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1.
In this article we analyze the relations between academic mathematical knowledge and the mathematical knowledge associated with issues mathematics school teachers face in practice, according to the specialized literature, and restricted to the theme “number systems”. We present examples that illustrate some areas of conflict between those forms of knowledge. We point out some implications of our study for teacher education, such as: 1) the importance of making conflicts explicit and of discussing them with prospective teachers in order to develop a professionally relevant perception of academic mathematics; 2) the relevance of further research in order to better understand the extent of those conflicts and their effects on the process of integrating, in a body of professional knowledge, the different kinds of mathematical knowledge presented to prospective teachers.
Plinio C. MoreiraEmail: Email:
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2.
Elementary teachers are typically hesitant to teach science. While a limited knowledge of science content is a reason for this, limited science pedagogical content knowledge (PCK) has emerged as another reason in recent research. This study constitutes two case studies of a professional development program for elementary teachers involving mentoring by a university professor. The mentor took the role of a critical friend in joint planning and teaching of science. The study examines the nature of the mentoring relationship and reports the type of teacher learning that occurred, with a particular focus on the teachers’ development of science PCK.
Ken AppletonEmail:
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3.
Student teaching (guided teaching by a prospective teacher under the supervision of an experienced “cooperating” teacher) provides an important opportunity for prospective teachers to increase their understanding of mathematics in and for teaching. The interactions between a student teacher and cooperating teacher provide an obvious mechanism for such learning to occur. We report here on data that is part of a larger study of eight student teacher/cooperating teacher pairs, and the core themes that emerged from their conversations. We focus on two pairs for whom the core conversational themes represent disparate approaches to mathematics in and for teaching. One pair, Blake and Mr. B., focused on controlling student behavior and rarely talked about mathematics for teaching. The other pair, Tara and Mr. T., focused on having students actively participating in the lesson and on mathematics from the students’ point of view. These contrasting experiences suggest that student teaching can have a profound effect on prospective teachers’ understanding of mathematics in and for teaching.
Steven R. WilliamsEmail:
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4.
Ireland has two official languages—Gaeilge (Irish) and English. Similarly, primary- and second-level education can be mediated through the medium of Gaeilge or through the medium of English. This research is primarily focused on students (Gaeilgeoirí) in the transition from Gaeilge-medium mathematics education to English-medium mathematics education. Language is an essential element of learning, of thinking, of understanding and of communicating and is essential for mathematics learning. The content of mathematics is not taught without language and educational objectives advocate the development of fluency in the mathematics register. The theoretical framework underpinning the research design is Cummins’ (1976). Thresholds Hypothesis. This hypothesis infers that there might be a threshold level of language proficiency that bilingual students must achieve both in order to avoid cognitive deficits and to allow the potential benefits of being bilingual to come to the fore. The findings emerging from this study provide strong support for Cummins’ Thresholds Hypothesis at the key transitions—primary- to second-level and second-level to third-level mathematics education—in Ireland. Some implications and applications for mathematics teaching and learning are presented.
John O’DonoghueEmail:
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5.
The main goal of the study reported in our paper is to characterize teachers’ choice of examples in and for the mathematics classroom. Our data is based on 54 lesson observations of five different teachers. Altogether 15 groups of students were observed, three seventh grade, six eighth grade, and six ninth grade classes. The classes varied according to their level—seven classes of top level students and six classes of mixed—average and low level students. In addition, pre and post lesson interviews with the teachers were conducted, and their lesson plans were examined. Data analysis was done in an iterative way, and the categories we explored emerged accordingly. We distinguish between pre-planned and spontaneous examples, and examine their manifestations, as well as the different kinds of underlying considerations teachers employ in making their choices, and the kinds of knowledge they need to draw on. We conclude with a dynamic framework accounting for teachers’ choices and generation of examples in the course of teaching mathematics.
Orit ZaslavskyEmail:
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6.
This study investigated 481 in-service elementary teachers’ level of mathematical content knowledge, attitudes toward mathematics, beliefs about the effectiveness of inquiry-based instruction, use of inquiry-based instruction and modeled the relationship among these variables. Upper elementary teachers (grades 3–5) were found to have greater content knowledge and more positive attitudes toward mathematics than primary teachers (grades K-2). There was no difference in teachers’ beliefs about effective instruction, but primary level teachers were found to use inquiry-based instruction more frequently than upper elementary teachers. Consistent with Ernest’s [Ernest (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15(1), 13–33] model of mathematics teaching, content knowledge, attitudes, and beliefs were all found to be related to teachers’ instructional practice. Furthermore, beliefs were found to partially mediate the effects of content knowledge and attitudes on instructional practice. Content knowledge was found to be negatively related to beliefs in the effectiveness of inquiry-based instruction and teachers’ use of inquiry-based instruction in their classrooms. However, overall, teachers with more positive attitudes toward mathematics were more likely to believe in the effectiveness of inquiry-based instruction and use it more frequently in their classroom. Teacher beliefs were found to have the strongest effect on teachers’ practice. Implications for the goals and objectives of elementary mathematics methods courses and professional development are discussed.
Jesse L. M. WilkinsEmail:
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7.
Preservice teachers’ knowledge of proof by mathematical induction   总被引：2，自引：1，他引：1  
There is a growing effort to make proof central to all students’ mathematical experiences across all grades. Success in this goal depends highly on teachers’ knowledge of proof, but limited research has examined this knowledge. This paper contributes to this domain of research by investigating preservice elementary and secondary school mathematics teachers’ knowledge of proof by mathematical induction. This research can inform the knowledge about preservice teachers that mathematics teacher educators need in order to effectively teach proof to preservice teachers. Our analysis is based on written responses of 95 participants to specially developed tasks and on semi-structured interviews with 11 of them. The findings show that preservice teachers from both groups have difficulties that center around: (1) the essence of the base step of the induction method; (2) the meaning associated with the inductive step in proving the implication P(k) ⇒ P(k + 1) for an arbitrary k in the domain of discourse of P(n); and (3) the possibility of the truth set of a sentence in a statement proved by mathematical induction to include values outside its domain of discourse. The difficulties about the base and inductive steps are more salient among preservice elementary than secondary school teachers, but the difficulties about whether proofs by induction should be as encompassing as they could be are equally important for both groups. Implications for mathematics teacher education and future research are discussed in light of these findings.
George N. PhilippouEmail:
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8.
In modern mathematical teaching, it has become increasingly emphasized that mathematical knowledge should be taught by problem-solving, hands-on activities, and interactive learning experiences. Comparing the ideas of modern mathematical education with the development of ancient Chinese mathematics, we find that the history of mathematics in ancient China is an abundant resource for materials to demonstrate mathematics by hands-on manipulation. In this article I shall present two cases that embody this idea of a hands-on approach in ancient Chinese mathematics, at the same time offering an opportunity to show how to utilize materials from the history of Chinese math in modern mathematical education.
Youjun WangEmail:
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9.
The first author, a student in a graduate children’s literature class, designed a project to locate “good” mathematics-based children’s literature selections. However, the reference tools usually consulted (e.g., Books in Print) to locate books by topic were of little help, and those she located under individual mathematics topics were mostly traditional mathematics books rather than good read-aloud selections. Consequently, she perused the university library’s sizeable juvenile collection to find books that would meet her selection criteria. This article describes the influence of two landmark documents for mathematics teaching and learning—Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989) and Principles and Standards for School Mathematics (NCTM, 2000)—as she engaged in the process.
Eula Ewing MonroeEmail:
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10.
This article arises from a study whose overall purpose was to investigate the relationship between Colombian mathematics teachers’ conceptions of beginning algebra and their conceptions of their own teaching practices. The teachers’ understandings of their teaching practices were explored with a view to unravelling their conceptions of change in their teaching. Focusing on the perspectives of teachers afforded opportunities that exposed the powerful role that the teachers’ conceptions of social/institutional factors of teaching played in their conceptions of their practices. The degree to which the teachers attributed these (external) factors as crucial reasons for what they do in their teaching was the basis of a categorisation of their conceptions of the crucial determinants of their teaching practices into three types. The findings are particularly relevant to our understanding of the stability of mathematics teaching approaches in the Colombian context but have likely implications for a range of international education contexts. Specific implications for the development of the research into teachers’ conceptions of mathematics and its teaching, and for teacher education programmes are presented.
Alan J. BishopEmail:
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11.
There is an increasing awareness of the social dimension in mathematics teacher education. Collaboration and co-operation are regarded as key factors in professional development. In this paper I will analyse some tensions that might arise when the professional development of mathematics teachers is considered a collective enterprise. I will present phenomenological group interview as a method that is designed to reveal the collective character of teacher development. Some primary teachers’ collective reflections on an ongoing professional development process will be interpreted by focusing on the concepts of routine and collective orientation. The discussion is centred on the ambivalence of routines, as facilitators of practice, and collective orientations, as socially-agreed-upon knowledge base, for mathematics teachers’ professional development.
Uwe GellertEmail:
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12.
This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.
Andreas J. StylianidesEmail:
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13.
The focus of this study was to understand knowledge flows among teachers by examining what types of knowledge was shared by teachers, as well as what motivates or hinders teachers to share knowledge online. We examined an electronic mailing list (listserv) supporting a community of practice of literacy teachers. Data were gathered on the teachers in the listserv through online observations. Additional data were collected through semi-structured telephone interviews with 20 teachers. Findings suggest that two motives of community involvement––collectivism, and principlism appear to be the main motivators for knowledge sharers to share knowledge, while lack of knowledge and competing priority appear to be the main barriers. Practical implications for knowledge sharing and suggestions for future research are discussed. The findings of this study inform teachers, listserv moderators, teacher associations, as well as researchers of educational technology who are interested in knowledge sharing among teachers within communities of practice mediated by computer networks.
Noriko HaraEmail:
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14.
It is generally perceived that Chinese elementary teachers have a profound understanding of the school mathematics they teach. This perception has led to further interest in understanding teacher education practices in China. As some dramatic changes in elementary teacher preparation have taken place in China over the past decade, this article aims to outline these changes with a focus on curriculum provided in the new 4-year bachelor preparation programs. Sample mathematics teacher educators in China were also surveyed to gather insiders’ views about teacher preparation practices and to identify relevant issues. We believe that elementary teacher preparation and its changes in China can provide an important case for mathematics teacher educators around the world to reflect on teacher education practices in their own systems.
Yeping LiEmail:
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15.
The doctoral advisor is said to be one of the most important persons—if not the single most critical person—with whom doctoral students will develop a relationship during their doctoral degree programs (Baird 1995). However, we have limited knowledge regarding how doctoral advisors see their roles and responsibilities as advisors. Therefore, through in-depth interviews, we explored the perceptions of 25 exemplary doctoral advisors, who have graduated a large number of doctoral students, about their roles and responsibilities as advisors. We conclude this article with implications for doctoral education.
Ann E. AustinEmail:
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16.
Debates persist over the knowledge needed to teach elementary reading effectively. In one commonly held view verbal ability is what matters most and the best approach to improving teacher quality is to recruit teachers who themselves are good readers. Others argue that teachers need special forms of professional knowledge that differ substantially from common adult reading and verbal ability. These different assumptions about what teachers need to know are directly relevant to whether teaching reading demands specialized professional knowledge and they have lead to radically different policy recommendations for both teacher preparation and induction. This study presents preliminary evidence that elementary reading teachers can hold a special knowledge of language, text, and reading process that differs substantially from common reading and verbal ability. Implications for the measurement and study of teacher quality and related implications for teacher evaluation and teacher development are discussed.
Geoffrey PhelpsEmail:
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17.
The study presented in this article investigates forms of mathematical interaction in different social settings. One major interest is to better understand mathematics teachers’ joint professional discourse while observing and analysing young students mathematical interaction followed by teacher’s intervention. The teachers’ joint professional discourse is about a combined learning and talking between two students before an intervention by their teacher (setting 1) and then it is about the students learning together with the teacher during their mathematical work (setting 2). The joint professional teachers’ discourse constitutes setting 3. This combination of social settings 1 and 2 is taken as an opportunity for mathematics teachers’ professionalisation process when interpreting the students’ mathematical interactions in a more and more professional and sensible way. The epistemological analysis of mathematical sign-systems in communication and interaction in these three settings gives evidence of different types of mathematical talk, which are explained depending on the according social setting. Whereas the interaction between students or between teachers is affected by phases of a process-oriented and investigated talk, the interaction between students and teachers is mainly closed and structured by the ideas of the teacher and by the expectations of the students.
Heinz SteinbringEmail:
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18.
In this response we address some of the significant issues that Tony Brown raised in his analysis and critique of the Special Issue of Educational Studies in Mathematics on “Semiotic perspectives in mathematics education” (Sáenz-Ludlow & Presmeg, Educational Studies in Mathematics 61(1–2), 2006). Among these issues are conceptualizations of subjectivity and the notion that particular readings of Peircean and Vygotskian semiotics may limit the ways that authors define key actors or elements in mathematics education, namely students, teachers and the nature of mathematics. To deepen the conversation, we comment on Brown’s approach and explore the theoretical apparatus of Jacques Lacan that informs Brown’s discourse. We show some of the intrinsic limitations of the Lacanian idea of subjectivity that permeates Brown’s insightful analysis and conclude with a suggestion about some possible lines of research in mathematics education.
Luis RadfordEmail:
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19.
Video has assumed an increasingly prominent role in teacher education, particularly in the form of the viewing of videotaped class lessons by preservice teachers. Yet there is little research that confirms whether preservice teachers attend to the aspects of the video(s) that teacher educators anticipate or desire. This article explores this issue and reports on the impact of video viewing as a means to improve teachers’ ability to be observers of classroom practice. We utilized a pre- and post-test design to measure the quantity and type of classroom events that preservice mathematics teachers noticed before and after a teaching methods course where improving observation skills was an explicit goal. The results of the pre-assessment suggest that preservice teachers generally do not enter teaching methods courses with well-developed observation skills. The post-assessment indicates that the course led to significant increases in preservice teachers’ observation skills, particularly in teachers’ ability to notice features of the classroom environment, mathematical content of a lesson, and teacher and student communication during a lesson.
Jon R. StarEmail:
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20.
This empirical paper considers the different purposes for which teachers use examples in elementary mathematics teaching, and how well the actual examples used fit these intended purposes. For this study, 24 mathematics lessons taught by prospective elementary school teachers were videotaped. In the spirit of grounded theory, the purpose of the analysis of these lessons was to discover, and to construct theories around, the ways that these novice teachers could be seen to draw upon their mathematics teaching knowledge-base in their lesson preparation and in their observed classroom instruction. A highly-pervasive dimension of the findings was these teachers’ choice and use of examples. Four categories of uses of examples are identified and exemplified: these are related to different kinds of teacher awareness.
Tim RowlandEmail:
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