Early childhood teachers' mathematical content knowledge

The process of becoming numerate begins in the early years. According to Vygotskian theory (1978), teachers are More Knowledgeable Others who provide and support learning experiences that influence children’s mathematical learning. This paper reports on research that investigates three early childhood teachers mathematics content knowledge. An exploratory, single case study utilised data collected from interviews, and email correspondence to investigate the teachers’ mathematics content knowledge. The data was reviewed according to three analytical strategies: content analysis, pattern matching, and comparative analysis. Findings indicated there was variation in teachers’ content knowledge across the five mathematical strands and that teachers might not demonstrate the depth of content knowledge that is expected of four year specially trained early years’ teachers. A significant factor that appeared to influence these teachers’ content knowledge was their teaching experience. Therefore, an avenue for future research is the investigation of factors that influence teachers’ content numeracy knowledge.

Objectifying early-number : a visual nomenclature to express mathematical domain knowledge

To address issues of divisive ideologies in the Mathematics Education community and to subsequently advance educational practice, an alternative theoretical framework and operational model is proposed which represents a consilience of constructivist learning theories whilst acknowledging the objective but improvable nature of domain knowledge. Based upon Popper’s three-world model of knowledge, the proposed theory supports the differentiation and explicit modelling of both shared domain knowledge and idiosyncratic personal understanding using a visual nomenclature. The visual nomenclature embodies Piaget’s notion of reflective abstraction and so may support an individual’s experience-based transformation of personal understanding with regards to shared domain knowledge. Using the operational model and visual nomenclature, seminal literature regarding early-number counting and addition was analysed and described. Exemplars of the resultant visual artefacts demonstrate the proposed theory’s viability as a tool with which to characterise the reflective abstraction-based organisation of a domain’s shared knowledge. Utilising such a description of knowledge, future research needs to consider the refinement of the operational model and visual nomenclature to include the analysis, description and scaffolded transformation of personal understanding. A detailed model of knowledge and understanding may then underpin the future development of educational software tools such as computer-mediated teaching and learning environments.

Future directions and perspectives for problem solving research and curriculum development

Since the 1960s, numerous studies on problem solving have revealed the complexity of the domain and the difficulty in translating research findings into practice. The literature suggests that the impact of problem solving research on the mathematics curriculum has been limited. Furthermore, our accumulation of knowledge on the teaching of problem solving is lagging. In this first discussion paper we initially present a sketch of 50 years of research on mathematical problem solving. We then consider some factors that have held back problem solving research over the past decades and offer some directions for how we might advance the field. We stress the urgent need to take into account the nature of problem solving in various arenas of today’s world and to accordingly modernize our perspectives on the teaching and learning of problem solving and of mathematical content through problem solving. Substantive theory development is also long overdue—we show how new perspectives on the development of problem solving expertise can contribute to theory development in guiding the design of worthwhile learning activities. In particular, we explore a models and modeling perspective as an alternative to existing views on problem solving.

Leading learning within a PLC : implementing new mathematics content

This paper does two things. Firstly, it examines the literature that coalesces around theoretical models of teacher professional development (PD) within a professional learning community (PLC). Secondly, these models are used to analyse support provided to two year 3 teachers, while implementing the draft Queensland mathematics syllabus. The findings from this study suggest that the development of this small PLC extended the teachers’ Zone of Enactment which in turn led to teacher action and reflection. This was demonstrated by the teachers leading their own learning as well as that of their students.

Will they know enough? Pre-service primary teachers' knowledge base for teaching intergrated social sciences

A significant issue in primary teacher education is developing a knowledge base which prepares teachers to teach in a range of subject areas. In Australia, the problem in primary social science education is compounded by the integrated nature of the key learning area of Studies of Society and Environment (SOSE). Recent debates on teaching integrated social sciences omit discussions on the knowledge base for teaching. In this paper, a case study approach is used to investigate primary pre-service teachers’ approaches to developing a knowledge base in designing a SOSE curriculum unit. Data from five teacher-educators who taught primary SOSE curriculum indicates that novice teachers’ subject content knowledge, as revealed through their curriculum planning, lacked a disciplinary basis. However, understanding of inquiry learning, which is fundamental to social science education, was much stronger. This paper identifies a gap in the scholarship on teaching integrated social science and illustrates the need to support and develop primary teachers’ disciplinary knowledge in teacher education.

Curriculum design innovation in flexible science teaching

In this paper you will be introduced to a number of principles which can be used to inform good teaching practice and rigorous curriculum design. Principles relate to: * Application of a common sequence of events for how learners learn; * Accommodating different learning styles; * Adopting a purposeful approach to teaching and learning; * Using assessment as a central driving force in the curriculum and as an organising structure leading to coherence of teaching and learning approach; and * The increasing emphasis that is being placed on the development of generic graduate competencies over and above discipline content knowledge. The principles are particularly significant in relation to adult learning. The paper will use three specific applications as illustrations to help you to learn how these principles can be applied. The illustrations are taken from a second year subject in supercomputing that uses scientific case studies. The subject has been developed (with support from Silicon Graphics Inc. and Intel) to be taught entirely via the Internet.

Students as decoders of graphics in mathematics

This paper reports on students’ ability to decode mathematical graphics. The findings were: (a) some items showed an insignificant improvement over time; (b) success involves identifying critical perceptual elements in the graphic and incorporating these elements into a solution strategy; and (c) the optimal strategy capitalises on how information is encoded in the graphic. Implications include a need for teachers to be proactive in supporting students’ to develop their graphical knowledge and an awareness that knowledge varies substantially across students.

A visual description of early-number counting, addition and subtraction knowledge

Early-number is a rich fabric of interconnected ideas that is often misunderstood and thus taught in ways that do not lead to rich understanding. In this presentation, a visual language is used to describe the organisation of this domain of knowledge. This visual language is based upon Piaget’s notion of reflective abstraction (Dubinsky, 1991; Piaget, 1977/2001), and thus captures the epistemological associations that link the problems, concepts and representations of the domain. The constructs of this visual language are introduced and then applied to the early-number domain. The introduction to this visual language may prompt reflection upon its suitability and significance to the description of other domains of knowledge. Through such a process of analysis and description, the visual language may serve as a scaffold for enhancing pedagogical content knowledge and thus ultimately improve learning outcomes.

Facilitating growth in prospective teachers’ knowledge : teaching geometry in primary schools

This paper reports on a study that focused on growth of understanding about teaching geometry by a group of prospective teachers engaged in lesson plan study within a computer-supported collaborative learning (CSCL) environment. Participation in the activity was found to facilitate considerable growth in the participants’ pedagogical-content knowledge (PCK). Factors that influenced growth in PCK included the nature of the lesson planning task, the cognitive scaffolds inserted into the CSCL virtual space, the meta-language scaffolds provided to the participants, and the provision of both private and public discourse spaces. The paper concludes with recommendations for enhancing effective knowledge-building discourse about mathematics PCK within prospective teacher education CSCL environments.

Education for sustainability : a case study of pre-service primary teachers' knowledge and efficacy

This study investigated the relationships between knowledge and efficacy for teaching sustainability in a sample of 266 pre-service primary teachers at a large, metropolitan university in Australia. A survey gathered information about the participant’s attitudes and self-efficacy for education for sustainability, along with their perceived and actual knowledge of environmental sustainability issues. The participants typically believed they were confident in their abilities to engage with education for sustainability with self-efficacy increasing with increased levels of perceived knowledge. However no relationship was found between perceived knowledge and actual knowledge which suggests that the participants either do not feel constrained by their lack of knowledge, or are perhaps unaware of their actual knowledge of sustainability issues. This lack of relationship may have implications for the development of pedagogical content knowledge with pre-service teachers potentially developing shallow, tokenistic approaches to Education for Sustainability.

Reshaping children’s knowledge through film making: What to film, how to film and why

How can teachers reinvigorate content area knowledge and representation through filmmaking? We give examples of what to film, how to film, and why, drawing on our visual ethnographic research with Year 5 students in a working class suburb of Logan, Queensland. The unit developed content knowledge of Indigenous places and practices through sensitising activities in nature. Valuing students’ funds of knowledge, we interpreted local places through epistemologies of different cultures. Through filmmaking workshops by a digital artist, students filmed community members in a local shopping mall about their perceptions of health and happiness in local places. Students were positioned as future community leaders, presenting their films at a national conference. To conclude, we map the dominant and marginalised, local and specialised, and print and visual forms of knowledge that were interwoven, reshaped, and shared through multimodal design.

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 paper provides a bank of practical strategies for mentoring pedagogical knowledge practices to assist a preservice teacher’s development.