Understanding a Teacher's Reflections: A Case Study of a Middle School Mathematics Teacher

The purpose of this research was to understand how one teacher reflected on different classroom situations and to understand whether the teacher's approach to these reflections changed over time. For the purposes of this study, we considered reflection as the teacher's act of interpreting her own practices and students' thinking to make sense of student understanding and how teaching might relate to that understanding. We investigated a middle school mathematics teacher's reflection on her students while watching videotapes of her classroom and categorized the reflection as Assess, Interpret, Describe, Justify, and Extend. The results show a higher percentage of Extend instances in later interviews than in earlier ones indicating the teacher's increasing attention to her own teaching in how her students developed their understanding. In addition, her reflection became clearer and better integrated as defined by the Cohen and Ball's triangle of interactions.     

Using Sociocultural Theory to Teach Mathematics: A Vygotskian Perspective

This study describes an elementary teacher's implementation of sociocultural theory in practice. Communication is central to teaching with a sociocultural approach and to the understanding of students; teachers who use this theory involve students in explaining and justifying their thinking. In this study ethnographic research methods were used to collect data for 4 1/2 months in order to understand the mathematical culture of this fourth‐grade class and to portray how the teacher used a sociocultural approach to teach mathematics. To portray this teaching approach, teaching episodes from the teacher's mathematics lessons are described, and these episodes are analyzed to demonstrate how students created taken‐as‐shared meanings of mathematics. Excerpts from interviews with the teacher are also used to describe this teacher's thinking about her teaching.     

Using Teacher Portfolios to Enrich the Methods Course Experiences of Prospective Mathematics Teachers

This paper illustrates ways to employ teacher portfolios to improve the quality of methods course experiences for prospective mathematics teachers. Based upon research conducted in an undergraduate teacher preparation program, this case study describes how the author used teacher portfolios to mentor prospective teachers in new ways. The case describes the author's experiences through a case study of his assessment of and response to one prospective teacher's portfolio. This portfolio illustrated themes that were present in other teachers' portfolios, but did so in ways that highlighted strategies for change to the methods course. Through the lens of this teacher's portfolio the author identified specific ways that the prospective teacher's beliefs were impacting her teaching practice, a result that enabled him to better help all of the teachers in the methods course reflect on their teaching. By providing a detailed account of the feedback process that led to this result, this paper illustrates how mathematics teacher educators can use prospective teachers' portfolios to enrich the quality of their methods courses.     

Connections and confusion: teaching perimeter and area with a problem-solving oriented unit

Problem-solving-oriented mathematics curricula are viewed as important vehicles to help achieve K-12 mathematics education reform goals. Although mathematics curriculum projects are currently underway to develop such materials, little is known about how teachers actually use problem-solving-oriented curricula in their classrooms. This article profiles a middle-school mathematics teacher and examines her use of two problems from a pilot version of a sixth-grade unit developed by a mathematics curriculum project. The teacher's use of the two problems reveals that although problem-solving-oriented curricula can be used to yield rich opportunities for problem solving and making mathematical connections, such materials can also provide sites for student confusion and uncertainty. Examination of this variance suggests that further attention should be devoted to learning about teachers' use of problem-solving-oriented mathematics curricula. Such inquiry could inform the increasing development and use of problem-solving-oriented curricula.     

Conflicting Beliefs: A Story of a Chemistry Teacher's Struggle with Change

One teacher's struggle to develop and implement a curriculum focused on student understanding of chemistry is explored in this case study of a high school chemistry teacher. Conflicting beliefs about her roles as a teacher in the classroom and her professional responsibilities are addressed. Three primary conflicts that emerged from data collected over a two year period include, (a) conflicts between state curriculum mandates and individual student understanding; (b) conflicts between theoretical and applicable chemistry content knowledge, and (c) conflicts between the students' goals and the teachers' goals for the course. The impact of the research process on the teacher's change process included reconceptualization of constraints and development of confidence in her professional judgment. The case study provides insights into contextual problems teachers face as they attempt to change practices.     

Teaching Mathematics with a New Curriculum: Changes to Classroom Organization and Interactions

This report describes a high school mathematics teacher's decisions about classroom organization and interactions during his first two years using a new curriculum intended to support teachers' development of student-centered, contributive classroom discourse. In year one, the teacher conducted class and interacted with students primarily in small groups. In year two, he conducted more whole-class instruction. In both years, teacher-student interactions contained univocal and contributive discourse, but in year two the teacher sustained contributive discourse with students for longer periods. The teacher facilitated the most significant changes to classroom discourse in the instructional format with which he had the greatest experience (whole-class instruction). Over the period of this study, two key factors appeared to affect the teacher's decisions about classroom organization and interactions: his perception of students' expectations about mathematics classroom roles and activity, and his own discomfort associated with using a new curriculum. These areas are important candidates for future research about teachers' use of innovative mathematics curricula.     

Progressive, classical or balanced— A look at mathematical learning environments in Swiss-German lower-secondary schools

In a national supplement to TIMSS, lower-secondary school teachers (N=102) and their students (N=975) reported on mathematics instruction by means of a teacher questionnaire (teaching-learning methods, instructional sub-goals, facilitated student activities, achievement assessment, teacher role) and a student questionnaire (teachers' instructional proficiency, classroom climate). A cluster analysis performed on the ratings of teaching-learning methods yielded a solution with three clusters referred to as progressive, classical, and balanced learning environment. Cluster-related differences in facilitated student activities, achievement evaluation and preferred teacher role were found but not in instructional sub-goals. Students from different learning environments equally approved teachers' instructional proficiency and classroom climate and also had similar TIMSS mathematics scores. The results of this study provide empirical evidence that in addition to classical teacher-centered learning environments there seem to exist more diversified and studentcentered learning environments that address the needs for students to direct their own learning, communicate and work with others, and develop ways of dealing with complex problems. In line with the research literature it was also found that high mathematics achievement is not restricted to a certain type of learning environment.     

The mathematics textbook at tertiary level as curriculum material – exploring the teacher's decision-making process

This paper reports on a study about how the mathematics textbook was perceived and used by the teacher in the context of a calculus part of a basic mathematics course for first-year engineering students. The focus was on the teacher's choices and the use of definitions, examples and exercises in a sequence of lectures introducing the derivative concept. Data were collected during observations of lectures and an interview, and informal talks with the teacher. The introduction and the treatment of the derivative as proposed by the teacher during the lectures were analysed in relation to the results of the content text analysis of the textbook. The teacher's decisions were explored through the lens of intended learning goals for engineering students taking the mathematics course. The results showed that the sequence of concepts and the formal introduction of the derivative as proposed by the textbook were closely followed during the lectures. The examples and tasks offered to the students focused strongly on procedural knowledge. Although the textbook proposes both examples and exercises that promote conceptual knowledge, these opportunities were not fully utilized during the observed lectures. Possible reasons for the teacher's choices and decisions are discussed.     

Rural teacher leadership in science and mathematics

This study adds to our understanding of science and mathematics teacher leadership in rural schools. Through In Vivo and Concept coding of teacher interviews, we investigated 20 secondary science and mathematics teachers' perceptions of rural teacher leadership during their participation in a three-year professional development program. As the teachers developed as teacher leaders, they broadened their focus from improving their own students' learning to sharing new knowledge learned through the program with other teachers both informally and formally. We compared our program components to the Teacher Leader Model Standards and added an emphasis on the importance of disciplinary content knowledge. We also identified patterns in science and mathematics teacher leadership that are contextually connected to teachers' instruction in rural high poverty schools. Rural teacher leadership included the importance of building strong teacher–student relationships, providing new academic opportunities for students, encouraging students' success, and building community connections.     

Teachers' Considerations of Students' Thinking During Mathematics Lesson Design

Teachers' abilities to design mathematics lessons are related to their capability to mobilize resources to meeting intended learning goals based on their noticing. In this process, knowing how teachers consider Students' thinking is important for understanding how they are making decisions to promote student learning. While teaching, what teachers notice influences their decision‐making process. This article explores the mathematics lesson planning practices of four 4th‐grade teachers at the same school to understand how their consideration of Students' learning influences planning decisions. Case study methodology was used to gain an in‐depth perspective of the mathematics planning practices of the teachers. Results indicate the teachers took varying approaches in how they considered students. One teacher adapted instruction based on Students' conceptual understanding, two teachers aimed at producing skill‐efficient students, and the final teacher regulated learning with a strict adherence to daily lessons in curriculum materials, with little emphasis on student understanding. These findings highlight the importance of providing professional development support to teachers focused on their noticing and considerations of Students' mathematical understandings as related to learning outcomes. These findings are distinguished from other studies because of the focus on how teachers consider Students' thinking during lesson planning. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

The Teacher's Discourse Moves: A Framework for Analyzing Discourse in Mathematics Classrooms

In this paper a framework is proposed for analyzing the deliberate actions taken by a teacher to participate in or influence the discourse in mathematics classrooms, and such actions are referred to as the teacher's discourse moves. This work synthesizes elements of several other discourse frameworks, including those of Richards, Sfard, Cobb, and Knuth and Peressini. Expanding on the improvisational dance metaphor of Heaton's, the framework views the teacher in the additional multiple roles as a Choreographer/Stage Manager/Director of classroom discourse. Several research applications of the discourse framework to collegiate mathematics education are discussed, including discourse around collaborative problem solving in Treisman Emerging Scholars workshops, a video‐based study of a college‐level geometry course for teachers, discourse in wireless networked classrooms, and the asynchronous discourse in an online statistics course.     

The Development of Mathematical Discussions: An Investigation in a Fifth-Grade Classroom

Promoting discussion and argumentation of mathematical ideas among students are aspects of the vision for communication in recent school mathematics reform efforts. Having rich mathematical discussions, however, can present a variety of classroom challenges. Many factors influence classroom discussions and need to be addressed in ways that will assist teachers in creating more inquiry-based mathematics classrooms. The study presented here examined the development of mathematical discussions in a fifth-grade classroom over the course of a school year. Various aspects of the participants' interactions, teacher's pedagogy, and the classroom microculture were investigated. One major result is the evolution of student participation from nonactive listening to active listening and use of others' ideas to develop new conjectures. These changes were paralleled by changes in the teacher's role in the classroom and the nature of her questions, in particular.     

Why it is important for in-service elementary mathematics teachers to understand the equality .999… = 1

Researchers conducted semi-structured interviews with in-service fifth grade teachers. The purpose of these interviews was to examine teachers’ reactions to arguments that .999… = 1. Previously reported results indicate that some pre-service elementary school teachers possess misunderstandings about mathematical issues related to decimals with single repeating digits. This research investigates whether some in-service teachers possess misunderstandings about mathematical issues related to .999…. This paper reports on one instance of a teacher whose responses indicate that the teacher's sense of number and sense of measurement are intertwined, resulting in fragile understanding of repeating decimals. These data present evidence that teachers continue to develop repeated decimal understandings and misunderstandings throughout their careers, and that the curriculum, everyday experience, and perceptions of student learning combine to form or reinforce these understandings. Because decimals with a single repeating digit (e.g. .333… and .666…) are an integral part of the elementary mathematics curriculum, we argue that it is important that in-service elementary mathematics teachers have a clear understanding of concepts related to the concept of infinity as they emerge through the study of the equality .999… = 1.     

Teacher-student interaction: A game theoretic extension of the economic theory of education

This paper employs the ‘economic theory of education’ to consider the joint interactive choices of student and teacher. Game theoretic analysis is applied extending the work begun by Correa (1974). It is shown that the relative degree of substitutability in the utility and achievement functions determines whether a student responds positively or negatively to the teacher's greater effort or harder grading. Conditions for the existence and stability of a non-cooperative equilibrium are investigated. Due to the public good nature of student achievement, the non-cooperative equilibrium will result in insufficient academic effort being allocated to academic achievement for Pareto optimality; and therefore, there is a need for binding cooperative agreements.     

Opportunity to learn in the preparation of mathematics teachers: its structure and how it varies across six countries

Cross-national research studies such as the Program for International Student Assessment and the Third International Mathematics and Science Study (TIMSS) have contributed much to our understandings regarding country differences in student achievement in mathematics, especially at the primary (elementary) and lower secondary (middle school) levels. TIMSS, especially, has demonstrated the central role that the concept of opportunity to learn plays in understanding cross-national differences in achievement Schmidt et al., (Why schools matter: A cross-national comparison of curriculum and learning  2001). The curricular expectations of a nation and the actual content exposure that is delivered to students by teachers were found to be among the most salient features of schooling related to academic performance. The other feature that emerges in these studies is the importance of the teacher. The professional competence of the teacher which includes substantive knowledge regarding formal mathematics, mathematics pedagogy and general pedagogy is suggested as being significant—not just in understanding cross-national differences but also in other studies as well (Hill et al. in Am Educ Res J 42(2):371–406, 2005). Mathematics Teaching in the 21st Century (MT21) is a small, six-country study that collected data on future lower secondary teachers in their last year of preparation. One of the findings noted in the first report of that study was that the opportunities future teachers experienced as part of their formal education varied across the six countries (Schmidt et al. in The preparation gap: Teacher education for middle school mathematics in six countries, 2007). This variation in opportunity to learn (OTL) existed in course work related to formal mathematics, mathematics pedagogy and general pedagogy. It appears from these initial results that OTL not only is important in understanding K-12 student learning but it is also likely important in understanding the knowledge base of the teachers who teach them which then has the potential to influence student learning as well. This study using the same MT21 data examines in greater detail the configuration of the educational opportunities future teachers had during their teacher education in some 34 institutions across the six countries.     

Reductive-holistic cycle: A model for the study of the didactic procedure

The concept ofreality level may be useful as a catalyst among several systems in the area of knowledge. This concept is leading us to ask, if we can make a reduction from a reality level to another, that is to the problem ofreductionism. Relative to it is the problem ofholism. At the end these concepts are connected to the category theory and adjoint functors. Within the framework of this aspect we set up a model for the study of the didactic procedure. This model is a feedback system between two reality levels or categories, these of the teacher and of the student. So, the article seeks to enhance and improve the teaching of mathematics by its attempt to understand both student's and teacher's knowledge in the same terms.     

A Teacher's Conception of Communication in Geometry Proofs

Mathematical proof has many purposes, one of which is communication of the reasoning behind a mathematical insight. Research on teachers' views of the role that proof plays as mathematical communication has been limited. This study describes how one teacher conceptualized proof communication during two units on proof (coordinate geometry proofs and Euclidean proofs). Based on classroom observations, the teacher's conceptualization of communication in written proofs is recorded in four categories: audience, clarity, organization, and structure. The results indicate differences within all four categories in the way the idea of communication is discussed by the teacher. Implications for future studies include attention to teachers' beliefs about learning mathematics in the process of understanding teachers' conceptions of proof as a means of mathematical communication.     

Real-World Contexts,Multiple Representations,Student-Invented Terminology,and Y-Intercept

One classroom using two units from a Standards-based curriculum was the focus of a study designed to examine the effects of real-world contexts, delays in the introduction of formal mathematics terminology, and multiple function representations on student understanding. Students developed their own terminology for y-intercept, which was tightly connected to the meaningfulness and implicit/explicit temporality of the contexts that students investigated as part of their classroom activities. This terminology held great promise for promoting the concept of y-intercept within a multiple representation environment. However, the teacher's interpretation of different activities and his assumptions about the transparency of different representations, as well as students' past experiences left the student-generated terminology and the concept of y-intercept disconnected from one another. This resulted in student-generated terminology that had limited applicability, a fragile understanding of y-intercept within different representations, and for some students, interference between their invented terminology and the concept of y-intercept itself.     

The potential of case studies of exemplary mathematics teaching

Because past research often highlights the problems of mathematics education, case studies of exemplary teachers were conducted in order to emphasize positive aspects and to stimulate and improve mathematics education. Observations of two primary mathematics teachers nominated as exemplary by their peers provided some marked contrasts. One teacher subscribed to a constructivist philosophy, provided students with opportunities for active mental engagement and had an energetic monitoring style which ensured high levels of time on task. The other teacher emphasized whole‐class activities and gave students limited opportunities to learn in meaningful ways. This teacher's style, however, was undergoing changes which were facilitated by the support of the researchers.     

The Detailed Analysis of an Established Teacher's Non-Traditional Lesson

This article presents the fine-grained analysis of an experienced teacher conducting a highly interactive, non-standard lesson of his own design. The analysis, often carried out on a line-by-line level, seeks to explain how and why the teacher made the decisions he did while interacting with his students. The analysis indicates that much of the lesson, in which the teacher is truly responsive to the ideas generated by the students, can be modeled closely using a small number of contingency-based teaching strategies. Even in a case where a student makes a rather unusual comment, the model—which includes the detailed characterization of the teacher's knowledge, goals, and beliefs—is capable of predicting with some precision the nature of the teacher's response.