Will Teachers Create Opportunities for Discussion when Teaching Proof in a Geometry Classroom?

This study presents an analysis of whether geometry teachers create opportunities for student discussion when engaging students in proving as revealed through the use of a multimedia survey instrument. We presented 42 secondary mathematics teachers with 8 multimedia narratives set in the situation of doing proofs in high school geometry and asked them to choose what they would do next, presenting options that included a normative instructional action which closed off discussion and less typical actions that encouraged student discussion. Our analysis provides insight into the professional obligations that teachers use to justify their departure from the norm in order to encourage student talk as well as the background variables that are associated with such decisions. We found that while secondary mathematics teachers frequently chose to promote discussion in their classrooms, the rationale that they chose for this decision and the reasons they might choose not to immediately encourage discussion differed according to the amount of experience they had teaching geometry. We use these differences to illustrate how the professional obligations can be used to better understand how teacher decisions are rationalized at the level of the instructional situation.     

First‐year secondary school mathematics students’ conceptions of mathematical proofs and proving

The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students (at the same time student teachers). The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for exams based on imitative reasoning which can be described as a type of reasoning built on copying proof, for example, by looking at a textbook or course notes proof or through remembering a proof algorithm. Moreover, they addressed to the differences between mathematics taught in high school and university as the main cause of their difficulties in proof and proving.     

Could Elementary Mathematics Textbooks Help Give Attention to Reasons in the Classroom?

Trainee teachers, new and non-specialist teachers of elementary mathematics have a tendency to avoid thought about reasons in mathematics. Instead, they tend to favour the development of computational skill through the rote application of procedures, routines and algorithms. Could elementary mathematics textbooks serve as models of practice and support the professional development of these teachers? Eighteen such textbooks for use by 7 to 11-year-old children in England were examined for their potential to help teachers attend to reasons. It was found that they were unlikely to point teachers towards reasons for patterns and procedures. Although some had the potential to help in introducing and structuring a lesson, they were unlikely to induce the teacher to address matters of understanding in their discourse. Their implicit message tended to be that mathematics education is about computational skill development through routines, algorithms and practice.     

Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving

In this study, we report on what types of mathematical knowledge for teaching teachers (MKTT) mathematics teacher educators (MTEs) use and develop when they work together and reflect on their teaching in a Community of Practice while helping prospective primary teachers (PTs) generate their own mathematical knowledge for teaching in learning mathematics via problem solving. Two novice MTEs worked with an experienced MTE and reflected on the process of learning to teach via problem solving and supporting PTs in developing deep understandings of foundational mathematical ideas. Taking a position of inquiry as stance, we examined our experiences teaching mathematics content courses for PTs via problem solving. We found that all of the MTEs used and developed some MKTT through (a) understanding and deciding on the mathematical goals of both the individual lessons and the two-course sequence as a whole, (b) choosing and facilitating tasks, and (c) using questions to scaffold PTs learning and engage them in mathematical processes such as making conjectures, justifying their reasoning, and proving or disproving conjectures.     

Learning to prove by investigations: a promising approach in Dutch secondary education

The Mathematics Department of the University of Nijmegen in collaboration with the Graduate School of Education has developed a math course in the field of fractals and dynamic processes for volunteer students in the second phase of secondary education in the Netherlands. The students, of approximately 16 years of age, show a special interest in, and an aptitude for mathematics and informatics. One of the main goals of the course was to highlight the deductive aspect of mathematics, an aspect that is neglected in the ordinary math curriculum of secondary education in the Netherlands. That goal was pursued by giving the students ample opportunity to conduct investigations on their own and in a way that they would be responsible for judging the correctness of their arguments in making mathematical deductions. In that way proving is imbedded in a larger structure and becomes a tool for the students to convince each other. During the courses we searched for the right way for teachers to coach these investigations, that is: to find a balance between mere concentration on guiding the process of the students' investigations and active intervention in the learning process of proving. In this article we illustrate with two examples – the first from the '95–'96 course and the second from the '96–'97 course – in what way we adjusted our coaching. Our results are explorative, but our approach appears promising and we are convinced that investigations as learning environment for proving would also be a valuable part for the regular secondary math education in the Netherlands. This revised version was published online in July 2006 with corrections to the Cover Date.     

西藏地区中小学数学教师认识信念的调查研究

本研究选取西藏地区353名中小学数学教师作为研究对象,通过问卷调查、访谈的方法,采用描述性统计、独立样本t检验和单因素方差分析等探索西藏中小学数学教师认识信念的现状和背景因素差异。研究发现:西藏中小学数学教师的数学认识信念总体上呈进步倾向,在教师自身认识信念上则存在显著的学校位置差异,并且职称对数学教师的数学认识信念、教龄对数学教师的学生信念、学历对数学教师的数学学习信念和数学教学信念均产生一定影响。     

A situational study of the meanings of personal excellence for secondary mathematics teachers and students

Excellence in education has become the watchword in the past decade. This paper describes aspects of research conducted with two secondary mathematics teachers and seven senior students, which explored their perceptions of mathematics as they have experienced it, and their beliefs about personal excellence in school mathematics. Three categories of excellent students are described as identified by the participants. While the teachers endeavoured to extend the meaning of excellence beyond test achievement alone, the system within which they function largely supports a technical (excellence as marks) conception of education. I argue that this has led teachers and students to a restricted view of the potential for excellence in the secondary mathematics classroom. In particular, it limits the possibility for a strong understanding of personal excellence as an inseparable intellectual and moral journey of becoming, fostered in the environment of a genuine community of mathematics learners.An earlier draft of this paper was presented at the Canadian Society for the Study of Education Annual Meeting, Carleton University, June 10, 1993.     

Science and Mathematics Teachers Working Toward Equity Through Teacher Research: Tracing Changes Across Their Research Process and Equity Views

We investigated secondary science and mathematics teachers engaged in a two-and-a-half-year professional development effort focused on equity. We examined how teachers conducting research on their own instructional practices—a central learning strategy of the professional development project—informed and/or constrained their views related to three strands of equity: teachers and teaching, students and learning, and students’ families and communities. Data collected included recordings of professional development seminars and school-site meetings, three sets of individual interviews with teacher researchers, and drafts and final products of the classroom research teachers conducted. From our qualitative analyses of data, we found that most teachers addressed at least two of the three equity strands in researching their own practice. We also found that most transformed their understandings of teachers and students as a result of their teacher research process. However, teachers’ views of families and communities changed in less substantive ways. We close with recommendations for other researchers and professional developers intent on supporting science and mathematics teachers in using teacher research to work toward equity.     

Domestic and international power relations in a Cameroonian mission school system

This paper is a critical ethnography of mathematics teachers in a Cameroonian private, mission school system. Findings from in-depth interviews with secondary mathematics teachers indicate that most of these teachers perceive their educational situation as disconnected, in several ways, from their cultures. However, most participants look to western processes as models for their own development. An analysis of state and international postcolonial hegemonies which influence these teachers is presented, including allocation of resources, curricular agendas, and professional development opportunities.     

Reflections on the Dilemmas and Tensions in Mathematics Education Courses for Student Teachers

The paper outlines the dilemmas and paradoxes faced by lecturers and student teachers as they interact in a mathematics education subject that deals with both mathematics as a discipline and as a language, and with appropriate pedagogies for the teaching and learning of mathematics in primary schools. For the lecturer there is a tension between comforting and challenging the students. Are they to be wooed into a more positive attitude to mathematics, at the cost of avoiding the complexity of the discipline; or are they to be challenged by the unique character of mathematics, at the risk of alienation and exclusion? The latter often returns students to their original perception of maths as a harsh and unforgiving subject which is beyond their capabilities as they struggle with unfamiliar concepts and the discomfort of ‘not knowing’. For student teachers there is, paradoxically, a desire to ‘instil understanding’ when they themselves may not fully understand. They often idealise what is good practice but deny it in their own learning.     

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.     

Research mathematicians as teacher educators: focusing on mathematics for secondary mathematics teachers

We believe that professional mathematicians who teach undergraduate mathematics courses to prospective teachers play an important role in the education of secondary school mathematics teachers. Thus, we explored the views of research mathematicians on the mathematics that should be taught to prospective mathematics teachers, on how the courses they teach can serve teachers in their work with school students, and on the changes they would implement if their courses were designed specifically for prospective teachers. We constructed profiles of the four mathematicians based on their responses to a clinical interview. We employed the construct of mathematics teacher-educators’ triad in the reflective analysis of our findings and extended the construct based on the results of this study. In conclusion, we commented on potential ways to draw stronger connections between university mathematics and the mathematics taught in schools.     

初中农村教师数学知识状况及来源的调查研究——以勾股定理为例

初中农村教师所给出勾股定理的证明方法或思路都是正确的,但是这些证明方法或思路大多来自于教材;"教科书"和"自身的教学经验和自我反思"是初中农村教师发展数学知识的两个主要来源,其次是"作为学习者的经验",接下来是"专业书刊"、"和同事的日常交流"、"课堂听课和教研活动"、"教学辅导资料"、"在职培训"、"网络资源"和"职前教育".促进初中农村教师数学知识发展需要做到:充分利用发展数学知识的一些常见途径,不断扩充发展数学知识的途径.     

Where's the Math? Prospective Teachers Visit the Workplace

The integration of academic and vocational subject matter is offered in response to efforts to make the study of mathematics meaningful and engaging for all students,as well as aid in the preparation of a mathematically literate workforce. Yet,teachers often come to mathematics education with more ‘pure’ than ‘applied’ backgrounds making it difficult for them to draw upon their own experiences to make subject matter meaningful. This paper analyses prospective teachers' opportunities to connect subject matter with workplace contexts. It examines the degree of importance prospective teachers place on workplace connections and the ways in which they incorporate these connections in classroom lesson plans. Results suggest that given opportunities to visit workplace sites, it is not a trivial task for prospective teachers to: 1) make the mathematics in work explicit, and 2) keep the mathematics contextualized when designing activities and problems for students. These results have implications for teacher education and the support prospective teachers require in building networks connecting mathematics, pedagogy,and work. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Teacher as Examiner: the case of mathematics coursework

Recent developments in curriculum and assessment in the UK have led to increased involvement of teachers in high‐stakes summative assessment of their own students. Case studies of experienced secondary school mathematics teachers reading and assessing General Certificate of Secondary Education (GCSE) coursework texts show that they experience tensions between their various roles and responsibilities as teachers and as examiners. Moreover, different teachers appear to resolve these tensions in different ways, adopting various positions in relation both to the content of the written coursework texts and to the student‐authors of the texts. Variations in teachers’ approaches to reading and assessing mathematics coursework may lead to differences in the ranks or grades allocated and, even where they do not, the meanings of the grades given by different teachers may be substantially different. Through examining teachers’ assessment practices, questions are raised both about the validity of such assessments and about the compatibility of the coursework examination system with the aims of the curriculum development which gave rise to it.     

Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Understanding the knowledge bases of mathematics teachers is an important task in working towards the construction of adequate models for: (i) teacher education and development, and (ii) teacher operations in the classroom. To date, little systematic attention has been focused on this task. The primary aim of this study is to obtain a view from the field of mathematics teacher knowledge with respect to content knowledge in mathematics, content‐specific pedagogical knowledge in mathematics and curriculum knowledge relevant to teaching tasks. This study has used data obtained from a survey of primary teachers and secondary mathematics teachers. Analysis of the results has indicated that less than half of the teachers in the study believed that they were sufficiently prepared in mathematics content, and that almost two‐thirds of the teachers in the sample believed that their level of knowledge in contemporary teaching methodologies in mathematics is not sufficient for their role as school teachers. Key differences emerge between the primary and secondary sectors and also within the secondary sector. Implications for preservice and in‐service mathematics teacher education are drawn.     

The Influence of Teachers’ Efficacy and Beliefs Regarding Mathematics Instruction in the Early Childhood Classroom

This was a correlational study conducted with a population of prekindergarten educators from a large, metropolitan school district. The purpose was to examine if there were relationships between and among early childhood teachers’ sense of self-efficacy, their beliefs about the importance of mathematics, and their mathematics instructional practices. Examining teachers’ efficacy and beliefs can inform educational practice and help to differentiate between more and less successful instructional practices when teaching mathematics in the early childhood classroom. Data were collected on teacher efficacy and teacher beliefs about the importance of mathematics with two self-report questionnaires. The hypothesis that the teachers higher in efficacy will rate the importance of mathematics higher on the teacher-belief scale than the teachers with lower efficacy was found to be true with this sample, but the correlation was weak. The level of efficacy of the early childhood teachers in this sample confirmed that in assessing their capabilities, they rate themselves high in instructional strategies, classroom management, and student engagement. The early childhood teachers did not rate their belief in mathematics as high as their efficacy. Observations of mathematics instructional practices were conducted with twenty teachers. It was hypothesized that the combination of high teacher efficacy and high teacher mathematics beliefs would show alignment with the presence of standards-based mathematics instructional practices. The results were not statistically significant. No correlation signals a need for more research to explore what other personal or external factors relate to mathematics instructional practices in the early childhood classroom. The research may inform pre kindergarten teachers and teacher educators about effective instructional strategies and knowledge needed to launch early childhood students on a developmentally appropriate pathway to mathematical literacy.     

Developing culturally responsive mathematics teachers: secondary teachers’ evolving conceptions of knowing students

Research advances in teaching, learning, curriculum, and assessment have not changed the continued underperformance of marginalized students in mathematics education. Culturally responsive teaching is a means of addressing the needs of these students. It is sometimes challenging, however, to convince secondary mathematics teachers about the importance of culture in mathematics education. To contribute to what is known about supporting secondary mathematics teachers in developing a culturally responsive teaching practice, we studied the impacts of a graduate course called Culture in the Mathematics Classroom on 13 teachers enrolled in the course. The course was designed to guide secondary mathematics teachers in understanding and growing their capacity to enact culturally responsive teaching in their classrooms. The purpose of our research was to explore how teachers’ perceptions changed as a result of their engagement in the class with respect to understanding the role of culture in knowing and being responsive to their students. Specifically, we examined how each of the four course projects seemed to individually and collectively influence teachers’ thinking. Overall, teachers appeared to expand their cultural awareness and dispositions for cultural responsiveness that would support them in knowing and supporting their students in the manner of a culturally responsive teacher. Teachers did not, however, develop some more “advanced” understandings related to power and privilege in society. This study provides researchers and mathematics teacher educators with a potential analytic framework for understanding teacher change with respect to culturally responsive teaching.