Mathematics Pedagogical Content Knowledge Research Articles

Developing Distinct Mathematical and Scientific Pedagogical Content Knowledge in an Early Childhood Dual-Content Methods Course: An Alternative to Integration

ABSTRACTMethods for teaching early childhood mathematics and science are often addressed in a single, dual-content course. Approaches to teaching this type of course include integrating the content and the pedagogy of both subjects, or keeping the subject areas distinct. In this article, the authors discuss and illustrate their approach to such a combined course. They emphasize the development of preservice teachers’ distinct pedagogical content knowledge in mathematics and in science rather than an integration of the two. The authors do this by concentrating their course development on a central unifying theme or idea within each content-area. Through course readings, media, in-class activities and discussions, and fieldwork, the authors highlight the similarities and differences between early childhood mathematics and science methods, thus offering preservice teachers a conceptual infrastructure that empowers them to make informed choices about their own mathematics and science programs.

The Relationship between Early Childhood Teachers’ Pedagogical Content Knowledge in Mathematics and Attitude towards Mathematics

The Relationship between Early Childhood Teachers’ Pedagogical Content Knowledge in Mathematics and Attitude towards Mathematics

Investigation on the Effects of Mathematics Professional Learning Community for Remedial Instruction

The purpose of study was to investigate the effects of mathematics professional learning community (PLC) for remedial instruction. The participant of the PLC was teachers from one primary school. They were invited to join the project of mathematics remedial instruction. This project was organized by the researcher who integrated components of effective mathematics instruction so as to help teachers carry out remedial mathematics instruction. The PLC teachers in the research are primary school teachers of fourth grade in Taiwan. In order to promote their knowledge of mathematics remedial instruction, they were engaged in the workshops of lesson studies and organized as school-based professional learning community. Lesson studies consisted of components to improve effectively remedial mathematics instruction. These components included life mathematics, scaffolding for mathematics reading comprehension, strategies for problem solving, guidance of representation and schema-based instruction. All the resources and instruction materials were established in the knowledge management platform in which teachers could access and share the resources. This longitudinal study was held for one year. The students of these PLC teachers took three waves of mathematics assessments so as to detect the effects of mathematics professional learning community. Results showed these students’ mathematics achievement improves gradually. It was concluded that workshops of mathematical pedagogical content knowledge and establishment of knowledge management platform could help teachers for mathematics instruction. Finally, based on the findings, some suggestions for practical instruction and further research were also discussed.

فاعلیة تخطیط الدرس فی تنمیة المعرفة البیداغوجیة للمحتوى الریاضی لدى الطلبة / المعلمین

هدفت الدراسة إلى استقصاء فعالية تخطيط الدرس في تنمية المعرفة البيداغوجية للمحتوى الرياضي لدى الطلبة/المعلمين. اهتم الباحثون في العقدين الأخيرين بمعرفة المعلم المهنية، ومن هنا ظهر الاهتمام بالمعرفة البيداغوجية للمحتوى کمکون أساسي لمعرفة المعلم. المعرفة البيداغوجية للمحتوى الرياضي هي مزيج من معرفة المعلمين العميقة لمحتوى الرياضيات، ومعرفتهم لخصائص المتعلمين فيما يتعلق بالمحتوى الرياضي، ومعرفتهم لأصول التدريس الضرورية لمساعدة المتعلمين على تعلم المحتوى الرياضي. في الدراسة الحالية تم مطالبة الطلبة المعلمين بتحضير سبعة دروس وفقا لنموذج مقترح لتخطيط الدرس. شارک في الدراسة ٣٠ طالبا معلما من طلاب الفرقة الثالثة شعبة الرياضيات. تم استخدام المنهج شبه التجريبي في إعداد أدوات المعالجة التجريبية للدراسة وتطبيقها باستخدام تصميم المجموعة الواحدة ذات القياس القبلي البعدي. تم جمع البيانات من خلال مقياس المعرفة البيداغوجية للمحتوى الرياضي واختبار المعرفة الرياضية اللازمة لتدريس الجبر وتحليل تخطيط درس رياضيات. أظهرت نتائج الدراسة فعالية تخطيط الدرس في تنمية المعرفة البيداغوجية للمحتوى الرياضي لدى الطلبة/المعلمين. وفي ضوء نتائج الدراسة، تم تقديم التوصيات المناسبة. This study aimed at investigating the effectiveness of lesson plan in the development of mathematical pedagogical content knowledge of students / teachers. In the last two decades, researchers interested in the professional knowledge of teachers, therefore pedagogical content knowledge became an essential ingredient for the teacher's knowledge. Mathematical pedagogical content knowledge is a combination of teachers’ deep knowledge of math content, their knowledge of the characteristics of learners in respect of the content, and their knowledge of pedagogy. In the current study, students/teachers were required to prepare seven lesson plans according to a suggested model. The study group consisted of 30 pre-service math teachers. The quasi-experimental approach was used in the preparation of the experimental treatment study tools and applied using the design of one group with pre and post test. Data were collected through the mathematical pedagogical content knowledge questionnaire, mathematical knowledge test, and analysis of lesson plans. Study results showed the effectiveness of lesson plan in the development of mathematical pedagogical content knowledge of students / teachers. In light of the results of the study, appropriate recommendations were presented.

Supporting Middle School Teachers’ Implementation of STEM Design Challenges

We describe and analyze a professional development (PD) model that involved a partnership among science, mathematics and education university faculty, science and mathematics coordinators, and middle school administrators, teachers, and students. The overarching project goal involved the implementation of interdisciplinary STEM Design Challenges (DCs). The PD model targeted: (a) increasing teachers’ content and pedagogical content knowledge in mathematics and science; (b) helping teachers integrate STEM practices into their lessons; and (c) addressing teachers’ beliefs about engaging underperforming students in challenging problems. A unique aspect involved low‐achieving students and their teachers learning alongside each other as they co‐participated in STEM design challenges for one week in the summer. Our analysis focused on what teachers came to value about STEM DCs, and the challenges in and affordances for implementing DCs. Two significant areas of value for the teachers were students’ use of scientific, mathematical, and engineering practices and motivation, engagement, and empowerment by all learners. Challenges associated with pedagogy, curriculum, and the traditional structures of the schools were identified. Finally, there were four key affordances: (a) opportunities to construct a vision of STEM education; (b) motivation to implement DCs; (c) ambitious pedagogical tools; and, (d) ongoing support for planning and implementation.This article features a Research to Practice Companion Article. Please click on the supporting information link below to access.

Incorporating practitioner inquiry into an online professional development program: the Prime Online experience

Engagement in practitioner inquiry by classroom teachers is a promising mechanism for teacher professional learning. While much has been learned about the positive role inquiry can play in traditional professional development efforts, we know less about the impact of inquiry in a rapidly advancing technological age that includes the proliferation of online professional development institutes and programs. The purpose of this article is to explore the role of inquiry in online professional development by providing a detailed description of a year-long program called Prime Online. Supported by US federal funding, this program was designed to help teachers develop the mathematics content and pedagogical content knowledge necessary to implement research-based practices in inclusive elementary classrooms within a framework of practitioner inquiry. A detailed description of this program is presented followed by an examination of the inquiry projects completed by the participants. Implications of this effort for teacher learning and online professional development endeavors are offered.

Clustering Analysis on Teachers’ Perceptions of Mathematics Pedagogical Content Knowledge

The purpose of this study is to cluster the perceptions of mathematics pedagogical content knowledge (MPCK) for teachers. The subject is 259 primary school teachers in Taiwan. This study constructs dimensions of MPCK according to the review and conclusions of literature. The MPCK assessment includes six dimensions, which are mathematics content knowledge (MCK), students’ cognition knowledge (SCK), mathematics instruction knowledge (MIK), mathematics instruction practice (MIP), mathematics assessment knowledge (MAK) and teacher professional responsibility (TPR). The MPCK questionnaire is 4-points Likert scale and its reliability and validity are acceptable. Fuzzy clustering is adopted to cluster the subject based on these six dimensions. Results show that all teachers could be properly classified into six clusters. Each cluster has its own features of mathematics pedagogical content knowledge. There are also significantly differences in the dimensional scores among clusters. Besides, teachers who have more years of in-service tend to have higher dimensional scores on MPCK. These results could provide references for cultivating pre-service teachers and professional promotion for in-service teachers. Based on the findings of this study, some suggestions and recommendations are discussed for future research.

An exploration of acceptance of innovative computer use in teaching mathematics among pre-service class teachers and mathematics teachers

Innovative computer use enables the strengthening and transformation of teaching practice. Notwithstanding, numerous studies have indicated that teachers do not use computers in teaching mathematics to a sufficient extent. This study was aimed at exploring the reasons for insufficient computer use by teachers, i.e. at examining the variables which may affect the acceptance of innovative computer use in teaching mathematics. Hence, the intention to use computers in teaching mathematics in primary school was explored on the sample of 455 pre-service class teachers and mathematics teachers from Serbia. The technology acceptance model was extended by external variables and the following were observed as the predictors of intention to use computers in teaching mathematics: students? attitudes towards computers, their perception of usefulness of computers in teaching mathematics, perceived ease of use, technological pedagogical content knowledge in mathematics, experience with use, subjective norms, technological complexity and content knowledge in mathematics. Structural equation modelling has shown that the proposed model had a good fit and that the selected variables were significant predictors of the intention to use computers. The proposed model explained 23.7% of variance in the intention to use computers. It has been shown that the intention to use computers is directly predicted by dominant technological pedagogical content knowledge in mathematics, students? attitudes towards computers and their experience, while it is indirectly predicted by perceived usefulness, subjective norm, technological complexity and content knowledge in mathematics. In accordance with the presented findings, the final part provides recommendations that may be beneficial for advancing the education of pre-service class teachers and mathematics teachers.

How Field Experiences Prepare Pre-service Teachers in Teaching Elementary Mathematics

Field experience is a foundation among teacher education programs in which student-teachers are provided with excellent role models willing to engage in a reflective practice of the teaching profession. This descriptive-narrative-phenomenological study is anchored on the theory that field experiences contributes to the mathematical pedagogical content knowledge of pre-service teachers. Responses from 60 elementary education pre-service teachers revealed that field experiences significantly affect their preparation for teaching elementary mathematics not only in the areas of content and pedagogy but also in management and social relationship. The study thus concludes that field experiences materialized the desire of every teacher education institution to model teacher candidates in all aspects of pedagogical content knowledge though intense exposure that enrich and enhance their capability in teaching elementary mathematics.

The Integration of Teacher’s Pedagogical Content Knowledge Components in Teaching Linear Equation

<p class="apa">This qualitative research aimed to explore the integration of the components of pedagogical content knowledge (PCK) in teaching Linear Equation with one unknown. For the purpose of the study, a single local case study with multiple participants was used. The selection of the participants was made based on various criteria: having more than 5 years of experience teaching mathematics and possessing high and low level of knowledge in algebra and high general PCK in mathematics. Six teachers were selected to be the respondents in this research. The data were collected using i) stimulated recall interview, ii) observation, and iii) analysis of documents. Nvivo 8 software was used to help the researcher organize and analyze the data. Kohen Kappa reliability values obtained from three experts were very good, exceeding 0.8. The findings showed that the integration of content knowledge component of pedagogical implications (CàP) is the most frequently used component in the teaching of Linear Equations with one unknown. This indicates that teachers focus mostly on the components of content knowledge and knowledge about students. The teachers were least likely to integrate content knowledge with pedagogical knowledge, which has implications on the knowledge of the students (C&PàS). The study suggests the need to improve teacher’s knowledge through collaborative partnership with colleagues and through courses.</p>

Epistemological beliefs of prospective preschool teachers and their relation to knowledge, perception, and planning abilities in the field of mathematics: a process model

Teacher competence is a multi-dimensional construct that includes beliefs as well as knowledge. The present study investigated the structure of prospective preschool teachers’ mathematics-related beliefs and their relation to content knowledge and pedagogical content knowledge. In addition, prospective preschool teachers’ perception and planning skills were assessed as indicators of teacher performance. Questionnaires were used to measure beliefs, paper-and-pencil based achievement tests to measure knowledge, and a video-based test to measure perception and planning abilities. Confirmatory factor analyses revealed that prospective preschool teachers’ mathematics-related beliefs can be distinguished into a static, a process, and an application orientation. Structural equation modeling revealed that an application orientation and mathematics pedagogical content knowledge can predict the ability to perceive mathematical learning situations and to plan adequate actions. Conclusions for preschool teacher education are drawn from the results.

Primary Teachers’ Knowledge for Teaching Ratio and Proportion in Mathematics: The Case of Indonesia

In this study, we aimed at examining Indonesian In-Service primary Teachers‟ Mathematics Content Knowledge (MCK) and Mathematics Pedagogical Content Knowledge (MPCK) for teaching ratio and proportion. The instruments were administered to 271 in-service primary teachers with various education background. There were three underlined factors on MCK and MPCK instrument classified by item analysis and factor analysis. Three teachers‟ categories (Good, Middle, and Low) were established by cluster analysis methodology. The most teachers‟ challenge on MCK was figural representation. For MPCK, Indonesian teachers need more opportunity to learn the factor of Knowing students conceptual understanding. The state of teachers‟ MCK and MPCK and related components are worthy to suggest the design of Indonesian In-service Teachers Professional Development Program.

Understanding Malaysian Pre-Service Teachers Mathematical Content Knowledge and Pedagogical Content Knowledge

Background:This article seeks to present findings from the analysis of the TEDS-M reports on the mathematical content knowledge (MCK) and pedagogical content knowledge (PCK) of the pre-service teachers in MalaysiaMaterials and methods:The main objective of this study was to investigate the level of teaching knowledge attained by the Malaysian pre-service primary and secondary teachers. Some 576 primary level pre-service teachers and 389 secondary level pre-service teachers participated in the TEDS-M study. The significance of this study is pertinent as the level of teachers' knowledge influences students’ achievement in schoolsResults:Results of analysis show that Malaysian pre-service teachers at the primary and secondary performed below the international average for both mathematics content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK).Conclusions:The article suggested several steps on how teacher educators could improve the PCK and MCK of pre-service teachers in Malaysia.

About the Complexities of Video-Based Assessments: Theoretical and Methodological Approaches to Overcoming Shortcomings of Research on Teachers’ Competence

Research on the evaluation of the professional knowledge of mathematics teachers (comprising for example mathematical content knowledge, mathematics pedagogical content knowledge and general pedagogical knowledge) has become prominent in the last decade; however, the development of video-based assessment approaches is a more recent topic. This paper follows the call for more situated and performance-related ways to assess teacher competence. We discuss the theoretical and methodological challenges connected to the development of such instruments and exemplify these by an instrument developed within the follow-up study of the international “Teacher Education and Development Study in Mathematics (TEDS-M)”, called TEDS-FU. Drawing on the novice-expert framework from cognitive psychology allows analysing the structure and development of mathematics teachers’ professional competence. More recent concepts on teacher noticing of classroom situations and students’ activities are incorporated into this video-based evaluation instrument, which is described in detail in this paper, by assessing perceptual, interpretative and decision-making skills. Reliability and validity concerns remain an issue of such assessments for which solutions are proposed. Overall, the paper shows that a more comprehensive evaluation of teachers’ competence comprising cognitive-affective and situated facets is possible and has been achieved.

The effect of teacher education programs on future elementary mathematics teachers’ knowledge: a five-country analysis using TEDS-M data

This article addresses the problem of how opportunities to learn in teacher education programs influence future elementary mathematics teachers’ knowledge. This study used data collected for the Teacher Education and Development Study in Mathematics (TEDS-M). TEDS-M measured the mathematics content knowledge (MCK) and the mathematics pedagogical content knowledge (MPCK) of future teachers in their final year in teacher preparation programs. The purpose of this study is to explore whether elementary teaching candidates’ MCK and MPCK are associated with their opportunities to learn in mathematics courses and mathematics methods courses in five countries. The results showed that opportunities to learn in some teacher preparation components are more important than in other components.

Norwegian mathematics teachers’ and educational researchers’ perception of mpck items used in the teds-m study

This paper presents how Norwegian teachers and educational researchers categorised a collection of items used in the international TEDS-M study. The teachers cate- gorised the items according to their own prototype understanding of mathematical content knowledge (MCK) and mathematics pedagogical content knowledge (MPCK), while the researchers categorised them after discussing how to understand MCK and MPCK and agreeing on what categorisation criteria to use. The results show that the item categorisation depended on the item characteristics. For example, multiple-choice items were associated with MCK and items asking the respondents to rewrite or reword a mathematical task were associated with MPCK. Furthermore, the results indicate a common Norwegian understanding of MPCK as the teachers’ and researchers’ categorisation largely coincided.

Developing a model of pedagogical content knowledge for secondary and post-secondary mathematics instruction

The accepted framing of mathematics pedagogical content knowledge (PCK) as part of mathematical knowledge for teaching has centered on the question: What mathematical reasoning, insight, understanding, and skills are required for a person to teach elementary mathematics? Many have worked to address this question in K-8 teaching. Yet, there remains a call for examples and theory in the context of teachers with greater mathematical preparation and older students with varied and complex experiences in learning mathematics. In this theory development report we offer background and examples for an extended model of PCK – as the interplay among conceptually-rich mathematical understandings, experience in and of teaching, and multiple culturally-mediated classroom interactions.

A comparison of three frameworks for measuring knowledge for teaching mathematics

This paper presents a comparison of three different frameworks used in research projects aimed at measuring knowledge for teaching mathematics. As the included cases all build on Shulman’s theoretical framework for teacher knowledge, in which the categories subject matter content knowledge (CK) and pedagogical content knowledge (PCK) are central, his framework was used as a reference. To enable comparison across the frameworks, each framework’s categories were analysed and organized taxonomically. The results indicate agreement on a superordinate level. However, important differences were found in the operationalisation of the basic level categories mathematics CK and mathematics PCK. As the basic level normally represents clear communication of categories, this paper suggests that more attention to the operationalisation of basic level categories is needed.

Toward Improving the Mathematics Preparation of Elementary Preservice Teachers

Research suggests the importance of mathematics knowledge for teaching (MKT) for enabling elementary school teachers to effectively teach mathematics. MKT involves both mathematical content knowledge (M‐CK) and mathematical pedagogical content knowledge (M‐PCK). However, there is no consensus on how best to prepare elementary preservice teachers (PSTs) to achieve M‐CK and M‐PCK. This study builds on research related to MKT by investigating influences of mathematics content courses designed specifically for elementary PSTs (IMPACT courses—Impact of Mathematics Pedagogy and Content on Teaching) on their attitudes (i.e., confidence and motivation) toward M‐CK and M‐PCK. Results suggest that the PSTs who participated in these IMPACT courses not only acquired high levels of confidence and motivation toward M‐CK, but also showed significant and greater gains in attitudes toward M‐PCK, after taking the required mathematics methods course, than their counterparts. Further, the findings suggest that these IMPACT courses provided a mathematical foundation that allowed the PSTs to engage in mathematics teaching methods better than those PSTs who did not have such a foundation. These results suggest potential course experiences that may enhance M‐CK and M‐PCK for elementary PSTs.

Resolving the chicken-or-egg causality dilemma: The longitudinal interplay of teacher knowledge and teacher beliefs

To examine the longitudinal relation between knowledge and beliefs and to determine cause and effect, 183 mathematics teachers were assessed three times during their first years of teacher education on their mathematics pedagogical content knowledge (MPCK) and their beliefs about teaching and learning. The data revealed that prior MPCK predicted later achievement. Prior beliefs also determined later ones. In addition, MPCK affected later beliefs: Higher MPCK at the first measurement resulted in more constructivist beliefs at later time points. By contrast, beliefs did not predict later MPCK. If constructivist teacher beliefs are to be fostered, teacher education should strengthen MPCK.