Investigating the development of prospective mathematics teachers’ pedagogical content knowledge of generalising number patterns through school practicum

The purpose of this study is to investigate the development of three prospective elementary mathematics teachers?? pedagogical content knowledge (PCK) of number patterns with its two components: (1) knowledge of students?? understanding and difficulties and (2) knowledge of topic-specific strategies and representations. For our investigation, we use Radford??s (ZDM Math Educ 40:83?C96, 2008) framework called ??architecture of algebraic pattern generalisation?? to redefine these two components of PCK in the context of patterns. A case study was conducted to explore the development of PCK through a school practicum course and to examine how observation of number pattern lessons in schools and discussions of these observation contribute to prospective teachers?? PCK. Analysis of the data indicated that observations and discussions of number pattern lessons resulted in considerable change in the way prospective teachers take into account students?? understanding of and difficulties with patterns, as well as in the way they use pattern-specific strategies such as algebraic pattern generalisation during their lessons.     

The nature and development of middle school mathematics teachers’ knowledge

In this article, we report on the use of a teacher profiling instrument with 62 middle school teachers at the start of a 3-year professional learning programme. The instrument was designed to assess the aspects of teachers’ knowledge identified by Shulman (1987) refined by Ball et al. (2008) and extended to include teachers’ confidence to use and teach various topics in the middle school mathematics curriculum and their beliefs about mathematics teaching and learning. Based on a hierarchical coding of items, the application of the partial credit Rasch model revealed that the profile items were measuring a single underlying construct and suggested that the various facets of teacher knowledge develop together. We describe the characteristics of four levels of the hierarchical construct measuring teacher knowledge and understanding for teaching mathematics in the middle years of schooling, and discuss the unique affordances of a holistic view of teacher knowledge in contrast to considerations of multiple knowledge categories.     

Geography pre-service teachers’ pedagogical content knowledge

This paper reports on an exploratory study that investigated the pedagogical content knowledge (PCK) of four Geography pre-service teachers as they completed their final practicum of their Bachelor of Secondary Education programme. It uses Shulman’s (1986, 1987) two categories of PCK as a conceptual framework to identify the participants’ PCK of their Geography teaching. The first type of PCK is teachers’ representations of content that is pedagogically powerful for students; the second type of PCK refers to teachers’ understanding of what makes the learning of specific topics easy or difficult for students. Historically, researchers have tended to focus on the teachers’ representations component of PCK while little evidence-based research has been reported on the second category of PCK. Findings from this study show that all participants demonstrated both types of PCK. This study sheds light on the importance of teacher-educators modelling both types of PCK as both are equally important for pre-service teachers.     

Supporting teachers’ technological pedagogical content knowledge of fractions through co-designing a virtual manipulative

This study explores the impact that co-designing a virtual manipulative, Fractions Lab, had on teachers’ professional development. Tapping into an existing community of practice of mathematics specialist teachers, the study identifies how a cooperative enquiry approach utilising workshops and school-based visits challenged 23 competent primary school teachers’ technological, pedagogical and content knowledge of fraction equivalence, addition and subtraction. Verbal and written data from the workshops alongside observations and interviews from the school visits were analysed using the technological, pedagogical and content knowledge (TPACK) framework. The findings show that the assumptions of even experienced teachers were challenged when Fractions Lab was shared as an artefact on which they were asked to co-design and subsequently interact with, using it in subsequent phases of the cooperative inquiry process. Two original aspects of successful co-design of virtual manipulatives through communities of practice are identified and offered to others: (1) careful bootstrapping of the first design iteration that gathers intelligence about the content area and the technological affordances and constraints available; and (2) involvement of highly motivated teachers who perceive themselves as agents of change in the domain area.     

Developing biology teachers’ pedagogical content knowledge through learning study: the case of teaching human evolution

This work explores how pedagogical content knowledge (PCK) on evolution was modified by two biology teachers who participated in a professional development programme (PDP) that included a subsequent follow-up in the classroom. The PDP spanned a semester and included activities such as content updates, collaborative lesson planning, and the presentation of planned lessons. In the follow-up part, the lessons were videotaped and analysed, identifying strategies, activities, and conditions based on student learning about the theory of evolution. Data were collected in the first round with an interview before the training process, identifying these teachers’ initial content representation (CoRe) for evolution. Then, a group interview was conducted after the lessons, and, finally, an interview of stimulated recall with each teacher was conducted regarding the subject taught to allow teachers to reflect on their practice (final CoRe). This information was analysed by the teachers and the researchers, reflecting on the components of the PCK, possible changes, and the rationale behind their actions. The results show that teachers changed their beliefs and knowledge about the best methods and strategies to teach evolution, and about students’ learning obstacles and misconceptions on evolution. They realised how a review of their own practices promotes this transformation.     

The Lesson Preparation Method: a way of investigating pre-service teachers’ pedagogical content knowledge

Teacher education should build on the knowledge base that pre-service teachers already have on account of their lifelong learning in and outside of school. An international team of mathematics and science teacher educators developed a method of investigating the pedagogical content knowledge parts of this knowledge base. They asked pre-service teachers to prepare a lesson about a topic, e.g. area, as if they had to teach that lesson the next day. In this article the main aspects of the research environment are described and their effects on the research outcomes are evaluated.     

Pre-service secondary teachers’ pedagogical content knowledge for the teaching of mathematical modelling

Educational Studies in Mathematics - The article deals with the pedagogical content knowledge of mathematical modelling as part of the professional competence of pre-service teachers. With the help...     

Prospective secondary mathematics teachers’ pedagogical knowledge for teaching the estimation of length measurements

Prospective secondary mathematics teachers’ pedagogical knowledge for teaching the estimation of length measurements was investigated by examining their personal benchmarks for measurement estimation. Benchmarks for measurement estimation are the meaningful representations of units that serve to increase one’s understanding of measurement and one’s ability to estimate measurements. Data included electronic journal responses, observation and verbal data, and work samples. Thematic analysis revealed that prospective teachers possessed various benchmarks for measurement estimation that enabled them to estimate length measurements, but these benchmarks for measurement estimation were not evident in participants’ pedagogical knowledge for teaching the estimation of length measurements. Participants’ pedagogical knowledge for teaching the estimation of length measurements was instead based on the belief that hands-on activities were the only way to teach the estimation of length measurements.     

Development of a performance assessment task and rubric to measure prospective secondary school mathematics teachers’ pedagogical content knowledge and skills

The purpose of this paper is to share a performance assessment task and rubric designed to assess secondary school mathematics preservice teachers’ pedagogical content knowledge and skills. The assessment task and rubric were developed in collaboration with five education faculty, four arts and sciences faculty, and four high school teachers over a period of one year. Over this period, the group met more than 15 times and designed an assessment task and rubric to measure secondary school preservice teachers’ pedagogical content knowledge and skills. The assessment task and rubric were piloted at the end of the year and extensively revised using the pilot data. The redesigned task and rubric were successfully carried out in the second and subsequent years. Other teacher education programs can benefit from this assessment of teacher candidates’ pedagogical content knowledge and skills.     

Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge

This study examines the influence of reading and planning from two differently organized mathematics textbooks on prospective high school mathematics teachers’ pedagogical content knowledge and content knowledge of exponential functions. The teachers completed a pretest and two posttests. On the pretest, the teachers possessed an incomplete understanding of content and pedagogical content knowledge related to exponential functions. The teachers’ understanding of how to translate from table to closed-form and recursive equations grew as a result of their use of the Mathematics: Modeling Our World textbook, while the Discovering Algebra textbook appeared to be more beneficial in terms of pedagogical content knowledge. Teachers read from the student lessons in both textbooks, but read differently from the sections of both textbooks intended for the teacher. They focused more on the purpose of the Mathematics: Modeling Our World lesson and more on the places where students might experience difficulties in the margins of the Discovering Algebra lesson. The teachers’ learning was influenced by their own personal characteristics (e.g., previous textbook experiences) as well as textbook qualities (e.g., organization).     

Stimulating pre-service teachers’ content and pedagogical content knowledge on rational numbers

This study explored the perspectives of mathematics teacher educator-researchers (MTE-Rs) on the use of theory in facilitating teacher growth. We adopted the framework of the theory-centered scholarship triangle (Silver & Herbst, 2007) to probe the underlying meanings of perspectives on theory use expressed by MTE-Rs. Qualitative analysis revealed three distinct types of perspectives: a perspective focusing on research, a perspective focusing on practice, and a perspective on the connection between research and practice. For the perspective on the connection between research and practice, two sub-categories were identified: connection with consideration of context and that without consideration of context. Different perspectives that MTE-Rs possess may influence their actions taken in professional development and consequently influence teachers’ learning of theory. Specifically, the cyclic process of decontextualizing and recontextualizing theory between research and practice domains is the key to the development of MTE-Rs’ educative power so that they can better facilitate teachers in learning theory.     

Student teachers’ pedagogical content knowledge for teaching systems thinking: effects of different interventions

ABSTRACT

Systems’ thinking has become increasingly relevant not only in education for sustainable development but also in everyday life. Even if teachers know the dynamics and complexity of living systems in biology and geography, they might not be able to effectively explain it to students. Teachers need an understanding of systems and their behaviour (content knowledge), and they also need to know how systems thinking can be fostered in students (pedagogical content knowledge (PCK)). But the effective development of teachers’ professional knowledge in teaching systems thinking is empirically uncertain. From a larger study (SysThema) that investigated teaching systems thinking, this article reports the effects of the three different interventions (technical course, didactic course and mixed course) in student teachers’ PCK for teaching systems thinking. The results show that student teachers’ PCK for teaching systems thinking can be promoted in teacher education. The conclusion to be drawn from our findings is that a technically orientated course without didactical aspects seems to be less effective in fostering student teachers’ PCK for teaching systems thinking. The results inform educators in enhancing curricula of future academic track and non-academic track teacher education.     

Validation and profile of Chinese pre-service teachers’ technological pedagogical content knowledge scale

Researchers state that teachers in different contexts reported different technological pedagogical content knowledge (TPACK). This phenomenon may partially be explained by cultural differences. Based on this consideration, the development and validation of the Chinese pre-service teachers’ technological pedagogical content knowledge (CTPCK) scale is described in this article. The sample was split into two subsamples on random basis (n1 = 229, n2 = 207) for the purpose of conducting (1) exploratory factor analysis (EFA) and (2) confirmatory factor analysis (CFA), respectively. After the EFA, the CTPCK scale excluded six items and included eight factors with 42 items. Reliability and correlations were discussed. The findings revealed that the CTPCK scale was a valid and reliable instrument for measuring the TPACK of Chinese pre-service teachers’ knowledge with or without linking educational technology.     

Examining the impact of educational technology courses on pre-service teachers’ development of technological pedagogical content knowledge

The purpose of this qualitative study was to examine the impact of educational technology courses on pre-service teachers’ development of knowledge of technology integration in a teacher preparation program in the USA. The present study was conducted with eight pre-service teachers enrolled in the elementary teacher education program at a large university in the mid-western USA. Data sources included interviews, documents, and observations. The findings identified knowledge of technology integration the pre-service teachers developed and identified knowledge of technology integration needed in the technology integration courses. The present study provided a number of suggestions on different activities that could be built into educational technology courses to better prepare pre-service teachers to teach with technology.     

The building of pre-service primary teachers’ knowledge of mathematics teaching: interaction and online video case studies

This study explores how preservice primary teachers became engaged in meaning-making mathematics teaching when participating in online discussions within learning environments integrating video-clips of mathematics teaching. We identified different modes of participation in the online discussions and different levels of knowledge-building. The data can be explained by the type of task set in the learning environments and by a gradually improved shared understanding of the teaching. The structure of the learning environments supported inquiry into meaningful topics and knowledge-building through activities like identifying, interpreting and designing. These findings suggest that the different types of task and conditions of online discussion in the learning environments influence the nature of the interaction showing the correspondence between synergistic interaction and higher-order thinking.     

Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions

The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs’ performance on tasks to elaborate upon PSTs’ conceptions of the operation of partitive division. This study contributes new insights into PSTs’ conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.     

Does teachers’ pedagogical content knowledge affect their fluency instruction?

The relation is studied between teachers’ pedagogical content knowledge of reading and the quality of their subsequent classroom behaviour in teaching fluent reading. A confirmatory factor analysis model with two latent variables is tested and shows adequate goodness-of-fit indices. Contrary to our expectations, the results of structural equation modelling reveal a small but significant γ-value of .29, indicating that only 8% of the variance in teachers’ classroom behaviour in teaching fluent reading is accounted for by teachers’ pedagogical content knowledge of reading. Presumably teacher knowledge is not as stable and conclusive as one might think. More research is needed in determining the factors that work restricting for teachers in putting their knowledge into classroom practice. It is recommended that preservice and in-service teacher training should not be limited to transfer of knowledge, but should also assist teachers in designing and performing effective fluent reading instruction.     

Exploring challenges of assessing pre-service science teachers’ pedagogical content knowledge (PCK)

The purpose of this paper is to report on the challenges we faced in designing an instrument aimed at measuring pre-service science teachers’ topic-specific pedagogical content knowledge (PCK). After the instrument was conceptualised and developed, it was administered to 30 pre-service chemistry teachers. The findings suggest that the instrument can be used as a teaching tool to enhance pre-service science teachers’ topic-specific PCK along with some limitations. The discussion focuses on the implications of the aforementioned PCK assessment tool for enhancing pre-service science teachers’ topic-specific PCK and the challenges associated with measuring and enhancing pre-service science teachers’ PCK. Finally, our discussion focuses on the ways in which science-teacher educators can engage in transforming the concept of PCK and its use for research and professional development.     

Professionalizing in- service teachers’ focus on technological pedagogical and content knowledge

In Ethiopia, primary school teachers of science and mathematics are encouraged to integrate Information and Communication Technology (ICT) into their teaching as a means to improve the quality of education. However, there has not been the same emphasis placed on providing professional learning opportunities for teachers on how to use ICT in their teaching. The present study investigated how a group of practising primary school science and mathematics teachers developed the skills needed to integrate ICT into their teaching. The study employed a combination of qualitative and quantitative research methods within an action research approach. The teachers took part in activities from a Technological Pedagogical and Content Knowledge (TPACK) based professional learning workshop, including designing lessons, classroom instruction, and reflection activities in teams. A lesson evaluation sheet, questionnaire, observation checklist, and logbook were used to gather data. The results showed that the teachers acquired an improved competency to integrate available ICT into their teaching through the intervention activities.