by the type of support provided. The main findings are that the extent of mathematics learning support provision is largely independent of mission group and the dominant provision is drop-in support.