Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving

In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.     

Understanding Teacher Educators and Their Pre-Service Teachers through Multi-Media Case Studies of Practice

The challenges facing those who seek to prepare mathematics teachers are well established in the literature. Most of the research to date has focused on the perceptions and understandings of pre-service teachers, but not on the perceptions and understandings of teacher educators. In this study, we explore how four teacher educators understand their pre-service secondary teachers as the pre-service teachers attempt to make sense of teaching through the investigation of a multimedia case study of practice. We found that the teacher educators adopted two different implementation strategies: one strategy tended to be open-ended and exploratory; the other was more focused on the teacher educators' goals of anticipating student understanding and developing mathematical content knowledge for teaching. We also found that, in using the case study, teacher educators elicited pre-service teachers' thinking about the complexities of the teacher's role in small group work, about the value of explicitly revealing the teacher's reflections on the lessons, about the role of planning and preparation, and about the limits of pre-service teachers' abilities to understand and appreciate students' thinking and to extend lesson ideas. Both teacher educators and their pre-service teachers gained perspectives on the role of a teacher's mathematical content knowledge. These results imply that multimedia case studies of practice can serve as vehicles for revealing the knowledge and practice of teacher educators, as they engage in supporting the professional development of pre-service teachers.     

Finding out more about teacher candidates' prior knowledge: implications for teacher educators

While it is recognised that prior knowledge is a key factor in determining future learning, its influence on learning to teach is less well known. This study investigates two cohorts of teacher candidates studying for a one-year, graduate qualification for primary teaching, who completed two tasks at entry to their initial teacher education programme: a task in writing and a task in mathematics. The tasks focused on teacher candidates' ability to recognise the key features of a piece of student work. The teacher candidates' responses to the tasks highlighted the diverse nature of the prior knowledge that underpinned their responses. The study raises questions about the pedagogy of initial teacher education, particularly in relation to the assumptions teachers educators make about the candidates they teach. The findings suggest that the prior knowledge that students bring to initial teacher education is both a resource and a challenge for teacher educators.     

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.     

A study of participatory action research as professional development for educators in areas of educational disadvantage

The main aim of this article is to explore the ways in which teacher educators can improve levels of reflection in postgraduate student teachers. The author argues that postgraduate student teachers are able to reflect on their practice in schools, and that the insights gained are useful in clarifying their own beliefs about teaching and learning. The outcome of such reflection at an early stage of their career may have implications for their ability to engage with the development opportunities made available to them as they move through their teaching careers. Using a qualitative approach, data from the learning journals and reflective writing assignments of student teachers, and a focus group interview, are examined in order to evaluate students' ability to engage in reflective writing at various stages over a 1 year course. The support given to students over the course of the year is examined with a view to evaluating its impact on students' abilities to reflect on their practice and their own learning. A number of recommendations are made for possible improvements to supporting student teachers in this important area of work.     

Learning as boundary-crossing in school–university partnership

This paper points out that globalization has raised fundamental questions about knowing and learning and that it is essential for educators to engage in collective knowledge generation by crossing community boundaries. Drawing on the theoretical framework of Activity Theory, this paper reports on a study on the expansive learning that was afforded by a school–university partnership as university tutors, mentor teachers and student teachers engaged in a new activity system mediated by lesson study. The study showed that in the course of resolving contradictions that were inherent in the boundary zone, they negotiated the mediating tool and consequently, the activity system was transformed from helping student teachers learn to teach into learning for all participants. This paper concludes that it is essential for teachers and teacher educators to develop the capability to engage in expansive learning through tackling ill-defined problems in boundary zones.     

Systematically Improving Lessons in Teacher Education: What's Good for Prospective Teachers is Good for Teacher Educators

Researchers have advocated building a knowledge base for teacher education based on practitioner knowledge. In this article, the authors, two teacher educators, present knowledge gained from their systematic study of two lessons to help prospective teachers learn to study teaching. They offer a detailed look at the lessons' successes and shortcomings as well as potential revisions. They also reflect on their use of a model for systematically studying lessons, sharing details about their learning during this process. This article attempts to provide a visible, tangible product for other teacher educators interested in helping prospective teachers examine evidence of student learning.     

Characterizing pivotal teaching moments in beginning mathematics teachers’ practice

Although skilled mathematics teachers and teacher educators often “know” when interruptions in the flow of a lesson provide an opportunity to modify instruction to improve students’ mathematical understanding, others, particularly novice teachers, often fail to recognize or act on such moments. These pivotal teaching moments (PTMs), however, are key to instruction that builds on student thinking about mathematics. Video of beginning secondary school mathematics teachers’ instruction was analyzed to identify and characterize PTMs in mathematics lessons and to examine the relationships among the PTMs, the teachers’ decisions in response to them, and the likely impacts on student learning. These data were used to develop a preliminary framework for helping teachers learn to identify and respond to PTMs that occur during their instruction. The results of this exploratory study highlight the importance of teacher education preparing teachers to (a) understand the mathematical terrain their students are traversing, (b) notice high-leverage student mathematical thinking, and (c) productively act on that thinking. This preparation would improve beginning teachers’ abilities to act in ways that would increase their students’ mathematical understanding.     

Secondary mathematics cooperating teachers’ perceptions of the purpose of student teaching

This article reports on the results of a survey of 45 secondary mathematics cooperating teachers’ perceptions of the primary purposes of student teaching and their roles in accomplishing those purposes. The most common purposes were interacting with an experienced, practising teacher, having a real classroom experience, and experiencing and learning about classroom management. The most common roles were providing the space for experience, modeling, facilitating reflection, and sharing knowledge. The findings provided insights into the cooperating teachers’ perceptions about both what should be learned through student teaching and how it should be learned. These findings paint a picture of cooperating teachers who do not see themselves as teacher educators—teachers of student teachers. Implications for mathematics teacher educators are discussed.     

Is It DAP? American Preschool Teachers’ Views on the Developmental Appropriateness of a Preschool Math Lesson from China

Planned instruction is an important teaching form in early childhood mathematics education, but some educators are concerned about its developmental appropriateness. The present study was undertaken to address this issue. One hundred and seven American preschool teachers participated in the study: They watched a video-taped math lesson that was conducted in a 4-year-olds’ classroom in China and answered video-cued questions in a questionnaire. A mix of qualitative and quantitative methods was used in analyzing the data. Research Findings: Results show that on average, participants rated the quality and developmental appropriateness of the lesson as high. They acknowledged children’s active learning, teacher’s careful preparation, and use of different learning methods. But, they also identified inadequate elements such as lack of peer interactions and lesson differentiation. Practice or Policy: A relief of American preschool teachers’ current beliefs regarding the developmental appropriateness of math lessons provides critical reference for professional preparation/development programs. American preschool teachers’ high value of children’s active learning in a planned math lesson and their appreciation of teacher’s skillful guidance imply an imperative need to promote American preschool teachers’ pedagogical knowledge and skills in teaching mathematics to young children.     

Teachers as Designers of Learning Environments

Abstract

This study examined how six Singapore teachers approached the design and implementation of a unit of work (topic) to demonstrate exemplary classroom practices that engage learners and use ICT in knowledge-generative rather than presentational activities. After a reflection and feedback session on the first lesson observation involving the researcher and the teacher, the teacher redesigned the lesson to enhance ICT use and involve students more actively in their learning. Our study revealed that there is a difference between students' physical engagement and cognitive engagement in a task and that the teacher, as a designer of the learning environment, needs to make explicit the cognitive processes involved in using the tool to ensure students' effective use of ICT. The teachers' understanding of what constitutes effective learning and their roles in students' learning determine how they design the learning environment. In essence, it is the teacher's skill in managing the “tripartite” partnership of IT tool, learning task, and teacher support that brings about higher levels of student engagement.     

The Lesson Preparation Method: a way of investigating pre-service teachers’ pedagogical content knowledge

Teacher education should build on the knowledge base that pre-service teachers already have on account of their lifelong learning in and outside of school. An international team of mathematics and science teacher educators developed a method of investigating the pedagogical content knowledge parts of this knowledge base. They asked pre-service teachers to prepare a lesson about a topic, e.g. area, as if they had to teach that lesson the next day. In this article the main aspects of the research environment are described and their effects on the research outcomes are evaluated.     

Using the Nuthall lens on learning to sharpen perceptions of teacher education practice

In this article I consider what teacher educators might learn about our own teaching practice from Nuthall's work on learning in elementary school classrooms. Using key ideas from Nuthall's theory of student learning applied to data from two prior studies of my own teacher education practice, I illustrate how his work might provide a productive way for teacher educators to analyze and critique the tasks they use to promote the learning of prospective teachers.     

The Dialectic Relationship between Research and Practice in Mathematics Teacher Education

This paper addresses issues linking research in student teacher learning with reflection on practice in mathematics teacher education. From a situated perspective on learning and practice, we explore our own practice as teacher educators while researching student teacher learning in our classrooms. We describe a study on student teacher learning, considering student teacher learning as a “process of becoming”, and how the results of this research have affected our development as mathematics teacher educators and members of a community of inquiry. Our work shows how in the mathematics teacher education context the relationship between theory and practice becomes an element of both teacher educator and researcher development.     

Revisiting Reflection: Utilizing Third Spaces in Teacher Education

Much has been written about the importance of reflective practice. What is missing is reflective work on the part of teacher educators to address the mismatch between university-based methods courses and the realities of classroom life. With examples from a third grade mathematics classroom as well as a university-based mathematics methods course, this article explores ways educators can employ third space theory as a way to engage in purposeful reflection into their teaching practices.     

跨界课例研究中的教师学习

目前我国中小学教师的学习大都停留在被专家灌输知识或同行低水平交流的状态,如何创设跨界学习情境,引发教师对自己思维和行为习惯的深度反思,形成新的对教育教学的理解和行动,是教师学习的一大难题。鉴于此,有必要从跨界学习的视角,探索教师在跨界课例研究中其学习是如何发生的。基于一个小组合作学习案例,分析了教师如何通过与外来学者合作,根据具体情境对理论进行再工具化,改进教案设计和教学实施,在提高学生学习素养的同时生成了自己新的实践性知识。研究发现,教师跨界学习的意义协商、视角再造、实践重构三个机制,对教师专业学习具有重要的指导意义。     

Learning environment,self-efficacy for teaching mathematics,and beliefs about mathematics

Learning environment research has a long history of significant relationships between the learning environment and student outcomes. This study investigated relationships between the learning environments of mathematics courses in a teacher education program and two outcomes, mathematics teaching self-efficacy and beliefs about mathematics. These two outcomes have been repeatedly shown to influence the future teaching practices of preservice teachers but, to date, their relationships with learning environment have been investigated neither with preservice teachers nor at all in the United Arab Emirates or wider Gulf region. The learning environment was found to be significantly related to preservice teachers’ mathematics teaching self-efficacy and beliefs about mathematics. Learning environments perceived more favourably by preservice teachers were associated with higher self-efficacy for teaching mathematics, but also with more-traditional beliefs, making these findings important for higher education institutes and teacher educators.

     

Developing prospective elementary teachers’ abilities to identify evidence of student mathematical achievement

This study investigated the effects of a classroom intervention on prospective elementary teachers’ ability to evaluate evidence of student achievement of mathematical learning goals. The intervention was informed by a framework for teacher education which aims to provide prospective teachers (PTs) with the skills needed to systematically learn from their own teaching practice. Prospective teachers (N = 160) participated in an intervention aimed at addressing their misconceptions about evidence of student learning. Results revealed that after the intervention, PTs were less likely to consider teacher behaviors to be evidence of student learning and more likely to discount student responses that were irrelevant to a specified learning goal. However, PTs were still likely to take procedural fluency as evidence of conceptual understanding and may have become overly skeptical of student understanding. Implications of the study suggest new ways of developing prospective teachers’ ability to systematically study and improve their teaching.