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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]

This paper reports on initial findings from the Visible Maths Pedagogy research project, a collaboration between an academic researcher and two teacher researchers (the paper's authors). The aim of the project was to explore the effects of making pedagogy more visible on students' success in school mathematics. We adopted a Participatory Action Research methodology to plan and evaluate five strategies used alongside 'progressive' teaching approaches to make the teacher's pedagogical rationale more visible to learners. Our findings show that students, particularly those from disadvantaged backgrounds, were initially prone to misinterpret the intentions of the teacher. However, the five strategies helped students gain a greater appreciation of the teacher's pedagogical rationale and how to respond appropriately. We discuss the implications of these findings for enabling all students to access the benefits of progressive teaching approaches and for opening up to scrutiny what it means to be a successful learner of mathematics. [ABSTRACT FROM AUTHOR]

Windmill images and shapes have a long history in geometry and can be found in problems in different mathematical contexts. In this paper, we share and discuss various problems involving windmill shapes and solutions from geometry, algebra, to elementary number theory. These problems can be used, separately or together, for students to explore rotational symmetry and connect seemingly unrelated problems. [ABSTRACT FROM AUTHOR]

In this paper, Harel and Sowder's proof schemes of taxonomy were applied for identifying eighth grade students' learning of proof and proofing. This study aimed to examine what kind of proof schemes are used in geometry and measurement learning areas and student's skills in giving proof and their opinions about proof. Qualitative methods were used. The first part consisted of 40 students', were selected by using convenient sampling, were administered the proof skills inventory. The second part consisted of semi-structured interviews with six students who were randomly selected through a convenient sampling. Proof skill inventory was administered to 40 students and semi-structured interviews with randomly selected 6 students were done. The results of the study indicated that eighth grade students had intermediate level proof skills. In-depth analysis of the students confirmed that the students who got high scores used the axiomatic proof schema, those who had a moderate score used basic proof examples, and those who had low scores used the authoritarian proof schemas. [ABSTRACT FROM AUTHOR]

This paper describes the main findings of a national action-research project, promoted by Regional School Office of Campania (South of Italy), aimed at improving numeracy and mathematical literacy through supporting schools' activities. The main focus is on processes underlying the creation of capabilities in representation, reasoning and communication. They have been investigated through creation of appropriate activities on the management of different semiotic systems and argumentation. About 400 fifteen-years- old students have been involved in the experimentation, divided into 20 high schools classes, in the Campania Region (South of Italy). The activities were designed to improve students' capabilities to solve real world problems, according to the OECD's Programme for International Student Assessment (PISA) framework. We investigated if the experimental activities might influence the acquisition of representation and communication skills and improve the PISA levels. We showed the project's rationale, its methodological choices, and theoretical framework. A protocols' analysis on conversion between various semiotic systems is also shown, mainly in two different schools' types (a Professional Institute and a Scientific High School). Finally, a quantitative analysis of the improvement of the PISA levels has been carried out, together with a discussion about the connection between the improvement of the mathematical skills and the realized activities. [ABSTRACT FROM AUTHOR]

The nine-point circle theorem is one of the most beautiful and surprising theorems in Euclidean geometry. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. This paper describes a set of instructional activities that can help students discover the nine-point circle theorem through investigation in a dynamic geometry environment, and consequently prove it using a method of guided discovery. The paper concludes with a variety of suggestions for the ways in which the whole set of activities can be implemented in geometry classrooms. [ABSTRACT FROM PUBLISHER]

Epidemiology is the study of the distribution of disease in human populations. This means that authentically teaching primary data collection in epidemiology is difficult as students cannot easily access suitable human populations. Using an action research methodology, this paper studied the use of a virtual human population (called The Island) to enable students to experience many features of authentic primary data collection in epidemiological research. The Island was used in a course introducing epidemiology and biostatistics for students in non-quantitative disciplines. This paper discusses how The Island was introduced into the course, and then evaluates the change. Students were highly engaged, and students and teaching staff responded favourably to the use of The Island, with 70% of students agreeing or strongly agreeing that The Island was easy to use, and 64% agreeing or strongly agreeing that the use of a virtual population was beneficial to their understanding of epidemiology. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider analogues of the twin-prime conjecture in various classes within modular rings. [ABSTRACT FROM AUTHOR]

This research investigates the influence that gender, single-sex and co-educational schooling can have on students’ mathematics education in second-level Irish classrooms. Although gender differences in mathematics education have been the subject of research for many years, recent results from PISA (Programme for International Student Assessment) show that there are still marked differences between the achievement and attitude of male and female students in Irish mathematics classrooms. This paper examines the influence of gender in more detail and also investigates the impact of single-sex or co-educational schooling. This is a follow on study which further analyses data collected by the authors when they designed a pedagogical framework and used this to develop, implement and evaluate a teaching intervention in four second-level Irish schools. The aim of this pedagogical framework was to promote student interest in the topic of algebra through effective teaching of the domain. This paper further analyses the quantitative data collected and investigates whether there were differences in students’ enjoyment and achievement scores based on their gender and whether they attended single-sex or co-educational schools. [ABSTRACT FROM PUBLISHER]

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students’ development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived. [ABSTRACT FROM AUTHOR]

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

This paper presents some critical perspectives regarding pedagogical approaches to the method of reversing the order of integration in double integrals from prevailing educational literature on multivariable calculus. First, we question the message found in popular textbooks that the traditional process of reversing the order of integration is necessary when solving well-known problems. Second, we illustrate that the method of integration by parts can be directly applied to many of the classic pedagogical problems in the literature concerning double integrals, without taking the well-worn steps associated with reversing the order of integration. Third, we examine the benefits and limitations of such a method. In our conclusion, we advocate for integration by parts to be a part of the pedagogical conversation in the learning and teaching of double integral methods; and call for more debate around its use in the learning and teaching of other areas of mathematics. Finally, we emphasize the need for critical approaches in the pedagogy of mathematics more broadly. [ABSTRACT FROM PUBLISHER]

The article reports on the results of two case studies on the impact of the regular use of puzzles as a pedagogical strategy in the teaching and learning of engineering mathematics. The intention of using puzzles is to engage students’ emotions, creativity and curiosity and also to enhance their generic thinking skills and lateral thinking ‘outside the box’. Students’ attitudes towards this pedagogical strategy are evaluated via short questionnaires with two groups of university students taking a second-year engineering mathematics course. Students’ responses to the questionnaire are presented and analyzed in the paper. [ABSTRACT FROM AUTHOR]

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]

We began by observing a variety of properties related to the tangent and normal lines of three conic sections: a parabola, an ellipse, and a hyperbola. Some of these properties include specific relationships between the x- and y-intercepts of the tangent and normal lines. Using these properties, we were able to form several differential equations. Afterwards, we took an arbitrary differentiable function satisfying those differential equations and attempted to solve them in order to find whether or not the function represents a conic section. Some of these differential equations were nontrivial, but to our surprise, we were still able to solve a few of them using a combination of standard methods. The main highlight of the paper was the discovery of a brand-new closed curve, which was obtained by mimicking a certain property satisfied by a normal line of a parabola. Another highlight of the paper was the creation of a novel set of unified definitions to describe the three conic sections using differential equations. These results are interesting because the traditional definitions involve the concepts of eccentricity and directrix, or sections of a double cone by a plane, while ours only deals with the x-intercepts of the tangent lines. Mathematica® was also used to create several animations that illustrate some of the discoveries in this paper. [ABSTRACT FROM AUTHOR]

An important and interesting area in the study of triangle geometry is the related issue of extrema problems and inequalities. These problems play a significant role in the mathematics study program in high school. In tasks such as these, the difficulty level is high when one does not know in advance what the expected answer is. When one knows what to prove, the difficulty level is lower and most of the effort is aimed at attaining a proof of the expected answer. This can be done using dynamic geometric software. The possibility of making frequent changes to the geometric objects and the ability of dragging objects, contributes to the process of deducing properties, checking hypotheses and generalizing. In this paper, eight investigative tasks in Euclidean geometry are presented together with the applets developed for carrying out the dynamic investigation. Some of the tasks are well known, while others are almost unknown and are worthy of presentation as enrichment for those interested in the subject. The tasks were given to preservice teachers of mathematics as part of an advanced course for integrating technological tools in the teaching of mathematics. [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]

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics. [ABSTRACT FROM PUBLISHER]

This paper reports on lessons in which 18–19 years old high school students modelled random processes with Excel. In the first lesson, 26 students formulated a hypothesis on the area of ellipse by using the analogy between the areas of circle, square and rectangle. They verified the hypothesis by the Monte Carlo method with a spreadsheet model developed in the lesson. In the second lesson, 27 students analysed the dice poker game. First, they calculated the probability of the hands by combinatorial formulae. Then, they verified the result with a spreadsheet model developed in the lesson. The students were given a questionnaire to find out if they found the lesson interesting and contributing to their mathematical and technological knowledge. [ABSTRACT FROM PUBLISHER]

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]

Some research studies, many of which used quantitative methods, have suggested that graphics calculators can be used to effectively enhance the learning of mathematics. More recently research studies have started to explore students’ styles of working as they solve problems with technology. This paper describes the use of a software application that records the keystrokes made by students as they use calculators, in order to enable researchers to gain better insights into students’ working styles. The recordings obtained from this software can be replayed to observe how students have actually used their calculator in tackling a problem. The paper describes three pilot studies from quite different contexts, in which the software reveals how the calculators have been used by the students. In all of these studies the software provides insights into the working that would have been very difficult to obtain without the record of the keystrokes provided by the software. [ABSTRACT FROM AUTHOR]

Offering perspectives in the teaching of statistics assists students, immersed in the study of detail, to see the leading principles of the subject more clearly. Especially helpful can be a perspective on the logic of statistical inductive reasoning. Such a perspective can bring to prominence a broad principle on which both interval estimation and hypothesis testing are designed, and so can unify these seemingly disparate techniques in students' minds. In this paper it is shown how to construct an illuminating perspective over this unifying principle, that experience shows is valued by intellectually lively students. To aid readers attracted by the idea of employing this perspective in their teaching, the paper includes a skeletal version of a recommended line of presentation. [ABSTRACT FROM AUTHOR]

This paper reports on a practitioner study of a first-year undergraduate service course that aligns a web-based, student-lecturer communication system with the mathematical curriculum. The report presents and analyses data from students and from the lecturer and outlines the nature of the technical interface. The paper indicates how communication of the students' affective learning needs had a positive influence on the professional development of the lecturer himself. Furthermore, it is claimed that the success of mathematics/statistics teaching that integrates emotional responses from learners is intimately related to the knowledge, skills, beliefs and values of the lecturer. [ABSTRACT FROM AUTHOR]

In this paper, we present, analyse and critique an episode from a secondary school lesson involving an introduction to the definition of the scalar product. Although the teacher attempted to be explicit about the difference between a definition and a theorem, emphasizing that a definition was just an arbitrary assumption, a student rejected the teacher's definition in favour of his own alternative. With reference to this particular case, we seek to explore some ways in which teachers can introduce mathematical definitions to students so as to support, rather than attempt to circumvent, their mathematical sense making. In this regard, we believe that it is important to develop learning opportunities for students which help them to gain some appreciation of important structural and historical reasons that underpin the definitional choices made. [ABSTRACT FROM AUTHOR]

In this paper, we give a new and straightforward method to solve the non-homogeneous second-order linear difference equations with constant coefficients. It is new because it does not require the uniqueness theorem of the solution of the problem of initial values. Neither does it require a fundamental system of solutions, nor the method of variation of parameters. Moreover, we get a unique formula that expresses the general solution independently of the multiplicities of the roots of the characteristic equation. [ABSTRACT FROM PUBLISHER]

This paper has originated from our interest in approaching mathematical concepts starting from people's common-sense intuitions and building up from there. This goal is challenging both in designing the didactical transposition and sequencing of the mathematical subject matter, and in adopting the open and interactive teaching approach needed to achieve students' active participation and reflection. To demonstrate these challenges, and our experience of trying to cope with them, we have chosen the concept of ‘inverses’ as used in group theory, and its common-sense precursor ‘opposites’. We present our approach via a series of workshop iterations, which summarizes our experience in the many actual workshops we ran in Israel and in Denmark. [ABSTRACT FROM PUBLISHER]

This paper gives an account of an experiment in which 33 high school students of age 16–19 acquired the principles of the integral calculus through applying the rectangle method in spreadsheet to calculate the area of a planar figure bounded by the graph of a function and the volume of a body created by the rotation of the graph. A questionnaire survey was carried out to find out whether the students found the lesson interesting, contributing to their mathematical and technological knowledge and motivating to continue with more complicated tasks. [ABSTRACT FROM PUBLISHER]

The aim of this paper is to present and discuss some of the evidence regarding the resources that students use when they establish relationships between a contextual situation and an ordinary differential equation (ODE). We present research results obtained from work by seven students in a graduate level course in mathematics education, where they were involved in solving tasks based on the study of ODEs. The students’ mastery of digital tools allowed them to use and articulate different mathematical representations to comprehend how phenomena develop. The use of digital tools helped to enhance the students’ interpretation of the relationships between the context and the mathematical model associated with it. We also found that difficulties related to interpretation are grounded in the literal relationship that students make between their mental model of how a phenomenon develops and its mathematical representations. [ABSTRACT FROM PUBLISHER]

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

Finance threshold concepts are the essential conceptual knowledge that underpin well-developed financial capabilities and are central to the mastery of finance. In this paper we investigate threshold concepts in finance from the point of view of students, by establishing the extent to which students are aware of threshold concepts identified by finance academics. In addition, we investigate the potential of a framework of different types of knowledge to differentiate the delivery of the finance curriculum and the role of modelling in finance. Our purpose is to identify ways to improve curriculum design and delivery, leading to better student outcomes. Whilst we find that there is significant overlap between what students identify as important in finance and the threshold concepts identified by academics, much of this overlap is expressed by indirect reference to the concepts. Further, whilst different types of knowledge are apparent in the student data, there is evidence that students do not necessarily distinguish conceptual from other types of knowledge. As well as investigating the finance curriculum, the research demonstrates the use of threshold concepts to compare and contrast student and academic perceptions of a discipline and, as such, is of interest to researchers in education and other disciplines. [ABSTRACT FROM PUBLISHER]

Past research has investigated students’ epistemologies while they were taking courses that required an integrated understanding of mathematical and scientific concepts. However, past studies have not investigated students who are not currently enrolled in such classes. Additionally, past studies have primarily focused on individuals who are native English speakers from Western cultures. In this paper, we aim to investigate whether Hammer and his colleagues’ claims concerning learners’ epistemologies could be extended to individuals who lack advanced mathematics and science training, have had different cultural and learning experiences, and have grown up speaking and learning in another language. To this end, we interviewed a participant with these characteristics about her understandings of the concept of speed. Our findings show that previous theoretical frameworks can be used to explain the epistemologies of the individual examined in this study. The case suggests that these theories may be relevant regardless of the learner's mathematics and science background, language, educational experience, and cultural background. In the future, more cases should be examined with learners from different academic backgrounds and cultures to further support this finding. [ABSTRACT FROM PUBLISHER]

This paper explored variation of strategy use in pattern generalization across different generalization types and across grade level. A test was designed to assess students’ strategy use in pattern generalization tasks. The test was given to a sample of 1232 students from grades 4 to 11 from five schools in Lebanon. The findings of this study showed that the frequency of student strategy use differed according to the generalization type and that the functional and recursive strategies were frequently used in each of the generalization tasks. The use of the recursive strategy decreased, whereas, the use of the functional strategy increased, as the demands of the task changed from constructing a step-by-step solution to finding a general formula. The findings also showed that the use of the recursive strategy increased across grades 4–8 and then it decreased across grades 8–11; whereas the use of the functional strategy increased across grades 4–11. [ABSTRACT FROM PUBLISHER]

The Feigenbaum constant δ is frequently met by students in a first course on chaos, and discussed with reference to the period doubling within the logistic map. The details of the actual calculation of δ are, however, nontrivial, and form the basis for an undergraduate project which may be used to develop skills in discrete maps and numerical methods. This paper considers how the Ricker map can be used to evaluate δ, and also suggests a number of problems which can be solved by students along the way. [ABSTRACT FROM AUTHOR]

This paper reports a study on retention of differential and integral calculus concepts of a second-year student of physical chemistry at a Danish university. The focus was on what knowledge the student retained 14 months after the course and on what effect beliefs about mathematics had on the retention. We argue that if a student can quickly reconstruct the knowledge, given a few hints, this is just as good as retention. The study was conducted using a mixed method approach investigating students’ knowledge in three worlds of mathematics. The results showed that the student had a very low retention of concepts, even after hints. However, after completing the calculus course, the student had successfully used calculus in a physical chemistry study programme. Hence, using calculus in new contexts does not in itself strengthen the original calculus learnt; they appeared as disjoint bodies of knowledge. [ABSTRACT FROM PUBLISHER]

In this paper we describe a method for teaching students to prove some mathematical statements independently, by using specially designed auxiliary assignments. The assignments are designed as homework problems and can be adapted for online learning. We illustrate our method using examples from calculus and differential equations. [ABSTRACT FROM AUTHOR]

This paper analyses the strategies used by pre-service primary school teachers for solving simple addition problems involving negative numbers. The findings reveal six different strategies that depend on the difficulty of the problem and, in particular, on the unknown quantity. We note that students use negative numbers in those problems they find easy and resort to other strategies in more complex problems. Furthermore, the problem-solving strategies reveal lingering difficulties involving negative numbers, such as mistakes with arithmetic rules and a lack of rigour in the mathematical notation used in the operations. [ABSTRACT FROM AUTHOR]

Little is known about how students learn the basic ideas of ring theory. While the literature addressing student learning of group theory is certainly relevant, the concepts of zero-divisor and, more generally, elements with no multiplicative inverse are among those for which group theory has no analogue. In order to better understand how students come to understand the corresponding ideas of units and zero-divisors, this paper presents results from a study that investigated how students can capitalize on their intuitive notions of solving equations to reinvent the definitions of ring, integral domain, and field. In particular, the emergence and progressive formalization of the concept of zero-divisor at various stages of the reinvention process are detailed and discussed. Findings include a conceptual framework characterizing the emergence of the concept of zero-divisor and unit. This framework, in addition to documenting the emergence of these concepts, suggests that the concept of zero-divisor arises in direct contrast to the concept of unit. [ABSTRACT FROM AUTHOR]

This design-based research project focuses on documenting statistical learning among 16–17-year-old Finnish upper secondary school students (N= 78) in a computer-supported collaborative learning (CSCL) environment. One novel value of this study is in reporting the shift from teacher-led mathematical teaching to autonomous small-group learning in statistics. The main aim of this study is to examine how student collaboration occurs in learning statistics in a CSCL environment. The data include material from videotaped classroom observations and the researcher's notes. In this paper, the inter-subjective phenomena of students’ interactions in a CSCL environment are analysed by using a contact summary sheet (CSS). The development of the multi-dimensional coding procedure of the CSS instrument is presented. Aptly selected video episodes were transcribed and coded in terms of conversational acts, which were divided into non-task-related and task-related categories to depict students’ levels of collaboration. The results show that collaborative learning (CL) can facilitate cohesion and responsibility and reduce students’ feelings of detachment in our classless, periodic school system. The interactive .pdf material and collaboration in small groups enable statistical learning. It is concluded that CSCL is one possible method of promoting statistical teaching. CL using interactive materials seems to foster and facilitate statistical learning processes. [ABSTRACT FROM PUBLISHER]

This paper describes an action-research experience carried out with second year students at the School of Engineering of the National University of Entre Ríos, Argentina. Vector calculus students played an active role in their own learning process. They were required to present weekly reports, in both oral and written forms, on the topics studied, instead of merely sitting and watching as the teacher solved problems on the blackboard. The students were also asked to perform computer assignments, and their learning process was continuously monitored. Among many benefits, this methodology has allowed students and teachers to identify errors and misconceptions that might have gone unnoticed under a more passive approach. [ABSTRACT FROM AUTHOR]

In this paper, we use a property of the centroids to study two general types of geometric loci: one in the plane and one in the space. Then from this result, we examine in detail the case of quadrilaterals and tetrahedra and, in particular, we determine some properties of the nine-point circle of a triangle. The study of those geometric loci is carried out by means of the homothetic transformations that facilitate and make more immediate the proofs of the properties. [ABSTRACT FROM AUTHOR]

Basic calculus skills that are prerequisites for advanced mathematical studies continue to be a problem for a significant proportion of higher education students. While there are many types of revision material that could be offered to students, in this paper we investigate whether short, narrated video recordings of mathematical explanations (screencasts) are a useful tool to enhance student learning when revisiting prerequisite topics. We report on the outcomes of a study that was designed to both measure change in student performance before and after watching screencasts, and to capture students’ perception of the usefulness of screencasts in their learning. Volunteers were recruited from students enrolled on an entry module for the Mathematics Master of Science programme at the Open University to watch two screencasts sandwiched between two online calculus quizzes. A statistical analysis of student responses to the quizzes shows that screencasts can have a positive effect on student performance. Further analysis of student feedback shows that student confidence was increased by watching the screencasts. Student views on the value of screencasts for their learning indicated that they appreciated being able towatch a problem being solved and explained by an experienced mathematician;hear the motivation for a particular problem-solving approach;engage more readily with the material being presented, thereby retaining it more easily. The positive student views and impact on student scores indicate that short screencasts could play a useful role in revising prerequisite mathematics. [ABSTRACT FROM PUBLISHER]

Mathematical problem solving is the mainstay of the mathematics curriculum for Singapore schools. In the preparation of prospective mathematics teachers, the authors, who are mathematics teacher educators, deem it important that pre-service mathematics teachers experience non-routine problem solving and acquire an attitude that predisposes them to adopt a Pólya-style approach in learning mathematics. The Practical Worksheet is an instructional scaffold we adopted to help our pre-service mathematics teachers develop problem-solving dispositions alongside the learning of the subject matter. The Worksheet was initially used in a design experiment aimed at teaching problem solving in a secondary school. In this paper, we describe an application and adaptation of the MProSE (Mathematical Problem Solving for Everyone) design experiment to a university level number theory course for pre-service mathematics teachers. The goal of the enterprise was to help the pre-service mathematics teachers develop problem-solving dispositions alongside the learning of the subject matter. Our analysis of the pre-service mathematics teachers’ work shows that the MProSE design holds promise for mathematics courses at the tertiary level. [ABSTRACT FROM PUBLISHER]

Research has shown that students have difficulties with understanding the process of determining whether an object is speeding up or slowing down, especially when it is applied to the analysis of motion in the negative direction. As inductively organized learning through its scaffolding sequencing supports the process of knowledge acquisition using scientific inquiry, this study hypothesized that this type of reasoning would help learners develop a deeper understanding of this concept and provide a better gateway for applying this understanding to real-life situations. Such a constructed lesson was conducted with a group of calculus students (N= 22), and a computerized simulation was used as a medium of instruction. The post-test results evidenced that the students’ perception of object rate of speed was enriched, as the majority of students correctly described the process applied to conclude the object's speed rate of change and displayed correct insight into the underlying principles of distilling rate of change of velocity from rate of change of speed. Thus, it is hypothesized that the inductively organized lesson along with the utilized interactive medium had a positive effect on students’ knowledge acquisition. This paper describes the emergence of the conduct of the study along with a detailed description of the treatment applied. [ABSTRACT FROM PUBLISHER]

This paper provides a method for analysing undergraduate teaching of proof-based courses using Toulmin's model (1969) of argumentation. It presents a case study of one instructor's presentation of proofs. The analysis shows that the instructor presents different levels of detail in different proofs; thus, the students have an inconsistent set of written models for their work. Similarly, the analysis shows that the details the instructor says aloud differ from what she writes down. Although her verbal commentary provides additional detail and appears to have pedagogical value, for instance, by modelling thinking that supports proof writing, this value might be better realized if she were to change her teaching practices. [ABSTRACT FROM PUBLISHER]

The study aims to identify areas of difficulty in learning about volumes of solids of revolution (VSOR) at a Further Education and Training college in South Africa. Students’ competency is evaluated along five skill factors which refer to knowledge skills required to succeed in performing tasks relating to applications of the definite integral, in particular to VSOR. The paper reflects on reasons for the difficulties that students experience in this topic. The study reveals that many students are not competent in drawing graphs and in interpreting the region bounded by the given graphs. If the graphs are given, students have difficulty in selecting the representative strip that is used in approximating the bounded region. Although many students are able to produce the correct formula to calculate the volume, be it a disc, washer or shell, they find it problematic to draw the three-dimensional (3D) representation of the rotated strip and the generated solid of revolution. Students seem to succeed better with tasks requiring simple manipulation skills. The study illustrates how a measure (the skill factors) can be put into practice for establishing exactly where the problems lie when students under-perform in the topic of VSOR. The results can serve as guide on how conclusions can be drawn by assessing the problematic situation through breaking it down along the framework of skill factors. [ABSTRACT FROM PUBLISHER]

This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different presentations will be discussed. Finally, several results will be presented and some conclusions proposed. [ABSTRACT FROM AUTHOR]

In this paper, we present some results of an exploratory study performed with students aged 16-17. We investigate the different uses that these students make of terms such as ‘to approach’, ‘to tend’, ‘to reach’, ‘to exceed’ and ‘limit’ that describe the basic notions related to the concept of the finite limit of a function at a point. We use the interpretive framework of conceptual analysis to infer the meanings that students associate with these specific terms in connection with the effective use of terms in their answers. [ABSTRACT FROM PUBLISHER]

Students have a tendency to see integral calculus as a series of procedures with associated algorithms and many do not develop a conceptual grasp giving them the desirable versatility of thought. Thus, instead of a proceptual view of the symbols in integration, they have, at best, a process-oriented view. On the other hand, it is not surprising that many students find concepts such as the integral difficult when they are unable to experience these processes directly in the classroom. With a view towards improving this situation, constructing the integral concept on the basis of the idea of accumulation has been proposed (Educ Stud Math. 1994;26:229–274; Integral as accumulation: a didactical perspective for school mathematics; Thessaloniki: PME; 2009. p. 417–424). In this paper, we discuss a curriculum that is based on this idea and a design for curriculum materials that are intended to develop an improved cognitive base for a flexible proceptual understanding of the integral and integration in high school. The main focus is on how we (mathematics teachers and mathematics educators) might teach the integral concept in order to help high school students to construct meaningful knowledge alongside acquiring technical abilities. [ABSTRACT FROM PUBLISHER]

In this paper a two-stage research study is described focused on problem solving relevant to place-value and on the use of the operations within the calculator environment. The findings show that in this specific environment and via appropriate tasks teachers are provided with a context to better understand what year 5 or 6 pupils know or do not know about place-value and how they are able to apply their understanding of place-value and the use of operations in this particular context. [ABSTRACT FROM AUTHOR]

Representations of mathematical concepts play an important role in understanding: both in helping learners understand the to-be-learned material and in facilitating teachers’ understanding of pedagogical processes which, in turn, are involved in developing learners’ understanding. In this paper, we report on work with a cohort of pre-service primary teachers, with the aim of developing their understanding of mathematics, their confidence in their subject knowledge and their confidence in teaching mathematics. This was attempted through the introduction and use of a ‘representational approach’ to the teaching of the mathematical concepts required of teachers training to teach in primary schools in the UK. We present the results of attitude measures and a follow-up qualitative questionnaire in identifying whether and how the use of this representational approach supported pre-service teachers’ understanding and their confidence in teaching mathematics. The results suggest that the representational approach used had a positively significant impact on the attitudes towards studying and teaching mathematics. [ABSTRACT FROM PUBLISHER]