Teaching teachers to teach Boris: a framework for mathematics teacher educator pedagogical content knowledge

The notion of pedagogical content knowledge (PCK) was posited in the context of school teaching and the knowledge used by teachers teaching school students. It has been examined for a number of discipline areas, notably mathematics. There are, however, other teaching contexts, including those of teacher educators, whose students are pre-service teachers (PSTs). The content these teacher educators teach is not subject discipline knowledge (or not solely), but the PCK for teaching a subject discipline. What knowledge do teacher educators use as they teach PCK? This paper presents a framework for the PCK required of mathematics teacher educators as they work to develop PSTs’ PCK for teaching mathematics. The framework builds on existing research into PCK and categorises aspects of the work of teacher education. The framework’s usefulness is examined by studying the PCK used by the first author in building PSTs’ understanding of mathematics teacher PCK.     

Children's self‐regulated learning profile in language and mathematics: The role of task value beliefs

This study explored the self‐regulated learning (SRL) profile of upper elementary (fifth and sixth grade) school children who were differentiated in their task value beliefs (low and high) in language and mathematics. Students' SRL profile involved their teachers' ratings of achievement outcomes and SRL behaviors. The subscale of task value beliefs from the Motivational Self‐Regulated Learning Questionnaire (MSLQ) was administered to the children, and their teachers completed a battery of scales and measurements concerning students' achievement and SRL behaviors. The results confirm previous evidence indicating that SRL involves high levels of motivation, metacognition, and strategic action. Furthermore, the results support the domain‐specific character of task value beliefs. Differences in teachers' evaluations about the achievement outcomes and SRL behaviors regarding the two groups of students (low and high in task value beliefs) were found significant mostly in the domain of mathematics. Students with high value beliefs in mathematics were described as more cognitively, metacognitively, and motivationally competent learners as compared to students with lower value beliefs. The results suggest that future intervention studies should focus on strengthening task value beliefs in “threatening” school subjects, such as mathematics, from the elementary school years. © 2010 Wiley Periodicals, Inc.     

An ideology critique of the use-value of mathematics

The idea that mathematics is needed for our mundane everyday activities has raised the question of how people deal with mathematics outside the school walls. Much has been written in mathematics education research about the possibility of transferring knowledge from and into school. Whereas the majority of this literature commends the possibility of transfer, thus assuming both the desirability of transfer and the importance of school mathematics for the professional and mundane lives of individuals, I am interested in developing an ideology critique on the beliefs underpinning the research on this issue. It will be argued that the use-value attributed to school mathematics disavows its value as part of a political and economic structure, which requires school mathematics to perform other roles than the one related with utility. This critique will be illustrated through the exploration of a typical transfer situation between school and workplace.     

Discipline,control, and the school mathematics tradition

Classroom management and discipline are primary concerns of many teachers, especially beginners. The current reform movement in mathematics education advocates a classroom atmosphere and teaching strategies that are substantially different than those that characterize the traditional approach to school mathematics. This paper considers the problems involving discipline and control that arose in the course of a beginning high school teacher's acculturation into the school mathematics tradition. The analysis relates these discipline and control problems to certain tensions and contradictions in the school mathematics tradition and suggests how efforts to cope with these problems serve to reflexively validate the tradition. In addition, the paper suggests some possible reasons for why discipline and control, and not pedagogical content knowledge, are primary concerns for many teachers. Finally, the paper considers how the conditions that foster the emphasis on discipline and control in the school mathematics tradition stand as obstacles in the face of the current reform movement.     

Modeling Change in Student Attitude Toward and Beliefs About Mathematics

Abstract

The authors used data from the Longitudinal Study of American Youth to investigate variables related to change in students' attitude toward and beliefs about mathematics in middle school and high school. Using hierarchical linear modeling techniques, the authors modeled variation in students' rate of change with variables associated with student characteristics, instructional experiences, and environment. They also identified variables that differentially affect change at different levels of secondary school (i.e., middle school vs. high school) and for different affective dimensions (i.e., attitude toward mathematics, beliefs about the social importance of mathematics, and beliefs about the nature of mathematics). Results showed a substantial negative change in students' attitudes toward and beliefs about the social importance of mathematics throughout secondary school. However, students' notions of the nature of mathematics did not change throughout secondary school. The authors identified variables related to change and found that they differed according to the level of secondary school and affective dimensions.     

Worldviews,religions, and beliefs about teaching and learning: perception of mathematics teachers with different religious backgrounds

Beliefs about mathematics education and their influences on teaching practices have been widely investigated in recent decades. There have been numerous empirical studies on the influences of religions on teachers’ and students’ beliefs about subjects such as sciences and language. However, the influences of worldviews in general and religions in particular, as one of the major sources of beliefs in relation to mathematics education, are under-researched. The current study is a first step to unpacking the relationship between teachers’ religions and their beliefs about mathematics teaching and learning. By means of semi-structured interviews with mathematics teachers of different religious backgrounds, teachers’ perceptions on the connection between their personal religious beliefs and their beliefs about teaching and learning are investigated. In-depth analyses of the perceptions of three mathematics teachers reveal the complex relationship between teachers’ religious beliefs and their teaching beliefs. First, there are some common values shared by different religions, which influence the beliefs about mathematics teaching and learning as well as education in general. Second, religion is a rich belief system, and the teachers appear to apply only a portion of their religious beliefs to guide their teaching. It is also possible that a teacher is influenced by more than one religion or cultural tradition. Despite its subtleties, our study provides evidence to support the alignment between teachers’ personal religious beliefs and their beliefs about mathematics teaching and learning.     

Changing perspectives: Examining the potential for advanced mathematical studies to influence pre-service teachers' beliefs about mathematics

This study considers the potential for advanced mathematical studies to impact pre-service teachers' beliefs about mathematics. Results show that, after completing a degree which includes advanced mathematical studies, many prospective teachers' beliefs still reflect limited interpretations of key terminology and do not value the theoretical and conceptual network underpinning the rules and procedures of secondary mathematics. Many of their beliefs about the nature of mathematics also fail to recognise its capacity to stimulate analytical thought and creativity. In cases where pre-service teachers showed evidence of well-developed beliefs, the study explores the role of their advanced mathematical studies in this development.     

The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice

Case studies were conducted to investigate the conceptions of mathematics and mathematics teaching held by three junior high school teachers. Examination of the relationship between conceptions and practice showed that the teachers' beliefs, views, and preferences about mathematics and its teaching played a significant, albeit subtle, role in shaping their instructional behavior. Differences among the teachers in their conceptions and practices are explained followed by a discussion of properties of their conceptual systems.This paper is based on the author's doctoral dissertation which was directed by Professor Thomas J. Cooney of the University of Georgia and was partly supported by a grant from San Diego State University. I wish to express my appreciation to the teachers who participated in the study and their administrators.     

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.     

I actually started to scream: emotional and mathematical trauma from doing school mathematics homework

Mathematics homework is an activity done by large numbers of students across the world. However, it is not without controversy, with concerns being raised about its academic value and whether parents have the appropriate resources to actively support or teach their children. In this article, we use the narratives of two 10-year-old girls to consider how emotional and mathematical trauma can arise from doing mathematics homework with family help. This is often the undiscussed outcome of homework interactions, but one that can have profound implications for relationships between children, their parents, the school and mathematics as a discipline. The way that the children described their and other participants’ actions in the narratives provided information about the children’s agency whilst doing school mathematics in the home. We discuss the opportunities and constraints on children doing homework as a consequence of the social and institutional relations that they operate within. The constraining influence of schooling over the opportunities provided within the home situations was the main determiner of the emotional and mathematical trauma experienced by the children.     

Motivational and affective determinants of self-regulatory strategy use in elementary school mathematics

The aim of the study was to investigate the relationships between elementary students’ reported use of self-regulatory strategies in mathematics and their motivational and affective determinants. Participants of the study were 344 fifth- and sixth-grade Greek students. Students were asked to complete self-reported measures regarding the strategies they use to self-regulate mathematics learning, their achievement goals in relation to mathematics, their self-efficacy concerning mathematics learning and achievement, the value they attribute to mathematics as a subject domain and their enjoyment of mathematics learning. Structural equation modelling confirmed a mediation model, that is, students’ mathematics self-efficacy, value beliefs about mathematics and enjoyment mediated the effects of achievement goals on reported strategy use. Results are discussed in terms of implications for elementary students’ self-regulated learning skills.     

Special education pre-service teachers' interest,subject knowledge,and teacher efficacy beliefs in mathematics

Teacher efficacy beliefs is an important characteristic to predict instructional quality and the level of cognitive activation and educational support. Since teacher efficacy beliefs are context and domain specific, this study focuses on how special education pre-service teachers' individual interest and subject knowledge in mathematics predict their efficacy beliefs in teaching mathematics. Data were collected from 57 special education pre-service teachers. The results indicated that the individual interest of pre-service teachers has a strong effect on teacher efficacy beliefs, while subject knowledge has only an indirect effect.     

Learning environment,self-efficacy for teaching mathematics,and beliefs about mathematics

Learning environment research has a long history of significant relationships between the learning environment and student outcomes. This study investigated relationships between the learning environments of mathematics courses in a teacher education program and two outcomes, mathematics teaching self-efficacy and beliefs about mathematics. These two outcomes have been repeatedly shown to influence the future teaching practices of preservice teachers but, to date, their relationships with learning environment have been investigated neither with preservice teachers nor at all in the United Arab Emirates or wider Gulf region. The learning environment was found to be significantly related to preservice teachers’ mathematics teaching self-efficacy and beliefs about mathematics. Learning environments perceived more favourably by preservice teachers were associated with higher self-efficacy for teaching mathematics, but also with more-traditional beliefs, making these findings important for higher education institutes and teacher educators.

     

A follow-up assessment of a second-grade problem-centered mathematics project

Five second-grade classes in two schools participated in a project that was generally compatible with a constructivist theory of knowing. At the end of the school year, the students in these classes and their peers in six non-project classes in the same schools were assigned to ten textbook-based third-grade classes on the basis of reading scores. The two groups of students were compared at the end of the third-grade year on a standardized achievement test and on instruments designed to assess their conceptual development in arithmetic, their personal goals in mathematics, and their beliefs about reasons for success in mathematics. The levels of computation performance on familiar textbook tasks were comparable, but former project students had attained more advanced levels of conceptual understanding. In addition, they held stronger beliefs about the importance of working hard and being interested in mathematics, and about understanding and collaborating. Further, they attributed less importance to conforming to the solution methods of others.     

The gap between expectations and reality: integrating computers into mathematics classrooms

As a result of dramatic changes in mathematics education around the world, in Turkey both elementary and secondary school mathematics curriculums have changed in the light of new demands since 2005. In order to perform the expected change in newly developed curriculum, computer should be integrated into learning and teaching process. Teachers’ beliefs play a key role in this integration process. Negative beliefs against using computer in mathematics teaching may lead to failure of this process. With the help of this study, it is aimed to detect mathematics teachers’ beliefs concerning Computer Assisted Mathematics Instruction (CAMI). Within the scope of this aim, the conducted questionnaire (The opinions of teachers about using computer in Mathematics Instruction) has been carried out on 91 mathematics teachers in the city of Trabzon. The acquired results have shown that mathematics teachers have developed negative opinions against CAMI. This state has revealed that there is a huge inconsistency between curriculum’s positive expectations arising from computer usage and teachers’ convictions.     

Special education and subject teachers’ self‐perceived readiness to teach mathematics to low‐performing middle school students

Educational reforms during the last decade have led to a more inclusive environment for students with different needs and have placed demands on teachers’ readiness to instruct diverse students in the general classroom. Previous research has ascertained that student achievement is correlated with teacher quality and teachers’ efficacy beliefs. Today, basic competence in mathematics is more important than ever for managing routine day‐to‐day activities and therefore, identifying and educationally supporting students with low performance in mathematics is necessary. The aim of the study was to investigate the perceived teacher efficacy beliefs of special education and mathematics teachers when teaching mathematics to low‐performing middle school students. Results indicated that special education teachers had higher teacher efficacy beliefs than mathematics teachers. Teacher experience, certification or gender had no effect on teachers’ efficacy beliefs. Furthermore, subject teachers reported high pedagogical knowledge for teaching low‐performing students, and special education teachers reported having moderate mathematical subject knowledge.     

Smart girls traversing assemblages of gender and class in Australian secondary mathematics classrooms

I examine experiences of former Australian schoolgirls in relation to mathematics during secondary school. This research scrutinises misunderstandings about success and impact on subject choice that can result in post-schooling trajectories that limit what girls can do in their lives beyond school. I examine ways affective relationality, as a sense of embodied belonging, may influence participation in subjects. I frame the discussion using the Baradian concept of intra-action, a co-production that engages an ethic of non-coincidence. For these participants, a reductive high-stakes testing environment and aspirations to become a master subject evoke a powerful not good enough assemblage. The responsibility to achieve enough success incites a soliciting of a particular self in affective regulation. The dread of not excelling in mathematics was often too much to endure thus participants chose to discontinue studying mathematics. They understood this as a sensible solution to prevent vulnerability, as not good enough.     

The effect of an historical perspective on prospective teachers’ beliefs in learning mathematics

Many learners hold traditional beliefs about mathematics that can hamper their learning in the discipline. To address this issue, a “history-based” intervention program entailing problem-solving and writing activities that instigate cognitive conflict was implemented. Data sources were pretest and posttest scores of a 12-theme questionnaire designed for this study called “Prospective teachers’ beliefs questionnaire about mathematics learning” and written reflections of prospective teachers. The survey of the prospective teachers beliefs related to the nature of mathematics and the way it is learned, taught, and practiced showed a great majority of them failed to hold progressive beliefs. Fortunately, the intervention program has helped the prospective teachers revise and correct their beliefs, thoughts, and understandings. Study of the prospective teachers’ written reflections and observations of their oral presentations during whole-class discussions strengthen the results of the quantitative study.     

Content and form in mathematics

The present article is based on the premise that mathematical activity exhibits an important duality. On the one hand pre-school children do mathematics spontaneously; the mathematics they do is often called informal arithmetic. On the other hand, the mathematics we teach is a cultural product developed by generations of mathematicians (Byers and Herscovics, 1977). Whatever premium school mathematics puts on individuality and original thought, it seems obvious that the mathematics done in school is neither spontaneous nor informal; the bulk of it is formal mathematics. We expect our pupils to do mathematics in a conventionally acceptable manner. In short, mathematics is both an individual and a social activity. We shall endeavour to show that duality in mathematical activity reflects a duality of mathematics as a discipline. The latter is a duality between content and form; in fact, it is the importance of form in this discipline that resulted in mathematics being classified as a formal science.