Impact of teachers’ beliefs on mathematics education

ABSTRACT

The objective of the current study was to analyse the impact of teachers’ personal theory and beliefs (PTB) towards Mathematics teaching. A total of 183 respondents were involved in this study, using the stratified random sampling method with Cronbach's alpha of 0.87. Due to the objective of the research and the hypothesis, it was positioned within the positivist paradigm following a survey design, concentrating on Bachelor of Education and Postgraduate Certificate Education students, with a quantitative approach. The instrument of data collection was a structured questionnaire schedule. All selected respondents (using a questionnaire) were students of University of Fort Hare, enrolled in either the Bachelor of Education (BEd) or Postgraduate Certificate of Education (PGCE) programme. Using analysis of variance (ANOVA), the results revealed that there was no significant effect of age and race on learning of Mathematics amongst BEds and PGCEs candidates.     

Understanding the influence of two mathematics textbooks on prospective secondary teachers’ knowledge

This study examines the influence of reading and planning from two differently organized mathematics textbooks on prospective high school mathematics teachers’ pedagogical content knowledge and content knowledge of exponential functions. The teachers completed a pretest and two posttests. On the pretest, the teachers possessed an incomplete understanding of content and pedagogical content knowledge related to exponential functions. The teachers’ understanding of how to translate from table to closed-form and recursive equations grew as a result of their use of the Mathematics: Modeling Our World textbook, while the Discovering Algebra textbook appeared to be more beneficial in terms of pedagogical content knowledge. Teachers read from the student lessons in both textbooks, but read differently from the sections of both textbooks intended for the teacher. They focused more on the purpose of the Mathematics: Modeling Our World lesson and more on the places where students might experience difficulties in the margins of the Discovering Algebra lesson. The teachers’ learning was influenced by their own personal characteristics (e.g., previous textbook experiences) as well as textbook qualities (e.g., organization).     

Development of a performance assessment task and rubric to measure prospective secondary school mathematics teachers’ pedagogical content knowledge and skills

The purpose of this paper is to share a performance assessment task and rubric designed to assess secondary school mathematics preservice teachers’ pedagogical content knowledge and skills. The assessment task and rubric were developed in collaboration with five education faculty, four arts and sciences faculty, and four high school teachers over a period of one year. Over this period, the group met more than 15 times and designed an assessment task and rubric to measure secondary school preservice teachers’ pedagogical content knowledge and skills. The assessment task and rubric were piloted at the end of the year and extensively revised using the pilot data. The redesigned task and rubric were successfully carried out in the second and subsequent years. Other teacher education programs can benefit from this assessment of teacher candidates’ pedagogical content knowledge and skills.     

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.     

A longitudinal study of effects of a developmental teacher preparation program on elementary prospective teachers’ mathematics beliefs

The universal emphasis in mathematics education on teaching and learning for understanding can require substantial paradigmatic shifts for many elementary school teachers. Consequently, a pressing goal of teacher preparation programs should be the facilitation of these changes during program experiences. This longitudinal, mixed methods study presents a thorough investigation of the effects of a distinctive teacher preparation program on important constructs related to prospective teacher preparedness to teach mathematics for understanding, including mathematics pedagogical and teaching efficacy beliefs, mathematics anxiety, and specialized content knowledge for teaching mathematics. The results indicate that the programmatic features experienced by the prospective teachers in this study, including a developmental two-course mathematics methods sequence and coordinated developmental field placements, provided a context supporting teacher change. These shifts are interpreted through the nature and timing of the experiences in the program and a model of teacher change processes. The findings provide insights for mathematics educators as to the outcomes of these programmatic features.     

An investigation of prospective secondary mathematics teachers’ conceptual knowledge of and attitudes towards statistics

The development of statistical literacy is fast becoming the focus of a large part of mathematics instruction at primary, secondary and tertiary levels. This broadening of the mathematics curriculum to encompass a focus on statistics makes considerable demands on teachers. Most mathematics teachers acknowledge the practical importance of statistics and are willing to give more relevance to the teaching of statistics; however, many mathematics teachers do not consider themselves well prepared to teach statistics. The aims of this study were to investigate the conceptual understanding of statistics of prospective secondary mathematics teachers; the nature of their attitudes towards statistics and if there was a relationship between attitude towards statistics and conceptual understanding of statistics. Conceptual understanding was measured using a standard assessment instrument (comprehensive assessment of outcomes in a first statistics course) which allows comparison across other disciplines. Despite being very mathematically able and confident, the prospective mathematics teachers in this study do no better in the assessment than students from other (mostly non-quantitative) disciplines. This, perhaps, gives further evidence that statistical thinking is different from mathematical thinking and that a strong background in mathematics does not necessarily translate to statistical thinking. Conceptual knowledge was poor in some fundamental areas of statistics such as being able to properly describe the distribution of a quantitative variable and data production. The attitudes of these teachers towards statistics were measured using a widely used instrument (survey of attitudes towards statistics). The results indicate generally positive attitudes but an acknowledgement that statistics is not a subject quickly learned by everyone and requires discipline to learn. No strong correlation was found between attitudes and conceptual knowledge. It is recommended that in order to improve teacher knowledge, teacher education programmes must include tailored modules in statistics and highlight the differences between mathematical and statistical thinking.     

Providing written feedback on students’ mathematical arguments: proof validations of prospective secondary mathematics teachers

Mathematics teachers play a unique role as experts who provide opportunities for students to engage in the practices of the mathematics community. Proof is a tool essential to the practice of mathematics, and therefore, if teachers are to provide adequate opportunities for students to engage with this tool, they must be able to validate student arguments and provide feedback to students based on those validations. Prior research has demonstrated several weaknesses teachers have with respect to proof validation, but little research has investigated instructional sequences aimed to improve this skill. In this article, we present the results from the implementation of such an instructional sequence. A sample of 34 prospective secondary mathematics teachers (PSMTs) validated twelve mathematical arguments written by high school students. They provided a numeric score as well as a short paragraph of written feedback, indicating the strengths and weaknesses of each argument. The results provide insight into the errors to which PSMTs attend when validating mathematical arguments. In particular, PSMTs’ written feedback indicated that they were aware of the limitations of inductive argumentation. However, PSMTs had a superficial understanding of the “proof by contradiction” mode of argumentation, and their attendance to particular errors seemed to be mediated by the mode of argument representation (e.g., symbolic, verbal). We discuss implications of these findings for mathematics teacher education.     

Exploring early childhood preservice teachers’ mathematics anxiety and mathematics efficacy beliefs

Early childhood preservice teachers participated in a qualitative multiple case study to explore and examine the effectiveness of reform-based constructivist methods used in a mathematics methods course to change their mathematics anxiety, mathematics self-efficacy, and mathematics teachers’ efficacy beliefs. Findings indicated that instructor’s use of a variety of reform-based strategies to teach and model concepts were effective in reducing their mathematics anxiety and improving their mathematics self-efficacy and mathematics teaching efficacy beliefs. Based on these findings, it is recommended that mathematics methods course instructors use reform-based constructivist methods in their courses as outlined by the NCTM’s (2014) principles. Teacher educators must also consider carefully their attitudes and disposition toward mathematics along with the type of classroom and learning environment they establish in mathematics methods courses. They must emphasize conceptual understanding during mathematics methods courses, understand the connection between preservice teachers’ mathematics anxiety and mathematics efficacy beliefs, and integrate field experiences as well as peer teaching opportunities into mathematics methods courses.     

The curricular importance of mathematics: a comparison of English and Hungarian teachers’ espoused beliefs

This paper reports an interview study of 45 English and 10 Hungarian teachers of mathematics. The semi‐structured interviews focused on the teachers’ professional life‐histories and invited them to discuss their beliefs about the necessary subject content for the teaching and learning of mathematics. Substantial differences emerged between the two cohorts, which accord with well‐defined national perspectives on education in general and mathematics education in particular. They reflect, at national rather than individual levels, the expectations of the curricular frameworks within which teachers operate. English teachers tended to view mathematics as applicable number and the means by which learners are prepared for a world beyond school. Hungarian teachers privileged mathematics as problem‐solving and logical thinking.     

History of mathematics in secondary school teachers’ training: towards a nonviolent mathematics education

In the context of mathematics teachers’ training, the concept of dépaysement épistémologique (epistemological disorientation) emphasizes that the contact with the history of mathematics, particularly with the use of original sources, pushes aside commonplace students’ perspectives about the discipline and offers them a critical look towards mathematics’s historical, social and cultural aspects. Conceptually supported by the theory of objectivation, an emergent sociocultural theory in mathematics education, this study describes the dépaysement épistémologique lived by future mathematics teachers engaged in the reading of historical texts. A phenomenological approach allowed us to clarify various meanings associated with students’ lived experiences and a dialogical perspective provides a way to get these meanings in tension through a polyphonic narration. Our reading of this polyphonic narration suggests that dépaysement épistémologique associated with the reading of historical texts encouraged empathy from students towards the authors and their future learners, opening up the possibility for a nonviolent mathematics education.     

The impact of field experiences on prospective preservice teachers’ technology integration beliefs and intentions

This study investigated how field experiences impacted the technology integration beliefs and intentions of prospective preservice teachers. Using structural equation modeling (SEM), a three-way interaction between the type, frequency, and quality of field technology observations predicted changes in beliefs and intentions. Beliefs about the utility of technology directly predicted intentions to use technology and intentions to use Meaningful Learning approaches to technology integration. Additionally, beliefs about technology’s importance in education indirectly predicted both variables. Positive impacts of field experiences on beliefs and intentions only existed when prospective preservice teachers saw technology used frequently by skilled teachers using Meaningful Learning approaches.     

The effect of a ‘Science,Technology and Society’ course on prospective teachers’ conceptions of the nature of science

The aim of this study is to assess prospective teachers’ views of some aspects of the nature of science (NOS) and the effects of a “Science, Technology and Society” (STS) course embedded with scientific investigation (SI) on these views. A questionnaire consisting of 13 items was given to 212 prospective teachers enrolled in a STS course before and after teaching. During the semester, participants were engaged in a specially designed pilot SI combined with explicit NOS instruction. Majority of the participants held traditional views of the target NOS aspects at beginning of the study. After the course there were significant changes in the conceptions of prospective teachers in majority target aspects of NOS as the results of a Sign test indicate. We suggest that the more suitable SI, performed in an active learning environment, related to target aspects of NOS must be selected to develop the more constructivist views about the NOS.     

The nature of prospective mathematics teachers’ pedagogical content knowledge: the case of multiplication of fractions

The purpose of this study was to investigate prospective mathematics teachers’ knowledge of common conceptions and misconceptions that sixth and seventh grade students had about multiplication of fractions. In addition, prospective teachers’ knowledge of the sources of these misconceptions and strategies they knew to overcome these misconceptions was examined. Data were collected from 17 Turkish prospective teachers at the end of the spring semester of the 2004–2005 academic year. A case study design was used in which data were collected through the Multiplication of Fractions Questionnaire and semi-structured interviews. The prospective teachers suggested many difficulties that elementary grade level students may have and stated that these difficulties stemmed from students’ lack of formal knowledge and rote memorization of the algorithms. In addition, the prospective teachers suggested many strategies that could be used to overcome these misconceptions or difficulties. These strategies could be grouped under three headings: strategies based on teaching methods, strategies based on formal knowledge of fractions, and strategies based on psychological constructs.     

Impact of religion on Turkish prospective early childhood teachers’ judgments of fact

The purpose of this study was to examine the impact of religion on prospective early childhood teachers’ judgments of fact and reasoning. Six hundred and sixteen prospective early childhood teachers from seven different cities and universities in Turkey participated in the study. Participants read the religious story of ‘King Solomon and the Ant’ and responded to the following questions: (1) Can Solomon converse with ants? (2) If yes, why, or no, why not? Please explain your answer with one sentence. Findings revealed that 520 of the participants (84%) responded to the first question affirmatively and 96 (16%) of the participants responded negatively. Among positive responders, 494 (80%) provided religious reasoning, while 83 (13.5%) laid out natural reasoning; 29 (4.7%) resorted to literary reasoning; and 10 (1.6%) offered uninformative explanations for their responses. Analysis revealed that participants’ responses did not differ by their class level at the university. Recommendations are provided for the relevance of these findings for teacher education.     

Greek-Cypriot elementary teachers’ epistemological beliefs about mathematics

Drawing on data from the Republic of Cyprus, this paper uncovers elementary teachers' epistemological beliefs about mathematics. Twenty-two experienced teachers were invited to individual semi-structured interviews. Thematic data-driven analyses identified three themes and eight sub-themes, which I discuss, taking their socio-cultural context into consideration. This study suggests that applying predetermined frameworks directly taken from the literature when examining teachers’ epistemological beliefs in mathematics can be problematic, as they might hinder other culturally specific beliefs from emerging. In closing, this paper presents some implications for the results on teacher education and professional development, as well as ideas for future research.     

The effect of video-based approach on prospective teachers’ ability to analyze mathematics teaching

This is an intervention study that explored the effect of using video lesson analysis methodology (VLAM) on the ability of prospective middle/high school mathematics teachers to analyze mathematics teaching. The sample of the study consisted of 26 female prospective mathematics teachers enrolled in a methods course at the United Arab Emirates University. The participants were divided equally into two groups, experimental and control. The experimental group was involved in video lesson analysis where they analyzed ten video lessons throughout the semester. The group members interacted via discussion forums through Blackboard technology. Both groups wrote analyses of two video lessons, one before the intervention program and another at the end of it. It was found that the intervention remarkably improved the ability to analyze mathematics teaching of the experimental group while little improvement occurred to the control group. Implications for teacher education programs are discussed.     

Impact of satisfaction and commitment on teachers’ organizational citizenship

This study examined the impact of job satisfaction and organizational commitment on teachers’ organizational citizenship behavior (OCB) in a structural equation model. The study was employed to a group of teachers and their supervisors. The results indicated that job satisfaction and commitment to the school had an impact on OCBs of the teachers and organizational commitment mediated the relation between job satisfaction and OCB.     

Investigating the development of prospective mathematics teachers’ pedagogical content knowledge of generalising number patterns through school practicum

The purpose of this study is to investigate the development of three prospective elementary mathematics teachers?? pedagogical content knowledge (PCK) of number patterns with its two components: (1) knowledge of students?? understanding and difficulties and (2) knowledge of topic-specific strategies and representations. For our investigation, we use Radford??s (ZDM Math Educ 40:83?C96, 2008) framework called ??architecture of algebraic pattern generalisation?? to redefine these two components of PCK in the context of patterns. A case study was conducted to explore the development of PCK through a school practicum course and to examine how observation of number pattern lessons in schools and discussions of these observation contribute to prospective teachers?? PCK. Analysis of the data indicated that observations and discussions of number pattern lessons resulted in considerable change in the way prospective teachers take into account students?? understanding of and difficulties with patterns, as well as in the way they use pattern-specific strategies such as algebraic pattern generalisation during their lessons.     

Preservice science teachers’ collective efficacy in a science methods course

This study investigated how preservice teachers develop collective efficacy when preparing lesson plans for a science methods course, and how this collective efficacy changed over time. The data were collected from four preservice science teachers working in a group to fulfil the requirements of the course. Findings revealed that four factors, including collaborative work, sharing the same goal, attitudes towards group work and group cohesion, played an important role in the development of collective efficacy among preservice teachers. The study also showed that collective efficacy improved continuously over the semester.