Search

This paper introduces the model of instrumental meta-orchestration (IMO), as an extension of the model of instrumental orchestration. This IMO model is defined as a systematic and intentional monitoring, by a teacher educator, of artefacts and teachers (or pre-service teachers) for facing a Metasituation, defined as a composition of situations of different natures and difficulty levels. An IMO, in itself, is a composition of instrumental orchestrations (sequenced or interwoven). This paper presents and discusses an instrumental meta-orchestration experienced in a class of undergraduate mathematics teachers. The development of the initial model results in: new concepts, such as ad hoc reaction, didactic meta-configuration, exploitation modes and didactic meta-performance; new features, such as flexibility and interactivity; and new phenomena, such as cascade effects. It also reveals the importance of unexpected events that occur between orchestrations. Moreover, it expanded the forms of destination for evaluating specific online documents, which are named webdocs. Finally, the paper discusses the contribution to mathematics teacher education of such an extended model. [ABSTRACT FROM AUTHOR]

This paper states three claims dealing with the relationship between mathematics education researchers and mathematics teachers: (1) Mathematics education research is a highly diverse field; (2) Teachers have various roles as stakeholders in mathematics education research; (3) Regarding teachers as stakeholders in mathematics education research affords reflecting some (fruitful) „cultural differences”. The paper claims the necessity to regard researchers as key stakeholders in practice, and teachers as key stakeholders in research. [ABSTRACT FROM AUTHOR]

This paper identifies the knowledge that emerges from collaborative situations, in an initial teachers' education with integration of digital technology for teaching of function, in a blended and collaborative modality course, aiming to emerge Technological, Pedagogical and Content Knowledge (TPACK) in pre-service teachers. The method comprised the design of a teachers' collaborative training experiment. The experiment had five phases, from a theoretical discussion to plan an Instrumental Orchestration to teach function with digital resources. Collaboratively, students worked in groups. The participants were students from a Mathematics teachers' education. The experiment took place in the Teaching Methodology of Mathematics discipline at a public university in Brazil. This article focuses on the analysis of students' knowledge in the planning stage, carried out collaboratively. The data analysis pointed out the three types of Technological, Pedagogical and Content knowledge were identified in different phases of the experiment, sometimes collectively, sometimes individually. In addition, we also identified that some intersections of TPACK emerged from the interactions; despite being in the initial stage of training, the students show pedagogical knowledge linked to knowledge of the content they had at the time of the training experiment. [ABSTRACT FROM AUTHOR]

This study investigates 48 preservice middle school teachers' formal textbook definitions, formulas, and graphs of the directly and inversely proportional relationships. The connections among these three types of representations and how preservice teachers' representations differ according to the two relationships are also examined. The data of the study included the preservice teachers' responses to a paper-and-pencil test with two items. An explanatory research design model was followed when developing this study. The study findings indicated the preservice teachers' over-attention to the simultaneous increases and/or decreases when representing the directly and inversely proportional relationships. Only four preservice teachers stated in their definitions that the ratio was constant in the directly proportional relationship. Whereas none of them stated constant product in their inversely proportional definition. Although the preservice teachers were slightly better at writing the direct proportion formula than the inverse proportion formula, this difference was much greater in terms of drawing the direct and inverse proportion graphs. Sixty percent of the preservice teachers drew their inverse proportion graphs as linear with negative slopes. Moreover, the analysis showed that the preservice teachers' definitions were not well-linked with their graphs and formulas. On the other hand, the preservice teachers who were better at providing correct formulas were also better at drawing the correct graphs. Implications for teaching and future study suggestions are discussed. [ABSTRACT FROM AUTHOR]

This paper provides a historical account of the development of algorithmic complexity in a form that is suitable to instructors of mathematics at the high school or undergraduate level. The study of algorithmic complexity, despite being deeply rooted in mathematics, is usually restricted to the computer science curriculum. By providing a historical account of algorithmic complexity through a mathematical lens, this paper aims to equip mathematics educators with the necessary background and framework for incorporating the analysis of algorithmic complexity into mathematics courses as early on as algebra or pre-calculus. [ABSTRACT FROM AUTHOR]

The e-textbook is one of the main resources that distance learning institutions make available in the virtual learning environment (VLE) for teachers and students. According to the Brazilian References for Distance Learning, the authors should follow some methodological, epistemological and political principles for its production: to allow students and teachers (educator and tutor) to construct knowledge through interaction with peers. In this paper we seek to map the enrichment of the content in DE carried out by teachers' educator, from the use of e-textbook, in the light of the analysis of the teacher's instrumental orchestrations. The theoretical framework comprised Luc Trouche's Instrumental Orchestration, focused on understanding the didactic configuration. Three trainers of Analytical Geometry from two courses of Mathematics teachers' initial education of Brazilian Open University composed our subjects. As for the method, we adopted some techniques used in the documentational approach to didactics, introduced by Ghislaine Gueudet and Luc Trouche. We made a virtual visit to the teachers' virtual classroom protocols in VLE, and then an interview with each teacher was undertaken. Finally, we asked them to draw a map of the resources they use in class. Our results revealed the need to add other content resources, during didactic mediation forcing the teacher to change his didactic configurations, and the need to think about a collective construction of the e-textbook, discussing in parallel with the main authors of distance learning, teachers and tutors of the performers. [ABSTRACT FROM AUTHOR]

Over the last forty years, standards and recommendations for teachers and learners of K-12 mathematics in the US have evolved to highlight mathematical practices (e.g., Common Core State Standards of Mathematics, Standards for Mathematical Practice [SMPs]). Practice standards (i.e., SMPs) describe mathematical competencies that should be developed in learners of mathematics at all levels. National organizations (e.g., Conference Board of Mathematical Sciences) have specifically called for attention to be given to SMPs in collegiate mathematics content courses for prospective elementary (ages 5-12) teachers (PTs). The goal of this paper is to help instructors of such courses, especially those new to the field of mathematics education, gain familiarity with the organizations and documents that support the development of these practices and conceptualize ways in which they might engage PTs in their content courses in SMPs. First, we synthesize the evolution of mathematics standards for K-12 learners and teachers in the US. Second, we report results from an investigation into the ways in which mathematics teacher educators (MTEs) are addressing SMPs in their content courses for PTs. In this study, SMP3: Construct viable arguments and critique the reasoning of others was reported by MTEs as being addressed in their courses more than any other SMP. This finding precipitated a qualitative analysis of the ways in which PTs were being provided opportunities to engage in SMP 3 within the descriptions and samples of tasks provided by the MTEs. We will share and discuss example tasks that provided opportunities for PTs to analyze others' thinking. Lastly, we consider the potential benefits of leveraging children's thinking in SMP 3-related tasks for PTs and provide resources for MTEs who are interested in utilizing samples of children's thinking in their classes. [ABSTRACT FROM AUTHOR]

Prospective Teachers often enter their mathematics content courses believing that they know enough mathematics to teach elementary school. However, research has shown that much of PTs' knowledge is procedurally based and lacks depth and conceptual understanding. One job of Mathematics Teacher Educators (MTEs) in mathematics content courses is to help PTs become more mathematically proficient, by relearning the mathematics that they believe they already know in deeper, more connected ways. We suggest that one way for MTEs to do this is to incorporate Transformative Learning Theory (TLT) into their mathematics content courses. TLT is an application of andragogy, which are the methods or techniques used to teach adults. Through TLT, learners participate in a process where they are presented with a disorienting dilemma that perturbs their prior understandings. Learners work through the dilemma by critically reflecting on their prior understandings and participating in rational discourse with others. Ultimately, learners are tasked with making connections between their prior understandings and their new knowledge. In this paper, we describe a cycle of transformative learning theory and give examples of incorporating TLT into mathematics content courses for PTs through lessons on proportional reasoning and whole number concepts. We conclude by discussing general considerations and resources for incorporating TLT into mathematics content courses and how this helps PTs develop the five strands of mathematical proficiency (National Research Council, 2001). [ABSTRACT FROM AUTHOR]

In order to be effectively prepared by a teacher education program, prospective elementary teachers (PTs) need to experience high quality mathematics instruction in their mathematics content courses. The instructors of these courses typically consist of individuals (mathematicians and mathematics educators) with ranging experiences, from tenured faculty members to first-year assistant professors or graduate students. This paper explores how to support novice mathematics teacher educators (MTEs) who are teaching elementary content coursework for PTs for the first time. We detail and describe how to implement three systems for supporting novice MTEs: working with a mentor, being provided with educative curriculum materials, and working in a collaborative teaching environment. We close by discussing specific challenges associated with these supports, and call for more institutions to share how they have successfully implemented systems to support novice MTEs. [ABSTRACT FROM AUTHOR]

This paper considers ways in which our positionality as mathematics education research (MER) scholars leads to contradictions between our theory and practice in relation to equity. I draw from Gramsci's theorizing about traditional intellectuals and hegemony to ground the relationships between MER scholars, teachers and students in an understanding of U.S. class relations and the function of the State in mediating them. Constraints and contradictions in achieving equity are related to broader roles of education reform in maintaining the State as well as local antagonisms between MER scholars, teachers and students. For Gramsci, consent to class relations is secured in large part through culture and ideas. Thus, I consider how the dominant theory of learning and pedagogical framework in MER - constructivism - facilitates consent to the relationships between MER scholars and the labor of teachers and students. [ABSTRACT FROM AUTHOR]

This study aims to exemplify a framework based on an empirical study and the empirical study is to explore how a mathematics expert teacher makes sense of multiplication. The study was conducted through interviewing a secondary school mathematics expert teacher with over 30 years of mathematics teaching experience. The theoretical framework used in this paper is proposed by Chin and Tall (2012) and further developed by Chin (2013) and Chin (2014). The collected data are anaylsed qualitatively using this framework in order to find out what are the supportive and problematic conceptions involved in making sense of multiplication and consequently investigate how do these conceptions affect the sense-making of multiplication across different contexts. The findings suggested that when changes of meaning occur, supportive or problematic conceptions might arise and these either support or inhibit learners from building a coherent understanding. This study also showed that supportive conceptions might contain problematic aspects and problematic conceptions might contain supportive aspects. The respondent strived to build a coherent understanding by removing the problematic aspect, however when the problematic aspect cannot be removed then he had no choice but to accept the new meaning of multiplication. It is the teachers' responsibility to guide students in realising their existing knowledge which may not be appropriate in a new context. Thus, this study offers a foundation for teachers to sense where, when and how the changes of meaning of multiplication take place which in turn can help them to facilitate students effectively in making sense of multiplication. [ABSTRACT FROM AUTHOR]

Mathematics teachers in many parts of the world have been facing many challenges in recent years. In Iceland teachers in lower secondary school have been implementing a new reform oriented curriculum. Many researchers claim teachers need considerable support if they are to meet the demands of the current reforms in mathematics education. This paper explores whether mathematics teachers at lower secondary level in Iceland are given good opportunities to develop professionally during the last five years and if not what can possibly be done to improve the situation. My conclusion is that the opportunities given are limited and do not meet features that characterize effective professional development. That specially applies for duration and coherence which are considered very important features along with, focus on content, active learning and collective participation. The organisation and funding of professional development in Iceland does not seem to allow for continuation and progression. One important step to deal with this problem would be to make it easier for teachers to attend courses at the universities and to arrange courses in such a way that teachers and student teachers can collaborate and form a learning community. [ABSTRACT FROM AUTHOR]

In this paper, we report on our experiences with professional development at the Calgary Girls School in Alberta, Canada. In particular we reflect on factors such as mentoring and teacher coaching that contribute to higher student achievement as well as a culture of co-operation and collaboration in the context of the Alberta Initiative for School Improvement (AISI). [ABSTRACT FROM AUTHOR]

This paper addresses questions of mathematics teachers' professional development. My goal is not to provide "answers," although I have worked for some years to enhance teachers' capacity to create rich learning environments for their students. Rather, my goal is to problematize the issue, to ask: How do we frame questions of professional development in ways that are theoretically grounded? What theories do you need to know, in order do a good job of professional development? In the light of this kind of theoretical framing, I will discuss two related attempts at supporting teachers in their work. [ABSTRACT FROM AUTHOR]

In this paper the nature of learning progressions and related concepts are discussed. The notions of learning progressions and learning trajectories are conceptualized and their usage is illustrated with the help of examples. In particular the nuances of instructional interventions utilizing these concepts are also discussed with implications for the teaching and learning of mathematics. [ABSTRACT FROM AUTHOR]

The author reflects on the support of "The Mathematics Enthusiast" journal for the Psychology of Mathematics Education-North America (PME-NA). He mentions that the relationship of "The Mathematics Enthusiast" with PME-NA is intended to be anti-symbiotic, where the journal does not rely on PME-NA because they do well on their own. He states that the journal supports PME-NA is to offer a possible journal outlet for Institutions colleagues that do not acknowledge online conference proceedings.

Copyright of Mathematics Enthusiast is the property of Prof. Bharath Sriraman and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper deals with the sustainable impact of innovations and professional development programmes. While research on this issue is rather scarce in educational disciplines (and in particular in mathematics teacher education), some other domains like health care or development aid are well grounded in research results regarding this topic. This article gives an insight into the other disciplines' knowledge concerning the impact of innovations and professional development programmes and the respective fostering and hindering factors. Moreover, possible implications for (mathematics) teacher education are discussed. [ABSTRACT FROM AUTHOR]

In this theoretical essay we problematize the statement "To teach mathematics one must know a specific mathematics for teaching" of the discourse of specific mathematical knowledge to teach. The study was inspired by Foucaultian concepts and the analysis focused on studies in the area of Mathematical Education that mobilize discourses regarding Mathematical Knowledge for Teaching, Mathematics Teacher's Specialized Knowledge and Mathematics for Teaching. The analysis showed that the discourses operate under two truth regimes about the mathematical knowledge needed to teach. Analyzing the enunciative intertwining, we observe an approximation with: i) the educational discourse, which deals with the inseparability between theory and practice; and ii) the capitalist economic discourse, which associates of economic development with the mathematical proficiency of a country. [ABSTRACT FROM AUTHOR]

This article approaches the relationship between mathematics education and field education, based on the academic production of the research group on mathematics education in field education contexts (Gpemce) created in 2008 at the Federal University of Pernambuco. After presenting the repercussions and lines of research of the group, we reflect on the roots and principles of field education and some approximations of this domain with mathematics education. Then, we present a documentary study, whose corpus consists of 20 articles published in journals by researchers from Gpemce from 2009 to 2020. These articles are organised around four thematic categories that are associated with the research lines of the group: teaching and learning mathematics in field education contexts; relationships between curriculum contents, mathematical practices and productive field activities; resources in the teaching of mathematics in field schools and qualification of mathematics teachers of schools in field areas. The study highlights the diversity of educational subjects, themes, and sociocultural contexts contemplated in the research, and points out ways to conduct new investigations that contribute to deepen our understanding of the teaching and learning of mathematics in field schools and initial and continuing teacher education in field education contexts. [ABSTRACT FROM AUTHOR]

Research on technology-assisted teaching and learning has identified several families of factors that contribute to the effective integration of such tools. Focusing on one such family, affective factors, this article reports on a qualitative study of 30 prospective secondary school mathematics teachers designed to acquire insight into the affect associated with the visualization of geometric loci using GeoGebra. Affect as a representational system was the approach adopted to gain insight into how the use of dynamic geometry applications impacted students' affective pathways. The data suggests that affect is related to motivation through goals and self-concept. Basic instrumental knowledge and the application of modeling to generate interactive images, along with the use of analogical visualization, played a role in local affect and prospective teachers' use of visualization. [ABSTRACT FROM AUTHOR]

The field of the history of mathematics education, inscribed in the field of research on mathematics education in Brazil, has been developed as a result of the work of many groups of researchers, dedicated to varied themes under the light of various theoretical and methodological approaches. While it is impossible to describe in detail such studies in a single article, the objective of this text is to offer an overview of research results on a specific topic - mathematics in secondary education and training of teachers to teach at this level, over the years. The current situation of the country regarding education in general is related to the past and different Brazilian political periods: Colony, Empire and Republic. We will succinctly examine the beginnings of Brazilian basic school education before the Republic, seeking to highlight characteristics of secondary school studies, especially regarding the teaching of mathematics. Then, we will focus on the same themes during the period after the establishment of the Republican regime. Finally, we will discuss the historical trajectory of the training mathematics teachers who work on secondary education in the country. [ABSTRACT FROM AUTHOR]

This first of a pair of articles describes a professional development project that prepared four non-Indigenous mathematics teachers (Grades 5-12) to implement Canada's Truth and Reconciliation Commission's (TRC, 2016) notion of reconciliation: cross-cultural respect through mutual understanding. The researchers collaboratively mentored the teachers to enhance their mathematics teaching with Indigenous mathematizing3. The teachers' focus was on developing and revising lesson plans for other teachers to teach. This process explicitly and implicitly revealed precise supports that need to be in place for a teacher to succeed at innovating with this Indigenous culture-based school mathematics (ICBSM). Part I is a template for scaling up the development of much needed Indigenous resources for mathematics teachers. Part II reports on the research results of this year-long research project. [ABSTRACT FROM AUTHOR]

Mathematics teacher educators (MTEs) charged with the content preparation of prospective elementary teachers (PTs) receive widespread recommendations on a variety of issues, including which mathematical topics to include in their courses, the types of pedagogical strategies to use when working with PTs, and how to foster the development of particular mathematical practices. The nature of the mathematical tasks that MTEs can use to best support and extend PTs' content knowledge is another area that deserves attention. This article considers uncertainty as a task feature and explains how it can be incorporated into mathematical tasks in order to support PTs' specialized content knowledge. Using Zaslavsky's (2005) three types of uncertainty - competing claims, unknown path or questionable conclusion, and non-readily verifiable outcomes - we provide three classroom examples of ways in which MTEs incorporated uncertainty into their mathematical tasks. Each example provides detail about the task design, background research on PTs' knowledge of the underlying mathematical content, the type of uncertainty the task generated for PTs, and the ways in which that uncertainty may have contributed to PTs' understandings of the content. Each example concludes with recommendations for MTEs on specific ways to incorporate that particular type of uncertainty into their own teaching. [ABSTRACT FROM AUTHOR]

One responsibility of mathematics teacher educators is to provide opportunities for prospective teachers to develop the necessary knowledge to be highly qualified professionals and effective mathematics teachers. Hill, Ball, and Schilling (2008) delineated two major knowledge domains required for mathematics teachers to be effective: subject matter knowledge and pedagogical content knowledge. In this article, based on literature review, we explore specific recommendations for mathematics teacher educators, including mathematicians and novice mathematics teacher educators, on different ways to incorporate lesson ideas into mathematics content courses to help provide opportunities for grades K-8 (age 5-14) prospective teachers' development of pedagogical content knowledge. The work here provides a foundation for mathematics teacher educators' practice, encompassing classroom-based examples and strategies, and offers potential avenues for mathematics teacher educators to conduct their own research on the nature and effectiveness of their course activities that help to promote and develop prospective teachers' pedagogical content knowledge. [ABSTRACT FROM AUTHOR]

Research cites the need for developing teachers' mathematical knowledge for teaching (MKT) as well as for developing mathematics teacher educators' (MTEs) mathematical knowledge for teaching teachers (MKTT). Using the framework of lesson study: formulating goals and researching, planning, implementing and observing, and reflecting (Lewis & Hurd, 2011), a group of MTEs designed and analyzed a lesson on multiplication for prospective elementary teachers. A qualitative analysis of MTE journal reflections and prospective teacher work showed a greater understanding of MTEs' MKTT related to multiplication after completion of the lesson study. The authors recommend MTEs conduct lesson studies for other mathematics topics to further understand what MKTT MTEs need to develop to best support prospective teachers' MKT. [ABSTRACT FROM AUTHOR]

Attention to issues of social justice is becoming increasingly prominent in teacher education programs and in the field of mathematics education (e.g., White, Crespo, & Civil, 2016). Moreover, scholars have argued that instead of remaining isolated in a single course (typically the "multicultural" course), a focus on social justice must be integrated throughout teacher education programs (McDonald & Zeichner, 2009). This requires that mathematics teacher educators be prepared to address social justice in their mathematics content and methods courses. There are a variety of perspectives in the literature on social justice mathematics education (Wager & Stinson, 2012). In my work I focus on preparing teachers who can connect mathematics to real-world contexts that involve exploring, analyzing, and proposing solutions for current social and political issues (e.g., Gutstein, 2006)--which I refer to as sociopolitical mathematics teaching. By drawing on existing literature and my own experiences, I identify the knowledge bases that teachers and mathematics teacher educators must draw on in order to enact sociopolitical mathematics. I provide tools for engaging in sociopolitical mathematics teaching, discuss a concrete example from my own practice, and propose a framework for thinking about the knowledge needed for sociopolitical mathematics teaching. [ABSTRACT FROM AUTHOR]

There is a large body of research that documents mathematics anxiety among elementary prospective teachers (PTs) (e.g., Dutton, 1951; Gresham, 2007; Sloan, 2010). Anxiety about mathematics leaves many PTs with the belief that they are not good at doing mathematics, a fear of doing mathematics, and the response that they do not want to talk about or display this fear to others. This disposition is dangerous for PTs because it perpetuates a cycle of mathematics anxiety with their future students. The goal of this article is to encourage the use of research findings when mathematics teacher educators (MTEs) are working with PTs as a way to decrease PTs' mathematics anxiety levels prior to their entry into the teaching profession. Starting with a summary of research findings about the causes of PTs' mathematics anxiety, this article then presents eight research-based recommendations that MTEs can use to help reduce PTs' mathematics anxiety levels while teaching a mathematics content course designed for elementary PTs. It is important for MTEs to be aware of the prevalence of mathematics anxiety among their PTs, to know where that mathematics anxiety originates so they can avoid intensifying the anxiety, and to utilize research-based strategies for reducing the anxiety. This article presents research on PTs' mathematics anxiety in a way that is easily accessible for MTEs. [ABSTRACT FROM AUTHOR]

There is a long history of mathematics educators trying to answer the questions of what mathematics is essential to learn and why, including the mathematical education of prospective teachers. Determining the answer to these questions and conveying that answer convincingly are not always easy tasks for new mathematics teacher educators (MTEs). As an MTE's career progresses (or in some cases as novice MTEs), they must also decide how much content is possible to cover in the context of one, two, or three semesters. That is, how deep is deep enough, and how much breadth is feasible? And how does an MTE make these decisions? To address these questions, we provide potential answers from three perspectives: policy, research, and textbook analysis. We begin with a brief summary of three policy documents: The Mathematical Education of Teachers II (MET-II) (CBMS, 2012), Standards for Preparing Teachers of Mathematics (SPTM) (AMTE, 2017), and the Statistical Education of Teachers (SET) (ASA, 2016). Drawing on these policy documents, a textbook analysis, the Common Core State Standards for Mathematics (CCSSM) (National Governors Association, 2010), and relevant research, we provide insights into how MTEs might choose topics for content courses, with the aim of aligning their teaching objectives with answers to why more math matters. [ABSTRACT FROM AUTHOR]

This study aims to characterise the pedagogical practices of 45 observed primary mathematics lessons taught by 24 mathematics teachers in six national primary schools (SK) and six Chinese vernacular primary schools (SJKC). The data were collected using two video cameras, one focused on the teacher while the other camera focused on the pupils' activities. The qualitative data were analysed based on two main activities in the classroom, which are teacher's activities and pupils' activities. The findings show that mathematics lessons conducted by SK teachers tended to engage the pupils in individual seatwork so as to assess pupils' understanding. Conversely, SJKC teachers were focused more on explaining mathematical concepts to help the pupils build up their conceptual understanding. By characterising the pedagogical practices of mathematics lessons in various schools, the researcher hopes that the findings of this study will contribute to better understanding of the teaching and learning process in SK and SJKC mathematics classrooms. The results serve as a documentation of pedagogical practices in Malaysia to enable implementation of suitable programmes to help in improving teachers' pedagogical practices from different types of primary schools. While the results are interesting and provide some directions, a much larger study would be needed to determine if the results are due to the teachers' enthusiasm, geographical differences, cultural or social differences, or what is known as the Hawthorne Effect. [ABSTRACT FROM AUTHOR]

Do students see things the same as what mathematics teachers have anticipated? In a light-hearted and inquisitive mode, two fundamental mathematical tasks were administered to 147 pre-university students who newly joined a private college for enrolment into various programs. One task simply required the participants to state the variables in a linear equation and the other attempted to elicit what the participants would naturally perceive from a diagram featuring a straight line crossing the coordinate axes. This study aimed to examine the participants' perceptual flexibility with mathematical elements without explicit hints. The emergent textual responses were qualitatively classified and the frequencies of the various types of response evaluated. While the two tasks appear to be drearily ordinary to arouse excitement, the data startlingly revealed a wide range of valid and invalid responses from the participants, and some mathematical elements appeared to be more visually implicit or explicit than others to the participants. Specifically, few participants perceived composite functions embedded in the linear equation as variables and even fewer participants noticed a right-angled triangle emerging from the straight line crossing the coordinate axes. The results are discussed from the perceptual perspective of Gestalt psychology, which suggests the participants' lack of flexibility in mentally reconfiguring perceptual elements. The potential implications for mathematics learning associated with perceptual flexibility are discussed and the instructional efforts that may enhance students' perceptual flexibility recommended. In particular, we argue that a problem-solving process may require flexible perception of mathematical elements in order to gain access to learned concepts and hence particular solution choices. [ABSTRACT FROM AUTHOR]

Guided by the question "Which aspects could affect mathematics teachers' identity, especially in the context of a postgraduate course that includes teaching practices as mathematics teacher educators?" we analyzed the transition between actual and designated mathematics teachers' identity in a postgraduate training course. In particular, teaching practices during the course were oriented by emerging recommendations for mathematics teacher training. We concluded that the presence of an explicit intention to develop a specific practice plays a key role in the transition between actual and designated identity. In addition, the possession of methodological tools as a means for implementing renewed practices gave support and helped practitioners to plan classes promoting mathematical activity. In the process experienced by the practitioners, mentor teacher educators were clearly significant narrators because they reinforced or hinder the desire to reach the designated identity, through their coherence with practitioner's goals or in contradiction with them respectively. [ABSTRACT FROM AUTHOR]

The contribution introduces the concept of epistemological violence from critical psychology into the discourse of mathematics education research. The concept is specific to violence produced through 'knowledge'. It addresses the negative impact of research on the Other - the group being studied as distinct from the ones studying. It holds the possibility to link research ethics and the idea of scientific correctness to each other, by focussing on the relationship between theoretical propositions about the Other and practices of data interpretation in empirical research products. An exemplary study in the Journal of Mathematics Teacher Education illuminates how easily epistemological violence is (re-)produced in the dissemination of research results when it goes unreflected. Finally, the scope and limits of this concept in mathematics education research are discussed. [ABSTRACT FROM AUTHOR]

Herbert Tate was a Professor of Mathematics at McGill University (Montréal, Canada) from 1921 to 1964. As the author of four textbooks, and in his capacity as Chairman of the Department of Mathematics from 1948 to 1960, he played a key role in structuring the institution’s research and study programs in mathematics during an important period of growth. McGill’s current position as a hub of mathematical research owes much to him. In this interview given shortly after his retirement, Herbert Tate describes his career and shares some of his views about mathematics and related topics. Beyond its archival value, this interview reminds us of the extent to which infrastructures and mentalities have changed, at least in Canada, over the past century. [ABSTRACT FROM AUTHOR]

In this article, we explore notions of risk as perceived or experienced by individuals involved in mathematical education. We present this exploration in the form of vignettes, each illustrating a form of risk: a parent’s reaction to classroom “propaganda”; a teacher trying to do justice by her students; a teacher confronted by his administration; and a college professor who believes university policy to be unjust. Each vignette sheds light on areas in which teacher education may offer additional support in fostering the mathematical knowledge, pedagogical sensitivity, and social awareness required to foster, what are in our view, much needed risks in the mathematical (and otherwise) education of pupils. Following the vignettes, we offer a discussion of factors that contributed to the risks perceived or experienced by teachers: neoliberal discourses, and the powerful cultural scripts that leave teachers feeling that they must hold all control, authority, and knowledge. [ABSTRACT FROM AUTHOR]

The concept of angles is one of the foundational concepts to develop of geometric knowledge, but it remains a difficult concept for students and teachers to grasp. Exiting studies claimed that students' difficulties in learning of the concept of angles are based on learning of the multiple definitions of an angle, describing angles measuring the size of angles, and conceiving different types of angles such as 0-line angles, 1- line angles, and 2-line angles. This study was designed to gain better insight into pre-service secondary mathematics teachers' (PSMTs) mental constructions of the concept of angles from the perspective of Action-Process- Object-Schema (APOS) learning theory. The study also explains what kind of mental constructions of angles is needed in the right triangle context. The four PSMTs were chosen from two courses at a large public university in the Midwest United States. Using Clements' (2000) clinical interview methodology, this study utilized three explanatory interviews to gather evidence of PSMTs' mental constructions of angles and angle measurement. All of the interview data was analyzed using the APOS framework. Consistent with the existing studies, it was found that all PSMTs had a schema for 2-line angles and angle measurement. PSMTs were also less flexible on constructions of 1-line and 0-line angles and angle measurement as it applied to these angles. Additionally, it was also found that although PSMTs do not have a full schema regarding 0-line and 1-line angles and angle measurement, their mental constructions of 1-line and 0-line angles and angle measurement were not required in right triangles, and the schema level for 2- line angles was sufficient for constructions of right triangle context. [ABSTRACT FROM AUTHOR]

The article presents a study which explores the ways of mathematicians in learning, knowing, and researching mathematics. The study participants, mathematics lecturers from local universities, were interviewed. The study shows the perspective of students on the characteristics of mathematicians, mathematics learning experience in schools, and British and Malaysian mathematicians' way of knowing.

For more than a decade, researchers, math educators and professional developers from the Galileo Educational Network (Galileo) in the Faculty of Education at the University of Calgary, to which the two of us are associated, have worked to improve the teaching of mathematics. Our focus has always been twofold: to improve teacher knowledge of mathematics and the pedagogy of teaching mathematics. We report on the extensive work we have conducted with teachers with lesson study, classroom mentorships and math fairs. [ABSTRACT FROM AUTHOR]

Professional noticing of students' mathematical thinking in problem solving involves the identification of noteworthy mathematical ideas of students' mathematical thinking and its interpretation to make decisions in the teaching of mathematics. The goal of this study is to begin to characterize pre-service primary school teachers' noticing of students' mathematical thinking when students solve tasks that involve proportional and non-proportional reasoning. From the analysis of how pre-service primary school teachers notice students' mathematical thinking, we have identified an initial framework with four levels of development. This framework indicates a possible trajectory in the development of primary teachers' professional noticing [ABSTRACT FROM AUTHOR]