Knowing and teaching fractions: A cross-cultural study of American and Chinese mathematics teachers

Guided by Shulman, 1986 and Shulman, 1987 tripartate model of teacher expertise (subject matter knowledge (SMK), pedagogical content knowledge (PCK), and general pedagogical knowledge (GPK)), the present study examined 162 U.S. and Chinese 3rd grade mathematics teachers’ expertise in teaching fractions. Results show that U.S. teachers lag significantly behind Chinese teachers in SMK (concepts, computations, and word problems) and in some areas of PCK (e.g., such as identifying important points of teaching the fraction concepts and how to ensure students’ understanding). Although the Chinese teachers performed poorly in comparison to their U.S. counterparts on a test designed to measure GPK (e.g., psychological and educational theories and applications), its poor reliability calls these results into question. The implications of this study are discussed in terms of practical ideas for teacher training in the U.S. and China.     

Pedagogical Content Knowledge in Social Studies

Abstract: Gudmundsdottir, S. & Shulman, L. 1987. Pedagogical Content Knowledge in Social Studies. Scandinavian Journal of Educational Research 31, 59‐70. The role of teacher's pedagogical content knowledge in social studies is addressed through two case studies: a novice and a veteran teacher. We demonstrate that the important difference between the novice and the expert is manifested in a special kind of knowledge that is neither content nor pedagogy per se. It rests instead in pedagogical content knowledge, a form of teacher understanding that combines content, pedagogy and learner characteristics in a unique way.     

Structure and development of pre-service physics teachers’ professional knowledge

ABSTRACT

Teachers’ professional knowledge is considered one of the most important predictors of instructional quality. According to Shulman, such professional knowledge includes content, pedagogical content and pedagogical knowledge. Although recent research shed some light on the structure of the dimensions of professional knowledge, little is known how teacher education impacts pre-service physics teachers’ professional knowledge. In an effort to address this issue, we examined the content, pedagogical content and pedagogical knowledge of N?=?200 pre-service physics teachers enrolled in different years of teacher education at 12 major teacher education universities in Germany. We used structural equation modelling (1) to examine the relations amongst pre-service physics teachers’ content, pedagogical content and pedagogical knowledge, (2) to explore how the three kinds of knowledge and their relations differ across different stages of teacher education and (3) to identify factors affecting the level of each component of professional knowledge. Our findings suggest that content, pedagogical content and pedagogical knowledge represent distinct types of knowledge. Furthermore, our findings show that in the first years of professional education, pedagogical content knowledge is more closely related with general pedagogical knowledge while in later years, it is more closely related with content knowledge, suggesting that it develops from a general knowledge about teaching and learning into knowledge about the teaching and learning of specific content. Finally, beyond school achievement and years of enrolment as predictors, we find in particular the amount of classroom observations to have a positive impact on the professional knowledge of pre-service physics teachers.     

Probing the amalgam: the relationship between science teachers’ content,pedagogical and pedagogical content knowledge

ABSTRACT

This Special Issue aims to present evidence about the relationships between content knowledge (CK), pedagogical knowledge (PK) and pedagogical content knowledge (PCK); the development of these types of knowledge in novice and experienced secondary science teachers; and how CK, PK and/or PCK impact students’ learning. Since Shulman’s introduction of PCK as the feature that distinguishes the teacher from the content expert, researchers have attempted to understand, delineate, assess and/or develop the construct in pre- and in-service teachers. Accordingly, empirical findings are presented that permit further discussion. Outcomes permit post-hoc examination of a recent, collectively described, ‘consensus’ model of PCK, identifying strengths and potential issues. As we will illustrate, the relationship between CK, PK and PCK is central to this; that is, probing the hypothesis of pedagogical content knowledge as an ‘amalgam’ of content and pedagogical knowledge.     

Learning to Read and Learning to Teach Reading

The National Curriculum for Initial Teacher Education in English is specific and detailed about the knowledge expected of primary teachers. Shulman (1987) argued that teachers transform this sort of subject content knowledge into something accessible and meaningful to their pupils and this knowledge is described as ‘pedagogic content knowledge’. Medwell et al. (1998) found that effective literacy teachers only knew literacy in the way that they taught it. The research project underpinning this article aimed to explore student teachers' conceptions of the teaching of reading in order to find out what they thought they were teaching when they taught reading. It was thought that the personal reading histories of the students would impact on their developing conceptions of teaching reading. This article traces one student, Gordon, through the year of his PGCE course. In the form of dialogue between Gordon and the researcher developing understanding is articulated. Three different types of reading are described: decoding, making meaning and engaging. Reading is seen as a transformative process, where the reader is both within and outside the text. This has implications both for the conception of reading contained within the curriculum and the way it is implemented within the classroom. A teacher can only introduce children to experiences and ways of reading that are known to herself. It is argued, therefore, that student teachers need to extend the boundaries of their own reading and so appreciate the wide range of ways in which meaning is constructed and readers are created.     

Experienced Middle School Science Teachers’ Assessment Literacy: Investigating Knowledge of Students’ Conceptions in Genetics and Ways to Shape Instruction

Using a framework of assessment literacy that included teachers?? view of learning, knowledge of assessment tools, and knowledge of assessment interpretation and action taking, this study explored the assessment literacy of five experienced middle school teachers. Multiple sources of data were: teachers?? predictions about students?? ideas, students?? written and verbal responses to assessment tasks, teacher background questionnaire, and a videotaped teacher focus group. We investigated middle school teachers?? predictions, interpretations, and recommended actions for formative assessment in genetics. Results documented a variety of ways that teachers would elicit students?? ideas in genetics, focusing on discussion strategies. Findings showed how well teachers predicted student conceptions compared to actual student conceptions. We also found that teachers mostly described general topics they would use to address students?? alternative conceptions. Less often, they explained specific content they would use to challenge ideas or pedagogical strategies for conceptual change. Teachers also discussed barriers to addressing ideas. Teacher professional development should provide more support in helping teachers close the formative assessment cycle by addressing conceptions that are elicited with assessments.     

教师学科教学知识:本质、特征与结构

教师学科教学知识(Pedagogical Content Knowledge)是教师研究的新焦点。自这一概念提出以来(Shulman,1986),其内涵与结构不断演变。教师PCK是教师关于如何将自己所知道的学科内容以学生易理解的方式加工、转化给学生的知识。它具有与内容相关;基于经验反思;有实践性;个体性以及情境性的特征。结构上,研究者基于Shulman(1986)的两个主要成分予以扩展;它是一个综合体;与其他教师知识类型紧密关联。值得注意的是,特定教师群体的特定主题PCK是日后研究的生长点。     

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.     

Strategies for mentoring pedagogical knowledge

Fundamental for mentoring a preservice teacher is the mentor’s articulation of pedagogical knowledge, which in this research draws upon specific practices, viz.: planning, timetabling lessons, preparation, teaching strategies, content knowledge, problem solving, questioning, classroom management, implementation, assessment and viewpoints for teaching. Mentoring is haphazard; consequently, mentors need a pedagogical knowledge framework and a repertoire of pedagogical knowledge strategies to guide a preservice teacher’s development. Yet, what are strategies for mentoring pedagogical knowledge practices? This qualitative research investigates mentoring strategies assigned to pedagogical knowledge from 27 experienced mentor teachers. Findings showed that there were multiple strategies that can be linked to specific pedagogical knowledge practices. For example, mentoring strategies associated with planning for teaching can include co-planning, verbally reflecting on planning with the mentee and showing examples of the mentor teacher’s planning (e.g. teacher’s plans, school plans, district and state plans). This article provides a bank of practical strategies for mentoring pedagogical knowledge practices to assist a preservice teacher’s development.     

Understanding the Theoretical Framework of Technological Pedagogical Content Knowledge: A collaborative self-study to understand teaching practice and aspects of knowledge

This paper describes a self-study research project that focused on our experiences when planning, teaching, and evaluating a course in initial teacher education. The theoretical framework of technological pedagogical content knowledge (TPACK) was used as a conceptual structure for the self-study. Our understanding of the framework in relation to our teaching practice was in focus. The principal educational goal of the course was to develop the pedagogical use of web 2.0 resources to support learning in the preschool/school context. As a result, the focus, content, form of distribution, teaching, and assessment of the course went beyond what is common in initial teacher training in Sweden. The potential of the different digital tools was explored by situated use in the design and teaching of the course. Analysis highlights the challenges and opportunities that teacher educators and student teachers may encounter while working with, and learning about, information and communication technologies to support learning. Some of the findings discussed are related to the identified challenges and opportunities for both teachers and students to integrate content knowledge, pedagogical knowledge, and technological knowledge into a TPACK. Taken-for-granted organizational and institutional assumptions about teaching, learning, and assessment in teacher education were identified in the study.     

Emotional ecology: The intersection of emotional knowledge and pedagogical content knowledge in teaching

The purpose of this paper is to offer some theoretical as well as empirical examples that describe the interrelations between pedagogical content knowledge (PCK) and emotional knowledge in teaching and learning. The argument put forward is that there is a need to expand current conceptions of PCK and acknowledge the role of emotional knowledge. It is shown how a teacher's emotional knowledge about teaching and learning is an inextricable part of the ecosystem of teacher knowledge; this is called emotional ecology. Our research shows that the construct of emotional ecology occurs on different planes as there are different types of emotional knowledge that are aspects of PCK. The implications for pre-service and in-service teacher development are discussed.     

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.

From Standards to Licensure: Developing an Authentic Assessment for School Principals

The Interstate School Leaders Licensure Consortium funded the development of a licensure assessment based on its Standards for School Leaders. But moving from a set of professional standards to a licensure assessment poses many challenges and raises difficult questions: Which standards should be assessed? How authentic should an authentic assessment be? How can licensure assessment help lead the field to a new pedagogical paradigm, when licensure has typically been reserved for protecting the public? This article explains these challenges and documents the methods used to meet them in the development of the School Leaders Licensure Assessment.     

Developing teachers’ pedagogical practice in teaching science lessons with mobile phones

This paper presents the findings of an investigation carried out in Sri Lanka to explore how mobile phones can support science teachers’ pedagogical practices throughout the teaching cycle of planning, teaching and evaluation. Data were collected using observation supported by audio and video recordings from both continuing professional development workshops where a group of teachers planned four science lessons that integrated mobile phones and the subsequent implementation of those lessons. Thematic network analysis, carried out on the data with the help of NVivo8 qualitative analysis software, showed how mobile phones were used and the issues involved. It concluded that mobile phones can support science teaching in a variety of ways, in particular with communication during planning lessons, with relating subject knowledge to authentic locations and activities during teaching and with image and data capture to support assessment and post-lesson reflection. The additional pedagogical practices and knowledge base required when integrating mobile phones into science lessons are discussed in the light of Shulman’s original model.     

信息技术支持下的学科教学知识:缘起、演化、结构模型及其教育意蕴

信息技术支持下的学科教学知识(TPACK)是学科教学知识(PCK)在信息化时代的发展。技术知识(TK)、教法知识(PK)和内容知识(CK)是TPACK结构模型中的基础成分;学科教学知识(PCK)、技术支持下的内容知识(TCK)、技术支持下的教学知识(TPK)和技术支持下的学科教学知识(TPACK)是TPACK结构模型中的互动成份。TPACK框架在理论上、在教师教育、在教学实践上等诸方面为信息技术与学科课程整合提供了新思路,将有助于我们推动学科教学知识发展,将为学科教育的信息化提供有力的教学知识基础。     

The development and alignment of pedagogical conceptions of health education

This study examined the development and alignment of conceptions of health education as a subject, and of its teaching and learning, among Finnish health education student teachers (n = 20). Longitudinal phenomenographic data (essays, interviews) were collected at two time points during health education studies, and at one time point after the participants had gained 1–3 years of work experience. The proportion of participants expressing the most sophisticated pedagogical conceptions decreased during teacher training, but increased after the gaining of work experience. Moreover, fewer than half of the participants expressed pedagogical conceptions that advanced in broad alignment with regard to the subject, the teaching, and the learning. Some methodological considerations and suggestions for teacher training are presented.     

On the alignment of teachers’ mathematical content knowledge assessments with the common core state standards

Instruments designed to measure teachers’ knowledge for teaching mathematics have been widely used to evaluate the impact of professional development and to investigate the role of teachers’ knowledge in teaching and student learning. These instruments assess a mixture of content knowledge and pedagogical content knowledge. However, little attention has been given to the content alignment between such instruments and curricular standards, particularly in regard to how content knowledge and pedagogical content knowledge items are distributed across mathematical topics. This article provides content maps for two widely used teacher assessment instruments in the USA relative to the widely adopted Common Core State Standards. This common reference enables comparisons of content alignment both between the instruments and between parallel forms within each instrument. The findings indicate that only a small number of items on both instruments are designed to capture teachers’ pedagogical content knowledge and that the majority of these items are focused on curricular topics in the later grades rather than in the early grades. Furthermore, some forms designed for use as pre- and post-assessment of professional development or teacher education are not parallel in terms of curricular topics, so estimates of teachers’ knowledge growth based on these forms may not mean what users assume. The implications of these findings for teacher educators and researchers who use teacher knowledge instruments are discussed.

     

The ability to understand and use conceptual change pedagogy as a function of prior content learning experience

This research examined the relationship between content instruction and the development of elementary teacher candidates' understanding of conceptual change pedagogy. Undergraduate students (n = 27) enrolled in two sections of a science methods course received content instruction through either traditional or conceptual change methods, followed by instruction about conceptual change pedagogy. Candidates were interviewed pre- and postinstruction about their content and pedagogical knowledge and also wrote conceptual change lessons. Twelve of the 27 subjects were videotaped teaching in the field. Results indicate that prior to instruction, most candidates had weak content knowledge and held traditional pedagogical conceptions. After instruction, students in the conceptual change group had significantly larger gains in their content knowledge than those in the traditional group, gave qualitatively stronger pedagogical responses, and used conceptual change strategies more consistently in practice. These results indicate that personal experience of learning science content through conceptual change methods facilitated the development of understanding and use of conceptual change pedagogy in teaching practice. Thus if conceptual change methods are to be incorporated into teacher candidates' repertoire, science content courses that students take prior to teacher education should be taught using conceptual change pedagogy. In addition, courses in science education should use pedagogy more in line with that taught in methods courses.     

What Do Teacher Mentors Tell Student Teachers About Pupil Learning in Infant Schools?

Abstract

Recent emphases on experiential classroom learning in the induction into expertise of student teachers may over‐estimate the learning support that class based practitioners offer students. Current understandings of teachers' thinking and planning suggest that expert practitioners may not be best placed to provide student teachers with frameworks for understanding what Shulman (1987) categorises as general pedagogical knowledge. Data from a sample of 11 UK infant school teacher mentors in the form of 5 hours of taped planning and evaluatory conversations with 20 students have been content analysed. Teacher‐talk related to explanations of how pupils learn forms the focus of the paper. Implications for mentor training and school‐university partnerships are drawn from the analysis of the data. Finally the need to confront the language difference that obtains between schools and universities in order to develop the theoretical base of teacher education is asserted.     

TEACHERS’ KNOWLEDGE OF THE CONCEPT OF A FUNCTION: A THEORETICAL FRAMEWORK

In order to teach mathematics effectively, mathematics teachers need to have a sound mathematical knowledge, but what constitutes sound mathematical knowledge for teaching is subject to debate. This paper is an attempt to unpack what constitutes teacher knowledge of the concept of a function which is a unifying idea in the mathematics curriculum. The central components of the framework, which will be elaborated on in this paper, are: teachers’ subject matter knowledge, teachers’ pedagogical content knowledge, teachers’ technological pedagogical knowledge, technological content knowledge, and technological pedagogical content knowledge in relation to the concept of a function. The framework is informed by Shulman’s (Educational Researcher 15:4–14, 1986) Types of Teachers Knowledge Framework, Ball, Bass &; Hill 29:14–17, 20–22, 43–46 (2005) Mathematical Knowledge for Teaching Framework, and Mishra &; Koehler’s (Teachers College Record 108:1017–1054, 2006) Technological Pedagogical Content Knowledge (TPACK) framework.