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.     

Making Connections: Elementary Teachers' Construction of Division Word Problems and Representations

If teachers make few connections among multiple representations of division, supporting students in using representations to develop operation sense demanded by national standards will not occur. Studies have investigated how prospective and practicing teachers use representations to develop knowledge of fraction division. However, few studies examined primary (K‐3) teachers' learning of contextual division problems, making connections among representations of division, and resolving the ambiguity of representing quotients with remainders. A written post‐course assessment provided evidence that most teachers created partitive division word problems, used a set model without splitting the remainder, and wrote equations with limited success. Post‐course written reflections demonstrated that many teachers developed pedagogical knowledge for helping students make connections among multiple representations, and mathematical knowledge of unit fractions. These findings suggest two areas that have implications for mathematics teacher educators who design professional development courses to facilitate teachers' learning of mathematical content and pedagogical knowledge of division and fraction relationships.     

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.     

Turkish Prospective Chemistry Teachers' Alternative Conceptions about Acids and Bases

The purpose of this study was to obtain prospective chemistry teachers' conceptions about acids and bases concepts. Thirty‐eight prospective chemistry teachers were the participants. Data were collected by means of an open‐ended questionnaire and semi‐structured interviews. Analysis of data indicated that most prospective teachers did not have difficulties about macroscopic properties of acids and bases. However, despite chemistry instruction, most of the prospective teachers were found to have problems in understanding the neutralization concept, the distinction between strength and concentration of acids and linking the acids and bases topic to daily life. These findings have some implications for teacher education programs.     

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.     

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.     

A teacher education experiment to challenge conceptions and practices

This study describes a teacher education experience with grade 5–6 teachers, based on a calculator module within a national program for mathematics in-service teacher education. The aim was to challenge the teachers’ conceptions about the role of the calculator in mathematics teaching and to promote their reflection about professional practices. The research methodology was qualitative and interpretive, with data collection through interviews and observation of teacher education and classroom supervision sessions, as well as analysis of teachers’ portfolios. The results indicate that some teachers are clearly against the use of the calculator in the mathematics classroom, while others allow students to use it in a passive way and some others are very affirmative about its use. The teachers who argue against the use of the calculator seem to predominate, suggesting a great distance between the curriculum orientations and classroom practice. The methodology of the course, combining collective sessions and individual classroom supervision, proved to be fruitful, providing new information, practice and discussion that allowed teachers to analyze different kinds of tasks in which the calculator might be useful, experiment using them in the classroom and reflect about the students’ work. The no imposing and questioning approach used in collective discussions encouraged teachers to assume their own positions; sharing and discussing in the collective reflections during the course stimulated a deeper reflection of their practice. Therefore, in this course, in-service teacher education focused on practice contributed to teachers to reflect on their conceptions and practices.     

How Pythagoras' theorem is taught in Czech Republic,Hong Kong and Shanghai: A case study

This paper attempts to explore certain characteristics of the mathematics classroom by investigating how teachers from three different cultures, namely, the Czech Republic, Hong Kong and Shanghai, handle Pythagoras' theorem at eighth grade. Based on a fine-grained analysis of one lesson, from each of the three places, some features in terms of the ways of handling the same topic were revealed, as follows: the Hong Kong teacher and the Shanghai teacher emphasized exploring Pythagoras' theorem, the Shanghai teacher seemed to emphasize mathematical proofs, while the Czech teacher and the Hong Kong teacher tended to verify the theorem visually. It was found that the Czech teacher and the Hong Kong teacher put stress on demonstrating with some degree of student input in the process of learning. On the other hand, the Shanghai teacher demonstrated a constructive learning scenario: students were actively involved in the process of learning under the teacher's control through a series of deliberate activities. Regarding the classroom exercises, the Shanghai teacher tended to vary problems implicitly within a mathematical context, while the teachers in the other two places preferred varying problems explicitly within both mathematical and daily life contexts.     

Engineering design and the development of knowledge for teaching among preservice science teachers

The goal of this article is to inform professional understanding regarding preservice science teachers’ knowledge of engineering and the engineering design process. Originating as a conceptual study of the appropriateness of “knowledge as design” as a framework for conducting science teacher education to support learning related to engineering design, the findings are informed by an ongoing research project. Perkins’s theory encapsulates knowledge as design within four complementary components of the nature of design. When using the structure of Perkins’s theory as a framework for analysis of data gathered from preservice teachers conducting engineering activities within an instructional methods course for secondary science, a concurrence between teacher knowledge development and the theory emerged. Initially, the individuals, who were participants in the research, were unfamiliar with engineering as a component of science teaching and expressed a lack of knowledge of engineering. The emergence of connections between Perkins’s theory of knowledge as design and knowledge development for teaching were found when examining preservice teachers’ development of creative and systematic thinking skills within the context of engineering design activities as well as examination of their knowledge of the application of science to problem‐solving situations.     

Representing context in video records of practice for urban mathematics teacher education

Mathematics education research has not sufficiently theorized about mathematics teacher knowledge and practice, teacher learning, and teacher education in ways that are reflective of the specificities of the sociopolitical contexts of schooling. In the USA, this is particularly important for urban mathematics education. This paper examines the affordances and challenges of representing context in video records of practice, particularly in the urban context, for use in the preparation of mathematics teachers for urban settings. The discussion, grounded in current research and theory relevant to representations of teaching, urban education, and mathematics teacher education, takes up three key issues: how is a focus on the urban context relevant to the design of video records of practice for mathematics teacher education? How can video records support prospective teachers’ understandings of the sociopolitical contexts of mathematics teaching? How does a focus on the urban context impact the meaning teachers make of video records?     

Implementing Coteaching and Cogenerative Dialoguing in Urban Science Education

Over the past 7 years the authors have been involved in the development of a new model for the education of science teachers that has the potential to address teacher education in challenging urban settings characterized by problems such as teacher turnover and retention, low job satisfaction, and contradictions arising from cultural and ethnic diversity. An intensive research program accompanied the development effort; the research results were used as resources in redesigning the evolving model to make it more appropriate for the situations at hand. The science teacher education program at an urban university was built around a yearlong field experience, during which all prospective teachers learned to teach in an urban high school while coteaching, that is, while teaching at the elbow of a mentor teacher or one or more peers. Over this period, a number of different configurations of coteaching and the associated cogenerative dialoguing were tried, tested, and investigated. The paper describes the historical development of the different configurations of the model and the emergent contradictions that led the researchers to enact changes to their approach. The central idea in the development effort was the creation of an environment that (a) best affords the learning of how to teach in urban high schools, (b) decreases teacher isolation, (c) mitigates turnover and retention, and (d) addresses contradictions arising from the cultural and ethnic diversity of students and teachers. Most importantly, this model of teacher education and enhancement simultaneously multiplies the resources and opportunities to support the learning of students.     

The Emergence of Mathematical Modeling Competencies: An Investigation of Prospective Secondary Mathematics Teachers

A study with prospective teachers without prior mathematical modeling experience sheds light on how their newly developed conceptual understanding of modeling manifested itself in their work on the final task of a modeling module within a pedagogy course in secondary mathematics curriculum and assessment. The main purpose of the module was to provide opportunity for the prospective teachers to experience the Common Core Mathematical Practice Model with Mathematics and begin to develop competency in modeling. Their work and reflections displayed a range of proficiency in several competencies associated with the modeling process. Examples of their work illustrating these ranges are provided. The prospective teachers expressed both struggle and rewards during the process, and reflected on challenges for teaching modeling. The results suggest that infusing modules in existing courses can be an effective way to elevate prospective teachers from unfamiliarity with modeling to noticeable levels of proficiency in various modeling sub-competencies.     

Impacting prospective teachers’ beliefs about mathematics

This study investigated: (1) the changes in the beliefs about mathematics held by 25 prospective elementary teachers as they went through a university mathematics course that aimed, among other things, to promote a problem-solving view about mathematics; and (2) the possible factors that accounted for the observed changes. The course incorporated specific features that prior research suggested reflect successful mechanisms for belief change (e.g., cognitive conflict). The data included students’ reflections, and responses to prompts and interview questions. Analysis of the data revealed the following major trends: (1) a movement towards a problem-solving view from the more traditional Platonist and instrumentalist views; and (2) no change in students’ initial views. Activities creating cognitive conflict, as well as the implementation of instruction valuing group collaboration and explanations, appear to have played important roles in the process of belief change. The findings have implications for research on teacher beliefs and teacher education.     

Microcomputers and the Preparation of Secondary Science Teachers: An Eight Year Follow-Up

Science teacher educators from 205 colleges/universities completed a questionnaire concerning the nature and extent of microcomputer offerings in their secondary science teacher preparation programs. These data are reported and compared to a similar sample surveyed in 1984. Seventy-seven percent of reporting institutions now require either a microcomputer course or completion of a microcomputer competency within their secondary science certification program. The most common applications in these courses are simulations, word processing, databases, and spreadsheet use. Chi-square analysis revealed that more institutions in 1992 were requiring a microcomputer course than in 1984. However, only 23.4% of the institutions offered a microcomputer course designed solely for secondary science majors, and 10.6% of the institutions offered such a course taught by a science teacher educator. Preservice science teachers were also more likely in 1992 than in 1984 to be using microcomputers in instruction during supervised field experiences in secondary schools. Finally, college/university science teacher educators perceived that microcomputer use in secondary science classrooms has increased during the past five years.     

Prospective middle grade mathematics teachers’ knowledge of algebra for teaching

This study examined prospective middle grade mathematics teachers’ knowledge of algebra for teaching with a focus on knowledge for teaching the concept of function. 115 prospective teachers from an interdisciplinary program for mathematics and science middle teacher preparation at a large public university in the USA participated in a survey. It was found that the participants had relatively limited knowledge of algebra for teaching. They also revealed weakness in selecting appropriate perspectives of the concept of function and flexibly using representations of quadratic functions. They made numerous mistakes in solving quadratic or irrational equations and in algebraic manipulation and reasoning. The participants’ weakness in connecting algebraic and graphic representations resulted in their failure to solve quadratic inequalities and to judge the number of roots of quadratic functions. Follow-up interview further revealed the participants’ lack of knowledge in solving problems by integrating algebraic and graphic representations. The implications of these findings for mathematics teacher preparation are discussed.     

Case-based pedagogy for prospective teachers to learn how to teach elementary mathematics in Korea

Cases have been used in mathematics teacher education with increasing prominence. Yet, there is little research that confirms cases as pedagogical tools to improve prospective teacher expertise, specifically in Asian contexts. This article illustrates how a specific case-based pedagogy was developed and implemented in Korea to increase prospective elementary teacher expertise in terms of paying attention to the mathematics-specific features of a lesson. The results showed that the participant teachers’ analytic foci moved from general to substantive features of a mathematics lesson. This tendency was evident when they reflected on their own teaching and was confirmed by their self-assessment. Given this, issues and suggestions in teacher education programs to promote teacher expertise in terms of mathematics-specific analysis ability are discussed.     

What Is a Fraction? Developing Fraction Understanding in Prospective Elementary Teachers

Classroom teachers need a well‐developed deep understanding of fractions and pedagogic practices so they can provide meaningful experiences for students to explore and construct ideas about fractions. This study sought to examine prospective elementary teachers' understandings of fraction by focusing specifically on their use of fractions meanings and interpretations. Results indicated that prospective elementary teachers bring with them to their final methods course a limited understanding of fractions and that experiences in methods courses resulted only in minor improvement of those limited understandings. The limited part‐whole understanding of fractions that prospective elementary teachers entered the course with was resilient. The implications of this study suggest a need for prospective elementary teachers to continue to develop their conceptual understanding of fractions and for changes to the content and instructional strategies of mathematics content courses designed for prospective elementary teachers.     

Symbolic Drawings Reveal Changes in Preservice Teacher Mathematics Attitudes After a Mathematics Methods Course

A new method of analyzing mathematics attitudes through symbolic drawings, situated within the field of Jungian‐oriented analytical psychology, was applied to 52 preservice elementary teachers before and after a mathematics methods course. In this triangulation mixed methods design study, pretest images related to past mathematics experiences drawn by prospective teachers were 63.2% negative in tone, and listed associated emotions were 60.4% negative; on the posttest these changed significantly to 72.1% positive images, with 70.5% positive associated emotions. The qualitative analysis of images and preservice teacher interpretations of them indicate that mathematics anxiety decreased and motivation changed from extrinsic to intrinsic as a result of the course. Pretest images and interpretations focused primarily on grades, unhappiness, time and pressure, struggle, and lack of success. Posttest images and interpretations revealed (a) greater understanding of mathematical concepts through use of concrete materials; (b) greater engagement in mathematics through interesting activities and discourse with peers; and (c) a sense of accomplishment from teaching practicum lessons. Because the drawing exercise helped students connect with their previously unconscious images of mathematics, thereby helping to shift the mathematics anxiety complex toward a more positive affective state, it is recommended that these activities be part of mathematics methods courses.