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Internationally, questions about the perceived utility of university mathematics for teaching school mathematics pose an ongoing challenge for secondary mathematics teacher education. This special issue is dedicated to exploring this topic and related issues in the preparation of secondary mathematics teachers—by which we mean teachers of students with ages, approximately, of 12–18 years. This article introduces this theme and provides a semi-systematic survey of recent related literature, which we use to elaborate and situate important theoretical distinctions around the problems, challenges, and solutions of university mathematics in relation to teacher education. As part of the special issue, we have gathered articles from different countries that elaborate theoretical and empirical approaches, which, collectively, describe different ways to strengthen university mathematics with respect to the aims of secondary teacher education. This survey paper serves to lay out the theoretical groundwork for the collection of articles in the issue. [ABSTRACT FROM AUTHOR]

This paper reports from a case study which explores kindergarten children's mathematical abstraction in a teaching–learning activity about reflection symmetry. From a dialectical perspective, abstraction is here conceived as a process, as a genuine part of human activity, where the learner establishes "a point of view from which the concrete can be seen as meaningfully related" (van Oers & Poland Mathematics Education Research Journal, 19(2), 10–22, 2007, p. 13–14). A cultural-historical semiotic perspective to embodiment is used to explore the characteristics of kindergarten children's mathematical abstraction. In the selected segment, two 5-year-old boys explore the concept of reflection symmetry using a doll pram. In the activity, the two boys first point to concrete features of the sensory manifold, then one of the boys' awareness gradually moves to the imagined and finally to grasping a general and establishing a new point of view. The findings illustrate the essential role of gestures, bodily actions, and rhythm, in conjunction with spoken words, in the two boys' gradual process of grasping a general. The study advances our knowledge about the nature of mathematical abstraction and challenges the traditional view on abstraction as a sort of decontextualised higher order thinking. This study argues that abstraction is not a matter of going from the concrete to the abstract, rather it is an emergent and context-bound process, as a genuine part of children's concrete embodied activities. [ABSTRACT FROM AUTHOR]

This paper reports on concept images of 38 secondary school mathematics prospective teachers, regarding the evenness of numbers. Written assignments, individual interviews, and lesson transcripts uncover salient, erroneous concept images of even numbers as numbers that are two times "something" (i.e., 2i is an even number), or to reject the evenness of zero. The notion of concept image serves in the analysis of the findings, and the findings serve in offering two refinement notions: mis-in concept images that mistakenly grant non-examples the status of examples (e.g., 2i is an even number), and mis-out concept images that mistakenly regard examples as non-examples (e.g., zero is not an even number). We discuss possible benefits in distinguishing between these two refinement notions in mathematics education. [ABSTRACT FROM AUTHOR]

This paper examines prospective mathematics teachers' knowledge of asymptotes and asymptotic behaviour of functions in calculus. They are university students and future facilitators of knowledge in upper secondary education. We constructed a reference epistemological model to describe the knowledge about asymptotes for upper secondary and university education and explored prospective teachers' relations to targeted knowledge. The study was conducted within the Anthropological Theory of the Didactic as a suitable framework to analyse and interpret knowledge of mathematical notions. Prospective teachers participated in three questionnaires with open, non-routine questions on different aspects of the notion of asymptotes. An analysis showed their knowledge was fragmented and their work relied heavily on algebraic manipulation and memorized formulas from calculus. The results indicated that knowledge of asymptotes and asymptotic behaviour is a potentially powerful context for developing knowledge related to the limits of functions in calculus. [ABSTRACT FROM AUTHOR]

The work that mathematics teachers do is frequently mathematical in nature and different from other professions. Understanding and describing common ways that teachers draw upon their content knowledge in the practice of teaching is important. Building on the descriptions by McCrory et al. (Journal for Research in Mathematics Education 43(5) 584-615, ) of two primary ways teachers use their content knowledge-decompressing and trimming-this paper differentiates between micro- and macro-levels of mathematical complexity as a way to extend and further conceptualize the moves that teachers make at a more nuanced grain size. This paper explains and portrays their counterparts, micro-level trimming and macro-level decompressing, which we discuss as concealing and foreshadowing complexity, respectively. Support for their use in teachers' work is provided through specific examples. Implications for the mathematical preparation and professional development of teachers are discussed. [ABSTRACT FROM AUTHOR]

Technology is available and accessible in many mathematics classrooms. Adopting technology to support teaching and learning requires teachers to change their teaching practices. This paper reports the responses of a diverse cohort of 92 secondary mathematics teachers who chose to respond to an Australian state-wide survey (Mathematics with Technology Perceptions Survey) developed using a Theory of Planned Behaviour framework. The items discussed in this paper targeted mathematics teachers’ perceptions of possible barriers and enablers to their intention to use technology in their teaching. The responses are varied but, overall, strength of agreement with enablers outweighed agreement with perceived barriers. However, it is clear that despite an overall positive attitude towards the use of technology for teaching mathematics, some perceived barriers to change are notable. It is, therefore, helpful if those responsible for professional development, promoting the use of technology, recognise and address these barriers as well as working to strengthening enablers. [ABSTRACT FROM AUTHOR]

Girls' underrepresentation in high-level post-compulsory mathematics is a longstanding issue of concern in many Western nations, with innumerable efforts to increase their participation producing little impact. In this paper, we shed new light on girls' underrepresentation through a post-structural feminist investigation of mathematics teachers' discursive constructions of high-level senior secondary mathematics students. Our analysis of semi-structured interviews with 22 Australian mathematics teachers revealed gendered views that serve to exclude many students from the high-level mathematics student category. Most concerning was their recurring naturalised construction of successful high-level mathematics students as endowed with the right, invariably male, brain. In so doing, teachers repeatedly closed off the possibility of success to those lacking such a 'mathematics gift', effectively 'counting girls out'. We argue that increasing girls' participation in mathematics requires moving beyond current efforts to raise female interest and confidence to, more profoundly, disrupt enduring discourses of male superiority in mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we will examine the mathematical knowledge that prospective mathematics teachers draw upon when graphing function graphs and curves, with a special focus on the occurrence of asymptotes. Three tasks which involved a graph of a rational and exponential function and a hyperbola as a conic section were designed and administered to students. We performed this study within the framework of Anthropological Theory of the Didactic to examine the relationship of prospective mathematics teachers' available knowledge with the knowledge to be taught in upper secondary schools and scholarly knowledge relevant for teaching. By studying prospective mathematics teachers' knowledge, we aim to understand the feasibility of our proposed reference epistemological model for graphing functions and curves in the upper secondary school. Our findings reveal students' shortcomings with respect to the choice of the appropriate graphing praxeology for given tasks. Students' graphing strategies relied mostly on plotting points obtained by evaluating a formula, which is a dominant approach in the textbooks we analysed. Plotting points did not lead students to examine asymptotic behaviour, along with the observed monotonicity of a function. Their graphing strategies were found to be predominantly dependent on the particular setting in which the task was presented. Additionally, in our study, the idea of an asymptote as a tangent line at infinity in the geometric setting was questioned. [ABSTRACT FROM AUTHOR]

This paper indicates that prospective teachers’ familiarity with theoretical models of students’ ways of thinking may contribute to their mathematical subject matter knowledge. This study introduces the intuitive rules theory to address the intuitive, same sides-same angles solutions that prospective teachers of secondary school mathematics come up with, and the proficiency they acquired during the course “Psychological aspects of mathematics education”. The paper illustrates how drawing participants’ attention to their own erroneous applications of same sides-same angles ideas to hexagons, challenged and developed their mathematical knowledge. [ABSTRACT FROM AUTHOR]

The author introduces several papers on action research in teacher education. They include "A study of the field experience of pre-service secondary mathematics teachers" by Joanne E. Goodell, "Awareness of equivalence and relational thinking of pre-service elementary mathematics teachers" by Ana C. Stephens, and a book review by Stephen Lerman, analyzing the education system in the United Kingdom.

In this paper we describe learners being asked to generate examples of new mathematical concepts, thus developing and exploring example spaces. First we elaborate the theoretical background for learner generated examples (LGEs) in learning new concepts. The data we then present provides evidence of the possibility of learning new concepts through a symbiosis of induction and abduction from experience and deduction from the relationships generated in exemplification. In other words, experience can be organised in such a way that shifts of understanding take place as a result of learners’ own actions. Actions, in this context, include mental acts of organisational reflection. [ABSTRACT FROM AUTHOR]

Building on the papers in this special issue as well as on our own experience and research, we try to shed light on the construct of example spaces and on how it can inform research and practice in the teaching and learning of mathematical concepts. Consistent with our way of working, we delay definition until after appropriate reader experience has been brought to the surface and several ‘examples’ have been discussed. Of special interest is the notion of accessibility of examples: an individual’s access to example spaces depends on conditions and is a valuable window on a deep, personal, situated structure. Through the notions of dimensions of possible variation and range of permissible change, we consider ways in which examples exemplify and how attention needs to be directed so as to emphasise examplehood (generality) rather than particularity of mathematical objects. The paper ends with some remarks about example spaces in mathematics education itself. [ABSTRACT FROM AUTHOR]

This paper describes the implementation of extended analysis tasks (EATs) in a required mathematics course for prospective high school mathematics teachers, and investigates the mathematical discovery promoted through engagement with these tasks. Extended analysis tasks are designed to move students beyond the problem context to the underlying mathematical structure of the problem. The results of this study revealed that with careful implementation, these tasks were effective for generating powerful mathematical thinking in preservice mathematics teachers, improving their problem solving, and generally enhancing the habits-of-mind exhibited by successful mathematicians (Carlson & Bloom, ). [ABSTRACT FROM AUTHOR]

Our work is in keeping with the general framework of didactical research in France, where we both studied, and now teach. This paper describes a model training session designed for in-service mathematics teachers, whose purpose is to guide them in their understanding of the relationship between students’ tasks and activities in secondary school (from 11 to 18 years). We first explain the didactical tools that we consider necessary in order to address tasks analysis, and then, as an example of a training session, we provide a list of tasks on similar triangles illustrating the two kinds of analysis in which we engage participants. As a conclusion to this paper, we comment on our choices, and their potential influence on teachers’ practices. [ABSTRACT FROM AUTHOR]

This paper describes an approach for pre-service or in-service mathematics education that teaches sophisticated mathematics using only the tools of high school mathematics. The idea is to start with a standard problem from high school mathematics and let the solution to this problem serve as a platform for asking good mathematical questions and searching for deeper mathematical structure beyond the obvious “answer” to the question. Surprisingly, it turns out that high school mathematics is remarkably open to this sort of analysis, and the results are more interesting mathematically than one might initially think. [ABSTRACT FROM AUTHOR]

This paper addresses the accumulating knowledge of prospective teachers of secondary school mathematics and their acquired proficiency during the course “Psychological aspects of mathematics education,” in which we discussed theoretical models including the intuitive rules theory. Participants’ performances are examined by means of an extensive report of two episodes, one during the course and one afterwards. These episodes marked different stages in the prospective teachers’ analysis of their own and of students’ solutions, which led me to conclude that exposing prospective teachers to the intuitive rules theory is important, since their familiarity with the theory provided them with a tool to reflect on their own mathematical solutions (subject matter knowledge; SMK), on others’ solutions, and on the tasks (pedagogical content knowledge; PCK). [ABSTRACT FROM AUTHOR]

This paper reports the activity of three secondary school mathematics departments in England in self-initiated states of change that led to overall improvements in students' achievements when compared to previous cohorts. This took place without intervention and without their participation in external projects. They provide examples of departments that can work effectively on their own development, and hence, their work adds to our knowledge of the potential for development through collaboration. The departments were monitored over 3 years, and data were analysed using the lens of activity theory. In contrast to departments in many studies, these departments worked overtly on mathematics pedagogy through the shared production and discussion of resources, shared planning and task design. Also in contrast to several other studies, they developed distinct ways to handle differences of subject knowledge among the teachers in the department. Their focus changed during the study from developing resource banks to supporting students' learning through hybrid teaching. [ABSTRACT FROM AUTHOR]

This paper reports the design of an introductory probability module intended for implementation at scale within the English educational system. It forms part of the Effecting Principled Improvement in STEM Education ( epiSTEMe) programme of redesign research aimed at improving the teaching and learning of mathematics and science at early-secondary level. The approach taken by the module is informed by the research literatures on effective teaching (with a particular emphasis on blending teaching components and exploiting dialogic discussion) and probabilistic thinking (with a particular emphasis on triangulating epistemic approaches and deconstructing fallacious reasoning). Recognising that scalable innovation must take account of the current state and established norms of the educational system, module development was informed by such considerations. Advice and feedback from classroom teachers, as well as observation and recording of their lesson implementations, provided a basis for assessing the viability of proposed features of the module, and the adaptation required of teachers, so that guidance materials and professional development could be framed appropriately. [ABSTRACT FROM AUTHOR]

The survey described in this paper was developed in order to gain an understanding of culturally-based aspects of creativity associated with secondary school mathematics across six participating countries. All participating countries acknowledge the importance of creativity in mathematics, yet the data show that they take very different approaches to teaching creatively and enhancing students' creativity. Approximately 1,100 teachers from six countries (Cyprus, India, Israel, Latvia, Mexico, and Romania) participated in a 100-item questionnaire addressing teachers' conceptions about: (1) Who is a creative student in mathematics, (2) Who is a creative mathematics teacher, (3) In what way is creativity in mathematics related to culture, and (4) Who is a creative person. We present responses to each conception focusing on differences between teachers from different countries. We also analyze relationships among teachers' conceptions of creativity and their experience, and educational level. Based on factor analysis of the collected data we discuss relevant relationships among different components of teachers' conceptions of creativity as they emerge in countries with different cultures. [ABSTRACT FROM AUTHOR]

Individual mathematics teachers may value different aspects of teaching and learning mathematics, but at the same time their value systems are under the influence of socially shared values. This paper describes such values in Japanese mathematics education from a historical and normative perspective. After the introduction of Western mathematics into the modern school system in the Meiji period (1868-1912), the people of Japan struggled to adapt and absorb it onto the foundation of Japanese tradition. In the subsequent development of Japanese mathematics education, the integration of both practical and theoretical aspects have been issues, alongside changes in educational focus and in society at large, which are symbolically represented by the enrolment rates at all of elementary, secondary, and tertiary education levels. Mathematics education in Japan has also been subject to international influences, such as the reform movement and the modernization movement, at critical junctures in its development. Key concepts such as mathematical ideas, mathematical thinking, and mathematical activities are traces of such historical efforts by the Japanese mathematics education community, and represent their socially shared values. [ABSTRACT FROM AUTHOR]

It is widely known that students often treat examples that satisfy a certain universal statement as sufficient for showing that the statement is true without recognizing the conventional need for a general proof. Our study focuses on special cases in which examples satisfy certain universal statements, either true or false in a special type of mathematical task, which we term 'Is this a coincidence?'. In each task of this type, a geometrical example was chosen carefully in a way that appears to illustrate a more general and potentially surprising phenomenon, which can be seen as a conjecture. In this paper, we articulate some design principles underlying the choice of examples for this type of task, and examine how such tasks may trigger a need for proof. Our findings point to two different kinds of ways of dealing with the task. One is characterized by a doubtful disposition regarding the generality of the observed phenomenon. The other kind of response was overconfidence in the conjecture even when it was false. In both cases, a need for 'proof' was evoked; however, this need did not necessarily lead to a valid proof. We used this type of task with two different groups: capable high school students and experienced secondary mathematics teachers. The findings were similar in both groups. [ABSTRACT FROM AUTHOR]

We draw on two studies of mathematics departments in 11-18 comprehensive maintained schools in England to compare and contrast the insights provided by differing theoretical perspectives. In one study, activity theory was used to describe common features of the work of three departments. In the other, a mathematics department was viewed and analysed as a complex system. In both cases, it was the learning of the departments as systems rather than of individuals that was of interest. The affordances and limitations of the analytical perspectives are discussed. Taken as a whole, this paper opens up the workings of school mathematics departments in a country which has a strong department culture. [ABSTRACT FROM AUTHOR]

For many years, education researchers excited by the potential for digital technologies to transform mathematics teaching and learning have predicted that these technologies would become rapidly integrated into every level of education. However, recent international research shows that technology still plays a marginal role in mathematics classrooms. These trends deserve investigation in the Australian context, where over the past 10 years secondary school mathematics curricula have been revised to allow or require use of digital technologies in learning and assessment tasks. This paper reports on a survey of mathematics teachers' use of computers, graphics calculators, and the Internet in Queensland secondary schools, and examines relationships between use and teachers' pedagogical knowledge and beliefs, access to technology, and professional development opportunities. Although access to all forms of technology was a significant factor related to use, teacher beliefs and participation in professional development were also influential. Teachers wanted professional development that modelled planning and pedagogy so they could meaningfully integrate technology into their lessons in ways that help students learn mathematical concepts. The findings have implications not only for resourcing of schools, but also for designing professional development that engages teachers with technology in their local professional contexts. [ABSTRACT FROM AUTHOR]

In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant similarity to valid examples of the concept, and consequently are more often mistakenly identified as examples. We describe and discuss these notions and present a study regarding kindergarten children’s grasp of nonexamples of triangles. [ABSTRACT FROM AUTHOR]

This study is grounded in the theoretical position that solving problems in different ways creates mathematical connections when learning and teaching mathematics. It acknowledges the central role teachers play in providing students with learning opportunities, and it is based on the empirical finding that mathematics teachers are reluctant to solve problems in different ways in the classroom. In this paper we address the contradiction between theory-based recommendations and school mathematics practice. Based on analysis of individual interviews and two group meetings with 12 Israeli secondary school mathematics teachers, we demonstrate that in the context of multiple-solution connecting tasks this discrepancy is caused by the situated nature of the teachers’ knowledge. We also reveal the complex relationship between different types of teacher knowledge and argue the significance of developing a common language between members of the mathematics education community, including teacher educators and researchers. [ABSTRACT FROM AUTHOR]

This paper describes two high school teachers' conceptions of the cooperation and exploration components of a reform-oriented mathematics curriculum. Although the teachers appreciated the themes of cooperation and exploration in theory, their conceptions of these themes with respect to their implementations of the curriculum differed. One teacher viewed the curriculum's problems as open-ended and challenging for students, whereas the other teacher claimed that the problems were overly structured. Each teacher attributed difficulties with students' cooperative work to the amount of structure and direction (too little or too much) offered by the problems. Discussion of such similarities and differences in the teachers' conceptions emphasizes the dynamic, humanistic nature of curriculum implementation and gives rise to important implications for mathematics teacher development in the context of reform. [ABSTRACT FROM AUTHOR]

When language is defined narrowly in mathematics classrooms, racially and linguistically minoritized students in classrooms could be systematically positioned as "learners of deficiency." Recent scholarship calls for expanding the notion of language to emphasize embodied expression of mathematical ideas. Taking a critical perspective to understand racialized experiences of using languages in disciplinary learning spaces, this article proposes the reconceptualization of embodiment as a language for racialized multilingual learners. This study was conducted in a Grade 1 classroom in a linguistically and racially diverse school in Canada. Through a series of professional development sessions, we worked with an experienced teacher to redesign the normalized and institutionalized pedagogy toward greater mobility of racialized multilingual learners' bodies, which was intertwined with their intellectual liberation. Focusing on the spatiality of pedagogy, the previously restrictive areas in the school were transformed into a place that augments embodied expression of mathematical ideas and agentive participation of minoritized learners. The analysis focused on the embodied discourse that participating racialized multilingual students used to actively engage in mathematical discussion. Our findings show that the designed pedagogy, characterized by the spatial and temporal expansion of the learning environment, offered more spaces for uncertainty and spontaneity with the decreased control of the teacher as an explicator. Our article furthers anti-colonial approaches to understand the intersection of racialized bodies and language in mathematics education. [ABSTRACT FROM AUTHOR]

Researchers point out the importance of teachers' knowledge of student thinking and the role of examining student work in various contexts to develop a knowledge base regarding students' ways of thinking. This study investigated prospective secondary mathematics teachers' interpretations of students' thinking as manifested in students' work that embodied solutions of mathematical modelling tasks. The data were collected from 25 prospective mathematics teachers enrolled in an undergraduate course through four 2-week-long cycles. Analysis of data revealed that the prospective teachers interpreted students' thinking in four ways: describing, questioning, explaining, and comparing. Moreover, whereas some of the prospective teachers showed a tendency to increase their attention to the meaning of students' ways of thinking more while they engaged in students' work in depth over time and experience, some of them continued to focus on only judging the accuracy of students' thinking. The implications of the findings for understanding and developing prospective teachers' ways of interpreting students' thinking are discussed. [ABSTRACT FROM AUTHOR]

From 2012 to 2016 all teachers of mathematics, in primary through to upper secondary and adult education, in Sweden are to be given the opportunity of receiving state-coordinated professional development (PD), generally involving around one meeting per week for a year. We examine the ways in which this programme and its content are research-based by analysing the design principles of the PD and its content via a review of the underlying documents. We find that the research base for the design of the PD decisions is formulated only partly explicitly in the documentation, as is the case for the content. Nevertheless, the principles still largely conform to internationally recognised research-based models for PD and teacher change. In the discussion we propose that a relatively open design process, involving many stakeholders with knowledge of both theory and practice rather than an analytical examination of previous research, has contributed to the adoption of this format for the PD programme. [ABSTRACT FROM AUTHOR]

In this study, the authors examined a group of related literatures that used the same large-scale nationally representative dataset with algebra achievement as the outcome to explore why results from the same dataset may differ across studies. More specially, the authors synthesized the extant research literature that utilized data from the High School Longitudinal Study of 2009 (HSLS:09) to investigate the student-, teacher-, school-, and parent-level characteristics associated with algebra achievement and whether these relationships were consistent across students' backgrounds (i.e., race/ethnicity, gender, socio-economic status, and prior achievement). The authors have summarized the results of 21 studies across 2 outcome variables (ninth- and eleventh-grade algebra achievement) using optimal resource theory as a framework to summarize evidence-based practices for improving student outcomes. [ABSTRACT FROM AUTHOR]

Despite compelling arguments for the importance of revisiting school mathematics in pre-service mathematics teacher education, there is little indication of how such revisiting might be done, particularly at secondary level. We argue that revisiting provides three key opportunities for learning: (1) additional time on the mathematics that pre-service teachers will teach in schools; (2) attention to mathematical practices and (3) opportunity to learn about seminal constructs and findings from the mathematics education research literature. We illustrate these with three revisiting tasks and their enactment in a course where opportunities for revisiting were deliberately offered. We also discuss challenges that arise as teacher educators intentionally design opportunities for revisiting school mathematics. [ABSTRACT FROM AUTHOR]

Ancient Chinese mathematics has been the focus of many research studies and scholarly works from a historical perspective. However, no move has been made to investigate its role in the teaching and learning of mathematics. This pilot study examined the effects of an Ancient Chinese Mathematics Enrichment Programme (ACMEP) on the academic achievement of second year students from a secondary school in Singapore—a strand of a principal study which had the intent of investigating possible roles of ancient Chinese mathematics in the Singapore secondary school mathematics curriculum. Analysis of covariance was used to examine the difference in mean scores on a variety of formal assessments in mathematics between students who participated in the ACMEP and those who did not. In addition, scores on formal assessments of other relevant subjects were analyzed to further investigate ACMEP’s scope of influence. [ABSTRACT FROM AUTHOR]

Feedback provided by mathematics teachers usually addresses procedural skills and, to a much lesser extent, other competencies such as conceptual understanding or engagement in mathematical practices. As most previous literature has studied feedback provided on homework or video prompts, how teachers provide such feedback in the classroom is poorly understood. Here, sixteen lessons taught by five teachers were purposefully sampled from a larger video study (172 lessons) as lessons with high-quality feedback according to a standardized observation instrument. The analysis focused on the instructional situations in which teachers provided feedback. When teachers provided procedural feedback, the situations were orderly and effective. Feedback on conceptual understanding and mathematical practices was provided in situations when students were especially challenged, and entailed a series of complex decisions, thereby placing demands on the teachers to manage both the students' understanding and behavior. We argue that researchers should focus on how teachers and students negotiate the instructional situation to allow for feedback on conceptual understanding or mathematical practices. [ABSTRACT FROM AUTHOR]

Numerous studies have reported positive outcomes of noticing interventions on the development of prospective mathematics teachers' (PMTs) noticing of a range of important aspects of classroom instruction. Less is known, however, about whether noticing skills that are developed during an intervention transfer to support PMTs' in-the-moment noticing during their own teaching practice. This study compared PMTs' noticing while teaching a lesson during their student teaching internship of PMTs who participated in a noticing intervention to those who did not participate in the intervention to determine whether the two groups of PMTs noticed different aspects of instruction. The study documented very different noticing foci of the two groups of PMTs, suggesting that the noticing intervention did, in fact, support the PMTs' in-the-moment noticing of aspects of instruction that aligned with the goals of the noticing intervention. The findings of the study suggest that noticing skills developed during a university-based noticing intervention can transfer to support novice teachers' in-the-moment noticing during their classroom practice. [ABSTRACT FROM AUTHOR]

Despite the recognised importance of mathematical proof in secondary education, there is a limited but growing body of literature indicating how preservice secondary mathematics teachers (PSMTs) view proof and the teaching of proof. The purpose of this survey research was to investigate how PSMTs in Australia, the USA and Korea perceive of proof in the context of secondary mathematics teaching and learning. PSMTs were able to outline various mathematical and pedagogical aspects of proof, including purposes, characteristics, reasons for teaching and imposed constraints. In addition, PSMTs attended to differing, though overlapping, features of proof when asked to determine the extent to which proposed arguments constituted proofs or to decide which arguments they might present to students. [ABSTRACT FROM AUTHOR]

A broad categorisation of different reading styles for mathematics text is generated in this research. The styles derive from those found in literature around academic reading skills. These styles are inductively refined using video transcripts of five specially chosen students studying out loud from a prescribed mathematics textbook. The context is a self-study mathematics course directed at high school mathematics teachers with weak content knowledge. Reading is understood as a transaction (enacted curriculum) between text (written curriculum) and reader. Reading styles are characterised in terms of depth of reading, focus on different components of text or not, connections within text or to prior knowledge, and performance on exercises. Five different styles of reading mathematics text are identified: close reading with strong connections, close reading with some connections, scanning, skimming, and avoiding. The different reading styles are also interpreted in terms of structure, voice, and genre of the textbook. [ABSTRACT FROM AUTHOR]

Reform-based discourses in mathematics education have fabricated different subjectivities for teachers such as the "traditional" and the "new" teacher. Professional development programs are proposed as effective mechanisms to fabricate the "new" teacher. However, this teacher has proved hard to produce. Thus, the "resistor" teacher has emerged into the field as a way to explain failure within school mathematics reform. In this article, I assume that resistance is a consequential response against particular forms of subjectivation imposed on mathematics teachers. Using conceptual tools from Hall and Foucault, I explore the ways wherein a high school mathematics teacher reinvents meanings of being a mathematics teacher in the context of a professional development program aimed to implement problem-solving instruction. Against the myth of the resistor teacher unwilling to change, what emerges is a process of struggle over meaning. School mathematics reform, considered as an ideological event, becomes a site in which competing meanings about being a mathematics teacher are negotiated, contested, and resisted. [ABSTRACT FROM AUTHOR]

This mixed methods study examined an association between cognitive types of teachers' mathematical content knowledge and students' performance in lower secondary schools (grades 5 through 9). Teachers ( N = 90) completed the Teacher Content Knowledge Survey (TCKS), which consisted of items measuring different cognitive types of teacher knowledge. The first cognitive type (T1) assessed participants' knowledge of basic facts and procedures. The second cognitive type (T2) measured teachers' understanding of concepts and connections. The third cognitive type (T3) gauged teachers' knowledge of mathematical models and generalizations. The study comprised two levels of quantitative data analysis. First, we explored each cognitive type of teachers' content knowledge and the overall TCKS score as they related to student performance. Second, we studied the correlation between each cognitive type of teacher content knowledge to deepen the understanding of content associations. Results of the study show a statistically significant correlation between cognitive types T1 and T2 of teacher content knowledge and student performance ( p < .05). The correlation between cognitive type T3 and student performance was not significant ( p = .0678). The most substantial finding was the correlation between teachers' total score on the TCKS and student performance (Pearson's r = .2903, p = .0055 < .01). These results suggest that teachers' content knowledge plays an important role in student performance at the lower secondary school. The qualitative phase included structured interviews with two of the teacher participants in order to further elaborate on the nature of the quantitative results of the study. [ABSTRACT FROM AUTHOR]

In order to develop a deeper understanding of mathematics teaching expertise, in this study we use the Documentational Approach to Didactics to explore the resource systems of three Chinese mathematics 'expert' teachers. Exploiting the Western and Eastern literature we examine the notion of 'mathematics teaching expertise', as it is perceived in the East and the West. The data consist of two rounds of in-depth interviews, observations and teachers' representations of their resource systems, where teachers describe their resources connected to their practice, their perceptions of mathematics teaching expertise, and how to develop it. Subsequently, the data are analyzed with respect to the different facets of the notion of teaching expertise and related to the teachers' views and practices, in order to deepen our understandings of what proficiency in mathematics teaching might mean and how to develop it, seen through the lens of 'resources'. The significance of the study relates to the enhancement of mathematics teachers' expertise and capacity building when working in collectives (e.g., in teacher professional development), in order to develop a strong workforce for supporting and helping to improve pupil learning. [ABSTRACT FROM AUTHOR]

This report examines teachers' self-espoused attitudes and beliefs on proving in the secondary mathematics classroom. Conclusions were based on a questionnaire of 78 US mathematics teachers who had completed at least 2 years of teaching mathematics at the secondary level. While these teachers placed importance on proving as a general mathematical skill, when they discuss their own classrooms, procedural skill consistently surpasses proof-related activities in importance for a majority of high school teachers. Furthermore, teachers labeling their own past experiences in proving as causing anxiety are predictably more likely to put less value on proving. Interestingly, the quantity of past college mathematics courses is a reverse predictor indicating that further study should consider how students perceptions of proving change as they pass through a mathematics major. [ABSTRACT FROM AUTHOR]

For decades, educators have advocated using history of mathematics in mathematics classrooms. Empirical research on the efficacy of this practice, however, is scarce. A quasi-experiment was used to investigate the effects of using history as a tool to teach mathematics on grade 11 students' mathematics achievement. Effects in three affective domains (attitudes, anxiety, and motivation) were also measured. Four classes from a school in Singapore participated in this quasi-experiment. The experimental group ( n = 51) and control group ( n = 52) were each made up of two classes. Results indicated that using history as a tool to teach mathematics had a significant positive effect on students' mathematics achievement, in an initial posttest and in two retention tests taken 4 months and 1 year, respectively, after the last intervention session. Significant positive effects were also found on two subscales within the affective domain variables (perceived value of mathematics and introjection, a type of extrinsic motivation), but only at a posttest administered midway through the study. These results suggest that using history in mathematics classrooms have both immediate short- and long-term effects on students' achievement, but only short-term positive effects in the affective domains. These results were discussed using qualitative feedback obtained from the participants of this study. [ABSTRACT FROM AUTHOR]

What are the important attributes of good mathematics teaching? The answer depends on whom you are asking. Teachers, researchers, policymakers, administrators, and parents usually provide their own view on what they consider the important attributes of good mathematics teaching. The purpose of this study was to determine what beginning teachers report as being the most important attributes at the secondary level. This research explored whether there was a relationship between the demographics of the participants and the attributes of good teaching. A quantitative analysis was used to gather information from the research participants regarding their beliefs and classroom practices of good mathematics teaching. The three most important attributes in good teaching were classroom management, motivation, and strong in content knowledge. One interesting finding was the discovery of four groups of beginning teachers, and how they were associated with specific attributes of good mathematics teaching according to their demographics. [ABSTRACT FROM AUTHOR]

Insights into teachers’ planning of mathematics reported here were gathered as part of a broader project examining aspects of the implementation of the Australian curriculum in mathematics (and English). In particular, the responses of primary and secondary teachers to a survey of various aspects of decisions that inform their use of curriculum documents and assessment processes to plan their teaching are discussed. Teachers appear to have a clear idea of the overall topic as the focus of their planning, but they are less clear when asked to articulate the important ideas in that topic. While there is considerable diversity in the processes that teachers use for planning and in the ways that assessment information informs that planning, a consistent theme was that teachers make active decisions at all stages in the planning process. Teachers use a variety of assessment data in various ways, but these are not typically data extracted from external assessments. This research has important implications for those responsible for supporting teachers in the transition to the Australian Curriculum: Mathematics. [ABSTRACT FROM AUTHOR]

This article describes the collaborative inquiry activity of a group of high school mathematics teachers interested in increasing student engagement and problem solving in the classroom. Specific findings related to the nature of the teacher interactions and subsequent impacts on practice are discussed. The findings focus on (a) the nature of the interactions within the teacher group and their impact on instructional decision-making and theory-building, (b) the role of student work in their collaborative inquiry, and (c) the role and impact of the inquiry group’s facilitation processes. A description of the professional development provides context for the results. The teachers in this study were able to build individual and collective theories on learners and instruction. The limitations of their inquiry context and their own conversational norms allowed for deeper reflections on classroom practice than on learners and learning. Specific connections between the inquiry work and classroom practice are found. [ABSTRACT FROM AUTHOR]

This collective case study reports on an investigation into the relationship between mathematics teachers’ beliefs and their classroom practices, namely, how they organized their classroom activities, interacted with their students, and assessed their students’ learning. Additionally, the study examined the pervasiveness of their beliefs in the face of efforts to incorporate reform-oriented classroom materials and instructional strategies. The participants were five high school teachers of ninth-grade algebra at different stages in their teaching career. The qualitative analysis of the data revealed that in general beliefs were very influential on the teachers’ daily pedagogical decisions and that their beliefs about the nature of mathematics served as a primary source of their beliefs about pedagogy and student learning. Findings from the analysis concur with previous studies in this area that reveal a clear relationship between these constructs. In addition, the results provide useful insights for the mathematics education community as it shows the diversity among the inservice teachers’ beliefs (presented as hypothesized belief models), the role and influence of beliefs about the nature of mathematics on the belief structure and how the teachers designed their instructional practices to reflect these beliefs. The article concludes with a discussion of implications of teacher education. [ABSTRACT FROM AUTHOR]

Teaching is a field that is dynamic, with innovations necessitating upgrading of skills and education of teachers for the successful implementation of reforms. The behaviour and attitudes of teachers towards teaching and learning and their knowledge banks are the result of the impact of in-service training. This study investigated the perceptions of mathematics and science teachers in Botswana towards in-service provision by the Department of Mathematics and Science Education In-service Training unit (DMSE-INSET), whose mandate is to improve the quality of teaching by supporting teachers through training programmes that enable them to take ownership of their professional development. Data were collected from a sample of 42 senior Mathematics and Science secondary school teachers, using structured interviews with open-ended questions, which were analyzed qualitatively. The findings show that teachers’ concerns included the lack of impact of current in-service training programmes on the education system, no regular follow-up activities to support the one-off workshops and insufficient skills acquired to sustain the implementation of the strategies solicited by the workshops. [ABSTRACT FROM AUTHOR]

We investigate experienced high school geometry teachers’ perspectives on “authentic mathematics” and the much-criticized two-column proof form. A videotaped episode was shown to 26 teachers gathered in five focus groups. In the episode, a teacher allows a student doing a proof to assume a statement is true without immediately justifying it, provided he return to complete the argument later. Prompted by this episode, the teachers in our focus groups revealed two apparently contradictory dispositions regarding the use of the two-column proof form in the classroom. For some, the two-column form is understood to prohibit a move like that shown in the video. But for others, the form is seen as a resource enabling such a move. These contradictory responses are warranted in competing but complementary notions, grounded on the corpus of teacher responses, that teachers hold about the nature of authentic mathematical activity when proving. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate how a community of practice focused on becoming a teacher of secondary school mathematics emerged during a pre-service teacher education programme and was sustained after students graduated and began their first year of full-time teaching in schools. Bulletin board discussions of one pre-service cohort are analysed in terms of Wenger’s (1998) three defining features of a community of practice: mutual engagement of participants, negotiation of a joint enterprise, and development of a shared repertoire for creating meaning. Emergence of the online community was associated with our own role in facilitating professional dialogue, the voluntary and unstructured nature of participation, initial face-to-face interaction that created familiarity and trust, and the convenience of using email rather than logging on to a website. The study shows that the emergent design of the community contributed to its sustainability in allowing the pre-service and beginning teachers to define their own professional goals and values. [ABSTRACT FROM AUTHOR]

This article discusses the design of tasks for teacher education. It focuses on tasks that are used in a university course for pre-service secondary school mathematics teachers. Special attention is given to tasks that use analogical thinking in their construction or implementation. These tasks are categorized by type of teacher education goal and analyzed with respect to the use of analogical thinking. Short examples are presented for three of the goal categories, while an elaborated example is given for the fourth one. The detailed example describes the goals and design of a task sequence following an emergent pedagogical need. Using the ad hoc constructed task-sequence the teacher educator avoids “telling” while demonstrating an alternative instructional approach, and seizing the opportunity to bring up additional pedagogical issues. [ABSTRACT FROM AUTHOR]