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This paper critically examines the discrepancies among the pre-requisite fractional concepts assumed by a curricular unit on operations with fractions, the teacher's assumptions about those concepts and a particular student's understanding of fractions. The paper focuses on the case of one student (Tim) in the teacher's 6th grade class who was interviewed by one of the authors once a week during the teaching of the unit. The teaching materials and the teacher's instruction were based on the assumption that students understood the concept of a unit fraction as being one of several equal parts of a given whole. The teacher neither emphasized the need for equal parts nor the part-to-whole relation. The teacher's reasonable assumptions about her students' understanding of fractions were severely challenged by the cognitive constructs that Tim exhibited during his first two interviews. When she viewed tapes of the class instruction and the interviews with Tim she realized Tim lacked essential constructs to make sense of her instruction. She subsequently made adjustments in her instruction, making effective use of more appropriate representations based on tasks from the unit that we modified and used with Tim in our interviews. These adjustments helped Tim to construct partitioning operations and an appropriate unit fractional scheme. This study illustrates the importance of coming to understand a student's mathematical activity in terms of possible conceptual schemes and modifying instructional strategies to build on those schemes. The coordinated design of the research study facilitated these instructional modifications. (Contains 13 figures.)

This paper reviews the presentation of problem solving and process aspects of mathematics in curriculum documents from Australia, UK, USA and Singapore. The place of problem solving in the documents is reviewed and contrasted, and illustrative problems from teachers' support materials are used to demonstrate how problem solving is now more often treated as a teaching method, rather than a goal in itself. The paper also analyses how the curriculum documents describe the growth of students' abilities in the process areas of mathematics, and assesses the guidance that this provides for teachers. At each stage, the paper suggests directions for research that would be useful in assisting curriculum documents to promote the fundamental but elusive goal of making students better problem solvers. (Contains 7 figures.)

Presents the experiences of a Russian mathematician teaching in Russian and American Universities. Topics discussed include what was taught, the importance of grades, cultural and parental factors, mathematicians' attitudes toward teaching, and some problems in elementary mathematics. (MDH)

This paper describes a case study of a math teacher working with a math coach and the effects of their interaction. A guiding question was whether the coaching intervention had affected the teacher's classroom practices and, if so, in what way. The study utilized data from teacher/coach planning sessions, classroom lessons, follow-up debriefing sessions, and interviews with the teacher, coach and school principal. These data enabled the author to study the impact, if any, of the coaching on teacher beliefs and practices. An important outcome was that new beliefs about mathematics education emerged from the teacher and that these beliefs were mirrored in various teaching practices. (Contains 16 figures and 1 chart.)

This paper focuses on three junior high school mathematics teachers and their knowledge of the nature of definitions. The mathematical context of exponentiation is used as a springboard for discussing two aspects of definitions: their corresponding domains and the distinction and relationships between definitions, proofs, and theorems. Through interviews it was shown that some teachers are not aware that definitions and domains are intrinsically connected and some teachers believe that definitions may be proved. Findings also indicate that knowledge of the nature of definitions may be dependent on the context. (Contains 1 figure.)

Researchers conducted semi-structured interviews with in-service fifth grade teachers. The purpose of these interviews was to examine teachers' reactions to arguments that 0.999... = 1. Previously reported results indicate that some pre-service elementary school teachers possess misunderstandings about mathematical issues related to decimals with single repeating digits. This research investigates whether some in-service teachers possess misunderstandings about mathematical issues related to 0.999.... This paper reports on one instance of a teacher whose responses indicate that the teacher's sense of number and sense of measurement are intertwined, resulting in fragile understanding of repeating decimals. These data present evidence that teachers continue to develop repeated decimal understandings and misunderstandings throughout their careers, and that the curriculum, everyday experience, and perceptions of student learning combine to form or reinforce these understandings. Because decimals with a single repeating digit (e.g. 0.333... and 0.666...) are an integral part of the elementary mathematics curriculum, we argue that it is important that in-service elementary mathematics teachers have a clear understanding of concepts related to the concept of infinity as they emerge through the study of the equality 0.999... = 1. (Contains 2 illustrations.)

The paper examines the roles and purposes of proof mentioned by university research faculty when reflecting on their own teaching and teaching at their institutions. Interview responses from 14 research mathematicians and statisticians who also teach are reported. The results suggest there is a great deal of variation in the role and purpose of proof in and among mathematics courses and that factors such as the course title, audience, and instructor influence this variation. The results also suggest that, for this diverse group, learning how to prove theorems is the most prominent role of proof in upper division undergraduate mathematics courses and that this training is considered preparation for graduate mathematics studies. Absent were responses discussing proof's role in preparing K-12 mathematics teachers. Implications for a proof and proving landscape for school mathematics are discussed.

While the general notion of "metaphor" may offer a thoughtful analysis of the nature of mathematical thinking, this paper suggests that it is even more important to take into account the particular mental structures available to the individual that have been built from experience that the individual has "met-before." The notion of "met-before" offers not only a principle to analyse the changing meanings in mathematics and the difficulties faced by the learner--which we illustrate by the problematic case of the minus sign--it can also be used to analyse the met-befores of mathematicians, mathematics educators and those who develop theories of learning to reveal implicit assumptions that support our thinking in some ways and act as impediments in others. (Contains 5 figures.)

This study is a part of a research project that seeks to characterize the relationship between mathematics teachers' knowledge and their practice. In this paper, we focus on identifying the characteristics of subject matter knowledge and pedagogical content knowledge that two teachers integrate in decisions they make about the introduction of specific mathematical content. Then, we examine the changes that arise in their classrooms as their plans are put in action. Data were obtained through audiotapes of several semi-structured interviews, through observations, and through videotapes. Although the two teachers in this study had similar backgrounds and experiences, our analysis shows differences in the characteristics of the domains of knowledge they integrated in their planning as well as differences in the adaptations that each made in the classroom. In this sense, this study contributes to better understanding the complexity of teachers' professional practice.

The main assumption of this article is that learning mathematics depends on the student's participation in processes of collective argumentation. On the empirical level, such processes will be analyzed with Toulmin's theory of argumentation and Goffman's idea of decomposition of the speaker's role. On the theoretical level, different statuses of participation in processes of argumentation will be considered. By means of the method of comparative analysis, different grades of autonomy according to the interactional contribution of a student can be reconstructed. The paper finishes with remarks about consequences for improving mathematics teaching in schools and mathematics teacher education at university level.

Anyone involved in mathematics education will at some point encounter a classroom situation in which a student's statement provides an opportunity for the teacher to enhance students' mathematical understanding. This paper aims to characterize such classroom situations, known as critical events, as noticed by pre-service mathematics teachers during classroom observations and teaching. Analysis of 80 critical events reported by 20 pre-service mathematics teachers during a field-based teaching preparation program yielded a model that enables teacher-educators and researchers to characterize critical events while reflecting on the wide variations within these events. The model comprises three axes: the participants in the event, the focus of the event, and the dimensions in learning and teaching mathematics (i.e., cognitive, affective and social). The paper discusses the theoretical and empirical construction of the model and demonstrates its potential contribution using examples taken from our research. • Pre-service teachers identified critical events during classroom observations. • A model for characterizing critical events was constructed theoretically and empirically. • The model affords eliciting the multidimensional nature and complexity of critical events. • The model's potential contribution is presented using examples from our research. [ABSTRACT FROM AUTHOR]

• We present a case study of a PST reasoning about fraction comparisons. • We view her experience as exploring unfamiliar paths through familiar territory. • Our analysis illustrates 4 themes. • We discuss implications for mathematics teacher education. Preservice elementary teachers (PSTs) have been described as having difficulties with fractions, relying on standard procedures, and experiencing math anxiety. We are interested in productive ways in which PSTs can and do use their prior knowledge when exploring unfamiliar paths through familiar mathematical territory. We conducted interviews with PSTs in which we challenged them with various fraction comparison tasks and encouraged them to develop new strategies. In this paper, we present a case study focused on one PST who made considerable progress in her reasoning about fraction comparisons during such an interview. We use Greeno's (1991) environment metaphor to conceptualize number sense as situated knowing in a conceptual domain. This perspective helps us account for both cognitive and affective factors. We highlight 4 themes concerning features of the mathematical environment that Jennifer (pseudonym) appeared to inhabit during the interview: (a) the interview context created a safe space that emphasized the interviewer's interest in Jennifer's ideas, as opposed to correct answers; (b) Jennifer used her prior knowledge of parts and wholes to ground her arguments meaningfully and as building blocks to invent new strategies; (c) she used her prior knowledge of cross multiplication as an established path that provided reassurance and facilitated her exploration of unfamiliar paths; (d) it was beliefs and affective factors, not deficiencies in knowledge, that constrained Jennifer's exploration of unfamiliar paths through familiar mathematical territory. We discuss the implications of these findings for research concerning PSTs' mathematical thinking and learning and for mathematics teacher education. [ABSTRACT FROM AUTHOR]

• Belief structures provide more information than belief content for changing beliefs. • A new belief structure, adaptive idealist, is introduced. • An adaptive idealist interprets experiences and vocabulary to justify beliefs. One goal of teacher mathematics education is to transform teachers' beliefs to more closely align with beliefs that promote desired teaching practices. However, beliefs are difficult to change. A construct that helps to explain how and why mathematics teachers' beliefs change is that of belief structures (Cooney et al., 1998). We expand upon belief structures to add a new construct: adaptive idealist. In this case study, we used information gleaned from interviews and reflective papers to introduce a prospective secondary mathematics teacher, describe her beliefs and shifts in beliefs, identify a resistance to change that resonated with our other experiences, and conclude that this new belief structure allows us to explain her reactions and interpretations within our secondary mathematics education program. [ABSTRACT FROM AUTHOR]

Responding constructively ‘in-the moment’ to student offers is described as a critical, and yet difficult, aspect of skilful and responsive teaching. South African evidence points to limited evaluation of student offers in schools serving poor communities. In this paper, we present and discuss an ‘elaboration’ framework emerging from a grounded analysis of data drawn from video recordings of 18 mathematics lessons prepared and conducted by four in-service primary school teachers in South Africa. This analysis led to a categorization of the situations in which teacher responses to student offers occurred, and the nature and range of these responses. Three response situations are identified within the framework: breakdown, sophistication, and individuation/collectivization, with a range of response (and non-response) categories in each situation. Literature on responsive feedback is drawn in to explore hierarchies and relationships between the emergent categories within situations of elaboration. The elaboration framework provides a tool for lesson observation, and a model for thinking about developments in responsive teaching. [ABSTRACT FROM AUTHOR]

The focus of this paper is on what research suggests primary school teachers need to know about the specialized language of mathematics teaching and learning: the mathematics register. When teachers act effectively on this knowledge, students have multiple opportunities to construct mathematical understandings and to demonstrate what they know in the variety of tasks that are required by formal schooling. First, the national and policy context within which mathematics is taught and learned is considered. Secondly, the mathematics register is defined, followed by a consideration the significance for teaching, learning, and assessment in school; examples are provided. The educational implications of these analyses include guidelines for what teachers need to know and to do, so that their students appropriately utilize the mathematics register in the classroom. In so doing, students are provided with optimal circumstances to learn, where they may deploy all of their cognitive-linguistic resources to the particular mathematics tasks at hand. [ABSTRACT FROM AUTHOR]

This article provides an overview of the Special Issue of the Journal of Mathematical Behavior : An international perspective on the knowledge in teaching mathematics, with a focus on the primary and elementary levels. The work originated with presentations given at the International Congress on Mathematics Education (ICME) 13 in Hamburg, Germany, in the Summer of 2016. The papers were solicited from the participants in Topic Study Group (TSG) 45: Knowledge In/For Teaching Mathematics at Primary Level . A primary goal for TSG 45 was to advance theories about the teacher content knowledge and pedagogical content knowledge required for focusing instruction on building mathematical understanding and promoting student reasoning. This Special Issue emphasizes the role that teachers play in supporting that development. A significant corpus of research has identified major factors influencing how students build mathematical ideas and construct pathways of reasoning. However, our current understanding is that what is lacking is a conceptual framework that identifies and integrates factors regarding teachers’ roles in supporting students’ mathematical learning; and also, identifying the set of conditions and resources that optimally support that learning. We address this area of inquiry by organizing this overview article into the following sections: Cognitive perspectives; pedagogical perspectives; attending to students’ reasoning; language, communication and dialogue perspectives; and finally, we conclude with research-based implications for teaching practice and teacher professional learning. [ABSTRACT FROM AUTHOR]

This paper presents a study on employing different types of cooperative learning (CL) settings in mathematics teacher education based on multiple research data. The study analyzes mechanisms in which CL contributes to the development of teacher knowledge of three kinds: subject matter knowledge, pedagogical content knowledge, and curricular content knowledge. Additionally, it reflects on interactions between different kinds of teachers’ knowledge in the process of development. Two wholes, which are greater than the sum of their parts, are analyzed in this paper. The first is a course whose design combines different CL settings, hence enhances different mechanisms for teachers’ professional development. The second whole, teachers’ “collaborative mind,” is presented as an outcome of the first. To exemplify possible contributions of employing CL in teacher education, this paper first zooms in on the structure of one particular course for mathematics teachers and then focuses on one particular mathematical activity. [Copyright &y& Elsevier]

Mathematicians and mathematics educators agree that proof is an important tool in mathematics, yet too often undergraduate students see proof as a superficial part of the discipline. While proof is often used by mathematicians to justify that a theorem is true, many times proof is used for another purpose entirely such as to explain why a particular statement is true or to show mathematics students a particular proof technique. This paper reports on a study that used a form of inquiry-based learning (IBL) in an introduction to proof course and measured the beliefs of students in this course about the different functions of proof in mathematics as compared to students in a non-IBL course. It was found that undergraduate students in an introduction to proof course had a more robust understanding of the functions of proof than previous studies would suggest. Additionally, students in the course taught using inquiry pedagogy were more likely to appreciate the communication, intellectual challenge, and providing autonomy functions of proof. It is hypothesized that these results are a response to the pedagogy of the course and the types of student activity that were emphasized. [ABSTRACT FROM AUTHOR]

• Mathematical Knowledge for Teaching for secondary mathematics teachers. • Design of tasks for pre-service mathematics teachers. • Mathematics as a human activity. This paper describes theoretical design principles emerging from the development of tasks for standard undergraduate mathematics courses that address applications to teaching secondary mathematics. While researchers recognize that mathematical knowledge for teaching is a form of applied mathematics, applications to teaching remain largely absent from curriculum resources for courses for mathematics majors. We developed various materials that contain applications to teaching that have been integrated into four standard undergraduate mathematics courses. Three primary principles influenced the design of the tasks that prepare future teachers to learn and apply mathematics in a manner central to their future work. Additionally, this paper provides guidance for instructors desiring to develop or implement similar applications. The process of developing these tasks underscores the importance of key features regarding the roles of human beings in the tasks, the intentional focus on advanced content connected to school mathematics, and the integration of active engagement strategies. [ABSTRACT FROM AUTHOR]

The Common Core State Standards in Mathematics ( CCSSM, 2010 ) are organized around two types of standards: the standards for mathematical content and standards for mathematical practice. The central goal of this paper is to present cognitive and instructional analyses of standards for mathematical practice through a discussion of field-based activities with inservice secondary mathematics teachers and students. A potential value of the study is that it provides researchers with specific field-based hypotheses on advancing standards for mathematical practice. [ABSTRACT FROM AUTHOR]

In this paper we report findings from a two-year, large-scale research project that describes the work of middle school mathematics specialists (also referred to as mathematics coaches or instructional coaches) who served in 10 school districts. We use mixed methods to describe how mathematics specialists spent their time supporting teachers and how these supports contributed to meaningful changes that teachers made in their instructional practices. We also report results that correlate student achievement scores with whether or not teachers were highly engaged with the mathematics specialists. We coordinate these quantitative results with findings from several case studies to illustrate the qualitatively different ways that mathematics specialists supported teachers’ ongoing work with their students. We also account for why some teachers participated more fully than others. Importantly, because mathematics specialists’ work was situated in different school settings with different demands, resources and administrative supports, these constraints and affordances contributed in part to how they could effectively support teachers’ work with their students. [ABSTRACT FROM AUTHOR]

The Association of Mathematics Teacher Educators adopted Standards for Elementary Mathematics Specialists calling for structured preparation of math coaches, specialists, and instructional leaders across the country (AMTE, 2013). The purpose of this paper is to illustrate the structure and design of our fully online Elementary Mathematics Instructional Leader (EMIL) graduate program for inservice teachers aiming to answer the call. Our graduates are teacher leaders and coaches who are responsible for supporting effective mathematics instruction and student learning at the classroom, school, district, or state levels. We will review formative data from our first cohort of graduates, the benefits and drawbacks to providing online (and grant-funded) courses, and the impact of participant attrition. Our goal is to offer guidance and information to other organizations as they move to develop programs of their own. [ABSTRACT FROM AUTHOR]

Mathematics coaching, as typically practiced in US schools tends to be responsive and individually focused work in which coaches respond to invitations from individual teachers to help them improve their teaching. But what does the work of coaching look like if it is organized instead to engage teachers collectively in service of school-wide improvement? This is the question we take up in this paper through examining the case of one school-based elementary mathematics coach whose work more closely aligned with emerging findings in the field of instructional improvement about the power of coaching for school-wide reform. The coach helped to dramatically transform a recent history of poor performance and deficit-oriented narratives pertaining to the school and its children. Through a fine-grain analysis, we illustrate the coach’s work implicated in supporting groups of teachers to come to mutual understanding around and further development of shared high-quality instructional practices. The components of coaching that help support collective capacity are discussed. [ABSTRACT FROM AUTHOR]

The Virginia Mathematics Specialist Initiative (VMSI) is an ongoing collaborative effort among all components of the mathematics and mathematics education community of Virginia spanning more than two and a half decades. This paper describes the use of extensive collaborations to reach shared beliefs about teaching and learning mathematics with the goal of improving student learning; accomplished by preparing and supporting school-based mathematics specialists to coach K-8 teachers. A foundation built on shared beliefs enabled community members to reach consensus on this goal and continues to fuel the accomplishments of the VMSI. Several accomplishments are discussed including identification of the role of K-8 mathematics specialists, the attainment of a Virginia licensure add-on endorsement; and development of a model for a Virginia Mathematics Specialist Preparation Program (VMSPP). Also discussed are the annotated syllabi for core mathematics content and leadership courses in the VMSPP and tools to sustain and extend the VMSI. [ABSTRACT FROM AUTHOR]

In this paper, we examined 14 pre-service secondary mathematics teachers' (PSMTs) criteria used in evaluating the validity of self-constructed and student-generated arguments in the domains of algebra, geometry, and number theory. During the interviews, participants produced and analyzed their own proofs or counterexamples for the three tasks first and analyzed if six student-generated arguments could be counted as proofs next. The two criteria most frequently used by PSMTs in evaluating their own construction and a student-generated argument as a proof or not a proof were verification and clear details. Our finding showed that more PSMTs were inclined to pay attention to clear details presented in students' work than their own productions. Our finding also showed that more PSMTs focused on the verification feature in their self-constructions than in student-generated arguments. These results suggest that the criteria–verification and clear details–do not have the same weight for PSMTs when evaluating the validity of their own work and student-generated products. This study also has implications for proof-intensive courses, as PSMTs' conceptions of proofs are more likely affected by their instructors in those proof-intensive classes. • Determining if student-generated arguments qualify as proofs is a critical skill and challenge practice for pre-service secondary mathematics teachers to provide constructive feedback for students. • What criteria were used by pre-service secondary mathematics teachers when determining if their own arguments and students' productions of the same mathematical statements can be counted as proofs or non-proofs from a learner perspective and a teacher perspective beyond high school geometry. • The two criteria most frequently used by pre-service secondary mathematics teachers in evaluating their own argument as a proof or not a proof from a learner perspective were verification and clear details. • The two most important features used by pre-service secondary mathematics teachers regarding if students' arguments qualified as proofs or non-proofs from a teacher perspective were verification and clear details. [ABSTRACT FROM AUTHOR]

Students with learning disabilities are often excluded from conversations about high-quality mathematics instruction as well as conversations about equitable mathematics instruction. As such, they may not be provided many opportunities to engage in mathematics with teachers who implement practices that the field of mathematics education would consider "high-quality" or "equitable". In this paper, we examine how three students with learning differences participate in instruction that demonstrates aspects of high quality and equitable mathematics instruction. Our findings outline practices that appeared to support increased opportunities for students to demonstrate their own reasoning and sense making in mathematics. Contributions of this work include illuminations of specific pedagogical moves that may support students identified as having learning disabilities to engage in more student-centered, conceptually-oriented mathematics. • We examine how three students with learning disabilities participate when math instruction is designed to be student-centered. • Practices that aimed for equity & were conceptually-oriented appeared to improve sense making opportunities for the students. • This work outlines specific pedagogical moves that supported these students in using their own reasoning while "doing math. [ABSTRACT FROM AUTHOR]

Students’ ability to coordinate multiple levels of units constitutes a cognitive core in their mathematical development across several domains, including counting, whole number multiplication, integer addition, fractions concepts, and algebraic reasoning. Identifying a progression of students’ units coordination activity would help educators leverage and support student development within and across the three stages of units coordination. This paper contributes to that goal by describing a cross-section of units coordination activity observed during pairs of clinical interviews with 47 grade 6 students. Results include a 36-rank scale of units coordinating activity with accompanying characterizations and descriptors. [ABSTRACT FROM AUTHOR]

It is still largely unknown what are effective and efficient strategies for graphing formulas with paper and pencil without the help of graphing tools. We here propose a two-dimensional framework to describe the various strategies for graphing formulas with recognition and heuristics as dimensions. Five experts and three secondary-school math teachers were asked to solve two complex graphing tasks. The results show that the framework can be used to describe formula graphing strategies, and allows for differentiation between individuals. Experts used various strategies when graphing formulas: some focused on their repertoire of formulas they can instantly visualize by graphs; others relied on strong heuristics, such as qualitative reasoning. Our exploratory study is a first step towards further research in this area, with the ultimate aim of improving students’ skills in reading and graphing formulas. [ABSTRACT FROM AUTHOR]

Discussions where teachers engage students in the comparison of multiple solution strategies to a single problem have been recommended in curriculum policy documents, yet integrating these discussions into teachers’ normative routines is not widespread. In this paper, we begin to explore variations in teachers’ implementation of Algebra I curriculum materials specifically focused on comparison. We explore (via case studies) implementation of the curriculum materials by two teachers with similar teaching backgrounds. The case studies suggest that these two teachers’ implementation of the comparison materials differed markedly, raising questions about possible factors which may have contributed to implementation differences. [ABSTRACT FROM AUTHOR]

A group of prospective mathematics teachers was asked to imagine a conversation with a student centered on a particular proof regarding odd/even functions and produce a script of this imagined dialog. These scripts provided insights into the script-writers’ mathematical knowledge as well as insights into what they perceive as potential difficulties for their students. The paper focuses on the script-writers’ understandings of derivative and of even/odd functions. [ABSTRACT FROM AUTHOR]

This paper deals with the attempt to study four-year old children's potential to engage in the mathematical problem solving process and their ability to construct mathematical meaning through their engagement with the process. We studied: (a) whether teaching could contribute to the development of the essential cognitive skills for solving a mathematical problem of detecting all possible additive combinations giving a specific sum and (b) whether children's familiarization with graphical representation practices of solutions could give meaning to the mathematical problem solving process, in a way that children would benefit from it in their involvement and effectiveness concerning the process. Findings showed that four-year old children's consistent involvement with the mathematical problem solving process enabled them to develop, apply and demonstrate skills and mathematical conceptual understanding in order to solve the given mathematical problems. The role of the teacher, in organizing the classroom learning environment as well as carrying out the teaching intervention, was essential to the development of the children's skills, abilities and mathematical conceptual understanding. [ABSTRACT FROM AUTHOR]

• Learning is viewed as routinization and routines are seen as < task, procedure > pairs. • Classroom learning requires the students' task to become closer to the expected task. • We suggest differentiating between meta-level and object-level tasks. • We monitored a class of prospective mathematics teachers in three lessons. • Changes noted in flexibility, bondedness, substantiability and students' agency. This paper conceptualizes classroom learning and operationalizes the proposed conceptualization. We adopt the commognitive framework, according to which learning is thought of as routinization and routines are conceptualized in terms of pairs. To examine the usefulness of this conceptualization, we applied our operationalization to examples from a class of preservice elementary school teachers learning to solve a specific type of questions. We analyzed their suggested solutions during three lessons that were recorded and transcribed verbatim. Progress in classroom learning was evidenced by changes in the participants' tasks, which became closer to the tasks expected in the discourse into which they were being initiated. Progress was also evidenced by changes in participants' procedures for tackling the tasks. Finally, in examining the deritualization characteristics of flexibility, bondedness, substantiability and students' agentivity, we found changes in students' performance. [ABSTRACT FROM AUTHOR]

• Novice facilitators' changes in practices in two different countries are examined. • Despite the different contexts, changes in practices are found to be similar. • Practices are linked to facilitators' resources, orientations, goals and identity. • Managing mathematics teachers' PD discussions can be learned and developed. • Recommendations for supporting novice facilitators of are presented. Novice facilitators of professional development (PD) programs for mathematics teachers often face challenges in leading productive discussions and achieving the goals of these programs. Although research in this area is gradually accumulating, not much is known about how novice facilitators address these challenges and change their practices accordingly. This paper presents case studies of two novice facilitators of PD programs in two different countries. The analyses look at their work over one year, to illustrate the changes in their practices while managing discussions. The results show that although the facilitators operated in different contexts, their practices and their processes of change resembled, suggesting that these processes are not idiosyncratic. We argue that novice facilitators' changes in practices correspond to changes in their resources, orientations, goals, and identities and that PD program teams can support these changes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we compare how three teachers, one from each of Finland, Flanders and Hungary, introduce linear equations to grade 8 students. Five successive lessons were videotaped and analysed qualitatively to determine how teachers, each of whom was defined against local criteria as effective, addressed various literature-derived equations-related problems. The analyses showed all four sequences passing through four phases that we have called definition, activation, exposition and consolidation. However, within each phase were similarities and differences. For example, all three constructed their exposition around algebraic equations and, in so doing, addressed concerns relating to students’ procedural perspectives on the equals sign. All three teachers invoked the balance as an embodiment for teaching solution strategies to algebraic equations, confident that the failure of intuitive strategies necessitated a didactical intervention. Major differences lay in the extent to which the balance was sustained and teachers’ variable use of realistic word problems. [Copyright &y& Elsevier]

Abstract: Justification is a core mathematics practice. Although the purposes of justification in the mathematician community have been studied extensively, we know relatively little about its role in K-12 classrooms. This paper documents the range of purposes identified by 12 middle grades teachers who were working actively to incorporate justification into their classrooms and compares this set of purposes with those documented in the research mathematician community. Results indicate that the teachers viewed justification as a powerful practice to accomplish a range of valued classroom teaching and learning functions. Some of these purposes overlapped with the purposes in the mathematician community; others were unique to the classroom community. Perhaps surprisingly, absent was the role of justification in verifying mathematical results. An analysis of the relationship between the purposes documented in the mathematics classroom community and the research mathematician community highlights how these differences may reflect the distinct goals and professional activities of the two communities. Implications for mathematics education and teacher development are discussed. [Copyright &y& Elsevier]

Abstract: Researchers conducted semi-structured interviews with in-service fifth grade teachers. The purpose of these interviews was to examine teachers’ reactions to arguments that .999…=1. Previously reported results indicate that some pre-service elementary school teachers possess misunderstandings about mathematical issues related to decimals with single repeating digits. This research investigates whether some in-service teachers possess misunderstandings about mathematical issues related to .999…. This paper reports on one instance of a teacher whose responses indicate that the teacher''s sense of number and sense of measurement are intertwined, resulting in fragile understanding of repeating decimals. These data present evidence that teachers continue to develop repeated decimal understandings and misunderstandings throughout their careers, and that the curriculum, everyday experience, and perceptions of student learning combine to form or reinforce these understandings. Because decimals with a single repeating digit (e.g. .333… and .666…) are an integral part of the elementary mathematics curriculum, we argue that it is important that in-service elementary mathematics teachers have a clear understanding of concepts related to the concept of infinity as they emerge through the study of the equality .999…=1. [Copyright &y& Elsevier]

Abstract: This paper elaborates the notion of a personal example space as the set of mathematical objects and construction techniques that a learner has access to as examples of a concept while working on a given task. This is different from the conventional space of examples that is represented by the worked examples and exercises in textbooks. We refer to three studies spanning the age range of learners, from school-age learners to pre-service teachers learning maths and professional mathematicians. Their constructions of examples are used as evidence of their personal example spaces. From these, we identify characteristics of such spaces that provide insight into learning mathematics. This perspective informs teaching by giving access to how personal knowledge is structured and what might enhance that structure. [Copyright &y& Elsevier]

Abstract: The paper examines the roles and purposes of proof mentioned by university research faculty when reflecting on their own teaching and teaching at their institutions. Interview responses from 14 research mathematicians and statisticians who also teach are reported. The results suggest there is a great deal of variation in the role and purpose of proof in and among mathematics courses and that factors such as the course title, audience, and instructor influence this variation. The results also suggest that, for this diverse group, learning how to prove theorems is the most prominent role of proof in upper division undergraduate mathematics courses and that this training is considered preparation for graduate mathematics studies. Absent were responses discussing proof''s role in preparing K-12 mathematics teachers. Implications for a proof and proving landscape for school mathematics are discussed. [Copyright &y& Elsevier]

Abstract: Calls for reform in mathematics education around the world state that proofs should be part of school mathematics at all levels. Turning these calls into a reality falls on teachers’ shoulders. This paper focuses on one secondary school teacher''s reactions to students’ suggested proofs and justifications in elementary number theory. To determine whether the justifications are acceptable, the teacher used not only her SMK regarding mathematical aspects, but also her PCK about what a student giving this justification might know or not know. A discussion of the findings is followed by some questions that arise. [Copyright &y& Elsevier]

Abstract: This study is a part of a research project that seeks to characterize the relationship between mathematics teachers’ knowledge and their practice. In this paper, we focus on identifying the characteristics of subject matter knowledge and pedagogical content knowledge that two teachers integrate in decisions they make about the introduction of specific mathematical content. Then, we examine the changes that arise in their classrooms as their plans are put in action. Data were obtained through audiotapes of several semi-structured interviews, through observations, and through videotapes. Although the two teachers in this study had similar backgrounds and experiences, our analysis shows differences in the characteristics of the domains of knowledge they integrated in their planning as well as differences in the adaptations that each made in the classroom. In this sense, this study contributes to better understanding the complexity of teachers’ professional practice. [Copyright &y& Elsevier]

Abstract: The main assumption of this article is that learning mathematics depends on the student''s participation in processes of collective argumentation. On the empirical level, such processes will be analyzed with Toulmin''s theory of argumentation and Goffman''s idea of decomposition of the speaker''s role. On the theoretical level, different statuses of participation in processes of argumentation will be considered. By means of the method of comparative analysis, different grades of autonomy according to the interactional contribution of a student can be reconstructed. The paper finishes with remarks about consequences for improving mathematics teaching in schools and mathematics teacher education at university level. [Copyright &y& Elsevier]

Abstract: This paper illustrates the role of a Thinking-about-Derivatives task in identifying learners’ derivative conceptions and for promoting their critical thinking about derivatives of absolute value functions. The task included three parts: Define the derivative of a function f(x) at x = x 0, Solve-if-Possible the derivative of f(x)=|x| at x =2 and at x =0, and evaluate the correctness of suggested solutions in a Right-or-Wrong part. Three prospective teachers, Noa, Anat and Daniel were individually interviewed when solving the task. We found that while the participants correctly solved the Define part, they exhibited some erroneous images in the Solve-if-Possible part, and their work on the Right-or-Wrong part contributed to their critical thinking about functions and derivatives. All three participants expressed their appreciation of their work on the Right-or-Wrong part of the task. [Copyright &y& Elsevier]

• A capstone course aimed to prepare preservice secondary teachers to integrate logical aspects of proof in secondary mathematics classrooms. • Course design aligned with a framework of Mathematical Knowledge for Teaching Proof (MKT-P). • 60 Lesson plans and 13 videos of lessons enacted in secondary classrooms were analyzed in terms of prospective teachers' MKT-P. • PSTs were successful in designing and implementing proof-oriented lessons in a wide range of mathematical content areas. • Challenges to proof integration were PSTs' doubts about student ability to prove and difficulty adjusting their language to student grade level. Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs' self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs' mathematical knowledge for teaching proof. [ABSTRACT FROM AUTHOR]

• We present the results of presenting an instructional module connecting the teaching of real analysis to the teaching of secondary mathematics. • We focus on the relationship between continuity, injectivity, and invertibility and the use of the arcsine function. • Prospective teachers who completed this module were better prepared to teach inverse trigonometric functions. Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers' instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter. [ABSTRACT FROM AUTHOR]