Teacher knowledge: a complex tapestry

Teachers need to know a great deal, in many areas and in multiple ways. Teacher knowledge is a complex tapestry, and teachers must successfully weave the multiple threads. In this article, I present a conceptualisation of teacher knowledge that provides a framework for describing the complexity of teacher knowledge. The framework describes three ways of knowing: “knowing how,” “knowing why,” and “knowing what” and then applies these three knowledge discourses across six domains of teacher knowledge. The framework was developed from a study of 14 teachers in their first year of teaching, and in this article the framework is applied to their experiences to illustrate specific gaps in their teacher knowledge. It is proposed that this conceptualisation of teacher knowledge allows those involved in teacher education and induction to more clearly identify professional learning needs and develop their programmes with specificity.     

美国《中学数学教与学》教材的特点及其对中国数学教师教育的启示

美国《中学数学教与学》是初中年级数学教师教育"数学教学法"课程的教材,其特点表现为:单元主题以数学教学理论为线索呈现,单元内容以具体数学知识为载体展开并且以参与式活动的形式组织教学活动.该教材对我国的数学教师教育有如下启示:数学教师教育课程应以数学知识为载体学习数学教育理论;数学教师教育教材应突出教学方式的示范性,体现做中学,在数学教学活动中学习数学教学方法;数学教师教育应注重教师学科专业素质的提升.     

Pre-service teachers’ future-oriented mathematical identity work

Research on identity has been a growing domain in the contexts of teacher education and mathematics education; however, identity work has been explored to a much lesser extent, with a future orientation overlooked. In addition, earlier studies have not provided sufficient knowledge on how different elementary teacher education programs might facilitate pre-service teachers’ identity work. In this study, we compare future-oriented mathematical identity work through a narrative framework considering six pre-service teachers undergoing two different teacher education programs. All pre-service teachers reported having had negative experiences with mathematics during their school years. Based on the results we conclude that despite the striking similarities in pre-service teachers’ mathematical backgrounds, the ways in which these cases are conducting their identity work differ substantially. It seems that the main reasons for these differences are different emphases and pedagogical practices in mathematics education courses. Additionally, we further elaborate on our earlier conceptualisation of identity work.     

Promoting Metacognitive Decision-Making in Teacher Education

Effective teachers are characterized by their abilities to make thoughtful, deliberate, and informed adaptations while teaching (Hoffman & Pearson, 2000). These in-the-moment teaching decisions are guided by a complex web of teacher knowledge. Raising teachers' awareness of the decisions they make on a moment-by-moment basis may aid in thoughtful, more deliberate decision-making and more robust instruction. We charge that preservice teachers can acquire this skill, as well. This article highlights practices employed in preservice teacher education courses as a means to promote awareness of, and engagement with, in-the-moment teaching decisions. Using examples from mathematics, social studies, and literacy methods courses, we describe our processes to intentionally cultivate preservice teachers' adaptive teaching skills through metacognitive decision-making.     

Argumentation in Science Teacher Education: The simulated jury as a resource for teaching and learning

In this article, we focus on the contributions that a simulated jury-based activity might have for pre-service teachers, especially for their active participation and learning in teacher education. We observed a teacher educator using a series of simulated juries as teaching resources to help pre-service teachers develop their pedagogical knowledge and their argumentation abilities in a physics teacher methods course. For the purposes of this article, we have selected one simulated jury-based activity, comprising two opposed groups of pre-service teachers that presented aspects that hinder the teachers' development of professional knowledge (against group) and aspects that allow this development (favor group). After the groups’ presentations, a group of judges was formed to evaluate the discussion. We applied a multi-level method for discourse analysis and the results showed that (1) the simulated jury afforded the pre-service teachers to position themselves as active knowledge producers; (2) the teacher acted as ‘animator' of the pre-service teachers' actions, showing responsiveness to the emergence of circumstantial teaching and learning opportunities and (3) the simulated jury culminated in the judges’ identification of the pattern ‘concrete/obstacles–ideological/possibilities' in the groups’ responses, which was elaborated by the teacher for the whole class. Implications from this study include using simulated juries for teaching and learning and for the development of the pre-service teachers’ argumentative abilities. The potential of simulated juries to improve teaching and learning needs to be further explored in order to inform the uses and reflections of this resource in science education.     

The Impact of Reform: Challenges for Mathematics Teacher Preparation

The mathematics education community has been widely recognized as a leader in the standards-based reform movement. Despite the widespread interest and attention that reform documents such as the NCTM Standards have generated, what has yet to be fully understood is the impact of these types of reform documents on the preservice teacher preparation process. This paper examines the state of the standards-based reform effort in mathematics teacher education by presenting the results of a three-year study of six cohorts of secondary mathematics student teachers (n = 63). Examined in particular are the ways in which these student teachers' professed beliefs about and knowledge of the NCTM Standards contrast with their teaching practices. Possible explanations for the mismatch between belief and knowledge statements and teaching practices are examined, along with other implications for mathematics teacher preparation.     

Cognitive Scientists Prefer Theories and Testable Principles With Teeth

The visions of mathematics classrooms called for by current educational reform efforts pose great challenges for kindergarten through Grade 12 schools and teacher education programs. Although a number of colleges and universities throughout the country are making changes in their teacher education programs to reflect these reform recommendations, we have little systematic information on the nature of these programs or their impact on prospective teachers. These issues are of central concern in the study-Learning to Teach Secondary Mathematics in Two Reform-Based Teacher Education Programs-that we draw on in this article. The article focuses on 1 preservice teacher's (Ms. Savant) knowledge, beliefs, and practices related to proof, tasks, and discourse. A situative perspective on cognition and components of teachers' professional knowledge frame our research. We examined data on Ms. Savant's experiences in her teacher education program to understand the influences of teacher education on her development as a mathematics teacher. This research indicates that Ms. Savant's teacher education experiences did make a difference in her development as a teacher. Her mathematics methods course provided a large collection of tasks, engaged her and her preservice colleagues in discourse, and provided her with both formal and informal experiences with proof-all of these experiences reflecting reform-based visions of mathematics classrooms. The situative perspective on cognition directed our attention to issues of compatibility of goals and visions across the various university and kindergarten through Grade 12 classroom settings, and it helped us to understand why some aspects of reform-based pedagogy are more easily learned than others: Why some ideas and practices learned as a student in the university setting are more easily transported to the novice teacher's kindergarten through Grade 12 field setting. We conclude that compatibility of these settings on several key dimensions is essential for the settings to reinforce each other's messages, and thus work in conjunction, rather than in opposition, to prepare reform-minded teachers.     

Future Directions in Research on Mathematics-Related Teacher Identity

Mathematics education research has placed great emphasis on teacher identity, examining both pre- and in-service teachers, and within these cohorts, specialised mathematics teachers and non-specialists such as elementary teachers. Extensive research has already been done; hence, this paper discusses possible future directions for research on teacher identity in mathematics education. Among other issues, we highlight that general education research on identity has infrequently informed research on mathematics-related teacher identity. This not only limits the transfer of knowledge but also isolates mathematics education from general education research. We suggest that connecting these lines of research and their findings may not only strengthen mathematics education research and mathematics teaching and learning but also contribute to less isolation within the discipline as a whole.     

Computer,craft, complexity,change: explorations into science teacher knowledge

In this article we review the notion of teacher knowledge from four metaphorical standpoints – teacher knowledge as computer, as craft, as complexity and as change. We employ each of these metaphors to examine and analyse the work of one teacher of science. We draw implications from our analysis for teacher education, teacher development and the study of teaching.     

构建学生良好数学言语思维空间的教学途径

根植于文化传统的数学教育要实现现代化，就必须实现数学知识的学术形态向教育形态的转化．为此，在数学教学中应构建良好的数学言语思维空间，加强学生数学言语能力的培养．在具体数学课堂教学中，可以通过数事问题具体化、头脑风暴和数学冒险等途径拓宽学生的数学言语思维空间。     

Mathematics teacher educator knowledge: What do we know and where to from here?

The knowledge that mathematics teacher educators need has attracted limited but increasing attention in recent years. The papers in this special issue build on emerging themes from that work and raise additional questions that contribute towards a future research agenda in the field. Several of the articles develop conceptualisations of teacher knowledge to apply to mathematics teacher educators, introducing new aspects and drawing attention to unique characteristics of mathematics teacher educators in the process. To some extent the questions these papers raise reflect similar questions in mathematics education more generally but for which the implications for mathematics teacher educators warrant attention. In this paper we review the articles in this special issue and use them as a springboard to examine contemporary developments in the field and posit ways forward for research on mathematics teacher educators’ knowledge.     

Sociological tools in the study of knowledge and practice in mathematics teacher education

In this paper, we put Basil Bernstein’s theory of pedagogic discourse to work together with additional theoretical resources to interrogate knowledge and practice in mathematics teacher education. We illustrate this methodology through analysis of an instance of mathematics teacher education pedagogic practice. While the methodology itself is our focus, the particular example provides a compelling story at the heart of which is the problem of integration of knowledge(s) within a pedagogic practice. Here, a constructivist pedagogy is at work, but differentially with respect to teaching/learning mathematics and teaching/learning mathematics teaching. The example illuminates mathematics and teaching, and their co-constitution in a particular pedagogic context.     

Special education pre-service teachers' interest,subject knowledge,and teacher efficacy beliefs in mathematics

Teacher efficacy beliefs is an important characteristic to predict instructional quality and the level of cognitive activation and educational support. Since teacher efficacy beliefs are context and domain specific, this study focuses on how special education pre-service teachers' individual interest and subject knowledge in mathematics predict their efficacy beliefs in teaching mathematics. Data were collected from 57 special education pre-service teachers. The results indicated that the individual interest of pre-service teachers has a strong effect on teacher efficacy beliefs, while subject knowledge has only an indirect effect.     

TELPS: A method for analysing mathematics pre-service teachers' Pedagogical Content Knowledge

A pre-service teacher's Pedagogical Content Knowledge (PCK) and their personal constructs of teaching develop throughout their teacher education program. PCK integrates generic pedagogical knowledge, mathematical teaching methodology and knowledge of the discipline of mathematics and this paper reports on a survey that can be used to assess a pre-service teacher's PCK. TELPS (Teacher Education Lesson Plan Survey) was developed to determine the PCK of pre-service teachers during their teacher education program. TELPS is shown to analyse pre-service teachers' PCK with some indication that the pre-service teacher's development of PCK can be observed.     

高中教师代数教学知识的调查研究

代数是中学数学课程的重要部分.在高等代数知识方面,教师能理解一些基础而重要的概念,并掌握公式算法.但理解涉及逻辑知识的一些符号表示的复合命题时有困难.在学校代数知识方面,大多数教师显示出扎实的代数基础知识与技能,特别是常规问题的解法明确,运算准确.高中数学教师具体较为扎实的学校代数与高等数学中有关代数的知识,但在理解学生的学习困难,错误概念等方面有待提高。     

Toward a framework for the development of mathematical knowledge for teaching

Shulman (1986, 1987) coined the term pedagogical content knowledge (PCK) to address what at that time had become increasingly evident—that content knowledge itself was not sufficient for teachers to be successful. Throughout the past two decades, researchers within the field of mathematics teacher education have been expanding the notion of PCK and developing more fine-grained conceptualizations of this knowledge for teaching mathematics. One such conceptualization that shows promise is mathematical knowledge for teaching—mathematical knowledge that is specifically useful in teaching mathematics. While mathematical knowledge for teaching has started to gain attention as an important concept in the mathematics teacher education research community, there is limited understanding of what it is, how one might recognize it, and how it might develop in the minds of teachers. In this article, we propose a framework for studying the development of mathematical knowledge for teaching that is grounded in research in both mathematics education and the learning sciences.

Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Understanding the knowledge bases of mathematics teachers is an important task in working towards the construction of adequate models for: (i) teacher education and development, and (ii) teacher operations in the classroom. To date, little systematic attention has been focused on this task. The primary aim of this study is to obtain a view from the field of mathematics teacher knowledge with respect to content knowledge in mathematics, content‐specific pedagogical knowledge in mathematics and curriculum knowledge relevant to teaching tasks. This study has used data obtained from a survey of primary teachers and secondary mathematics teachers. Analysis of the results has indicated that less than half of the teachers in the study believed that they were sufficiently prepared in mathematics content, and that almost two‐thirds of the teachers in the sample believed that their level of knowledge in contemporary teaching methodologies in mathematics is not sufficient for their role as school teachers. Key differences emerge between the primary and secondary sectors and also within the secondary sector. Implications for preservice and in‐service mathematics teacher education are drawn.     

Teacher Effectiveness Research: towards describing user‐friendly classroom instruction

In this article we examine the teacher's role as a mediator of knowledge and cognitive learning in the classroom. We cite findings from the research areas of teacher effectiveness, teacher expertise, and curriculum knowledge. These data strongly support user‐friendly explicit methods of classroom teaching. We also discuss issues raised in conjunction with alternative conceptions of the teaching process. Finally, we note the importance of incorporating teacher effectiveness research findings into teacher education programmes, and of identifying the various misconceptions that have been used to criticise this body of information.     

Terry's Learning: some limitations of Shulman's pedagogical content knowledge

This exploratory article seeks to map out some possible practical limitations of Shulman's notion of pedagogical content knowledge and the associated concept of pedagogical reasoning. Interview extracts are used to illustrate the argument that the views and prior knowledge of a trainee teacher of mathematics may subtly shape what is learned in the context of training. It is suggested that representations of mathematics and mathematics teaching, originating in the trainee's own learning, might be concomitant with the development of a particular style of pedagogical reasoning which then determines the type of pedagogical content knowledge that is acquired. Shulman's framework is useful for understanding the trainee's learning, but insufficient to cover a broader range of pedagogical knowledge that might be deemed desirable in the trained teacher. It is suggested that a wider framework, extending pedagogical content knowledge to incorporate alternative forms of teaching, is needed.     

Prospective teachers' initial conceptions about pupils' understanding of science and mathematics

Earlier reports have shown that prospective teachers' conceptions about teaching science to a high degree are resistant and do not change substantially during the teacher‐training programme. In our investigation we elucidate the prospective teachers' initial conceptions about pupils' understanding of science and mathematics. We applied ‘The Lesson Preparation Method' and used a phenomenographic approach in order to reveal the range of conceptions that the prospective teachers hold. A third of the prospective teachers did not consider pupils' conceptions when planning lessons. The rest of the 32 participants expressed awareness; some of the prospective teachers even referred to subject‐specific teaching experience. Also regarding the prospective teachers' conceptions about pupils' knowledge and beliefs, as well as about pupils' difficulties, there was a significant diversity. By raising these issues about pedagogical content knowledge the prospective teachers' conceptions can be extended and developed during the education.