Mathematics teachers’ representations of authority

Issues of authority abound in education and schooling but have not been interrogated sufficiently. We describe a tool that we have developed to initiate dialogue with teachers about authority in their classrooms—using a diagram to represent authority in their classrooms. Our analysis of the diagrams mathematics teachers created and discussed in our work with them illustrates the importance of understanding teachers’ perspectives about authority. To understand better how mathematics teachers think about the authority in their classrooms, we investigated what sources of authority they represented in their diagrams, and how the teachers related these sources to each other. The diversity in the teachers’ representations exceeded our anticipations, indicating that research on authority in classrooms has merely scratched the surface of understanding the ways mathematics teachers think about authority in their classrooms.     

Research–Practice Partnerships to Support the Development of High Quality Mathematics Instruction for All Students

The gap in achievement in mathematics between at-risk students and their more advantaged counterparts is a persistent problem of the U.S. education system. Although some research-based curricula and pedagogy have demonstrated promise in supporting students from diverse backgrounds to develop conceptual understanding and procedural fluency in mathematics, scaling up instructional change across a district organization is a significant challenge. The Middle School Mathematics and the Institutional Setting of Teaching (MIST) Project is a research–practice partnership seeking to understand how large urban school districts can support the development of rigorous and equitable middle-school mathematics instruction at scale. This article enumerates the goals and design of this multiyear, multidistrict partnership, and describes one illustrative example of how our partnership activities informed and supported one district's efforts to improve mathematics instruction over multiple years. General recommendations for district–researcher partnerships are discussed.     

Personal understanding of assessment and the link to assessment practice: the perspectives of higher education staff

The study investigates how higher education staff understand assessment, and the relationship between these understandings and their assessment practices. Nine individuals attended a workshop that guided them through the creation of a concept map about assessment, which was subsequently discussed in one-to-one semi-structured interviews. We found considerable variation in understanding of assessment, both between and within participants, and this appeared to be a consequence of the varied contexts within which assessment operates. Some assessment practices were highly complex, and at times closely entwined with teaching. In addition, individuals’ practices helped to illuminate variation in how underlying concepts (e.g. assessment for learning) were understood. The approach supported the construction of the participants’ understanding of assessment, and enabled the exploration of the interplay between thinking and reported practice, which were closely aligned. It also drew attention to the need to further develop methodologies which capture both the complexity of thinking about assessment and real-world assessment practices.     

Parents' representations of their children's mathematics learning in multiethnic primary schools

There is a growing concern that governmental calls for parental involvement in children's school mathematics learning have not been underpinned by research. In this article the authors aim to offer a contribution to this debate. Links between children's home and school mathematical practices have been researched in sociocultural studies, but the origins of differences within the same cultural group are not well understood. The authors have explored the notion that parents' representations of school mathematics and associated practices at home may play a part in the development of these differences. This article reports an analysis of interviews with parents of 24 children of Pakistani and White origin enrolled in primary schools in England, including high and low achievers in school mathematics. The extent to which the parents represented their own school mathematics and their child's school mathematics as the ‘same’ or ‘different’ are examined. In addition, ways in which these representations influenced how they tried to support their children's learning of school mathematics are examined. The article concludes with reflections on the implications of the study for education policy.     

Troubling “understanding mathematics in-depth”: Its role in the identity work of student-teachers in England

In this paper, we focus on an initiative in England devised to prepare non-mathematics graduates to train as secondary mathematics teachers through a 6-month Mathematics Enhancement Course (MEC) to boost their subject knowledge. The course documentation focuses on the need to develop “understanding mathematics in-depth” in students in order for them to become successful mathematics teachers. We take a poststructural approach, so we are not interested in asking what such an understanding is, about the value of this approach or about the effectiveness of the MECs in developing this understanding in their participants. Instead we explore what positions this discourse of “understanding mathematics in-depth” makes available to MEC students. We do this by looking in detail at the “identity work” of two students, analysing how they use and are used by this discourse to position themselves as future mathematics teachers. In doing so, we show how even benign-looking social practices such as “understanding mathematics in-depth” are implicated in practices of inclusion and exclusion. We show this through detailed readings of interviews with two participants, one of whom fits with the dominant discourses in the MEC and the other who, despite passing the MEC, experiences tensions between her national identity work and MEC discourses. We argue that it is vital to explore “identity work” within teacher education contexts to ensure that becoming a successful mathematics teacher is equally available to all.     

The Emerging Pracitces of a Novice Teacher: The Roles of His School Mathematics Images

The relationship between teachers' beliefs about mathematics and its teaching and learning, and their classroom practices, has been investigated in many studies. The results of these studies are by no means unanimous, and the purpose of the present report is to contribute to a further understanding of the significance of one teacher's beliefs and images of school mathematics on his teaching practices. The article presents a novice teacher whose images of school mathematics were strongly influenced by the current reform of mathematics education. The study focuses on how the teacher coped with the complexities of the classroom. On the basis of the teacher's role within the classroom interactions, the construct of critical incidents of practice is developed, and some of the results of previous research in the field are re-examined. This revised version was published online in August 2006 with corrections to the Cover Date.     

PROFESSIONAL NOTICING PRACTICES OF NOVICE MATHEMATICS TEACHER EDUCATORS

The focus on professional noticing in mathematics education has recently gained increased interest as researchers work to understand how and what is noticed and how this translates into practice. Much of this work has focused on the professional noticing practices of inservice teachers and preservice teachers, with less attention focused on those educating teachers. This research explores how novice mathematics teacher educators professionally notice as they engage in teaching experiments and create models of student’s mathematical thinking. Findings indicate the novice teacher educators are including some evaluative comments in their professional noticing practices but lack in-depth interpretive analysis about student thinking and rarely make connections between student’s thinking and the broader principles of teaching and learning. These findings provide evidence for the importance of supporting teacher educators with developing their abilities to professionally notice.     

Developing the next generation of diverse computer scientists: the need for enhanced,intersectional computing identity theory

This theoretical paper explores the need for enhanced, intersectional computing identity theory for the purpose of developing a diverse group of computer scientists for the future. Greater theoretical understanding of the identity formation process specifically for computing is needed in order to understand how students come to understand themselves as computer scientists. To ensure that the next generation of computer scientists is diverse, this paper presents a case for examining identity development intersectionally, understanding the ways in which women and underrepresented students may have difficulty identifying as computer scientists and be systematically oppressed in their pursuit of computer science careers. Through a review of the available scholarship, this paper suggests that creating greater theoretical understanding of the computing identity development process will inform the way in which educational stakeholders consider computer science practices and policies.     

Mentor, Mentee and Mathematics: A Story of Professional Development

In this paper, I critically examine my role as a mentor to a group of mid-career mathematics teachers, to better understand how mentoring practices influence teacher learning. The teachers taught mathematics to Grades 5 or 6. They planned and taught lessons to promote conceptual understanding of mathematics through the use of discussions, and reflected on their own and pupils' learning in the process. The study lasted for a year. It draws on qualitative data from sources such as semi-structured interviews, participant observation, and teachers' reflective journals. As a mentor/researcher I found that in the case of mentees' perception of the mentor as a problem solver or evaluator of performance learning was inhibited, whereas mentor and mentee relationships established on mutual trust supported mentees' personal and professional growth. The close work with the teachers also gave me insights into aspects of learning influenced by teachers' beliefs and understanding of mathematics and of how learning takes place. The study has implications for other such teacher education initiatives, particularly in the context of developing countries like Pakistan. This revised version was published online in July 2006 with corrections to the Cover Date.     

Exploring Young Children’s Understanding About the Concept of Volume Through Engineering Design in a STEM Activity: A Case Study

This case study explores young children’s understanding and application of the concept of volume through the practices of engineering design in a STEM activity. STEM stands for science, technology, engineering, and mathematics. However, engineering stands out as a challenging area to implement. In addition, most early engineering education research centers on curriculum and instruction rather than students’ understanding and application of models and content knowledge to a design project (Johri and Olds, Journal of Engineering Education, 100(1):151–185, 2011). For this study, we created a play-based STEM activity which helps students understand and apply the concept of volume in creating clay boats through the practices of engineering design. Three students voluntarily participated in this study. Data sources included interviews, photocopies of boats, and observation field notes which were analyzed using a comprehensive cross-case analysis. Findings indicate that the common pattern is rather intuitive when students define engineering problems using different criteria. Also, students’ solutions to engineering problems are associated with their experiential ideas of reality. How students express their understanding of volume is correlated to the way of their structuring volume. Lastly, students understand the concept of volume gradually while going through the practices of engineering design. Implications are discussed in terms of how engineering education contributes to students’ understanding of volume.     

Linking reform-oriented experiences to teacher identity: The case of an elementary mathematics teacher

Novice teachers often have difficulty transferring what they learn in teacher education programs to classroom practice. This is especially true for elementary school teachers who are expected to teach mathematics with reform-oriented methods. The purpose of this longitudinal case study was to examine the experience of one novice elementary school teacher over a 3-year period, and understand the factors that supported her to enact a reform-oriented practice in mathematics as she transitioned from being a preservice to inservice teacher. Influential mathematics education models, her commitment to learning, and school-based contexts affected her teaching identity and practices. To conclude, an argument is made for teacher education experiences to explicitly address mathematics teacher identity to support the enactment of reform-oriented practices.     

Learning together: a transdisciplinary approach to student–staff partnerships in higher education

ABSTRACT

Partnership in higher education has gained prominence over recent decades, but recent studies have identified a lack of research exploring how partnership practices unfold in specific disciplinary contexts. This article explores how a transdisciplinary approach can be used to better understand and facilitate student–staff partnerships where staff and students have diverse disciplinary backgrounds and knowledge. We present a case study of the Bachelor of Creative Intelligence and Innovation at the University of Technology Sydney, focusing on the adaptation of our curriculum co-creation processes by drawing on multiple knowledge types through a reflexive process of mutual learning. We conclude that explicit consideration of these principles, which are common to both transdisciplinary and partnership frameworks, have the potential to enhance consideration of diverse perspectives and the roles played by worldviews, norms and values when building student–staff partnerships around curriculum co-creation.     

‘I’m not a “maths‐person”!’ Reconstituting mathematical subjectivities in aesthetic teaching practices

In this study I have investigated how alternative ways of teaching mathematics influence and affect Early Childhood Education (ECE) students’ attitudes towards maths and how they understand their own subjectivities as more or less mathematical during a 10‐week alternative maths course. The investigated course adopts a feminist post‐structural approach based on critical pedagogy and deconstructive theory and includes an interdisciplinary approach to investigative mathematics. The data used include the memory/narrative writings and process‐writings of 75 female teacher‐education students, collected from three different cohorts, in which the students describe their learning processes throughout the maths course. The study shows that, in the main, the students became much more positively inclined to the subject of mathematics after the maths course and agreed that this course had changed their understanding of their own mathematical subjectivity, albeit in different and varying ways.     

Validity in teachers’ summative assessments

This paper describes some of the findings of a project which set out to explore and develop teachers’ understanding and practices in their summative assessments. The focus was on those summative assessments that are used on a regular basis within schools for guiding the progress of pupils and for internal accountability. The project combined both intervention and research elements. The intervention aimed both to explore how teachers might improve those practices in the light of their re‐examination of their validity, and to engage them in moderation exercises within and between schools to audit examples of students’ work and to discuss their appraisals of these examples. This paper reports findings, arising from this work, of the research that aimed to study how teachers understand validity, and how they formulate their classroom assessment practices in the light of that understanding. The paper also considers how that understanding might be challenged and developed. It was found that teachers’ attention to validity issues had been undermined by the external test regimes, but that teachers could re‐address these issues by reflection on their values and by engagement in a shared development of portfolio assessments.     

Alignment, cohesion, and change: Examining mathematics teachers’ belief structures and their influence on instructional practices

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.     

Elementary teachers’ use of content knowledge to evaluate students’ thinking in the life sciences

Science learning environments should provide opportunities for students to make sense of and enhance their understanding of disciplinary concepts. Teachers can support students’ sense-making by engaging and responding to their ideas through high-leverage instructional practices such as formative assessment (FA). However, past research has shown that teachers may not understand FA, how to implement it, or have sufficient content knowledge to use it effectively. Few studies have investigated how teachers gather information to evaluate students’ ideas or how content knowledge factors into those decisions, particularly within the life science discipline. We designed a study embedded in a multi-year professional development program that supported elementary teachers’ development of disciplinary knowledge and FA practices within science instruction. Study findings illustrate how elementary teachers’ life science content knowledge influences their evaluation of students’ ideas. Teachers with higher levels of life science content knowledge more effectively evaluated students’ ideas than teachers with lower levels of content knowledge. Teachers with higher content exam scores discussed both content and student understanding to a greater extent, and their analyses of students’ ideas were more scientifically accurate compared to teachers with lower scores. These findings contribute to theory and practice around science teacher education, professional development, and curriculum development.     

The development of a mathematics teacher’s professional identity during her first year teaching

This study is focused on a period that poses several challenges for the development of mathematics teachers’ professional identities and agency: their first year of teaching in schools. During this period, beginning mathematics teachers confront tensions and contradictions among the principles, ideals, and experiences encountered during pre-service education and the demands, and restrictions of their teaching practice in schools. This article approaches this topic by developing an interpretative case study centered on one novice mathematics teacher, Sol. The aim is to describe and understand the development of Sol’s professional identity and agency during her first teaching year. Considering identity development as a diachronic phenomenon, we carry out a narrative analysis of the research data. The findings show that Sol developed her professional identity and agency through a process that gathered together the teaching practices possible inside her school, the positions she could negotiate as a newcomer inside the institution, and the cultural practices and discourses embodied during pre-service education. The results bring to the forefront the profound and tense interactions between the intimate and personal terrain of mathematics teachers and the social and cultural world of the schools where they work. Sol’s case also contributes to understanding the role that a robust pre-service education can play in the development of beginning teachers’ professional identities and in the possibility they could become agents of transformation in their schools.     

On the outskirts of education: the liminal space of rural teen pregnancy

The historical context of teen pregnancy in the US and its evolution to the present embodiment of the predominantly urban ‘welfare mother’ developed from specific socio-historical contexts that deemed early childbirth, especially illegitimate pregnancies, as morally and socially deviant. Two conceptual elements missing within the literature on teen motherhood are the geographically diverse locations of teen pregnancy and motherhood and the spatial boundaries that arise in the institutional interface of pregnancy and schooling. Since the rural teen mother is rooted within a different social-geographical context than her urban counterpart it becomes essential to examine ways in which rural teen mothers navigate their educational opportunities. Likewise, the literature does not address the spatial boundaries pregnant and mothering teens face as they negotiate their relationship with schools. This study addresses both questions in its attempt to understand how pregnant and mothering teens in rural communities navigate their education and how they perceive themselves within the overall institutional setting of the school.     

Conceptualizing Teachers' Ways of Knowing

This article addresses issues related to the ways teachers learn mathematics and the teaching of mathematics and the relevance of those ways to their professional development. Preservice teachers' understanding of school mathematics lacks sophistication, a situation that needs to be addressed in mathematics teacher education programs. What is critical is the means by which they encounter and explore the mathematics they will be teaching. Fundamentally, their mathematical experiences need to be congruous with the kind of teaching we would expect of a reflective, adaptive teacher. The article contains both practical and theoretical considerations of how these experiences might be structured. Theoretical orientations for conceptualizing teachers' belief structures are offered as a foundation for conceptualizing teachers' ways of knowing. The moral dimension of teacher education is considered as a backdrop for understanding how teachers come to know.This revised version was published online in September 2005 with corrections to the Cover Date.