Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving

This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.

Academic mathematics and mathematical knowledge needed in school teaching practice: some conflicting elements

In this article we analyze the relations between academic mathematical knowledge and the mathematical knowledge associated with issues mathematics school teachers face in practice, according to the specialized literature, and restricted to the theme “number systems”. We present examples that illustrate some areas of conflict between those forms of knowledge. We point out some implications of our study for teacher education, such as: 1) the importance of making conflicts explicit and of discussing them with prospective teachers in order to develop a professionally relevant perception of academic mathematics; 2) the relevance of further research in order to better understand the extent of those conflicts and their effects on the process of integrating, in a body of professional knowledge, the different kinds of mathematical knowledge presented to prospective teachers.

Facilitating preservice teachers’ development of mathematics knowledge for teaching arithmetic operations

Mathematical tasks, centered on arithmetic word problems, are discussed as the basis of an approach to facilitate preservice elementary teachers’ development of mathematical knowledge for teaching arithmetic operations. The approach consists of three groups of tasks that allow students to reflect on their initial knowledge, explore arithmetic word problem situations from multiple perspectives, and apply their resulting knowledge, respectively. Findings from the investigation of the approach indicate that the tasks can provide an effective and meaningful basis to help preservice teachers to develop deeper understandings of the arithmetic operations and their relationships and pedagogical-mathematical knowledge for teaching the arithmetic operations conceptually.     

A typological approach to investigate the teaching career decision: Motivations and beliefs about teaching of prospective teacher candidates

This study utilized cluster analysis to identify typologies of prospective teachers enrolled in a teacher education program in the U.S. based upon their self-report motivations for teaching. A three-cluster solution generated three distinctive typologies of prospective teachers (N’s of 93, 70 and 52), and further differences among typologies were investigated with respect to participants’ beliefs about teaching.Follow-up semi-structured interviews confirmed meaningful differences found between the clusters with regard to unique combination of motivations and beliefs about teaching relevant to each typology. Implications for teacher education and teacher development are further discussed in relationship with study findings.     

Video as a tool for fostering productive discussions in mathematics professional development

This article explores the use of classroom video as a tool for fostering productive discussions about teaching and learning. The setting for our research is a 2-year mathematics professional development program, based on the Problem-Solving Cycle model. This model relies on video from the teachers’ own classrooms and emphasizes creating a community in which members feel comfortable learning from video. We describe our experiences carrying out the Problem-Solving Cycle model, focusing on our use of video, our efforts to promote a supportive and analytical environment, and the ways in which teachers’ conversations around video developed over a 2-year period.     

Toward a framework for the development of mathematical knowledge for teaching

Shulman (1986, 1987) coined the term pedagogical content knowledge (PCK) to address what at that time had become increasingly evident—that content knowledge itself was not sufficient for teachers to be successful. Throughout the past two decades, researchers within the field of mathematics teacher education have been expanding the notion of PCK and developing more fine-grained conceptualizations of this knowledge for teaching mathematics. One such conceptualization that shows promise is mathematical knowledge for teaching—mathematical knowledge that is specifically useful in teaching mathematics. While mathematical knowledge for teaching has started to gain attention as an important concept in the mathematics teacher education research community, there is limited understanding of what it is, how one might recognize it, and how it might develop in the minds of teachers. In this article, we propose a framework for studying the development of mathematical knowledge for teaching that is grounded in research in both mathematics education and the learning sciences.

Changing perspectives: Examining the potential for advanced mathematical studies to influence pre-service teachers' beliefs about mathematics

This study considers the potential for advanced mathematical studies to impact pre-service teachers' beliefs about mathematics. Results show that, after completing a degree which includes advanced mathematical studies, many prospective teachers' beliefs still reflect limited interpretations of key terminology and do not value the theoretical and conceptual network underpinning the rules and procedures of secondary mathematics. Many of their beliefs about the nature of mathematics also fail to recognise its capacity to stimulate analytical thought and creativity. In cases where pre-service teachers showed evidence of well-developed beliefs, the study explores the role of their advanced mathematical studies in this development.     

The role of analogical thinking in designing tasks for mathematics teacher education: An example of a pedagogical ad hoc task

This article discusses the design of tasks for teacher education. It focuses on tasks that are used in a university course for pre-service secondary school mathematics teachers. Special attention is given to tasks that use analogical thinking in their construction or implementation. These tasks are categorized by type of teacher education goal and analyzed with respect to the use of analogical thinking. Short examples are presented for three of the goal categories, while an elaborated example is given for the fourth one. The detailed example describes the goals and design of a task sequence following an emergent pedagogical need. Using the ad hoc constructed task-sequence the teacher educator avoids “telling” while demonstrating an alternative instructional approach, and seizing the opportunity to bring up additional pedagogical issues.     

Knowledge foundations for teaching reading and spelling

Changes in education policy, the accumulation of research evidence that skilled instruction prevents and ameliorates reading failure, accountability requirements, and a new emphasis on multi-tiered interventions in schools are all causing a growing interest in improving teacher knowledge and skill in reading instruction. Consensus frameworks that explain reading development and individual differences provide an outline for what teachers need to know. The details of that content, however, including the English phonological system, the organization of English orthography, and the language structures that are processed during reading and writing, are challenging for teachers to learn. Recent studies are reviewed that investigate the relationship between teacher knowledge, practice, and student outcomes. The paper argues that teachers must have considerable knowledge of language structure, reading development, and pedagogy to differentiate instruction for diverse learners. Policy mandates for improvement of reading achievement should provide for more effective teacher education, as the knowledge base is not learned casually or easily. Research on how teachers best develop expertise should inform our licensing and professional development programs.     

Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks

This study is grounded in the theoretical position that solving problems in different ways creates mathematical connections when learning and teaching mathematics. It acknowledges the central role teachers play in providing students with learning opportunities, and it is based on the empirical finding that mathematics teachers are reluctant to solve problems in different ways in the classroom. In this paper we address the contradiction between theory-based recommendations and school mathematics practice. Based on analysis of individual interviews and two group meetings with 12 Israeli secondary school mathematics teachers, we demonstrate that in the context of multiple-solution connecting tasks this discrepancy is caused by the situated nature of the teachers’ knowledge. We also reveal the complex relationship between different types of teacher knowledge and argue the significance of developing a common language between members of the mathematics education community, including teacher educators and researchers. The names of the teachers have been changed to protect their privacy.     

Change in beliefs after first-year of teaching: The case of Turkish national curriculum context

The impact of national curriculum context on mathematics related beliefs of 12 beginning middle grades mathematics teachers was investigated after the teacher education program and first-year teaching through interviews and a combination of three belief frameworks. National curriculum requirements, lack of effective mentoring programs, and preservice tutoring experiences were detected as the major factors impacting beliefs. Participants simultaneously had teacher-centered and student-centered beliefs, causing inconsistency in beliefs and practices. While participants’ preservice and first-year experiences contrasted, they developed contextual beliefs to deal with the difficulties. Reflections on teacher education and beginning teacher support policies in national curriculum contexts were discussed.     

<Emphasis Type="Italic">Mathematical sophistication</Emphasis> among preservice elementary teachers

This study explores the ways in which eleven preservice elementary teachers used a web-based teacher resource to apply a mathematical definition, to correct a procedural error in arithmetic, and to make sense of a story requiring the multiplication of fractions. In our analysis we propose a framework to compare the behaviors and values expressed by our participants with the values and norms of the mathematical community. This analysis suggests that many preservice elementary teachers are profoundly mathematically unsophisticated. In other words, they displayed a set of values and avenues for doing mathematics so different from that of the mathematical community, and so impoverished, that they found it difficult to create fundamental mathematical understandings.     

Metacognitive monitoring: an obstacle and a key to effective teaching and learning

Actions speak louder than words. Teachers may verbally agree with curriculum change but seldom do their statements match up with their classroom practice and, as a result, the intended curriculum is never fully implemented. The study described below reports on how primary and secondary teachers’ statements of belief, otherwise referred to as expressed beliefs, influenced the implementation of a new science curriculum. Through the methodology of educational criticism the results revealed that teachers made use of a set of expressed beliefs to influence the direction of the science curriculum. Teachers’ expressed beliefs were categorized into four subsets: platonic expressed beliefs, organizational beliefs, associated beliefs and transitional beliefs. The outcome of this study provides leaders in educational reform with an understanding of the influence of teachers’ beliefs in the direction of curriculum change.     

Special education pre-service teachers' interest,subject knowledge,and teacher efficacy beliefs in mathematics

Teacher efficacy beliefs is an important characteristic to predict instructional quality and the level of cognitive activation and educational support. Since teacher efficacy beliefs are context and domain specific, this study focuses on how special education pre-service teachers' individual interest and subject knowledge in mathematics predict their efficacy beliefs in teaching mathematics. Data were collected from 57 special education pre-service teachers. The results indicated that the individual interest of pre-service teachers has a strong effect on teacher efficacy beliefs, while subject knowledge has only an indirect effect.     

Conditions for linking school mathematics and moral education: A case study

In recent years the mathematics education research community has undergone a social turn towards a greater interest in the values and broader educational purposes of mathematics education, including issues of social justice and citizenship education. Building on these developing interests, this paper presents a conceptual framework that links the teaching of school mathematics with moral education. Then, in a case study involving two countries, England and Canada, this framework is used to explore the affordances and constraints faced by mathematics teachers in those countries if they want to intentionally practice moral education in the classroom.     

Expanding the instructional triangle: conceptualizing mathematics teacher development

As mathematics educators think about teaching that promotes students’ opportunities to learn, attention must be given to the conceptualization of the professional development of teachers and those who teach teachers. In this article, we generalize and expand the instructional triangle to consider different interactions in a variety of teacher development contexts. We have done so by addressing issues of language for models of teachers’ professional development at different levels and by providing examples of situations in which these models can be applied. Through the expansion of our understanding and use of the instructional triangle we can further develop the concept of mathematics teacher development.     

Mathematics professional development for elementary teachers: Building trust within a school-based mathematics education community

Despite increased attention in professional development to building communities of teachers, few studies have investigated the development of trust in these communities. This paper presents issues related to trust in a professional development project at an elementary school. The project aimed for school-based elementary teachers and university-based mathematics educators to work together to improve mathematics instruction at the school by developing a mathematics education community of learners, where learners were conceived of as both the teachers and the mathematics educators. This paper addresses the following question: What factors in the project supported the development of trust among mathematics educators and teachers as the community was formed? More specifically, the paper is about one aspect of trust, namely, building school-based elementary teachers’ trust in the university-based mathematics educators. We point to three factors that helped the development of trust and we use care theory to further understand and discuss these factors in the context of in-service teacher education initiatives.     

Discourse communities: A framework from which to consider professional development for rural teachers of science and mathematics

This paper explores aspects of professional development for teachers of science and mathematics in schools in rural Australia. The study identified a range of issues including tensions between government, school and individual professional learning priorities and their respective responsibilities; the shaping of professional development by rurality; and issues of generic versus subject-specific professional learning. The paper will focus on the needs of teachers, particularly secondary teachers, for subject-specific professional development and the capacity of different professional development approaches for satisfying these needs. The notion of discourse communities is proposed as a framework for considering this issue.     

Looking through the lens of a teacher's life: The power of prototypical stories in understanding teachers' instructional decisions in mathematics

The focus of this narrative inquiry was to examine how teachers describe influences on their instructional decisions in mathematics. Using a lens provided by pivotal or prototypical stories teachers told about their past experiences, connections were suggested between these past experiences and their instructional decisions. These connections helped to illustrate how instructional decisions may be shaped by priorities and beliefs established earlier in life, as well as by current contexts. Further research on the use and characteristics of prototypical stories is suggested.