Beliefs and Perspectives of First‐Year,Alternative Preparation,Elementary Teachers in Urban Classrooms

Results from an earlier study ( Hart, 2002 ) suggested that a group of 14 teachers participating in an alternative preparation program for elementary teachers had developed beliefs that were consistent with current thinking in mathematics education. The current study follows 8 of those teachers into their first year of teaching in an urban classroom. Qualitative data were collected from three sources: reflection logs, mathematics case discussions, and field notes made during classroom observations. This provided a triangulation of perspectives: the teachers' views of themselves, the teachers' views of others, and the university faculty member's view of them. Also, teachers completed the Standards Belief Instrument (Zollman & Mason, 1996) at the end of Phase I and at the end of Phase II. Results from the instrument and the qualitative analysis suggest that the teachers maintained a strong reform perspective in their beliefs, but they were unable to consistently implement pedagogy that was consistent with those beliefs.     

The Road to Reform: A Grounded Theory Study of Parents' and Teachers' Influence on Elementary School Science and Mathematics

Though elementary teacher educators introduce new, reform‐based strategies in science and mathematics methods courses, researchers wondered how novices negotiate reform strategies once they enter the elementary school culture. Given that the extent of parents' and veteran teachers' influence on novice teachers is largely unknown, this grounded theory study explored parents' and teachers' expectations of children's optimal science and mathematics learning in the current era of reform. Data consisted of semi‐structured, open‐ended interviews with novice teachers (n = 20), veteran teachers (n = 9), and parents (n = 28). Researchers followed three stages of coding procedures to develop a logic model connecting participants' discrete designations of the landscape, regulating phenomena, contextual orientation, and desired outcomes. This logic model helped researchers develop propositions for future research on the interactive nature of parents' and teachers' influential role in elementary science and mathematics education. Implications encourage science and mathematics teacher educators—as well as school administrators—to explicitly develop and support novice teachers' ability to initiate and sustain parent/family engagement in order to create a school climate where teachers and parents are synergistically motivated to change.     

Reforming the NCTM standards in light of historical perspective: Premature changes

The mathematics education community finds itself in strange times; times of transition and continual change. The constant evolution of mathematics education reform is upon us. A full nine years into the publishing and partial implementation of the NCTM Curriculum and Evaluation Standards mathematics educators eagerly anticipate the evaluation, rewriting, and publication of reform Standards. However, a din of opposition has arisen: one, a call for reform from within, seeking to rewrite the current Standards, and call from without, calling for a return to “Back to Basics.” This parallels the reform antagonism and Back to Basics movement which followed the unsuccessful reform effort of the New Math Movement more than three decades ago. Therefore, since historical precedence exists, this present phenomenon must be thoroughly considered in light of past reform efforts.This paper defines some of the historically consistent events and situations surrounding both the New Math Movement and the NCTM Standards. These commonalities include: overestimating teachers; overly ambitious suggestions in curriculum modifications; delayed teacher training; philosophically inconsistent evaluatory methodologies; and insufficient programmatic assessment. Arguing from these commonalities, this paper reports that any immediate reform of the NCTM Standards is premature and not supported by empirical evidence.     

Meeting the Needs of Diverse Student Populations: Comprehensive Professional Development in Science,Math, and Technology for Teachers of Students With Disabilities

Professional development for teachers has become a key component for reform in teaching, learning, and curriculum change. This report describes a model of professional development designed to improve the skills and knowledge of teams of special education and regular education teachers in science, mathematics, and technology instruction. The comprehensive model included summer and academic year content and methodology-focused workshops and summer “practician” experiences. It was designed to link those factors impacting teacher practices and interventions with teachers' beliefs in instruction. The training component for teachers included opportunities for collaborative teaching, upgrading knowledge of math and science subject matter, and identifying, integrating, and practicing alternative approaches for teaching science and math that address the needs of special education students, with a focus on techniques for adapting instruction to specific disabilities. The program led to development of coping skills and persistence in the teaching of science and math for all students. As a result, strong efficacy expectations have been developed through repeated experiences of success with children in a classroom environment. Remaining issues still to be addressed include classroom management, teaching in a heterogeneous classroom, and further improvement of content expertise of teachers.     

Reforming mathematics learning in Indonesian classrooms through RME

This paper reports an experimental study on the development of exemplary curriculum materials for the teaching of fractions in Indonesian primary schools. The study’s context is the current reform movement adopting realistic mathematics education (RME) theory, known as Pendidikan Matematika Realistik Indonesia (PMRI), and it looked at the role of design research in supporting the dissemination of PMRI. The study was carried out in two cycles of teaching experiments in two primary schools. The findings of the design research signified the importance of collaboration between mathematics educators and teachers in developing RME curriculum materials. The availability of RME curriculum materials is an important component in the success of the PMRI movement, particularly in supporting students and teachers in activity-based mathematics learning. Most of the students and teachers in the two schools positively appraised teaching and learning with the developed materials. Since the teachers were actively involved in developing the materials, they felt a sense of ownership and recognised that their students’ classroom experiences of the materials helped them avoid standard difficulties. That appears to be a particular benefit of the bottom-up approach characteristic of the PMRI movement.     

The functions of function discourse – university mathematics teaching from a commognitive standpoint

This paper addresses a topic within university mathematics education which has been somewhat underexplored: the teaching practices actually used by university mathematics teachers when giving lectures. The study investigates the teaching practices of seven Swedish university teachers on the topic of functions using a discursive approach, the commognitive framework of Sfard. In the paper a categorization of the construction and substantiation routines used by the teachers is presented, for instance various routines for constructing definitions and examples, and for verifying whether an example satisfies a given definition. The findings show that although the overall form of the lectures is similar, with teachers using ‘chalk talk’, and overt student participation limited to asking and answering questions, there are in fact significant differences in the way the teachers present and do mathematics in their lectures. These differences present themselves both on the level of discursive routines and on a more general level in how the process of doing mathematics is made visible in the teachers’ teaching practices. Moreover, I believe that many of the results of the study could be relevant for investigating the teaching of other mathematical topics.     

Humans-with-media and continuing education for mathematics teachers in online environments

This paper begins by situating online mathematics education in Brazil within the context of research on digital technology over the past 25?years. I argue that Brazilian research on technology in mathematics education can be divided into four phases, and then present an example that ??blends?? aspects of the second and third phases. Phase two can be characterized by research with software designed to address traditional mathematics topics, such as functions, while the third phase is characterized by online courses. The data presented show creative solutions for a problem designed for collectives of humans-with-function-software. The paper is analyzed from a perspective that emphasizes the role of different technologies as teachers and professors collaborate to produce knowledge about the use of mathematical software in regular face-to-face classrooms. A model of online education is presented. Finally, the paper discusses how technology may change collaboration and teaching approaches in continuing education, as it allows for greater integration of online learning with teachers?? classroom activities in schools. In this case, the online platform plays an active role in the learning collective composed of humans-with-media.     

Observation of Reform Teaching in Undergraduate Level Mathematics and Science Courses

This paper reports on initial results from an ongoing evaluation study of a National Science Foundation project to implement reform‐oriented teaching practices in college science and mathematics courses. The purpose of this study was to determine what elements of reform teaching are being utilized by college faculty members teaching undergraduate science and mathematics courses, including a qualitative estimate of the frequency with which they are used. Participating instructors attended summer institutes that modeled reform‐based practices and fostered reflection on current issues in science, mathematics, and technological literacy for K‐16 teaching, with an explicit emphasis on the importance of creating the best possible learning experience for prospective K‐12 science and mathematics teachers. Utilizing a unique classroom observation protocol (the Oregon‐Teacher Observation Protocol) and interviews, the authors (a) conclude that some reform‐oriented teaching strategies are evident in undergraduate mathematics and science instruction and (b) suggest areas in which additional support and feedback are needed in order for higher education faculty members to adopt reform‐based instructional methodology.     

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM—The International Journal on Mathematics Education—by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers’ and didacticians’ engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice.     

Learning Trajectories,Lesson Planning,Affordances, and Constraints in the Design and Enactment of Mathematics Teaching

Recent reform efforts in mathematics education have stimulated a focus on learning trajectories. At the same time, a global increase in high-stakes testing has influenced instructional practices. This study investigated how four fourth grade teachers within a school planned and enacted lessons to understand what mediated their planning and teaching decisions. Findings reveal that three of these teachers, who were veteran teachers, used a testing trajectory approach with decisions mediated by preparing students for high-stakes tests. The fourth teacher, a novice, attempted to use a learning trajectory approach to support student understanding. Results reveal that high-stakes testing played a crucial role in teachers' instructional decisions. Based on the findings, we provide a framework for a testing trajectory approach that the veteran teachers used to make instructional decisions. Further research is needed to understand how to support teachers to prepare students for testing using effective teaching practices.     

The Influences of Three Interventions on Prospective Elementary Teachers' Beliefs About the Knowledge Base Needed for Teaching Mathematics

To meet the challenge to reform mathematics education, effective opportunities to learn are needed to promote prospective elementary school teachers' development of the knowledge base that supports teaching for mathematical proficiency. This article describes three professional development interventions and their influence on prospective teachers' beliefs about mathematics, how children learn mathematics, and mathematics teaching. The three interventions consisted of problem‐solving journals, structured interviews, and peer teaching that were integrated in a PreK‐6 mathematics methods course. Results of precourse and postcourse survey data are included that measured 24 prospective teachers' beliefs about the knowledge base needed to teach elementary school mathematics. Data indicated that using these interventions and other course experiences facilitated change in the prospective teachers' beliefs, with a shift toward reform‐oriented mathematics education perspectives.     

Multipurpose Professional Growth Sequence: The catwalk problem as a paradigmatic example

An important concern in mathematics teacher education is how to create learning opportunities for prospective and practicing teachers that make a difference in their professional growth as educators. The first purpose of this article is to describe one way of working with prospective and practicing teachers in a graduate mathematics education course that holds promise for positively influencing the way teachers think about mathematics, about student learning, and about mathematics teaching. Specifically, I use the “catwalk” task as an example of how a single problem can serve as the basis for a coherent sequence of professional learning experiences. A second purpose of this article is to provide background information that contextualizes the subsequent two articles, each of which details the positive influence of the catwalk task sequence on the authors’ professional growth.     

A Longitudinal Look at Attitudes and Perceptions Related to the Integration of Mathematics,Science, and Technology Education

The purpose of this study is to provide an in‐depth analysis of attitudes and perceptions related to the integration of mathematics, science, and technology education of preservice teachers preparing to teach STEM disciplines. Longitudinal data by individual cohort and across 7 years of the Integrated Mathematics, Science, and Technology (MSAT) Program are reported, analyzed, and interpreted to help design and improve preservice teacher education programs and improve teaching and learning in STEM classrooms. Results of quantitative analyses indicate that there was generally no change in preservice teacher attitudes and perceptions related to the value of the integration of mathematics, science, and technology education—they clearly valued integration at the onset and at the completion of the program. However, there was a significant change in preservice teacher attitudes and perceptions related to integration feasibility in terms of inefficiency and difficulty. Implications for teacher education programs include: (a) more exposure to concepts, processes, and skills in STEM that are similar, analogous, complementary, or synergistic; (b) familiarity with instructional strategies and access to resources; (c) deeper understanding of content across STEM; and (d) strategies for collaboration and team work to make integrated instruction time more efficient and less difficult to manage.     

Perceptions of Professional Growth: A Mathematics Teacher Educator in Transition

To meet the need for reform in mathematics teacher preparation courses, two cycles of changes made in an elementary mathematics methods course are presented. Using action research, teaching approaches were developed, implemented, and evaluated as a meaningful way to continue my professional development. Results suggested that I improved my teaching practices and focused more on teaching tasks that engaged my students to “think like teachers.” Three critical components of teacher preparation courses are identified that are important for teacher educators to acknowledge when implementing change: (a) using reflective verbal and written communication, (b) establishing a collaborative mathematical community, and (c) focusing on a narrower selection of mathematical content.     

Mathematics teaching in French écoles normales primaires, 1830–1848: social and cultural challenges to the training of primary school teachers

In the nineteenth century, the French education system, and teaching itself, was organized according to the social class the pupils came from: primary education was for the working classes, and secondary education was for the wealthier classes. In relation to this ??educational duality?? this paper looks at the mathematics teaching provided in the écoles normales primaires (primary teacher training colleges), which developed in France in the 1830s to train future (male) primary school teachers. What, precisely, was the content of this teaching? How was it organized? In what spirit and for what purpose was it provided? The aim of the paper is to show how these schools participated in the construction of a specifically primary mathematics culture for the education of the children from the lower classes, as distinct from the scholarly culture of secondary education.     

Proofs as bearers of mathematical knowledge

Yehuda Rav’s inspiring paper “Why do we prove theorems?” published in Philosophia Mathematica (1999, 7, pp. 5–41) has interesting implications for mathematics education. We examine Rav’s central ideas on proof—that proofs convey important elements of mathematics such as strategies and methods, that it is “proofs rather than theorems that are the bearers of mathematical knowledge”and thus that proofs should be the primary focus of mathematical interest—and then discuss their significance for mathematics education in general and for the teaching of proof in particular.     

Mathematics teachers’ views about the limited utility of real analysis: A transport model hypothesis

In the United States and elsewhere, prospective teachers of secondary mathematics are usually required to complete numerous advanced mathematics courses before obtaining certification. However, several research studies suggest that teachers’ experiences in these advanced mathematics courses have little influence on their pedagogical practice and efficacy. To understand this phenomenon, we presented 14 secondary mathematics teachers with four statements and proofs in real analysis that related to secondary content and asked the participants to discuss whether these proofs could inform their teaching of secondary mathematics. In analyzing participants’ remarks, we propose that many teachers view the utility of real analysis in secondary school mathematics teaching using a transport model, where the perceived importance of a real analysis explanation is dependent upon the teacher’s ability to transport that explanation directly into their instruction in a secondary mathematics classroom. Consequently, their perceived value of a real analysis course in their teacher preparation is inherently limited. We discuss implications of the transport model on secondary mathematics teacher education.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.