Developing mathematics teacher knowledge: the paradidactic infrastructure of “open lesson” in Japan

In this paper, we first present a theoretical approach to study mathematics teacher knowledge and the conditions for developing it, which is firmly rooted in a systemic approach to didactic phenomena at large, namely the anthropological theory of the didactic. Then, a case of open lesson is presented and analysed, using this theoretical approach, to show how the format of open lesson contributes to the construction and diffusion of didactic knowledge in the community of mathematics teachers in Japan. The basic idea of this format is that teachers from other schools are invited to observe a class taught by a teacher then participate in a discussion session with him on the details of the lesson. For the case study, we analyse the lesson plan prepared for an open lesson, the observed lesson and the teachers’ discussion. As a result, an open lesson session has been described as a specific form of post-didactic practice related directly to an actual observed lesson, and aiming specifically at elaborating the theoretical aspects of teacher’s didactic practice in the lesson and beyond it.     

Learning from teaching: pre-service primary teachers’ perceived learning from engaging in formal Lesson Study

While there is abundant research reporting the impact of Lesson Study as a professional development framework for qualified teachers, its potential within initial teacher education remains relatively unexplored. We present a particular model of formal Lesson Study which has been experienced annually for a decade by Irish pre-service primary teachers. This study examines the perceived effects on learning of this Lesson Study model on a large sample of pre-service primary teachers (N?=?225) who worked in 45 different lesson study groups over a 10 year period (2008–2017). Data analysis reveals that participants’ perceptions of learning fell within two distinct themes namely ‘An awakening regarding teacher content knowledge’ and ‘Cornerstones of learner-centred practice’, each of which consisted of rich inter-connected sub-themes. As a result of engaging in Lesson Study, participants report an increased awareness of the importance of deep content knowledge in addition to adopting a learner-centred approach to their practice. Participants also believed that Lesson Study enhanced their professional knowledge and skills in the curricular area of focus (mathematics, in this case). While this study focuses on self-report data, these opinions are fundamental given research which proposes that interventions will fail if participants judge them to be ineffective.     

Teachers’ use of their mathematical knowledge for teaching in learning a mathematics learning trajectory

In the past decade, research on learning trajectories has progressed from an agenda for research on student learning to include an agenda for research on teaching. In this paper, we use a design experiment to examine a professional development program created to support teacher learning of one learning trajectory. Using a re-interpretation of Ball and colleagues’ mathematics knowledge for teaching framework, we conducted a retrospective analysis to examine how a purposefully selected sample of three teachers used their knowledge to participate in discussions of the professional learning tasks designed to support teacher learning of the trajectory. Findings indicate that professional learning tasks focusing on pedagogical content knowledge present in learning trajectories also allow for teacher learning of subject matter knowledge; this learning is mediated by teachers’ prior mathematics knowledge for teaching. We conclude with considerations for researchers and professional developers seeking to support teachers in learning about learning trajectories.     

Lesson study in prospective mathematics teacher education: didactic and paradidactic technology in the post-lesson reflection

This paper presents a detailed analysis of the post-lesson reflection, carried out in the context of eight cases of lesson study conducted by teams of Danish, lower secondary prospective teachers and their supervisors. The participants, representing different institutions, were all new to the lesson study format. Nevertheless, it is demonstrated how their interaction shape the development of discourse about mathematical learning. The anthropological theory of the didactic is employed as the theoretical approach to analyse the mathematical and primarily didactical praxeologies developed and discussed during the meetings. The study investigates what happens when lesson study, a well-established Japanese “system” for professional teacher development, is transposed to another educational and cultural context, with the aim of enhancing prospective teacher learning during the practicum of a teacher education programme. The findings highlight significant different positions in the discourse during the post-lesson reflection. Specific practice-related knowledge is developed, to the benefit of prospective teachers, educators and researchers alike. This kind of knowledge is of interest and concern to the whole profession of mathematics teachers and the analysis adds to our insight into the potential of lesson study in prospective education as a meeting place where pertinent actors contribute to the expansion and dissemination of shared professional knowledge.     

Engaging Prospective Elementary Teachers in Lesson Study

This paper discusses the ways in which teacher educators can effectively engage prospective teachers (PTs) in lesson study during mathematics methods courses. Evidence from this work suggests that engagement in lesson study provides PTs with opportunities to strengthen their knowledge about mathematics, students, and pedagogy. More specifically, the results demonstrated that lesson study helped PTs to engage in deep and thoughtful discussions about mathematics, reflect on their teaching, and make effective changes to their lessons that showed noticeable improvements in student learning. Recommendations from this work provide specific methods-course suggestions on the implementation of lesson study as a model for PTs' professional and practitioner development, including strategies and guiding questions that teacher educators can utilize to help deepen PTs' knowledge about mathematics, teaching, and student learning.     

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.     

Provoking contingent moments: Knowledge for ‘powerful teaching’ at the horizon

Background: Teacher knowledge continues to be a topic of debate in Australasia and in other parts of the world. There have been many attempts by mathematics educators and researchers to define the knowledge needed by teachers to teach mathematics effectively. A plethora of terms, such as mathematical content knowledge, pedagogical content knowledge, horizon content knowledge and specialised content knowledge, have been used to describe aspects of such knowledge.

Purpose: This paper proposes a model for teacher knowledge in mathematics that embraces and develops aspects of earlier models. It focuses on the notions of contingent knowledge and the connectedness of ‘big ideas’ of mathematics to enact what is described as ‘powerful teaching’. It involves the teacher’s ability to set up and provoke contingent moments to extend children’s mathematical horizons. The model proposed here considers the various cognitive and affective components and domains that teachers may require to enact ‘powerful teaching’. The intention is to validate the proposed model empirically during a future stage of research.

Sources of evidence: Contingency is described in Rowland’s Knowledge Quartet as the ability to respond to children’s questions, misconceptions and actions and to be able to deviate from a teaching plan as needed. The notion of ‘horizon content knowledge’ (Ball et al.) is a key aspect of the proposed model and has provoked a discussion in this article about students’ mathematical horizons and what these might comprise. Together with a deep mathematical content knowledge and a sensibility for students and their mathematical horizons, these ideas form the foundations of the proposed model.

Main argument: It follows that a deeper level of knowledge might enable a teacher to respond better and to plan and anticipate contingent moments. By taking this further and considering teacher knowledge as ‘dynamic’, this paper suggests that instead of responding to contingent events, ‘powerful teaching’ is about provoking contingent events. This necessarily requires a broad, connected content knowledge based on ‘big mathematical ideas’, a sound knowledge of pedagogies and an understanding of common misconceptions in order to be able to engineer contingent moments.

Conclusions: In order to place genuine problem-solving at the heart of learning, this paper argues for the idea of planning for contingent events, provoking them and ‘setting them up’. The proposed model attempts to represent that process. It is anticipated that the new model will become the framework for an empirical research project, as it undergoes a validation process involving a sample of primary teachers.     

Teachers’ professional development: a theoretical review

Background and purpose:?The article reviews studies that focus on the professional development of teachers after they have completed their basic teacher training. Teacher professional development is defined as teachers’ learning: how they learn to learn and how they apply their knowledge in practice to support pupils’ learning. The research question addressed in the article is: How do experienced teachers learn?

Main argument:?The review is framed by theories within the constructivist paradigm. From this perspective, knowledge is perceived as the construction of meaning and understanding within social interaction. The social surroundings are seen as decisive for how the individual learns and develops. It is argued that courses and lectures, or ‘times for telling’, and teachers’ development of a metacognitive attitude are decisive factors for teachers’ learning within a constructivist frame of reference.

Sources of evidence and method:?To attempt to answer the research question, a search was conducted of the subject of pedagogy in the ISI WEB of Science (search undertaken 9 August 2011) using the search strings ‘teacher learning’, ‘teacher development’ and ‘teacher professional development’, and covering the period from 2009 to 2011 to probe the most recent decade of research. Articles that dealt with basic education, primary and secondary school, were selected, and articles that dealt with learning using digital tools and the internet and newly trained teachers were rejected. A set of 31 articles was selected from this search. To ensure width and depth of coverage, this was supplemented by a selection of review studies and research on further education in respect of teachers’ learning. The texts were analysed by means of open and axial coding, developing main and sub-categories.

Conclusions:?The review of articles shows that both individual and organisational factors impact teachers’ learning. Teacher co-operation has importance for how they develop, and some of the teachers can lead such learning activities themselves. Moreover, a positive school culture with a good atmosphere and understanding of teachers’ learning, in addition to co-operation with external resource persons, may impact the professional development of teachers. The article concludes with the reflection that learning in school is the best arena for further development of teachers.     

A CASE STUDY OF THE IMPLEMENTATION OF CHINESE KINDERGARTEN MATHEMATICS CURRICULUM

In 2001, the Chinese Ministry of Education issued Guidelines for Preschool Education (GPE) (trial version) to call on early childhood practitioners to use a child-centered and play-based approach to teaching and learning. The guidelines also include mathematics within the science domain and described its standards in a way that significantly lessens the teacher’s role as knowledge disseminator. Instead, the guidelines encourage mathematics learning and teaching through play, self-discovery, and problem solving. This qualitative case study examined one kindergarten teacher’s implementation of and associated challenges in delivering mathematics lessons from a newly developed integrated provincial curriculum (PC) and the quality of instruction as influenced by using a curriculum that is organized around thematic units. The analysis of the curriculum, ten videotaped lessons, two teacher interviews, and other supporting documents, such as student work and teacher lesson plans, indicated that the teacher encountered difficulties implementing the PC as intended due to a number of factors, including inadequate curricular resources, incoherency of the curriculum design, and a mismatch between the mathematics standards and curriculum goals and the teacher’s instructional style and teaching philosophy. As a result, the quality of mathematics instruction was poor, especially in the areas of (1) using effective teaching strategies, (2) providing clear and accurate instruction with a positive attitude, (3) engaging students in sustained interactions with mathematical ideas, and (4) offering on-going assessment and feedback. To close, recommendations for curriculum refinement and professional development are provided.     

Supporting children’s mathematical understanding: professional development focused on out-of-school practices

This study describes the Reflection Connection Cycle professional development designed to support teachers’ use and appreciation of students’ out-of-school practices related to school mathematics. The year-long program incorporated group lesson design, readings, and video analysis for 14 elementary school (ages 5–12) teachers. Analysis of lesson development, written reflections, and analysis of teacher talk revealed important patterns related to the difficulty in writing lessons that built on students’ informal understandings. While initial lessons focused solely on the context of practices like gardening and sports, subsequent lessons show a greater concern for the mathematics in which children were engaged within a practice. A Multi-approach Engagement Framework is presented both as a tool to support further professional development efforts and as a means to describe stability and change in teachers’ efforts to connect in-school and out-of-school mathematical understandings.     

Professional knowledge or motivation? Investigating the role of teachers’ expertise on the quality of technology-enhanced lesson plans

In an expertise study with 94 mathematics teachers varying in their relative teacher expertise (i.e., student teachers, trainee teachers, in-service teachers), we examined effects of teachers' professional knowledge and motivational beliefs on their ability to integrate technology within a lesson plan scenario. Therefore, we assessed teachers' professional knowledge (i.e., content knowledge, pedagogical content knowledge, technological knowledge), and their motivational beliefs (i.e., self-efficacy, utility-value). Furthermore, teachers were asked to develop a lesson plan for introducing the Pythagorean theorem to secondary students. Lesson plans by advanced teachers (i.e., trainee teachers, in-service teachers) comprised higher levels of instructional quality and technology exploitation than the ones of novice teachers (i.e., pre-service teachers). The effect of expertise was mediated by teachers' perceived utility-value of educational technology, but not by their professional knowledge. These findings suggest that teachers’ motivational beliefs play a crucial role for effectively applying technology in mathematics instruction.     

Beyond ‘what works’: understanding teacher identity as a practical and political tool

Drawing on previous research that focused upon the formation and mediation of teacher professional identity, this paper develops a model for conceptualising teacher professional identity. Increasingly, technical-rational understandings of teachers’ work and ‘role’ are privileged in policy and public discourse over more nuanced and holistic approaches that seek to understand teacher professional identity – what it is to ‘be’ a teacher. This article seeks to offer an alternative view, presenting the idea that an understanding of the processes by which teacher professional identity is formed and mediated is central to understanding the professional learning and development needs of teachers and advancing a richer, more transformative vision for education. I argue that instrumentalist notions of teachers’ work embedded in neo-liberal educational agendas such as those currently being advanced in many western countries offer an impoverished view of the teaching profession and education more broadly, and suggest that the concept of teacher professional identity holds the potential to work as a practical tool for the teaching profession and those who work to support them in the development of a more generative educational agenda.     

Promoting teacher learning: a framework for evaluating the educative features of mathematics curriculum materials

Educative curricula, curriculum materials that intentionally foster teacher professional development, can serve as a site for teacher learning through their use in daily instructional practices. The present article introduces a framework, Teacher Learning Opportunities in Mathematics Curriculum Materials (TLO-Math), for designing and evaluating mathematics curriculum materials’ educative features according to seven theoretically based variables: (1) mathematics content knowledge for teaching, (2) teacher knowledge of student thinking in mathematics, (3) teacher knowledge of disciplinary discourse in mathematics, (4) teacher knowledge of assessment in mathematics, (5) teacher knowledge of differentiated instruction in mathematics, (6) teacher knowledge of technology use in mathematics, and (7) teacher knowledge of mathematical community. Each variable is illustrated with a definition, guiding questions, discipline-specific literature, and examples from two sets of elementary mathematics curriculum materials. The development of the TLO-Math framework is a critical first step for further study of the use of mathematics curriculum materials as sites for teacher learning.     

Teachers’ Journeys: A Case of Teachers of Learners Aged Five to Six

South Africa’s esteem is low with regard to current learner achievement in mathematics. Results from international studies, such as TIMMS and SACMEQ, continuously indicate that South African learners perform below international benchmarks in mathematics. Research and evaluation studies assert that teacher practices and poor early mathematics stimulation influence this state of affairs. Hence, this article reports on a study that explored the developmental progression of Grade R teachers’ mathematics knowledge and general knowledge of teaching participating in a professional development intervention. In this article I report on data collected after a two-day workshop conducted with the teachers. The video recordings of 14 teachers’ self-selected lessons were collected and transcribed, using a lesson observation tool, and analysed by means of thematic analysis. The findings indicated that teachers’ knowledge of number sense forms a strong foundation for improving the quality instruction of number knowledge. The first step seen to be taken by these teachers to engage the learners was the question and answer approach. The lessons of those who tried to let learners engage lacked structure and resulted in chaos. Demonstration and drill were used to develop the learners’ content knowledge. The teachers’ practices were inadequate as far as developing conceptual understanding is concerned; and their classroom management skills were not sufficiently developed to include all learners in their mediation. In instances where all the learners participated, the teacher directed the participation and did not allow for learner-to-learner engagement. It is crucial to note that they showed potential for growth and willingness to learn despite the limited or minimal the training they received.     

Educational policy or practice? Traversing the conceptual divide between subject knowledge,pedagogy and teacher identity in England

This research paper is framed by concerns about recent UK Government policy regarding the training of mathematics and science teachers in England and discusses how two cohorts of pre-service teachers negotiated the development of a professional identity while undertaking subject-specific training. The data reported upon were garnered in two ways; through an evaluation survey that received quantitative and qualitative responses from 159 teacher trainees and through focus groups conducted with 40 trainees. In the paper, the authors take the concept of ‘participation in communities of practice’ as a departure point to explore how trainees demonstrate their development of professional identities as chemistry, maths or physics teachers. In the conclusion, the authors consider the implications of the findings for pre-service teachers and teacher trainers given the current education climate of financial austerity being experienced across Europe.     

在高职数学教学中关于操作性教学的研究报告

数学作为高职院校理工科重要的一门基础课,既为专业基础课和专业课提供了必要的数学知识和数学方法,又在培养具有良好数学素养的各类创新型人才、应用技术型人才方面起着特别重要的作用。针对高职学生的特点,展开关于＂在高职数学教学进行操作性教学＂的教学尝试,能够在有限的教学时间里,结合学生的个性特质,培养学生既能运用理论、定义等来解决问题,又能利用数学软件来解决数学问题。     

Finding Common Ground: A Synthesis of Science and Mathematics Teacher Educators’ Experiences with Professional Growth

The seven articles that comprise this Special Issue examine the professional growth of mathematics and science teacher educators across different contexts and different foci of who is the teacher educator being studied. Despite these differences, a common thread running throughout these seven articles is the need for learning to be situated in collaboration with others. In this final article, we examine the contribution of these articles through two perspectives: that of the collaborative contexts supporting the professional growth of mathematics and science teacher educators, and the role of disciplinary knowledge as part of the purpose for teacher educators’ professional growth. We notice that collaboration can take on very different structures in supporting teacher educators’ professional learning due to the different purposes and roles of the teacher educators in the studies. We also notice that while collaboration figures as an important component in all of the studies, the disciplinary specific aspects of collaboration, i.e., how collaboration might be negotiated differently by teacher educators in mathematics and science, is still not well understood. Overall, these articles provide important insights that help to shed new light on the complex and multifaceted nature of teacher educators’ learning and growth and provide productive avenues for future research.