Facebook as a mechanism for informal teacher professional learning in Indonesia

This study investigated the use of Facebook as a medium for, and process of, teachers’ informal professional learning regarding mathematical and pedagogical knowledge. Facebook responses to four inter-related posts about mathematics examples from Facebook users (N = 117) were captured and systematically analysed using content analysis. The data were examined through the lens of the five characteristics of the Effective Professional Development framework. The results identified different types of responses to the four Facebook posts including the opening of learning opportunities, rich mathematical and pedagogical discussions, respondents’ appreciations and direct impacts of the posts, as well as the acknowledgement of the misalignment between the proposed ideas and current teaching practices in Indonesia. These responses provided evidence that all five characteristics of the framework were apparent and highlighted the role of Facebook in supporting teacher informal professional learning within the Indonesian context. The limitations of such social networking-based research and implications for teacher professional development are provided.     

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.     

A discipline apart: the challenges of ‘Fostering a Community of Learners’ in a mathematics classroom

This is the fifth in a series that examines the challenges that teachers in different domains face as they attempt to implement the pedagogical reform ‘Fostering a Community of Learners’ (FCL). Here we focus on the relationship between FCL and the teaching of mathematics. We argue that it is possible to teach mathematics through the FCL pedagogy, but that doing so requires some rethinking of both mathematics instruction and FCL. In particular, we describe three shifts that aided a teacher's implementation of FCL pedagogy with mathematics: the teacher developed a new perspective on mathematics that emphasized the importance of having students learn both mathematical concepts and processes; the teacher developed a new understanding of the role of the teacher in mathematics‐education reform; and the teacher modified his understanding of FCL, coming to believe that a discourse community could be the basis for FCL pedagogy in a mathematics classroom.     

Teaching teachers to teach Boris: a framework for mathematics teacher educator pedagogical content knowledge

The notion of pedagogical content knowledge (PCK) was posited in the context of school teaching and the knowledge used by teachers teaching school students. It has been examined for a number of discipline areas, notably mathematics. There are, however, other teaching contexts, including those of teacher educators, whose students are pre-service teachers (PSTs). The content these teacher educators teach is not subject discipline knowledge (or not solely), but the PCK for teaching a subject discipline. What knowledge do teacher educators use as they teach PCK? This paper presents a framework for the PCK required of mathematics teacher educators as they work to develop PSTs’ PCK for teaching mathematics. The framework builds on existing research into PCK and categorises aspects of the work of teacher education. The framework’s usefulness is examined by studying the PCK used by the first author in building PSTs’ understanding of mathematics teacher PCK.     

傅种孙——中国现代数学教育的先驱

傅种孙先生(1898-1962)是一位博学多才、刚正不阿,接受过西方科学民主思想熏陶的儒家学者.他是第一个膊将数理逻辑引入中国的数学家,是第一个将西方的数学基础研究引入中国的数学家,是为中国的数学教育和普及奉献了一生的才智与心血的伟大的数学教育家.作为中国现代数学教育的先驱,傅种孙先生可谓呕心沥血,鞠躬尽瘁.他先后参与了教材的编写,课程标准的制订,致力于教师教育,并担任<数学通报>总编.傅种孙先生的数学教育思想将在中国永存.     

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.     

A Mathematics Teachers’ Perspective and its Relationship to Practice

In this paper we try to characterize the pedagogical approaches that mathematics teachers are developing to meet the challenges posed by education reforms. A key aspect is the identification of the perspectives that underlie those pedagogical approaches, using the term perspective to include a broad pedagogical structure composed of multiple conceptions that are related to some aspects of a teacher’s practice. Through the study of the practice of a secondary mathematics teacher, we try to explore how his/her pedagogical approaches on mathematics, mathematics learning, and mathematics teaching are related to the relational architecture that is established in the classroom during the development of an instructional unit of similarity at a secondary school level, and we examine if that relationship can be explained in terms of the underlying perspective. The results of the study have shown the characteristics of that relationship, and the important role that the teacher’s knowledge of the students’ difficulties plays both in making decisions and in developing the teachers’ actions.     

Teaching units as the integrating core of mathematics education

How to integrate mathematics, psychology, pedagogy and practical teaching within the didactics of mathematics in order to get unified specific theories and conceptions of mathematics teaching? This problem—relevant for theoretical and empirical studies in mathematics education as well as for teacher training—is considered in the present paper. The author suggests an approach which is based on teaching units (Unterrichtsbeispiele). Suitable teaching units incorporate mathematical, pedagogical, psychological and practical aspects in a natural way and therefore they are a unique tool for integration.     

Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency

In an innovative, progressive school, students were asked to solve a fairly routine mathematical problem using real money in a “real-world” scenario. Even though the school values students’ ideas, the reaction of the teacher to one student’s alternative modelling of the problem suggests that he was expecting a particular answer to be provided using routine mathematical models and thinking while not being interested in exploring the student’s unexpected alternative. We place his reasoning for doing so within broad pedagogical discourses that we think define the “allowable” responses of teachers and students in ways that inhibit meaning-making for both. These broad discourses are defined as the progressive constructivist approach, the scaffolding discursive approach, the situation modelling approach and the dialogic approach. We consider the advantages and the potential consequences each might bring to the case. We suggest that extensive consideration of pedagogical discourses in mathematics classes must be reconsidered both for how we understand students’ mathematical meaning-making and how we construct student agency in relationship to culture, whether as apprentices or authors.     

Being a Mathematics Teacher Educator in China: Challenges and Strategic Responses

In this exploratory study, we developed a portrait of the challenges and strategic responses of secondary mathematics teacher educators (MTEs) in Chinese universities. The MTEs reported encountering more challenges when teaching pedagogical courses and supervising student teachers than when teaching college mathematics courses and teaching mathematical problem-solving courses. This finding reflects a key difference between the content, goals, and teaching demands of these 2 elements of mathematics teacher education programs (acting in the role of an MTE versus the role of a mathematics teacher). In this study, we also analyzed the strategies that MTEs use to deal with the challenges that arise in their work and the suggestions they have for the training of future MTEs.     

Designing and implementing authentic investigative proportional reasoning tasks: the impact on pre-service mathematics teachers’ content and pedagogical knowledge and attitudes

In this study we created, implemented, and evaluated the impact of proportional reasoning authentic investigative tasks on the mathematical content and pedagogical knowledge and attitudes of pre-service elementary and middle school mathematics teachers. For this purpose, a special teaching model was developed, implemented, and tested as part of the pre-service mathematics teacher education programs conducted in Israeli teacher colleges. The conclusion of the study is that application of the model, through which the pre-service teachers gain experience of and are exposed to authentic investigative proportional reasoning tasks with incorporation of theory (reading and analyzing relevant research reports) and practice, leads to a significant positive change in the pre-service teachers’ mathematical content and pedagogical knowledge. In addition, improvement occurred in their attitudes and beliefs towards learning and teaching mathematics in general, and ratio and proportion in particular.     

Levels of use of Interactive Whiteboard technology in the primary mathematics classroom

Despite the availability of Interactive Whiteboard (IWB) technology in a large number of Australian primary schools, many teachers focus only on technical issues as opposed to pedagogical engagement in an attempt to incorporate the technology. Previous research suggests that the technology is being used for sophisticated transmission-style teaching as opposed to constructivist approaches. This article presents findings of a project that considered the implementation of IWB technology in three Victorian primary mathematics classrooms (5 to 12 years of age). The study analysed the teaching strategies adopted by three teachers as they embarked on the use of IWB technology as an integral component of mathematical activities with the support of professional development. Teacher use of IWB technology in the primary mathematics classroom was aligned against Beauchamp’s generic transitional framework for viewing the development of teacher use of IWB technology. Through this alignment, a transitional framework emerged which is specific to the introduction of IWB technology in the mathematics classroom.     

数学教学的三种水平及其理论分析

数学教学分为基本型教学、智慧型教学、创新型教学三个层次,三个层次分别依托数学教师的基本知识结构、学科教学能力(PCA)、创新能力。三个层次的教学即三种水平的教学,表现出自身相对独立的个性和特点,在教学理念、教学操作等方面存在差异。从基本型教学走向智慧型教学再走向创新型教学,是数学教学的应然追求。     

Becoming a Teacher Educator: A Self-Study of the Use of Inquiry in a Mathematics Methods Course

This article details the self-study of a beginning teacher educator in her first experience in teaching a mathematics methods course. The transition from teacher to teacher educator is explored through the experience of a course focused on inquiry. Inquiry is embedded within the course from two perspectives: mathematical inquiry and teaching as inquiry. The framework of teaching as inquiry is an important part of both the self-study and the mathematics methods course. Implications of the transition from teacher to teacher educator as well as the role of inquiry in mathematics teacher education are discussed.     

Special education and subject teachers’ self‐perceived readiness to teach mathematics to low‐performing middle school students

Educational reforms during the last decade have led to a more inclusive environment for students with different needs and have placed demands on teachers’ readiness to instruct diverse students in the general classroom. Previous research has ascertained that student achievement is correlated with teacher quality and teachers’ efficacy beliefs. Today, basic competence in mathematics is more important than ever for managing routine day‐to‐day activities and therefore, identifying and educationally supporting students with low performance in mathematics is necessary. The aim of the study was to investigate the perceived teacher efficacy beliefs of special education and mathematics teachers when teaching mathematics to low‐performing middle school students. Results indicated that special education teachers had higher teacher efficacy beliefs than mathematics teachers. Teacher experience, certification or gender had no effect on teachers’ efficacy beliefs. Furthermore, subject teachers reported high pedagogical knowledge for teaching low‐performing students, and special education teachers reported having moderate mathematical subject knowledge.     

An Ode to Imre Lakatos: Quasi-Thought Experiments to Bridge the Ideal and Actual Mathematics Classrooms

This paper explores the wide range of mathematics content and processes that arise in the secondary classroom via the use of unusual counting problems. A universal pedagogical goal of mathematics teachers is to convey a sense of unity among seemingly diverse topics within mathematics. Such a goal can be accomplished if we could conduct classroom discourse that conveys the Lakatosian (thought-experimental) view of mathematics as that of continual conjecture-proof-refutation which involves rich mathematizing experiences. I present a pathway towards this pedagogical goal by presenting student insights into an unusual counting problem and by using these outcomes to construct ideal mathematical possibilities (content and process) for discourse. In particular, I re-construct the quasi-empirical approaches of six!4-year old students’ attempts to solve this unusual counting problem and present the possibilities for mathematizing during classroom discourse in the imaginative spirit of Imre Lakatos. The pedagogical implications for the teaching and learning of mathematics in the secondary classroom and in mathematics teacher education are discussed.     

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.     

Collaborative Effort to Develop Middle School Preservice Teachers’ Mathematical Knowledge

Abstract

This article describes reflections of two mathematicians and a mathematics teacher educator who collaborated on the development and implementation of courses (probability and statistics connections, number concept connections, and middle school mathematics methods) for middle school mathematics preservice teachers. The instructors of the courses, two in mathematics and one in mathematics education, worked together to more explicitly link course materials, assignments, and the pedagogical approaches. Collectively, the courses were designed to address the five components of preservice teachers’ mathematical knowledge for teaching (PT-MKT), and to model effective teaching practices. Using their collective experiences co-planning and implementing these course adjustments were made in the subsequent year. The instructors were pleased by their implementation and student outcomes in all three courses.

We describe how each component of the PT-MKT framework was approached in these courses and discuss challenges experienced by the instructors, who were part of a larger effort to develop and implement a middle school teacher preparation program. The information shared is based on data collected as part of a program evaluation effort, and is bolstered by the instructors’ recollection of events. Overall, the instructors enhanced the curricula and their instructional practices and found that the attention placed on developing PT-MKT support the mathematical development of middle school mathematics preservice teachers.