Mathematical myths: Teacher candidates' beliefs and the implications for teacher educators

Abstract

For more than 2 decades, professional, state, and federal agencies have adopted educational standards aimed at improving mathematics instruction. One way of measuring the success of these adopted standards is to examine their impact on the mathematics attitudes and beliefs of students who received their education during this reform period. How teachers approach and implement these mathematics standards is heavily influenced by what they believe about mathematics content and pedagogy. For those who seek to improve mathematics education, examining beliefs regarding mathematics and the factors that influence those beliefs is imperative. In this paper we explore the persistence of teacher candidates' beliefs in myths despite changing educational standards. The beliefs of 76 current elementary teacher candidates were compared to beliefs of 131 elementary teacher candidates from 1990. The results confirm the pervasiveness and persistence of math myths among female elementary teacher candidates.     

Unpacking teacher candidates’ decision-making and justifications in dilemmatic spaces during the student teaching year

Learning to teach is a complex process that hinges on teacher candidates’ personal experiences, values, beliefs and characteristics, as well as the contexts they are exposed to. Student teaching as a transitional period may create dilemmas for teacher candidates, who are new to the profession and still learning about what it means to be a teacher. Thus, this study examined the dilemmas experienced by a group of Taiwanese teacher candidates during the student teaching phase. Using the framework of dilemmatic spaces, we identified the sources of their dilemmas and the decisions they made towards the dilemmatic situations and further examined how they justified their decisions. Our findings suggest that conflicting ideologies and teacher candidates’ identities as student teachers are the main sources of their dilemmas. In response to the dilemmas, teacher candidates showed different decision-making trajectories that revealed the different actions and justifications they took within the dilemmatic space.     

Preparing Elementary School Teachers to Learn From Teaching: A Comparison of Two Approaches to Mathematics Methods Instruction

Teacher preparation programs face a significant challenge in determining how to design learning experiences that develop the combination of knowledge, practices, and dispositions needed for effective classroom teaching. Time constraints and the theory–practice divide are two well-documented concerns. We introduce the conceptual framework and design elements of a video-enhanced mathematics methods course that targets these concerns. The course centers on systematic reflection and analysis of practice intended to foster career-long learning. We then examine the impact of this course on several facets of learning-from-teaching competencies, including teacher knowledge, beliefs, and practices. Sixty-two preservice teachers enrolled in a one-year post-bachelor elementary teacher preparation program were randomly assigned to attend this course or a more typical mathematics methods course. Findings suggest that teacher preparation experiences centered on systematic reflection and analysis create opportunities to develop certain aspects of learning-from-teaching competencies that remain otherwise underdeveloped. Implications for the design of teacher preparation include the integration in mathematics methods courses of cycles of analysis through video-enhanced discussions; collaborative planning, implementation, and reflection on teaching; and live observation and co-constructed interpretations and considerations of next steps.     

Undergraduates' attitudes and beliefs about subject matter and pedagogy measured periodically in a reform‐based mathematics and science teacher preparation program

This study describes the design and use of a valid and reliable instrument to measure teacher candidates' attitudes and beliefs about mathematics and science and the teaching of those subjects. The instrument, Attitudes and Beliefs about the Nature of and the Teaching of Mathematics and Science, was developed for the Maryland Collaborative for Teacher Preparation (MCTP), a statewide, standards‐based project in the National Science Foundation's Collaborative in Excellence in Teaching Preparation (CETP) Program. We report on two applications of the instrument: (a) a contrast between MCTP teacher candidates' and non‐MCTP teacher candidates' attitudes and beliefs about mathematics and science as they initially encountered reform‐based instruction in their undergraduate courses, and (b) a landscaping of how the MCTP teacher candidates' attitudes toward and beliefs about mathematics and science evolved over a 2.5‐year period. In support of current reform in science and mathematics teacher education, we determined that over an extended period the MCTP teacher candidates' attitudes and beliefs moved substantively and significantly in the direction intended. However, we also found that the non‐MCTP teacher candidates in the same reform‐based courses did not mirror this improvement in their attitudes and beliefs about mathematics and science or the teaching of those subjects. © 2002 Wiley Periodicals, Inc. J Res Sci Teach 39: 713–737, 2002     

CONNECTING UNIVERSITY SUPERVISION AND CRITICAL REFLECTION: MENTORING AND MODELING

This study reports on the experiences of supervisors in a university teacher preparation program regarding their critical reflection on their practice. This has an impact on the learning opportunities available for their student teachers. Findings include: (a) an understanding of critical reflection is something that builds over time for student teachers through exposure to their supervisor's practice; (b) explicitly modeling, guiding, and communicating the importance of critical reflection in teaching practice through supervisory stance helps teacher candidates develop critically reflective practices and understandings; (c) developing critical reflection in their individual and shared practices takes time for both parties.     

Change in beliefs after first-year of teaching: The case of Turkish national curriculum context

The impact of national curriculum context on mathematics related beliefs of 12 beginning middle grades mathematics teachers was investigated after the teacher education program and first-year teaching through interviews and a combination of three belief frameworks. National curriculum requirements, lack of effective mentoring programs, and preservice tutoring experiences were detected as the major factors impacting beliefs. Participants simultaneously had teacher-centered and student-centered beliefs, causing inconsistency in beliefs and practices. While participants’ preservice and first-year experiences contrasted, they developed contextual beliefs to deal with the difficulties. Reflections on teacher education and beginning teacher support policies in national curriculum contexts were discussed.     

Teacher candidates' growth in designing mathematical tasks as exhibited in their lesson planning

Abstract

In order to gain insight into preservice teachers' beliefs about planning for mathematics instruction, a study was carried out involving K‐8 teacher candidates enrolled in an elementary mathematics methods course. Doyle's (1992) notion of academic task and the research on pedagogical content knowledge served as the theoretical framework for this study. The teacher candidates submitted lesson plans at three intervals during a semester‐long methods course; the lesson plans were then coded based on candidates' planned uses of academic tasks. Analyses of the data revealed trends in these teacher candidates' design of academic tasks over the course of the semester. Recommendations and implications are pre‐sented highlighting the benefits of incorporating the knowledge base on academic task into a mathematics methods course as a means to con‐tribute to teacher candidates' developing pedagogical content knowledge via their designing of academic tasks in lesson planning.     

How do pre-service teachers’ reflective processes differ in relation to different contexts?

Reflective practice represents a central theme in teacher education. The focus of this study is to look at pre-service teachers’ reflection processes and their breadth and depth in four different contexts. Our research data consist of 53 pre-service teachers’ mathematics portfolios, from which three were selected for closer scrutiny. The chosen portfolios represent different reflection profiles. According to our findings, the breadth and the depth of the pre-service teachers’ reflection processes varied greatly, and their former experiences as learners of mathematics seemed to have a great impact on their reflection when teaching mathematics for the first time. However, through acquaintance with research articles, the pre-service teachers’ reflection deepened and, at the same time, seemed to broaden. The participants’ consideration of biographical contexts aided their understanding and their reflection of other contexts.     

A primary teacher’s mathematics teaching: the development of beliefs and practice in different “supportive” contexts

This article refers to a longitudinal case study of a primary school teacher over a period of 4 years. The focus is on the development of the teacher’s beliefs regarding mathematics teaching and learning from the last year of her university studies up to the third year of teaching mathematics in school. This development has been investigated within three different contexts, which have been distinguished in terms of the kind of support provided to this teacher. Two dominant beliefs emerged which have been traced through the period of the study from both the teacher’s reflections and actions. The first belief drew on the idea that what was considered an easy mathematical task by an adult could also be easily understood by children, while the second was that children learn mathematics through their actual involvement in a variety of teaching activities. The results indicate the way that teacher’s experiences from her university studies, actual classroom practice and inservice education interact and influence her beliefs and professional development.     

Autonomy,beliefs and the learning of elementary mathematics teachers

Seven elementary teachers participated in a project designed to help them learn to teach mathematics according to reform recommendations. Teachers were provided opportunities to learn through both private reflection and public inquiry about their teaching and children's learning. The teachers’ instruction, reflection, and beliefs were studied. All of the teachers adopted some reform-based procedures including having children report problem-solving strategies. However, only three of them developed more complex practice in which children were involved in inquiry into one another's strategies. The groups had different beliefs about the autonomy of children to construct mathematics and their own autonomy to make instructional decisions.     

Cognitive Scientists Prefer Theories and Testable Principles With Teeth

The visions of mathematics classrooms called for by current educational reform efforts pose great challenges for kindergarten through Grade 12 schools and teacher education programs. Although a number of colleges and universities throughout the country are making changes in their teacher education programs to reflect these reform recommendations, we have little systematic information on the nature of these programs or their impact on prospective teachers. These issues are of central concern in the study-Learning to Teach Secondary Mathematics in Two Reform-Based Teacher Education Programs-that we draw on in this article. The article focuses on 1 preservice teacher's (Ms. Savant) knowledge, beliefs, and practices related to proof, tasks, and discourse. A situative perspective on cognition and components of teachers' professional knowledge frame our research. We examined data on Ms. Savant's experiences in her teacher education program to understand the influences of teacher education on her development as a mathematics teacher. This research indicates that Ms. Savant's teacher education experiences did make a difference in her development as a teacher. Her mathematics methods course provided a large collection of tasks, engaged her and her preservice colleagues in discourse, and provided her with both formal and informal experiences with proof-all of these experiences reflecting reform-based visions of mathematics classrooms. The situative perspective on cognition directed our attention to issues of compatibility of goals and visions across the various university and kindergarten through Grade 12 classroom settings, and it helped us to understand why some aspects of reform-based pedagogy are more easily learned than others: Why some ideas and practices learned as a student in the university setting are more easily transported to the novice teacher's kindergarten through Grade 12 field setting. We conclude that compatibility of these settings on several key dimensions is essential for the settings to reinforce each other's messages, and thus work in conjunction, rather than in opposition, to prepare reform-minded teachers.     

Facilitating Professional Development during International Practicum: Understanding our Work as Teacher Educators through Critical Incidents

This collaborative self-study details the experiences of an Australian teacher educator and a Canadian teacher educator, who led teacher candidates on international practicum placements to the Cook Islands and Kenya respectively. Focusing on critical incidents, they collaboratively analyzed dilemmas that occurred when providing professional development sessions for local teachers during these placements. These dilemmas required the teacher educators to think deeply about their beliefs and practices in these contexts. Findings from the study included the teacher educator’s tendency to make assumptions about good teaching and learning practices that were reflective of their personal pedagogical values and beliefs; their discomfort with their perceptions of some neo‐colonial practices within these international practicum sites and uncertainty about how to navigate the resultant tensions; and, the need to view the work of teacher educators though a new cultural lens when working in transnational contexts. Implications for teacher educators working with local hosting teachers during international placements include the need to understand and acknowledge the complexity of this dimension of teacher educators’ work, and for teacher educators to engage in parallel learning journeys with the teacher candidates they accompany. This involves critically reflecting on the experiences, assumptions, and beliefs that they bring to their new contexts, as well as adopting a global perspective and a deep consciousness of how one is perceived by others who are culturally, racially, or linguistically different.     

A Holistic Model to Infer Mathematics Performance: The Interrelated Impact of Student,Family and School Context Variables

The present study aims at exploring predictors influencing mathematics performance. In particular, the study focuses on internal students' characteristics (gender, age, metacognitive experience, mathematics self-efficacy) and external contextual factors (GDP of school location, parents' educational level, teachers' educational level, and teacher beliefs). A sample of 1749 students and 91 teachers from Chinese primary schools were involved in the study. Path analysis was used to test the direct and indirect relations between the predictors and mathematics performance. Results reveal that a large proportion of mathematics performance can be directly predicted from students' metacognitive experiences. In addition, other student characteristics and contextual variables influence mathematics performance in direct or indirect ways.     

RETHINKING FOOD UTILIZED AS MATERIALS FOR ART

ABSTRACT

Teaching mathematics in an early childhood program requires mathematical content knowledge and teacher self-efficacy, yet research has shown that early childhood educators often have negative attitudes towards mathematics and feel underprepared to teach mathematical concepts. The study reported here documents the reconceptualization of a graduate, preservice teacher education program, a program designed to address teacher anxiety and increase capacity to teach mathematics in a play-based early childhood setting. The study aimed to investigate: (1) the effectiveness of the mathematics component of the course in equipping teacher candidates to teach mathematics in early childhood, and (2) whether participation in the mathematics component of the course changed teacher candidates’ self-efficacy regarding mathematics. Findings show that both self-efficacy and content knowledge improved when teacher candidates had the opportunity to engage with play-based learning experiences that embed mathematical concepts. Furthermore, the focus on a learning trajectories approach supports the identification of developmental progression points in children’s emerging mathematical understanding, assisting with teacher candidates’ fine-grained observations, assessment of children’s learning, and authentic, individualized planning for learning.     

Perceptions of Reform-Based Teaching and Learning in a College Mathematics Class

This study investigated, from the students',professor's, and researchers' perspectives, the effects of a reform-based introductory undergraduate mathematics course, and the efforts of a mathematics professor to teach such a course. The class had been designed for teacher candidates of middle school mathematics and science (Grades 4 to 8) but was open to all qualified students. We addressed the following research question: What perceptions about learning and teaching mathematics emerged through the participants' experiences in a reform-based mathematics course? Results of the analysis of the data suggested that the teacher candidates and the professor took an important first step toward enculturation into a reform-based vision of mathematics learning and teaching. Implications for mathematics faculty and teacher education faculty interested in promoting reform-based mathematics are presented.     

Finding out more about teacher candidates' prior knowledge: implications for teacher educators

While it is recognised that prior knowledge is a key factor in determining future learning, its influence on learning to teach is less well known. This study investigates two cohorts of teacher candidates studying for a one-year, graduate qualification for primary teaching, who completed two tasks at entry to their initial teacher education programme: a task in writing and a task in mathematics. The tasks focused on teacher candidates' ability to recognise the key features of a piece of student work. The teacher candidates' responses to the tasks highlighted the diverse nature of the prior knowledge that underpinned their responses. The study raises questions about the pedagogy of initial teacher education, particularly in relation to the assumptions teachers educators make about the candidates they teach. The findings suggest that the prior knowledge that students bring to initial teacher education is both a resource and a challenge for teacher educators.     

Looking through the lens of a teacher's life: The power of prototypical stories in understanding teachers' instructional decisions in mathematics

The focus of this narrative inquiry was to examine how teachers describe influences on their instructional decisions in mathematics. Using a lens provided by pivotal or prototypical stories teachers told about their past experiences, connections were suggested between these past experiences and their instructional decisions. These connections helped to illustrate how instructional decisions may be shaped by priorities and beliefs established earlier in life, as well as by current contexts. Further research on the use and characteristics of prototypical stories is suggested.     

Alignment, cohesion, and change: Examining mathematics teachers’ belief structures and their influence on instructional practices

This collective case study reports on an investigation into the relationship between mathematics teachers’ beliefs and their classroom practices, namely, how they organized their classroom activities, interacted with their students, and assessed their students’ learning. Additionally, the study examined the pervasiveness of their beliefs in the face of efforts to incorporate reform-oriented classroom materials and instructional strategies. The participants were five high school teachers of ninth-grade algebra at different stages in their teaching career. The qualitative analysis of the data revealed that in general beliefs were very influential on the teachers’ daily pedagogical decisions and that their beliefs about the nature of mathematics served as a primary source of their beliefs about pedagogy and student learning. Findings from the analysis concur with previous studies in this area that reveal a clear relationship between these constructs. In addition, the results provide useful insights for the mathematics education community as it shows the diversity among the inservice teachers’ beliefs (presented as hypothesized belief models), the role and influence of beliefs about the nature of mathematics on the belief structure and how the teachers designed their instructional practices to reflect these beliefs. The article concludes with a discussion of implications of teacher education.     

Foreseeing the Unforeseen through Collaborative Self-Study by a Teacher Educator and Two Teacher Candidates

This article reports the collaborative self-study of a teacher educator and two teacher candidates, focusing on the unforeseen negativity of experiences encountered by the three researchers in both a university methods course and student teaching settings. The implication of this study helped the researchers foresee new hypothetical learning trajectories for their respective students. Throughout the study, the university teacher educator and two teacher candidates tried to maintain a collegial relationship while analyzing their teaching and learning experiences. The researchers analyzed the effectiveness of their mathematics teaching strategies over an extended period of time. The changing roles of the researchers throughout the study encouraged them to examine not only their own teaching processes but also those of the other researchers. As critical colleagues they positioned themselves in different contexts. This study generates several insights for the improvement of the researchers' future teaching practices. The results suggest that collaborative self-study by a teacher educator and teacher candidates can generate effective learning experiences for all participants.