Learning study – promoting and hindering factors in mathematics teaching

This article focuses on presenting success factors for a group of teachers in carrying out a learning study in mathematics at their school. The research questions are: what are the actions of the school teaching community during development projects? What factors enable a group of teachers to carry out a learning study at their school? Activity theory provides a holistic framework to investigate relationships among the components present in a learning study. The results are based on analysis of interviews with teachers, students, principal organizers of schools and project coordinators, videotaped lessons, students’ tests and minutes taken at meetings of mathematics projects. The results show that the skills of facilitators, the time devoted to collaborative work, the link to learning theory and avoiding overly comprehensive content when teaching lessons are important promoting factors in mathematics teaching. The findings raise important questions about the way in which teacher work within universities.     

Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations.     

Moving Beyond “Those Kids”: Addressing Teacher Beliefs Regarding the Role of Culture Within Effective Science Pedagogy for Diverse Learners

This study focused on intensive work within a large, urban, low‐performing middle school in the southwest to address and transform teacher beliefs regarding the role of culture within their science pedagogy. Given the recent, rapid growth of numbers of students from Hispanic/Latino(a) backgrounds in the United States, it is critical that a paradigm shift takes place within schools and existing beliefs of science teachers to address the needs of a much changed classroom. This study details the journey of middle‐school teachers who were provided support and experiences targeted at addressing their existing beliefs regarding the role of culture and the growth and change that were achieved through a science education reform effort. Implications for further research will be discussed, including the need for attention to beliefs regarding culture are shared.     

Benefits of a teacher and coach collaboration: A case study

This paper describes a case study of a math teacher working with a math coach and the effects of their interaction. A guiding question was whether the coaching intervention had affected the teacher's classroom practices and, if so, in what way. The study utilized data from teacher/coach planning sessions, classroom lessons, follow-up debriefing sessions, and interviews with the teacher, coach and school principal. These data enabled the author to study the impact, if any, of the coaching on teacher beliefs and practices.     

Beginning Secondary Science Teachers' Conceptualization and Enactment of Inquiry‐Based Instruction

This study investigates the conceptions and use of inquiry during classroom instruction among beginning secondary science teachers. The 44 participants were beginning secondary science teachers in their first year of teaching. In order to capture the participants' conceptions of inquiry, the teachers were interviewed and observed during the school year. The interviews consisted of questions about inquiry instruction, while the observations documented the teachers' use of inquiry. All of the interviews were transcribed or coded in order to understand the conceptions of inquiry held by the teachers, and all of the observations were analyzed in order to determine the presence of inquiry during the lesson. The standard for assessing inquiry came from the National Science Education Standards. A quantitative analysis of the data indicated that the teachers frequently talked about implementing “scientific questions” and giving “priority to evidence.” This study found a consistency between the way new teachers talked about inquiry and the way they practiced it in their classrooms. Overall, our observations and interviews revealed that the beginning secondary science teachers tended to enact teacher‐centered forms of inquiry, and could benefit from induction programs focused on inquiry instruction.     

The Role of Teachers' Beliefs and Knowledge in the Adoption of a Reform‐Based Curriculum

Science as inquiry is a key content standard in the National Science Education Standards; however, few secondary science teachers successfully and consistently implement inquiry‐based instruction in their classrooms. This research examines the role of reform‐based curricular materials in influencing the classroom practices of 12 high school chemistry teachers and investigates the role of the teachers' knowledge and beliefs in their implementation of the reform‐based chemistry curriculum. Qualitative and quantitative data were collected in the form of beliefs interviews and classroom observations. The teachers' classroom practices were measured prior to and during the field test of the reform‐based chemistry curriculum. Analysis of the data revealed that teachers' classroom practice became more reform‐based in the presence of the new curriculum; however, the degree of change is related to the teachers' beliefs about teaching and learning, depth of chemistry knowledge, and years of teaching experience. Experienced, out‐of‐discipline teachers with transitional or student‐centered teaching beliefs demonstrated the most growth in reform‐based teaching practices. This study reinforces the need for reform‐based curriculum to assist teachers in implementing the intent of the National Science Education Standards.     

Enhancing elementary-school mathematics teachers' efficacy beliefs: a qualitative action research

Individuals and societies that can use mathematics effectively in this period of rapid changes will have a voice on increasing the opportunities and potentials which can shape their future. This has brought affective characteristics, such as self-efficacy, that affect mathematics achievement into focus of the research. Teacher efficacy refers to the extent to which a teacher feels capable to help students learn, influence students’ performance and commitment, and thus plays a crucial role in developing the student in all aspects. In this study, we used two sources of efficacy beliefs, mastery experiences and physiological and emotional states, in an interesting and challenging seven month workshop, as tools to foster teacher efficacy for six elementary-school teachers who were frustrated and wanted to leave their job. Our aim was to study the nature of these teachers’ efficacy in order to change it. In this qualitative action research, we used open interviews, non-participant observations and field notes. Results show that these teachers became efficacious, their students’ achievements and motivation were enhanced, and the school climate was changed. Qualitative inquiry of this construct sheds light on efficacy beliefs of mathematics teachers. Nurturing teacher efficacy has borne much fruit in the field of mathematics in school.     

Cross‐Cultural Issues Arising for Four Science Teachers During Their International Migration to Teach in U.S. High Schools

U.S. schools have experienced a perennial shortage of teachers. Recently, many school districts have been inviting foreign veteran teachers to help mitigate such teacher shortages. This study describes the initial cross‐cultural issues four international science teachers encountered when they immigrated to teach in U.S. high schools. In‐depth, semistructured interviews of four science teachers (from Ghana, Britain, and Germany) produced the main source of data. The international teachers faced a variety of support system problems, which were not directly classroom related, but nevertheless had an impact on their instructional effectiveness. They also faced teaching‐related issues, including differences in school organization and structure, assessment and philosophical beliefs, communication, textbooks, teaching methods, and teacher‐student relations. They all expressed a need to become active learners in order to function effectively in their new teaching contexts. The implications are discussed based on the findings.     

Shaping assessment through instructional innovation

How do changes in mathematics instruction shape the kind of assessment practices teacher use? This qualitative study examined the assessment beliefs and practices of four middle school mathematics teachers implementing a reformed mathematics curriculum for the first time. Through observations of classroom interaction, teacher interviews, and analysis of instructional documents, we charted the beginnings of change in the types of assessment information these teachers found valuable. Constraints on change in assessment included time available to develop, implement, and interpret alternative assessments, as well as parent and student beliefs about what it meant to do well in mathematics. Tightening the link between curriculum and assessment involves more than excellent curriculum materials—it will require support from multiple audiences and broadening beliefs about what it means to do and know mathematics.     

Inventing Creativity: An Exploration of the Pedagogy of Ingenuity in Science Classrooms

Concerns with the ability of U.S. classrooms to develop learners who will become the next generation of innovators, particularly given the present climate of standardized testing, warrants a closer look at creativity in science classrooms. The present study explored these concerns associated with teachers' classroom practice by addressing the following research question: What pedagogical factors, and related teacher conceptions, are potentially related to the demonstration of creativity among science students? Seventeen middle‐level, high school, and introductory‐level college science teachers from a variety of school contexts participated in the study. A questionnaire developed for this study, interviews, and classroom observations were used in order to explore potential areas of relatedness between pedagogical factors and manifestations of student creativity in science. Five categories ultimately emerged and described potential areas in which teachers would have to explicitly plan for creativity. These areas could inform the pedagogical considerations that teachers would have to make within their lesson plans and activities in order to support its manifestation among students. These provide a starting point for science teachers and science teacher educators to consider how to develop supportive environments for student creative thinking.     

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.     

Instructional identities of geometry students

We inspect the hypothesis that geometry students may be oriented toward how they expect that the teacher will evaluate them as students or otherwise oriented to how they expect that their work will give them opportunities to do mathematics. The results reported here are based on a mixed-methods analysis of twenty-two interviews with high school geometry students. In these interviews students respond to three different tasks that presented students with an opportunity to do a proof. Students’ responses are coded according to a scheme based on the hypothesis above. Interviews are also coded using a quantitative linguistic ratio that gauges how prominent the teacher was in the students’ opinions about the viability of these proof tasks. These scores were used in a cluster analysis that yielded three student profiles that we characterize using composite profiles. These profiles highlight the different ways that students can experience proof in the geometry classroom.     

An “inverse” relationship between mathematics identities and classroom practices among early career elementary teachers: The impact of accountability

This qualitative case study guided by portraiture examines the relationships between three early career elementary teachers’ beliefs about themselves in relation to mathematics (mathematics identities) and their classroom practices. Through autobiographical inquiry, reflective practice, classroom observations, interviews, and artifacts, findings show that all three second grade teachers appeared to have an “inverse” relationship between their mathematics identities and their classroom practices. In this relationship, as negative as they felt about themselves with regards to mathematics, they expended that much more effort to ensure that their students would have positive experiences with it and not be stigmatized by it as they had been. Accountability to schools, students, and parents, to increase student achievement appeared to play an important role in this relationship. Implications for preservice teacher education, inservice professional development, and research on beliefs and practices are discussed.     

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.     

Sociocultural perspectives in research on and with mathematics teachers: a zone theory approach

Sociocultural theories view teacher learning as changing participation in social practices that develop their professional identities rather than as acquisition of new knowledge or beliefs that are internal to the individual. Although sociocultural research on mathematics teacher education has tended to focus on understanding teachers’ learning, this article argues that sociocultural perspectives can also guide more interventionist research involving changing classroom practice. The approach illustrated here uses an adaptation of Valsiner’s zone theory to analyse teacher learning and development in two separate research studies. In one study the aim was to understand how teachers incorporated digital technologies into their practice, while the other study helped teachers implement an investigative approach to working mathematically consistent with a new syllabus. In both studies, productive tensions between teachers’ beliefs, contexts, and goals were a trigger for learning and development.     

Learning to Teach Inquiry: A Beginning Science Teacher of English Language Learners

Early career science teachers are often assigned to classrooms with high numbers of English language learners (ELLs). For the underprepared early career science teacher, these circumstances are challenging. This study examines the changes in beliefs and practices of an early career science teacher who taught high numbers of ELLs in an urban setting. Victoria participated in the Alternative Support for Induction Science Teachers (ASIST) program during her initial two years of teaching. Our research team followed her over a three‐year period, and the data collected included classroom observations and interviews about her beliefs and practices. In addition, documents such as teacher evaluations and classroom artifacts were collected periodically for the purpose of triangulation. The analysis of the data revealed that with the support of the ASIST program, Victoria implemented inquiry lessons and utilized instructional materials that promoted language and science competencies for her ELLs. Conversely, standardized testing and her teaching assignment played a role in constraining the implementation of inquiry‐based practices. The results of this study call for collaborative efforts between university science educators and school administrators to provide professional development opportunities and support to build the capacity of early career science teachers of ELLs.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: practitioners’ views

This study investigates the different ways by which secondary school mathematics teachers view how advanced mathematics studies are relevant to expertise in classroom instruction. Data sources for this study included position papers and written notes from a group interview of 15 Israeli teachers who studied in a special master’s program, of which advanced mathematics courses comprise a sizeable share. Data analysis was iterative and comparative, aiming at identifying and characterizing teachers’ different perspectives. Overall, all participating teachers thought that the advanced mathematics studies in the program were relevant to their teaching of secondary school mathematics. Moreover, teachers specifically mentioned the importance of studying contemporary mathematics from research mathematicians. All teachers pointed out at least one specific feature that they viewed as relevant to their work: advanced mathematics courses (1) as a resource for teaching secondary school mathematics, (2) for improving understanding about what mathematics is, and (3) for reminding teachers what learning mathematics feels like.