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.     

Non‐Traditional Preservice Teachers and Their Mathematics Efficacy Beliefs

In Florida, recent legislative changes have granted community colleges the ability to offer baccalaureate degrees in education, frequently to non‐traditional students. Based on information obtained from the literature covering preservice teachers' math knowledge, teachers' efficacy beliefs about math, and high‐stakes mathematics testing, a study examined a population of preservice teachers in a new Florida teacher preparation program. The research investigated relationships surrounding non‐traditional preservice teachers' characteristics such as: ages, high‐stakes math failures, lower division mathematics history, and math methods course performance, in relation to their efficacy beliefs about mathematics. Results revealed that preservice teachers' ages, lower division mathematics history, and math methods course performance, did have a significant relationship with their math efficacy beliefs, as measured by the Mathematics Teaching Efficacy Beliefs Instrument (MTEBI); the variable of high‐stakes math failures did not. Additionally, a multiple regression model including the aforementioned variables did predict preservice teachers' MTEBI scores, but did not generalize to the greater population. The findings from this study can assist new teacher preparation programs in isolating variables that identify preservice teachers who are at risk for poor mathematical attitudes; can posit avenues for fostering positive math beliefs in preservice teachers; and can recommend further research in this area.     

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 Relationship Between Teacher and Student Beliefs About Mathematics

In this study the relationship between teacher (n = 7) beliefs about mathematics, the learning and teaching of mathematics and their respective students' beliefs about mathematics (n = 158) are examined. The data were collected by means of two instruments specifically designed to measure belief systems about mathematics. Teacher scores were adjusted so that a higher score reflected beliefs in aligntnent with the National Council of Teachers of Mathematics (NCTM) Standards. Results indicated that the students of teachers whose beliefs were in alignment with the NCTM Standards had significantly different beliefs about factors that lead to success in mathematics than did other students. Specifically these students felt that working hard to solve problems and striving for understanding would lead to success. No student differences were found for subscales of ego orientation, competitiveness, interest and extrinsic factors such as neatness and cooperation. These findings suggest that this group of teachers practiced what they believed and that these practices affected what their students believed about mathematics. We suggest that using these two assessments in tandem give a clearer picture of the mathematical environment within a classroom and can be used in professional development workshops to initiate teacher reflection about classroom practices.     

Students’ ratings of teacher practices

In this paper, we explore a novel approach for assessing the impact of a professional development programme on classroom practice of in-service middle school mathematics teachers. The particular focus of this study is the assessment of the impact on teachers’ employment of strategies used in the classroom to foster the mathematical habits of mind and mathematical self-efficacy of their students. We describe the creation and testing of a student survey designed to assess teacher classroom practice based primarily on students’ ratings of teacher practices.     

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.     

Effectively coaching middle school teachers: A case for teacher and student learning

In this paper we report findings from a two-year, large-scale research project that describes the work of middle school mathematics specialists (also referred to as mathematics coaches or instructional coaches) who served in 10 school districts. We use mixed methods to describe how mathematics specialists spent their time supporting teachers and how these supports contributed to meaningful changes that teachers made in their instructional practices. We also report results that correlate student achievement scores with whether or not teachers were highly engaged with the mathematics specialists. We coordinate these quantitative results with findings from several case studies to illustrate the qualitatively different ways that mathematics specialists supported teachers’ ongoing work with their students. We also account for why some teachers participated more fully than others. Importantly, because mathematics specialists’ work was situated in different school settings with different demands, resources and administrative supports, these constraints and affordances contributed in part to how they could effectively support teachers’ work with their students.     

Beliefs and beyond: hows and whys in the teaching of proof

In this paper, we report on a study aimed at describing the way secondary school teachers treat proof and at understanding which factors may influence such a treatment. This study is part of a wider project on proof carried out for many years. In our theoretical framework, we combine references from research on proof with those from research on teachers in relation to their beliefs. The study was carried out through interviews with secondary school teachers aimed at learning how they describe their work with proof in the classroom, and to elicit beliefs and other factors that shape this work. Through the interviews we were able to detect reasons behind teachers’ choices in planning their work in the classroom. In the present paper, we concentrate on four cases that, among other factors, offer elements suitable to unravel the problem of inconsistencies using the construct of leading beliefs, i.e., beliefs (whose nature may vary from teacher to teacher) that seem to drive the way each teacher treats proof.     

Development of the Environmental Education Teaching Efficacy Belief Instrument

The increasing popularity of including environmental topics and issues in school curricula has created a need for effective environmental education teachers. One way to evaluate teacher effectiveness is through teacher efficacy, a belief measure that evaluates a teacher's perception that he/she can teach effectively. Research suggests that teachers’ instructional decisions are influenced by their beliefs, which are framed by their personal experiences. Because teacher efficacy is content specific, the purpose of this study was to develop a survey, the Environmental Education Teacher Efficacy Belief Instrument (EETEBI), to measure the teacher efficacy beliefs of preservice teachers as they relate to environmental education teaching strategies and outcomes.     

Developing teacher content understanding by integrating pH and logarithms concepts

Logarithms are notorious for being a difficult concept to understand and teach. Research suggests that learners can be supported in understanding logarithms by making connections between mathematics and science concepts such as pH. This study investigated how an integrated chemistry and mathematics lesson impacted 29 teachers’ understanding of the logarithmic relationship and pH. Pre- and post-test data indicated 23 teachers improved their understanding of logarithms and 28 improved their understanding of pH, suggesting that teacher educators in both science and mathematics context can use this approach to foster better understanding with their teachers and ultimately school students. Our analysis also identified professional development components and teacher characteristics associated with gains in understanding of pH and logarithms, which mathematics and science teacher educators can use to strategically adapt and implement the lesson within other teacher education settings.     

Facilitating parental engagement in school mathematics and science through inquiry-based learning: an examination of teachers’ and parents’ beliefs

This study examined teachers’ and parents’ beliefs on the implementation of inquiry-based modeling activities as a means to facilitate parental engagement in school mathematics and science. The study had three objectives: (a) to describe teachers’ beliefs about inquiry-based mathematics and science and parental engagement; (b) to describe parents’ beliefs about inquiry-based mathematics and science and their engagement in inquiry-based problem solving; and (c) to explore the impact of an inquiry-based learning environment comprising a model-eliciting activity and Twitter. The research involved three sixth-grade teachers and 32 parents from one elementary school. Teachers and parents participated in workshops, followed by the implementation of a model-eliciting activity in two classrooms. Three teachers and six parents participated in semi-structured interviews. Teachers reported positive beliefs on parental engagement in the mathematics and science classrooms and the potential positive role of parents in implementing innovative problem-solving activities. Parents expressed strong beliefs on their engagement and welcomed the inquiry-based modeling approach. Based on the results of this aspect of a four-year longitudinal design, implications for parental engagement in inquiry-based mathematics and science teaching and learning and further research are discussed.     

A Longitudinal Study of Elementary Pre‐service Teachers' Mathematics Beliefs and Content Knowledge

This study investigated the mathematics beliefs and content knowledge of 103 elementary pre‐service teachers in a developmental teacher preparation program that included a two course mathematics methods sequence. Pre‐service teachers' pedagogical beliefs became more cognitively‐oriented during the teacher preparation program with these changes occurring during the two methods courses. Pedagogical beliefs remained stable during student teaching. The pre‐service teachers also significantly increased their personal efficacy for teaching mathematics throughout the program with these shifts occurring across both methods courses and into student teaching. Pedagogical beliefs and teaching efficacy beliefs were not related at the beginning of the program, but, in general, were positively related throughout the program. In addition, the pre‐service teachers' pedagogical beliefs were positively related to their specialized content knowledge for teaching mathematics at the end of the program.     

Changing students’ beliefs about the relevance of mathematics in an advanced secondary mathematics class

ABSTRACT

This study shows that using authentic contexts for learning differential equations in a differentiation-by-interest setting can enhance students’ beliefs about the relevance of mathematics. The students in this study were studying advanced mathematics (wiskunde D) at upper secondary school in the Netherlands. These students are often not aware of the relevance of the mathematics they have to learn in school. More insights into the application of mathematics in other sciences can be beneficial for these students in terms of preparation for their future study and career. A course differentiating by student interest with new context-rich curriculum materials was developed in order to enhance students’ beliefs about the relevance of mathematics. The intervention aimed at teaching differential equations through guided small-group tasks in scientific, medical or economical contexts. The results show that students’ beliefs about the relevance of mathematics improved, and they appreciated experiencing how the mathematics was applied in real-life situations.     

Using Metaphors as a Tool for Examining Preservice Elementary Teachers' Beliefs About Mathematics Teaching and Learning

This study used metaphors as a tool to gain insight about preservice teachers' conceptualizations of the role of the teacher and the learner and held the view that the examination of these metaphors might provide an opportunity for teacher educators to reflectively and critically examine those beliefs. Thus, this research examined possible differences in the reflected beliefs of elementary preservice teachers as depicted in their metaphors about mathematics teaching and learning at three different points throughout their mathematics education methods courses. The results of this study indicated that elementary preservice teachers' beliefs primarily remained static throughout their mathematics methods courses despite ongoing experiences designed to challenge and extend those beliefs.     

Being a mathematics teacher in times of reform

Within research on mathematics teachers and/or their professional development, the concept of identity emerges as a critique of views of how teaching practice is related to teachers’ ‘internal states’ of knowledge and beliefs. Identity relates teachers’ professional lives to teaching practices and to the contexts in which the teaching and/or professional development occurs. However, what might count as the context still needs in-depth discussion. In order to contribute to the development of a theoretical framework for understanding mathematics teachers’ professional lives, we will draw on one remarkable teacher’s identity as a primary mathematics teacher in relation to one political, sociocultural, and pedagogical context. We use this teacher’s experience to discuss how education policies that create what Ball (2003) called ‘terrors of performativity’ tend to impede the formation of a balanced teacher identity.     

Australian teachers’ views of effective mathematics teaching and learning

Thirteen Australian teachers who had been nominated by their professional mathematics teachers’ associations as excellent teachers of elementary school mathematics were interviewed on their beliefs about mathematics, mathematics learning and mathematics teaching. In particular, they were asked to discuss the characteristics of effective teachers of mathematics and excellent mathematics lessons. In spite of their differences in location, experience and teacher education, the teachers displayed a lot of consistency in their responses and in their lists of characteristics. While this group of teachers cannot be claimed to be representative of Australian teachers, they have provided a snapshot of what is regarded as effectiveness in mathematics education in Australian elementary schools.     

Developing a reform mathematics curriculum program in Sweden: relating international research and the local context

This paper reports on a research-based mathematics curriculum program development project in Sweden, whose educational context is currently characterized by multiple reform initiatives. Current reforms include a repositioning of the teacher as central for students’ learning, but also a trend toward initiatives and teacher resources that are more directive than has been the case in the past 30 years. Collecting data from multiple sources, such as teacher log books, lesson observations and feedback meetings, we build on input from 11 elementary school teachers trying out our materials, including student texts and a teachers’ guide, during four trial rounds. We analyze how international research about curriculum programs and teachers’ use of these programs are interpreted and operationalized within the Swedish context. In particular, the two research questions guiding the study are: (1) “How do Swedish teachers interact with and reason about the reform-based classroom practices promoted by the curriculum program?” and (2) “How do Swedish teachers interact with and reason about their use of a teachers’ guide?” From our experiences in the Swedish educational context, we suggest the following contextual aspects to take into account when designing a curriculum program whose design is grounded in international research literature: characteristics of current classroom practices, teachers’ role in classrooms, the level of explicit/implicit support teachers are used to receiving, and teachers’ experiences using a teachers’ guide.     

Toward professional development for teachers grounded in a theory of decision making

There is now robust evidence that teachers’ and others’ in-the-moment decision making can be modeled and explained as a function of the following: their knowledge and other intellectual, social, and material resources; their goals; and their orientations (their beliefs, values, and preferences). The role of beliefs as they affect teachers’ behavior can thus be described at a level of mechanism—but of necessity in interaction with resources and goals. This paper outlines and exemplifies how resources, goals, and orientations shape teachers’ behavior. It indicates how they are interconnected, and why their evolution is necessarily slow. It then suggests how these understandings can be used as a foundation for mathematics teachers’ professional development, and describes how they are being used to shape a course of participatory professional development for middle school mathematics teachers.     

Knowledge and confidence of pre-service mathematics teachers: the case of fraction division

To make teacher preparation and professional development effective, it is important to find out possible deficiencies in teachers’ knowledge as well as teachers’ own perceptions about their needs. By focusing on pre-service teachers’ knowledge of fraction division in this article, we conceptualize the notion of pre-service teachers’ knowledge in mathematics and pedagogy for teaching as containing both teachers’ perceptions of their preparation and their mathematics knowledge needed for teaching. With specific assessment instruments developed for pre-service middle school teachers, we focus on both pre-service teachers’ own perceptions about their knowledge preparation and the extent of their mathematics knowledge on the topic of fraction division. The results reveal a wide gap between sampled pre-service middle school teachers’ general perceptions/confidence and their limited mathematics knowledge needed for teaching fraction division conceptually. The results suggest that these pre-service teachers need to develop a sound and deep understanding of mathematics knowledge for teaching in order to build their confidence for classroom instruction. The study’s findings indicate the feasibility and importance of conceptualizing the notion of teachers’ knowledge in mathematics and pedagogy for teaching to include teachers’ perceptions. The applicability and implications of this expanded notion of teachers’ knowledge is then discussed.     

Japanese and American teachers' evaluations of mathematics lessons: A new technique for exploring beliefs

In our study, we use a novel technique to explore the beliefs of Japanese and American elementary school teachers. Four American and four Japanese teachers watched a mathematics lesson—videotaped in either Nagano, Japan or Chicago, Illinois—and commented on the lesson's strengths and weaknesses. The major pedagogical issues that differentiated the teachers' comments were: what students should do during a lesson, how instructors should use language, how instructors should pace lessons and address ability differences, and how instructional materials should be used. The specific beliefs of the American and Japanese teachers in this study mapped easily onto common instructional practices in elementary school mathematics classes in the United States and Japan. We conclude that, at least for the teachers in this sample, beliefs are linked to practices and they may help to tie teachers to their culturally preferred method of mathematics instruction.