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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 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 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]

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]

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]

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]

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 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]

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]

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 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 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]

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]

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]

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]