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The current study compared the effects of technological environments with that of the paper-and-pencil environment on reasoning about the concept of derivatives in the context of maximum and minimum problems. The data consisted of clinical interviews conducted with three pre-service secondary mathematics teachers and a newly registered graduate student all of whom had quite a lot of mathematics courses in their repertoire. The study revealed that participants mostly depended on and were limited to analytical reasoning within paper-and-pencil environments, whereas they were able to refer to practical and creative reasoning with the help of the facilities technology environments provided. On the other hand, although participants made progress and used different reasoning types within technology environments, there were cases where they could not move beyond analytical reasoning even within the presence of technological tools. The reason for such a limitation seemed to be because of the way they treated technology as an analytical tool and they depended on 'learning from technology' instead of 'learning with technology' [Hanna, G., 1989, Proofs that prove and proofs that explain. In: G. Vernaud, J. Rogalski and M. Artigue (Eds), Proceedings of the International Group for the Psychology of Mathematics Education (Paris: ERIC), pp. 45-51]. [ABSTRACT FROM AUTHOR]

This note demonstrates multiple representations (numerical and graphical) of spreadsheet-based quadratic sequences together with prospective teachers' pencil-paper transformations of these numerical sequences into a corresponding symbolization as algebraic expressions. With the majority of prospective teachers, the experience of school mathematics is one of disaffection. They are in a teacher education programme that offers them certification to teach up to grade nine in the public schools. Their work illustrates processes, such as recognizing and extending patterns, by specializing and generalizing particular functional relationships. Insights gained from various methods used by them, together with the instructional affordances, i.e. the instructional benefits of spreadsheets, represent a curricular continuity between quadratic, numerical sequences and related algebraic expressions and suggest a possible change to the order in which the two are introduced. [ABSTRACT FROM AUTHOR]

This paper shows how inequalities can be utilized in reducing dimensionality of problems associated with the partition of unit fractions into the sums of two, three, and four like fractions. The need for such a reduction in this context stems from using a spreadsheet as a modelling tool. Activities described in this paper are grounded in standards for teaching and recommendations for teachers in North America. They are designed for prospective teachers of secondary mathematics. (Contains 4 figures.)

The research question studied in this paper is how teachers' task designing can be enhanced in professional development. We show that this can be done by scaffolding. Scaffolding is used in problem solving and we apply it by considering designing a task as a problem. In the paper, we describe a professional development of four teachers led by a mathematics educator. The professional development consisted of 10 meetings. The teachers were first exposed to a well-designed task in geometry as learners and then were asked to design new geometry tasks for their students. The mathematics educator helped them by supplying scaffolds such as modelling, reflection, articulation and worked examples. The data consisted of video-recordings of the workshops and a semi-structured interview with each of the teachers at the end of the project. The data analysis shows how the use of particular scaffolds was helpful while preserving the teachers' autonomy as task designers. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses articles in the issue on topics including relationship between research and scholarship; mathematics educators; and challenges of mathematics teaching at university level.

Offering mathematics majors the opportunity to engage in current, real-world applications can be an important enhancement to their undergraduate course curriculum. Instead of focusing on the traditional topic areas in pure and/or applied mathematics, one may structure a seminar course for senior mathematics majors by concentrating on a specific applications area. While there is value in seminars covering traditional topics such as applications of differential equations or current topics in topology, to mention just two areas, one might consider structuring a seminar where the focus is on the application, and mathematics is used as a tool. This paper describes an outline for presenting a one hour credit course in the subject area of modeling options and hedging in the securities industry, along with comments on the importance of various activities and discussions. Topics involving bond trading strategies and currency markets are not included.

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

The field of mathematics education has seen an increase in research focused on the concept of identity, with many researchers focusing on the role of teacher pedagogies in shaping learners' identities and their participation in and engagement with mathematics. In this paper, the actual and narrated practices of two teachers, which incorporate teachers' pedagogical approaches and social relationships with learners, are analysed in relation to how they provided opportunities for the construction of learner identities. The paper draws on data that were collected over two years in the form of videotaped lessons, field notes and audiotaped semi-structured interviews with teachers and learners. Data were analysed using qualitative data analysis techniques. The analysis suggests a relationship between the teachers' actual and narrated practices, with the teachers' practices influencing the kinds of identities constructed by learners in their classrooms, in terms of learners' enjoyment of learning mathematics and their willingness to participate in and become fully fledged members of their classroom community. [ABSTRACT FROM AUTHOR]

Mathematics teacher candidates enter teacher education with knowledge that could support or limit their learning. It is therefore important to gain insights of this knowledge to inform teacher education. This paper offers such insights for prospective mathematics teachers' initial knowledge of problem solving (PS) for teaching at the beginning of their teacher education programme. It reports on a study that investigated three categories of problem-solving knowledge for teaching: problem characterization, the PS process, and PS disposition. Participants were 109 prospective primary teachers at a university in Spain. Data sources were questionnaires designed for the study to explore the initial knowledge held by the participants for three categories of problem-solving knowledge. Findings indicated that the participants' knowledge was stronger regarding the PS process and disposition than for problem characterization but had limitations that needed to be addressed in their teacher preparation programme. [ABSTRACT FROM AUTHOR]

This study aims to analyse teachers' understanding of realistic mathematics education (RME) through a blended workshop on designing an RME-based learning trajectory. Data was collected via online pre-test and post-test, involving mathematics teachers of junior high schools in Banda Aceh, Indonesia. The tests were accompanied by interviews, chats on WhatsApp groups, and video recording. A descriptive analysis of the various sources of data indicated that the teachers made varying shifts in understanding the principles of RME. The teachers slightly altered their understanding of the intertwinement and level principles. However, their comprehension of the reality principle remained the same; they considered real problems as daily life problems only, even though some teachers had already written the meaning of real problems as problems that students can imagine. They also have already possessed a satisfactory understanding of the activity principle before the workshop. Moreover, nearly half of the teachers showed a good understanding of RME-based learning trajectories. The findings reported that teachers benefited from the blended workshop and obtained knowledge of the RME-based learning trajectory after participation. Finally, the implications for further research were discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper explores the specialized knowledge mobilized by a mathematics teacher in the selection and use of examples for teaching sequences. Taking an experimental case study approach, we analyse the examples deployed in a series of third-year secondary level lessons on sequences and identify the different knowledge subdomains activated according to the mathematics teachers' specialized knowledge analytical model. We will analyse active and passive examples, pointing out the mathematical entity that is being exemplified and the aspect of this entity which is being emphasized by the example. The results identify the different subdomains and categories which are drawn on in the selection and use of examples, along with the various interconnections across knowledge subdomains which interact in the process. [ABSTRACT FROM AUTHOR]

In this paper, we examine the relationships between teachers' subject matter preparation and experience in teaching and their performance on an instrument measuring mathematical knowledge for teaching Algebra 1. We administered the same instrument to two different samples of teachers−high school practicing teachers and community college faculty − who teach the same algebra content in different levels of institutions, and we compared the performance of the two different samples and the relationships between the measured knowledge and their educational and teaching background across the samples. The comparison suggested that the community college faculty possess a higher level of mathematical knowledge for teaching Algebra 1 than high school teachers. The subsequent analyses using the Multiple Indicator Multiple Causes (MIMIC) models based on our hypothesis on the factors contributing to the differences in the knowledge between the two teacher samples suggest that experience teaching advanced algebra courses has positive effects on the mathematical knowledge for teaching Algebra 1 in both groups. Highlighting the positive effect of algebra-based teaching experience on test performance, we discuss the implications of the impact of subject specific experience in teaching on teachers' mathematical content knowledge for teaching. [ABSTRACT FROM AUTHOR]

This article aims to delve into the meaning of school mathematical concepts by their semantic analysis. Concretely, this paper aimed primarily to describe and characterize the meaning of derivative of a function at a point expressed by mathematics pre-service teachers. In order to achieve this goal, we use a semantic framework, in which the meaning is composed of: (a) conceptual structure, (b) representation systems, and (c) sense and usage. This qualitative, descriptive study analysed the replies of 37 pre-service teachers to questions involving the definition, requirements, and structural concept of derivatives. It also explores the sense and usage attributed by participants to this mathematical concept. At least five profiles of meanings for derivatives could be constructed from the results. The most prominent findings included the prevalence of the geometric interpretation of the concept and participants' failure to mention elements and requisites essential to the definition of derivatives. [ABSTRACT FROM AUTHOR]

Providing students with the necessary skills to apply mathematical knowledge in real-life situations through mathematical modelling is a common goal of contemporary mathematics curricula. As such, it is important that prospective mathematics teachers develop the ability to design learning environments that serve this purpose in their future teaching. This paper presents an activity presented as part of a course that, among other outcomes, aimed to establish awareness in mathematics teacher candidates of the potential use of computers in mathematical modelling. The activity included an experimental investigation of Buffon's needle problem. The steps used to solve the problem using a GeoGebra environment are described in this article. [ABSTRACT FROM AUTHOR]

This paper describes a research project implemented in a course of Real Analysis II. The aim of this project is to analyse students' comprehension of the limiting behaviour of a function from R2 to R at a point focusing on the formal definitions. Alternative definitions and other statements are discussed, some of which meet both sufficient and necessary conditions, while others meet only sufficient conditions; some involve neighbourhoods and others 'paths'. Emphasis is placed on the criteria according to which students choose one of the given definitions, equivalence or not of these and their overall influence on the concept images formed by the students for the limit concept. In particular, the paper discusses how students come to a conclusion about the existence of the limit of a function in R2 through the consideration of contour curves. We also present and analyse data from a written test as well as from the conducted interviews. [ABSTRACT FROM AUTHOR]

The present paper aims at investigating the relationship between students' attitudes toward online homework use and mathematics anxiety among secondary school students. In this study, a model has been proposed by integrating the technology acceptance model (TAM) with mathematics anxiety theory. Three hundred and forty-five secondary school students participated in the instrument of the study that included 20-item based on a 5-point Likert scale. The subsequent analysis using Partial Least Squares Structural Equation Modeling (PLS-SEM) supported the negative relationship between attitude toward an online homework (OHW) tool and mathematics anxiety. The results revealed that perceived usefulness and perceived ease of use are predictors of attitude toward the use of OHW. This study maximized the capability of acceptance of the OHW tool in mathematics learning. The findings provided by the study will encourage mathematics educators to implement mathematics OHW tools in the learning process. [ABSTRACT FROM AUTHOR]

This paper, which describes a study undertaken with pairs of future primary teachers using GeoGebra software to solve geometry problems, includes a brief literature review, the theoretical framework and methodology used. An analysis of the instrumental genesis process for a pair participating in the case study is also provided. This analysis addresses the techniques and types of dragging used, the obstacles to learning encountered, a description of the interaction between the pair and their interaction with the teacher, and the type of language used. Based on this analysis, possibilities and limitations of the instrumental genesis process are identified for the development of geometric competencies such as conjecture creation, property checking and problem researching. It is also suggested that the methodology used in the analysis of the problem solving process may be useful for those teachers and researchers who want to integrate Dynamic Geometry Software (DGS) in their classrooms. [ABSTRACT FROM AUTHOR]

In this paper, a process is described that is used in professional development sessions in mathematics with pre-tertiary teachers referred to as leveraging teachers' mathematical ideas (LTMI). To examine LTMI and how it impacted participating teachers, qualitative research methodological approaches were used. Research literature is reviewed that demonstrates how student access to a demanding education in mathematics is related to class, race and ethnicity. Literature is also reviewed about mathematical identity. Sample instructional vignettes are given that illustrate LTMI and how teachers' solutions to mathematical problems are employed as a means to support their mathematical thinking, challenge deficit beliefs they may hold about teaching and their students, while also validating the mathematical ideas that they bring to the learning process. Descriptions are provided about how during LTMI episodes, an intentional effort is made to showcase the mathematical solutions of women and teachers of color in particular. Research findings are provided to lend insight into how teachers perceived the professional development impacted them. Through LTMI, a community of practice is validated as mathematically worthy, a place where the human dignity of the participants is respected. [ABSTRACT FROM AUTHOR]

Given the vast experiences pre-service mathematics teachers (PSMTs) have with functions, one may assume that they have existing personal definitions and concept images. The purpose of this paper is to build an understanding of the variety of images that one might associate with a definition of function that includes 'for each input there is one output' and how these images are coordinated with the definition to determine whether an object is a function or not. Two cases are presented to describe PSMTs who were working from the same personal definition of function, yet used that definition differently within a task designed around the function concept. The task itself was novel in that it is presented in a non-algebraic context, namely a vending machine. As such, PSMTs were required to coordinate their personal definition and images within the novel context. Attending to PSMTs' concept images revealed that though they were working from the same personal definition, images they drew upon to determine whether or not an object is a function were varied. [ABSTRACT FROM AUTHOR]

This paper aims to contribute to the growing field of research on teaching practices in higher education, with a focus on the prevalent yet unresearched practice of concept introduction in lectures. This small-scale case-study provides data which lend themselves to a comparative analysis, with a view to describing and accounting for the dispersion of practices which can be observed in spite of the similarity of teaching conditions: three teachers lecturing on the same formal definition of limits of sequences, in similar institutional conditions. A combination of an a priori mapped out space of possible actions, and concepts from the theory of teaching as decision-making allows for a fine-grained analysis of the data, consisting of video-recordings of the sessions, supplemented by pre- and post-interviews with the teachers. We identified two key factors – knowledge of action-plans and 'subjective expected utility' (Schoenfeld, A. H. (2013). Classroom observation in theory and practice. ZDM (Mathematics Education), 45, 607–621) – on the basis of which the instructors' engagement in teaching can be accounted for in a non-evaluative research perspective. [ABSTRACT FROM AUTHOR]

In application-oriented mathematics, particularly in the context of nonlinear system analysis, phase plane analysis through SageMath offers a visual display of the qualitative behaviour of solutions to differential equations without inundating students with cumbersome calculations of the plane-phase. A variety of examples is usually given to illustrate phase-plane behaviour. We approach these problems by considering a problem containing a single real parameter that exemplifies the various situations clearly and simply. We developed two computer programs in SageMath: one program calculates aspects of the Jacobian matrix and displays the phase plane portraits, the other determines the centre manifold. Computations and images generated with computer codes are useful in understanding dynamic models in biology, physics and engineering that involve planar non-linear autonomous differential equations. This paper covers analytical and computational skills that are helpful for students and teachers. [ABSTRACT FROM AUTHOR]

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

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

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

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

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]

In this paper we report on the outcomes of two surveys carried out in higher education institutions of Ireland; one of students attending first-year undergraduate non-specialist mathematics modules and another of their lecturers. The surveys aimed to identify the topics that these students found difficult, whether they had most difficulty with the concepts or procedures involved in the topics, and the resources they used to overcome these difficulties. In this paper we focus on the mathematical concepts and procedures that students found most difficult. While there was agreement between students and lecturers on certain problematic topics, this was not uniform across all topics, and students rated their conceptual understanding higher than their ability to do questions, in contrast to lecturers’ opinions. [ABSTRACT FROM AUTHOR]

This study investigates prospective elementary and secondary school mathematics teachers' ways of reasoning about differentiability at a point and corner points while working on a mathematical modelling activity. Adopting a multiple-case study design, the participants of the study were 68 prospective elementary school mathematics teachers enrolled in the 'Calculus-1' course (Case-1) and 20 prospective secondary school mathematics teachers enrolled in the 'Mathematics Teaching Methods' course (Case-2). We adapted a task from the well-known filling bottle problem and implemented it as a regular part of each course. Written group solutions, individual reflection papers and task-based group interviews were the data sources. Constant comparative analysis method was employed in analysing the data. The results from both cases indicated that prospective mathematics teachers frequently relied on roughly drawn graphs while deciding differentiability. They had difficulties in writing an algebraic model and comprehending the meaning of a corner point in the filling water context. Prospective mathematics teachers got confused when they got different results for differentiability at a critical point through the roughly sketched graph and algebraic model. The data also showed that thinking about these concepts while working on such dynamic situations fostered prospective mathematics teachers in developing more productive ways of thinking. [ABSTRACT FROM AUTHOR]

Mathematics education researchers are asking themselves about why technology has impacted heavily on the social environment and not in the mathematics classroom. The use of technology in the mathematics classroom has not had the expected impact, as it has been its use in everyday life (i.e. cell phone). What about teachers' opinions? Mathematics teachers can be divided into three categories: those with a boundless overflow (enthusiasm) who want to use the technology without worrying much about the construction of mathematical concepts, those who reject outright the use of technology because they think that their use inhibits the development of mathematical skills and others that reflect on the balance that must exist between paper-pencil activities and use of technology. The mathematics teacher, by not having clear examples that support this last option about the balance of paper-pencil activities and technology, opt for one of the extreme positions outlined above. In this article, we show the results of research on a methodology based on collaborative learning (ACODESA) in the training of mathematics teachers in secondary schools and implementation of activities in an environment of paper-pencil and CAS in the mathematics classroom. We also note that with the development of technology on the use of electronic tablets and interactive whiteboards, these activities will take on greater momentum in the near future. [ABSTRACT FROM AUTHOR]

This paper presents research into closed orbits parallel to quadrilaterals inscribed in various geometric shapes that can be represented by mathematical functions: straight lines, circles, ellipses, parabolas, and hyperbolas. Mathematical proofs have been given for the existence of an infinite number of parallel orbits for each of these shapes. Quadrilaterals parallel to a quadrilateral inscribed in a circle were found to have two interesting characteristics. The mathematics was conducted with student mathematics teachers and teacher trainees, accompanied by dynamic research using computer software. [ABSTRACT FROM AUTHOR]

This paper presents a case study of how a teacher and nine-year-old students deal with mathematics in a practical technology project. By analysing videotaped dialogues between teacher and a pair of students working on constructing a house model, we identify challenges of meaningful inclusion of mathematics in the project. The dialogues are analysed in terms of an analytical framework where four categories of interaction patterns are combined with two main paradigms of mathematics teaching: the exercise paradigm and landscapes of interaction. The project in itself has a potential for facilitating landscapes of investigation in mathematics teaching. However, we find that the teacher as well as students adheres to the exercise paradigm when mathematics is involved in the activity. Two illustrating episodes from the project are examined and presented in detail in this paper. The findings illustrate that the conceptions teachers and students hold of what mathematics teaching means can act as an obstacle in attempts to realize mathematics teaching in creative and meaningful contexts for young students. We suggest that making the various purposes of a project more explicit may help overcome this obstacle, and that the mathematics involved might be taught in separate sessions in order to form a constructive part of a cross-curricular project. [ABSTRACT FROM AUTHOR]

This paper describes students’ perspectives of a one-off flipped lecture in a large undergraduate mathematics service course. The focus was on calculating matrix determinants and was designed specifically to introduce debate and argumentation into a mathematics lecture. The intention was to promote a deeper learning and understanding through engagement with the added hope of instilling some passion for the subject. During the lecture, students were asked to vote with their feet, literally moving around the lecture theatre to form groups according to their shared favourite technique for calculating matrix determinants. Group discussions were then followed by a whole class debate facilitated by the lecturers, before they wrapped up the lecture by resolving the professional disagreements that had come to light during the debate. Following the lecture, data on student perspectives was gathered using both surveys and focus groups. Within this paper, we share the data and reveal the interesting results that emerged from our analysis. Despite remaining unconvinced as to whether flipped lectures are better for learning, students reported greater engagement and increased understanding of the material covered. [ABSTRACT FROM AUTHOR]

The school system in both the Republic of Ireland [ROI] and Northern Ireland [NI] is configured to accommodate children into discrete educational bands at different stages on their learning journey. Research has consistently shown that the transition from primary to post-primary, which typically occurs between the ages of 11 and 13, is the most challenging transition with the negative effects more pronounced for mathematics than any other subject. This cross-border study investigates this transition in mathematics education from the final year of primary school to first year of post-primary school from the perspective of teachers involved with these year groups. It examines teachers' knowledge of the mathematics curriculum and teaching strategies employed in the other phase. The majority of primary teachers reported being unfamiliar with the curriculum and teaching approaches employed in the first year of post-primary and the differences between the two jurisdictions were not significant. However, a difference was noted in the responses of second level teachers. Teachers in NI appeared more confident in their knowledge of the curriculum and teaching approaches adopted in the final year of primary school than their ROI colleagues. In this paper, the authors will elaborate on these findings; outline possible reasons for the differences across jurisdictions and share what impact teacher knowledge is having on classroom practices. [ABSTRACT FROM AUTHOR]

This study investigated how Turkish mathematics teachers evaluate the effectiveness of classroom teaching in terms of improving students' mathematical proficiency. To this purpose, teachers were asked to evaluate a mathematics lesson as presented them in a vignette. By means of cluster analysis, the participants' evaluations of the lesson were described in five thematic dimensions, which could be further assembled into two overriding categories: students' understanding of the subject, and teachers' classroom practices. The overall aim of the current paper is to propose a preliminary model of the framework that Turkish mathematics teachers use to evaluate a mathematics lesson. [ABSTRACT FROM AUTHOR]

This paper suggests that mathematics teacher educators should listen carefully to what their students are saying. More specifically, it demonstrates how from one pre-teacher's non-traditional geometric representation of a unit fraction, a variety of learning environments that lead to the enrichment of mathematics for teaching can be developed. The paper shows how new knowledge may be generated through an attempt to validate an intuitive idea; in other words, how the quest for rigour may serve as a catalyst for the growth of mathematical concepts in the context of K-16 mathematics. [ABSTRACT FROM AUTHOR]

This paper shows how inequalities can be utilized in reducing dimensionality of problems associated with the partition of unit fractions into the sums of two, three, and four like fractions. The need for such a reduction in this context stems from using a spreadsheet as a modelling tool. Activities described in this paper are grounded in standards for teaching and recommendations for teachers in North America. They are designed for prospective teachers of secondary mathematics. [ABSTRACT FROM AUTHOR]

Many students struggle with college mathematics topics due to a lack of mastery of prerequisite knowledge. Set theory language is one such prerequisite for linear algebra courses. Many students' mistakes on linear algebra questions reveal a lack of mastery of set theory knowledge. This paper reports the findings of a qualitative analysis of a group of linear algebra students' mistakes on a set of linear algebra questions. The paper also details an in-time intervention (a pedagogical approach) to enhance students' understanding of linear algebra concepts through advancing their set theory knowledge. Mathematics teachers can consider similar approaches to address their students' mistakes. [ABSTRACT FROM AUTHOR]

This paper analyses a lecture by an excellent teaching award winner professor of mathematics, given to high school mathematics teachers. The analysis is based upon two sources: (i) the lecture plan, as expressed in a series of 29 transparencies, prepared by the lecturer in advance; (ii) the actual implementation of the lecture, as transcribed from its video-taped record. Based on this analysis six principles for planning and giving a good lecture were developed. The paper provides the readers with full details of the content as well as non-verbal communication gestures exemplifying the employment of the six principles. [ABSTRACT FROM AUTHOR]

This paper shows how the computational and graphical capabilities of spreadsheets allow for interactive analytic and geometric constructions from numerical modeling of homogeneous Diophantine equations of the second order. Suggested activities, designed for prospective teachers of mathematics and made possible by what is referred to in the paper as the tool kit approach, enable one to revisit classic mathematics concepts within the framework of computerized mathematical experiment. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to examine the cognitive demand levels of tasks used by an in-service primary teacher during length measurement and perimeter instruction and to examine a possible link between these tasks and the teacher's mathematical knowledge in teaching. For this purpose, a case study approach was used and the data was drawn from classroom observations, semi-structured interviews, and field notes. Specific tasks from length measurement and perimeter instruction were presented and analyzed according to the Mathematical Tasks Framework. Then, how these tasks gave information about the teacher's mathematical knowledge in teaching in the length measurement and perimeter topics was examined according to the Knowledge Quartet model. According to the findings of the study, the tasks used during length measurement and perimeter instruction were mostly categorized as low-level tasks. In addition, teacher's mathematical knowledge in teaching affected the implementation of the tasks. [ABSTRACT FROM AUTHOR]

Paying attention to the planning of lessons represents a crucial teaching task. This paper analyses teachers' framing of lesson objectives and their descriptions of how they would introduce particular mathematics topics. Connections between the objectives and the instructional strategies employed, were also studied. The study involved 75 South African mathematics teachers who were teaching Grades 4-12, in the KwaZulu-Natal Province. These teachers completed a questionnaire asking them to reflect upon and describe their teaching practices with respect to a chosen topic. The findings revealed that setting up clear lesson objectives is not easily done with only 64.0% of the participants providing clear learning objectives. The findings also showed that teachers preferred introducing the topics using teacher-centred methods. It was also found that for more than half of the teachers, there were clear connections between lesson objectives and the introduction of the topic in the classroom whereas less than a quarter of them made connections between lesson objectives and teaching and assessment strategies. Teachers, as important drivers of education, should prioritize the setting of clear objectives around which coherent lessons can be developed so that their teaching can be more effective. [ABSTRACT FROM AUTHOR]

Measurement of the quality of teaching activities is becoming increasingly important since universities are rewarding performance in terms of promotion, awards and bonuses and research is no longer the only key performance indicator. Good teaching is not easy to identify and measure. This paper specifically deals with the issue of good teaching in mathematics at tertiary level-what good teaching is and how to measure it. [ABSTRACT FROM AUTHOR]

This paper describes a content upgrading course on Secondary School Algebra which was planned and conducted for in-service mathematics teachers who are trained to teach mathematics at lower secondary level (ages 13 to 14). In order to be an effective mathematics teacher, one needs to know a great deal of mathematical knowledge on top of good pedagogy. This course aims to "upgrade" the mathematics content of these teachers in order to enable them to have a holistic view of the entire Algebra syllabus in the secondary school mathematics curriculum. The details of the content, exercises and the assessments for the participants are discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper describes part of a research and development project carried out in public elementary schools. Its objective was to update the mathematical and didactic knowledge of teachers in two consecutive levels in urban and rural public schools of Region de Los Lagos and Region de Los Rios of southern Chile. To that effect, and by means of an advanced training project based on a professional competences model, didactic interventions based on types of problems and types of mathematical competences with analysis of contents and learning assessment were designed. The teachers’ competence regarding the didactic strategy used and its results, as well as the students’ learning achievements are specified. The project made possible to validate a strategy of lifelong improvement in mathematics, based on the professional competences of teachers and their didactic transposition in the classroom, as an alternative to consolidate learning in areas considered vulnerable in two regions of the country. [ABSTRACT FROM AUTHOR]

This paper reports on a study about how the mathematics textbook was perceived and used by the teacher in the context of a calculus part of a basic mathematics course for first-year engineering students. The focus was on the teacher's choices and the use of definitions, examples and exercises in a sequence of lectures introducing the derivative concept. Data were collected during observations of lectures and an interview, and informal talks with the teacher. The introduction and the treatment of the derivative as proposed by the teacher during the lectures were analysed in relation to the results of the content text analysis of the textbook. The teacher's decisions were explored through the lens of intended learning goals for engineering students taking the mathematics course. The results showed that the sequence of concepts and the formal introduction of the derivative as proposed by the textbook were closely followed during the lectures. The examples and tasks offered to the students focused strongly on procedural knowledge. Although the textbook proposes both examples and exercises that promote conceptual knowledge, these opportunities were not fully utilized during the observed lectures. Possible reasons for the teacher's choices and decisions are discussed. [ABSTRACT FROM PUBLISHER]

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

The paper describes a personal experience with using counterexamples as a pedagogical strategy in the teaching and learning of an introductory calculus course at a university of technology. [ABSTRACT FROM AUTHOR]

The changes in academic identity a teacher may undergo, as they modify their teaching practice, will vary depending on their experiences and the support they receive. In this paper, we describe the shifts in academic identity of two lecturers, a mathematician and a mathematics educator, as they both made changes to their teaching practice by implementing new questioning techniques in a large undergraduate mathematics course. Both the lecturers were members of the research group, which became their community of practice. Our findings recommend that lecturers endeavouring to step out and try changes to their teaching practice, particularly with large groups of students, belong to a community of practice. The community of practice provides a place for shared reflection, new learning, and opportunities to negotiate new identities. [ABSTRACT FROM PUBLISHER]

The paper presents findings of a small scale study of a few items related to problem solving with squares and roots, for different teacher groups (pre-service and in-service mathematics teachers: elementary and junior high school). The research participants were asked to explain what would be the units digit of a natural number to be squared in order to obtain a certain units digit as a result. They were also asked to formulate a rule – an algorithm for calculating the square of a 2-digit number which is a multiple of 5. Based on this knowledge and estimation capability, they were required to find, without using calculators, the square roots of given natural numbers. The findings show that most of the participants had only partial intuition regarding the units’ digit of a number which is squared when the units’ digit of the square is known. At the same time, the participants manifested some evidence of creativity and flow of ideas in identifying the rule for calculating the square of a natural number whose units digit is 5. However, when asked to identify, by means of estimation and based on knowledge from previous items, the square roots of three natural numbers, only few of them managed to find the three roots by estimation. [ABSTRACT FROM PUBLISHER]

In this paper, we explore a novel approach for assessing the impact of a professional development programme on classroom practice of in-service middle school mathematics teachers. The particular focus of this study is the assessment of the impact on teachers’ employment of strategies used in the classroom to foster the mathematical habits of mind and mathematical self-efficacy of their students. We describe the creation and testing of a student survey designed to assess teacher classroom practice based primarily on students’ ratings of teacher practices. [ABSTRACT FROM PUBLISHER]