Identifying and supporting teachers’ robust understanding of proportional reasoning

This case study uses the Framework for Teachers’ Robust Understanding of Proportional Reasoning for Teaching (Weiland et al., 2020) to characterize how 51 mathematics teachers solved a comparison proportional problem. We found 50 of the 51 teachers productively drew upon four knowledge resources: (1) proportional situation, (2) ratios as part: part or part: whole, (3) unit rates, and (4) ratio as measure. This study details these and teachers’ less commonly used knowledge resources, as well as counterproductive statements related to the knowledge resources. We analyze the structure of the comparison proportion problem and suggest why teachers drew on particular knowledge resources. Lastly, we highlight how counterproductive statements highlight areas of focus for mathematics teacher educators and extends the operationalizing of the robust proportional reasoning framework for mathematics education researchers.     

Teachers’ mathematical problem-solving knowledge: In what way is it constructed during teachers’ collaborative work?

This research aims to analyze the type of mathematics problem-solving knowledge for teaching used when working collaboratively in a Lesson Study (LS) process and examine how dialogic interactions contribute to knowledge construction. Five meetings during one LS cycle of a group of eight Swiss primary teachers were video recorded, transcribed and coded with the help of qualitative data analysis software. This analysis is conducted by crossing theoretical frameworks from two different fields in education, namely mathematics education and dialogic analysis. The mixed-method uses quantitative analysis with Markov chains and cross-tables, as well as qualitative analysis at micro-, meso- and macro-levels. This research suggests that participants collectively use their mathematics and their problem-solving content knowledge to focus on pedagogical problem-solving knowledge, that they navigate between different knowledge levels and that the roles of teachers and facilitators are differentiated but are also coequal.     

Lesson study in Asia Pacific classrooms: local responses to a global movement

Japanese Lesson Study is a model for teacher professional learning that has recently attracted world attention particularly within the mathematics education community. It is a highly structured process of teacher collaboration, observation, reflection and practice. The world focus has been mainly due to the work of American researchers such as Stigler and Hiebert (Am Educ Winter:1–10, 1998; The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. Free Press, New York 1999), Lewis and Tsuchida (Am Educ Winter:14–17; 50–52, 1998) and Fernandez [J Teach Educ 53(5):395–405, 2002]. These researchers have documented Lesson Study from the perspective of their social, cultural and educational contexts. In order to develop a deeper understanding of Lesson Study in a post-modern global world, there is a need to seek views beyond those presented from an American perspective. This paper will provide further additional perspectives from an Australian state view and a Malaysian state district view and a university view. The aim is to develop an understanding of how the different contexts have influenced the structure and implementation of the Japanese Lesson Study model.     

Mathematics and Science Integration: Models and Characterizations

The squeeze on instructional time and other factors increasingly leads educators to consider mathematics and science integration in an effort to be more efficient and effective. Unfortunately, the need for common understandings for what it means to integrate these disciplines, as well as the need for improving disciplinary knowledge, appears to continue to be significant obstacles to an integrated approach to instruction. In this study we report the results of a survey containing six instructional scenarios administered to thirty-three middle grades science and math teachers. Analysis of teacher responses revealed that while teachers applied similar criteria in their reasoning, they did not possess common characterizations for integration. Furthermore, analysis suggested that content knowledge serves as a barrier to recognizing integrated examples. Implications for professional development planners include the need to develop and provide teachers with constructs and parameters for what constitutes mathematics and science integration. Continued emphasis on improving teacher content knowledge in both mathematics and science is also a prerequisite to enabling teachers to integrate content.     

On the content-dependence of prospective teachers’ knowledge: a case of exemplifying definitions

In this article, we demonstrate that prospective teachers’ content knowledge related to defining mathematical concepts is dependent on content area. We use the example of generation (a research tool we developed in a previous study) to investigate prospective teachers’ knowledge. We asked prospective secondary mathematics teachers to provide multiple examples of definitions of concepts from different areas of mathematics. We examined teacher-generated examples of concept definition and analysed individual and collective example spaces, focusing on their correctness and richness. We demonstrate differences in prospective teachers’ knowledge associated with defining mathematical concepts in geometry, algebra and calculus.     

Using video as a tool for promoting inquiry among preschool teachers and didacticians of mathematics

This paper explores the use of video as a tool for promoting inquiry among preschool teachers and didacticians. In this case, the didacticians are teacher educators who are also mathematics education researchers. Preschool teachers recorded themselves with video implementing number and geometry tasks with children and shared these recordings with other teachers and didacticians. The session where the teachers and didacticians viewed and discussed these recordings was recorded and viewed later by a group of didacticians. The multiple uses of video led to inquiry on several levels. Teachers inquired into the practice of implementing tasks with children, evaluating children’s knowledge, and the practice of using video as a tool. Didacticians inquired into their practice of research with children, their practice as teacher educators, the use of video as a tool in professional development, and the use of video in their inquiry process. Teachers’ and didacticians’ inquiries led to increased appreciation for the practice of inquiry, belonging to a community of practice, and its role in promoting both teachers’ and didacticians’ knowledge for teaching.     

Learning to teach high school mathematics: Patterns of growth in understanding right triangle trigonometry during lesson plan study

“Lesson plan study” (LPS), adapted from the Japanese Lesson Study method of professional development, is a sequence of activities designed to engage prospective teachers in broadening and deepening their understanding of school mathematics and teaching strategies. LPS occurs over 5 weeks on the same lesson topic and includes four opportunities to revisit one's own ideas and the ideas of others. In this paper, we describe one prospective teacher's growth in understanding right triangle trigonometry as she participated in LPS. This study is part of a much larger study investigating how prospective secondary teachers learn to teach mathematics within the context of LPS. Results of this study indicate that Image Saying, an activity for growth in understanding from the Pirie-Kieren model [Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterize it and how can we represent it? Educational Studies in Mathematics, 26, 165-190], is critical to prospective teachers’ growth in understanding school mathematics. Multiple opportunities and contexts within which to share understanding of school mathematics led to significant growth in understanding of right triangle trigonometry which in turn led to growth in understanding of teaching strategies. That is, the results of this study indicate that growth in understanding school mathematics (what to teach) leads to growth in understanding teaching strategies (how to teach) as prospective teachers participate in LPS.     

Developing teacher capacity to explore non-routine problems through a focus on the social semiotics of mathematics classroom discourse

This paper reports on a research project exploring the social semiotics of mathematics teaching and learning in urban middle schools. Participating teachers attended a Lesson Study Group that focused on the linguistic and diagrammatic challenges of framing and solving non-routine mathematics problems. This paper describes key social semiotic concepts explored with the teachers during the lesson study activities, focusing on the complex conjunction of the mathematics register and everyday language. We use examples from the participants’ classrooms to show the relevance of these concepts in studying classroom discourse, focusing in particular on the complex conjunction of diagramming and language.     

Similarities and differences in teachers’ questioning in German and Japanese mathematics classrooms

This study aims to capture similarities and differences in teachers’ questioning in German and Japanese mathematics classrooms, specifically focusing on the stage of introducing new mathematical content. The author analyzed consecutive mathematics classes taught by experienced teachers in Germany and Japan, who were recruited based on their locally defined “teaching competence” in the Learner’s Perspective Study. The results revealed that even questions that required students to recall previously learned content or provide the results of a calculation, which were regarded as lower cognitive questions in previous studies, played key roles at the stage of introducing new mathematical content in both German and Japanese classrooms. Further, distinctive patterns in the sequences of teachers’ questioning were identified. These differences suggest what is valued as quality mathematics teaching in each educational system.     

Examining Secondary Mathematics Teachers' Opportunities to Develop Mathematically in Professional Learning Communities

To make progress toward ambitious and equitable goals for students’ mathematical development, teachers need opportunities to develop specialized ways of knowing mathematics such as mathematical knowledge for teaching (MKT) for their work with students in the classroom. Professional learning communities (PLCs) are a common model used to support focused teacher collaboration and, in turn, foster teacher development, instructional improvement, and student outcomes. However, there is a lack of specificity in what is known about teachers’ work in PLCs and what teachers can gain from those experiences, despite broad claims of their benefit. We discuss an investigation of the work of secondary mathematics teachers in PLCs at two high schools to describe and explicate possible opportunities for teachers to develop the mathematical knowledge needed for the work of teaching and the ways in which these opportunities may be pursued or hindered. The findings show that, without pointed focus on mathematical content, opportunities to develop MKT can be rare, even among mathematics teachers. Two detailed images of teacher discussion are shared to highlight these claims. This article contributes to the ongoing discussion about the affordances and limitations of PLCs for mathematics teachers, considerations for their use, and how they can be supported.     

Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

Examining novice teacher leaders’ facilitation of mathematics professional development

This paper reports on novice teacher leaders’ efforts to enact mathematics PD through an analysis of their facilitation in workshops conducted at their schools. We consider the extent to which teacher leaders facilitated the Problem-Solving Cycle model of PD with integrity to its key characteristics. We examine the characteristics they enacted particularly well and those that were the most problematic to enact. Facilitators were generally successful with respect to workshop culture and selecting video clips for use in the PD workshops. They had more difficulty supporting discussions to foster aspects of mathematics teachers’ specialized content knowledge and pedagogical content knowledge. We suggest a number of activities that may help to better prepare novice PD leaders to hold effective workshops. Furthermore, we conjecture that leaders of mathematics PD draw from a construct we have labeled Mathematical Knowledge for Professional Development (MKPD), and we posit some domains that may comprise this construct.     

Knowing more than their students: Characterizing secondary science teachers’ subject matter knowledge

Despite agreement among teacher educators, scholars, and policymakers on the importance of teachers’ subject matter knowledge (SMK), existing models provide limited information about the nature of this foundational component of teacher knowledge. The common assumption is that teachers need to know more about the science subject matter than their students are expected to learn, but what and how much more is underspecified. In order to more characterize science teachers’ SMK, we present the science knowledge for teaching (SKT) model, which has been adapted from the mathematics education literature to apply to science education. The SKT model includes three domains: core content knowledge, specialized content knowledge, and linked content knowledge. We used this model to explore the SMK new secondary chemistry teachers in South Africa and the United States drew on when they explained the conservation of mass and analyzed a related teaching scenario, two important tasks of teaching. Findings indicated these new teachers drew on knowledge from all three SKT domains in order to engage in these tasks of teaching. This result suggests the potential of the SKT model to characterize the nature of science teachers’ SMK and thereby better inform teacher preparation and professional development programs.     

The roles of tools and models in a prospective elementary teachers’ developing understanding of multidigit multiplication

It is important for prospective elementary teachers to understand multidigit multiplication deeply; however, the development of such understanding presents challenges. We document the development of a prospective elementary teacher’s reasoning about multidigit multiplication during a Number and Operations course. We present evidence of profound progress in Valerie’s understanding of multidigit multiplication, and we highlight the roles of particular tools and models in her developing reasoning. In this way, we contribute an illuminating case study that can inform the work of mathematics teacher educators. We discuss specific instructional implications that derive from this case.     

Secondary mathematics teachers’ pedagogical content knowledge and content knowledge: validation of the COACTIV constructs

Research interest in the professional knowledge of mathematics teachers has grown considerably in recent years. In the COACTIV project, tests of secondary mathematics teachers’ pedagogical content knowledge (PCK) and content knowledge (CK) were developed and implemented in a sample of teachers whose classes participated in the PISA 2003/04 longitudinal assessment in Germany. The present article investigates the validity of the COACTIV constructs of PCK and CK. To this end, the COACTIV tests of PCK and CK were administered to various “contrast populations,” namely, candidate mathematics teachers, mathematics students, teachers of biology and chemistry, and advanced school students. The hypotheses for each population’s performance in the PCK and CK tests were formulated and empirically tested. In addition, the article compares the COACTIV approach with related conceptualizations and findings of two other research groups.     

Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: the power of task-based, one-to-one assessment interviews

In this paper, we outline the benefits to teachers’ expertise of the use of research-based, one-to-one assessment interviews in mathematics. Drawing upon our research and professional development work with teachers and students in primary and middle years in Australia and the research of others, we argue that the use of the interviews builds teacher expertise through enhancing teachers’ knowledge of individual and group understanding of mathematics, and also provides an understanding of typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children, building both their pedagogical content knowledge and their subject matter knowledge.     

United States teachers’ views of effective mathematics teaching and learning

This study investigates US teachers’ cultural beliefs concerning effective mathematics teaching using semi-structured interviews with 11 experienced teachers. For US teachers, effective teaching is student-centered. Cognitively appropriate mathematical content should be understood through many hands-on activities that allow students to explore by themselves the relationship between mathematical knowledge and their life experiences. Correspondingly, the US teachers view an effective teacher as a facilitator who is sensitive to student social and cognitive needs and is skillful at organizing collaborative learning. The result of this study helps researchers and educators understand the student-centered learning model in US classrooms.     

Exploring Probability and Statistics with Preservice and Inservice Teachers

NCTM's mathematics curriculum and evaluation standards (1989) have provided educators with the challenge of revamping high school mathematics curricula as well as pedagogies by which content is taught. This article presents a lesson designed for preservice and inservice teachers that permits participants to: (a) strengthen their conceptual understanding, and (b) experience learning in a cooperative environment that encourages communication. The lesson engages participants in the collection and representation of probabilistic data using dice with 4, 6, 8, 10, 12, and 20 faces. Opportunities are provided for participants to discover patterns and construct mathematical knowledge concerning theoretical probability. Teacher educators can facilitate reform of mathematics education by developing and delivering such lessons.