Pre-service versus in-service mathematics teachers’ opinions of mathematics reform in post-primary schools in Ireland

A new approach to the teaching and learning of mathematics has been introduced on a phased basis in post-primary education in Ireland. As part of the OECD PISA 2012, a survey was undertaken of in-service mathematics teachers in Ireland, with the aim of gathering data on the opinions of teachers about this reform. Here, a modified version of the survey was distributed to pre-service teachers immediately before and after a four-month teaching placement to ascertain if there were differences in the views expressed by pre-service and in-service teachers in relation to the teaching and learning of mathematics, and also if there were differences expressed by the pre-service teachers before and after their placement in post-primary schools. Analysis of the results showed considerable levels of agreement between all groups, with some exceptions in relation to confidence in teaching the reformed curriculum, emphasis on various teaching and learning activities, and usage of ICT in the classroom.     

The effects of constructivist learning environment on prospective mathematics teachers＇ opinions

To explore the effects of constructivist learning environment on prospective teachers＇ opinions about ＂＇mathematics, department of mathematics, discrete mathematics, countable and uncountable infinity＂ taught under the subject of Cantorian Set Theory in discrete mathematics class, 60 first-year students in the Division of Mathematics Education at the Department of Science and Mathematics in Buca Education Faculty at Dokuz Eylul University were divided into two homogenous groups. In order to do this segmentation, Minimum Requirements Identification Test was developed and used by the researchers. This test includes concepts like ＂＇set＂, ＂correlation＂ and ＂function＂, which are required to understand Cantorian Set Theory. While the control group was taught by traditional methods, a teaching method based on a constructivist approach was applied to the experimental group. Data were gathered by an open-ended questionnaire administered to total 40 students, 20 from the each group. Collected data were evaluated through content analysis. In the end, despite the minor differences, no statistically significant difference was found between the opinions of control and experimental groups about mathematics （χ^2calculation=2.578, SD=3, p〉0.05）, department of mathematics （χ^2calculation=3.185, SD=3, p〉0.05） and discrete mathematics （χ^2calculation=4.935, SD=3, p〉0.05） after the instruction. However, opinions about Cantorian Set Theory were significantly differentiated between experimental and control groups after the instruction （χ^2 calculation= 13.486, SD=2, p〈0.05）.     

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).     

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.     

Formatively assessing prospective teachers’ skills in leading mathematics discussions

Educational Studies in Mathematics - Mathematics discussions are important for helping students to develop conceptual understanding and to learn disciplinary norms and practices. In recent years,...     

Investigating teachers’ images of mathematics

Research suggests that understanding new images of mathematics is very challenging and can contribute to teacher resistance. An explicit exploration of personal views of mathematics may be necessary for pedagogical change. One possible way for exploring these images is through mathematical metaphors. As metaphors focus on similarities, they can be used to express already-held perceptions about the nature of mathematics. In addition to providing a way of talking about current views of mathematics, the analogous dimensions of metaphors can prompt new ways of thinking about these images. In this article, I consider the use of metaphors as a strategy for explicating elementary teachers’ views of mathematics. I claim that the investigation of metaphors of mathematics helped create a shared communicative space and enhanced the quality of the discussion with the teachers. In particular, our exploration of the metaphor mathematics is a language encouraged a consideration of the humanistic dimensions of mathematics and contributed to a varied re-imaging of mathematics.     

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.     

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.     

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.     

Do prospective mathematics teachers teach who they say they are?

In this case study, the professional mathematics teacher identity (PMTI) of final year mathematics education students is investigated in terms of their self-perceived and actualised identity. These prospective teachers were required to discuss and describe their own PMTI in terms of three aspects: mathematics specialisation, teaching-and-learning specialisation, and caring. Subsequently, they were observed in the classroom, where the actualisation of their PMTI was considered in terms of the same three. The participants’ perceptions of their own PMTI and the actualisation of that PMTI in the classroom were found not to be congruent. While their self-perceptions regarding their prowess as Mathematics Specialists were accurate, since this is concretely tested as part of their studies, their self-perceptions as teaching-and-learning specialists and particularly as Carers, were not verifiable in their classroom practice. Espoused theory, theory that the individual perceives as true and valid, and which may thus be seen as intrinsic to their PMTI’s, is not necessarily enacted.     

Impact of a content and methods course sequence on prospective secondary mathematics teachers’ beliefs

In this article, we report on a study of beliefs about mathematics, teaching, and proof conducted with six prospective secondary mathematics teachers as they completed a two-semester sequence of a content course and a methods course. The initial beliefs of the participants were identified using interview and survey data, and potential shifts in beliefs were examined through further interview and survey data combined with classroom observations and written work. While their beliefs about mathematics and proof appeared to be relatively stable, their beliefs about teaching shifted from a more teacher-centered view to beliefs that foreground the activities and understandings of the students. These shifts are analyzed using the construct of belief structures, and activities and events from the courses that may have facilitated the shifts are identified. The results are consistent with the literature in some respects, such as the stability of the participants’ beliefs about mathematics. On the other hand, our results present new information about how prospective secondary mathematics teachers’ beliefs about teaching may be impacted.     

The role of subject knowledge in primary prospective teachers’ approaches to teaching the topic of area

The relationship between primary teachers?? subject knowledge and their approaches to teaching is an ongoing concern. This study reviews the relationship between prospective teachers?? subject knowledge in the topic of area and their approaches to teaching that topic. The research presents case studies of four primary prospective teachers on a 1-year postgraduate teaching course. The strengths and limitations of their subject knowledge are examined, in relation to their selection of teaching activities. The results suggest connections between these strengths and limitations, in relation to espoused teaching activities and pedagogical orientations. This questions the assumption that secure subject knowledge is necessarily transformed into effective teaching and concurs with other research that suggests other factors may be involved, such as knowledge of learners.     

What mathematics do teachers with contrasting teaching approaches address in probability lessons?

This paper (1) presents a conceptual framework for analyzing the mathematics addressed in probability lessons and (2) uses the framework to compare the mathematics that two teachers with contrasting teaching approaches addressed in class when teaching the topic of probability. One teaching approach aimed to develop understanding; the other emphasized mechanistic answer finding. Class work on 193 problems was analyzed qualitatively and quantitatively, showing some similarities and some differences in the mathematics that the two teachers offered to students. The differences found seemed to be linked to the teachers’ teaching approaches. The findings suggest that teachers who adopt different teaching approaches, to some extent, make available to learn different mathematics even when they use the same textbooks.     

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.     

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.     

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.     

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.     

An analysis of teachers’ self-reported competencies for teaching metacognition

For successful reading experiences in native and/or foreign/second language, individuals need to benefit from not only cognitive strategies but also metacognitive strategies. Although research found reading comprehension and performance increase following metacognitive trainings, such findings may not transfer into mainstream classrooms as easily for several reasons. This study, therefore, aimed to disseminate the phenomenon of teaching metacognition with an emphasis on teacher’s instrumental role during classroom learning. More specifically, it investigated language instructors’ metacognition and their self-reported competencies for teaching metacognition. It also examined whether and how self-reported competencies changed following a professional development (PD) module of teaching for metacognition. Utilising Metacognitive Awareness Inventory (MAI) and think aloud protocols for instructional planning, this study found most participants were either highly metacognitive or metacognitive individuals. It was also found that most participants were initially not knowledgeable about and/or competent in teaching metacognition. Following PD, highly metacognitive teachers developed authentic lesson plans manifesting metacognition instruction while metacognitive teachers adopted similar instructional designs presented during the PD. Besides, half of the participants appreciated teaching metacognitively following the PD. Finally, under the light of these findings, future research and policy adjustments were proposed.     

Didactic competencies of teachers from the learner’s viewpoint

Teacher competencies can be researched in many different ways. In the present article they are studied from the learner’s viewpoint. The article presents results of the extensive project “Teacher Education for New Competencies for the Knowledge Society and the Role of these Competencies in Educational Goal Attainment at School”, carried out at the Faculty of Arts of the University of Ljubljana, with the financial support of the Slovenian Ministry of Education and Sport. We present the results of the learner’s judgement of the teacher’s didactic competencies. We formed a questionnaire with which we operationalised teacher competencies with the purpose of measuring the teacher’s efficiency of instruction. In order to achieve the goals of modern instruction it is important to engage both models: learner‐centred and teacher‐centred education. The research showed that elements of the traditional instruction model are more frequently used than the model that demands a changed learner role.     

Elementary school teachers’ uses of mathematics curricular resources

With the adoption of new content standards, teachers are often left without adequate curriculum resources. This study examined how educators used their curricular resources to teach new mathematics standards in the USA. Analyses of open-ended survey responses from 257 teachers and teacher–leaders in Grades 3 through Grade 5 indicated that every educator reported supplementing their districts’ or schools’ primary curricular resources with other materials. These supplements primarily included resources found for free on websites and resources that claimed to be aligned to the new standards, but varied in terms of alignment to national standards for effective mathematics curriculum. Implications for this study include further research on how teachers make decisions regarding curriculum resources as well as increasing teachers’ access to quality curriculum materials that can support students’ mathematical learning.