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1.
In 1997, an interview-based study of 70 research mathematicians was undertaken with a focus on how they ‘come to know’ mathematics, i.e. their epistemologies. In this paper, I discuss how these mathematicians understand their practices, locating them in the communities of which they claim membership, identifying the style which dominates their organisation of research and looking at their lived contradictions. I examine how they talk about ‘knowing’ mathematics, the metaphors on which they draw, the empiricist connections central to the work of the applied mathematicians and statisticians, and the importance of connectivities to the construction of their mathematical Big Picture. I compare the stories of these research mathematicians with practices in mathematics classrooms and conclude with an appeal for teachers to pay attention to the practices of research mathematicians and their implications for coming to know mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

2.
The objective of this study was to examine gender differences in the relations between verbal, spatial, mathematics, and teacher–child mathematics interaction variables. Kindergarten children (N = 80) were videotaped playing games that require mathematical reasoning in the presence of their teachers. The children’s mathematics, spatial, and verbal skills and the teachers’ mathematical communication were assessed. No gender differences were found between the mathematical achievements of the boys and girls, or between their verbal and spatial skills. However, mathematics performance was related to boys’ spatial reasoning and to girls’ verbal skills, suggesting that they use different processes for solving mathematical problems. Furthermore, the boys’ levels of spatial and verbal skills were not found to be related, whereas they were significantly related for girls. The mathematical communication level provided in teacher–child interactions was found to be related to girls’ but not to boys’ mathematics performance, suggesting that boys may need other forms of mathematics communication and teaching.  相似文献   

3.
This article examines mathematics teacher collegiality by focusing on both the ways in which teachers interacted as critical colleagues in a long-term professional development project and the evolving role of the teacher–educator–researcher as the facilitator of this project. The professional development collaboration comprised two phases: one focused on reading classroom discourse literature and one focused on supporting each other through cycles of action research related to mathematics classroom discourse. Lord’s (1994) critical colleagueship framework is used to examine how a study group of middle-grades (ages 11–16) mathematics teacher–researchers took (or did not take) a more critical stance toward their own teaching practice and that of their colleagues. We found that challenging interactions were related to instances in which the teachers interacted as critical colleagues and were marked by particular features including the use of particular words and the use of personal experience as a form of evidence. We present the ways in which we came to understand what it might look like to scrutinize one’s practice and findings related to the development of this type of collegiality across the two different phases of this project. We end with a section in which the teacher–educator–researcher who facilitated the professional development project reflects on the ways in which the analysis caused her to reconsider both the nature of argumentation in mathematics study group settings and what implications this has with respect to her own practice as a facilitator.  相似文献   

4.
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5.
Is it possible that a meeting of mathematicians and primary school teachers will be productive? This question became intriguing when one professor of mathematics initiated a professional development course for practicing primary school teachers, which he taught alongside a group of mathematics Ph.D. students. This report scrutinizes the uncommon meeting of these two communities, who have very different perspectives on mathematics and its teaching. The instructors had no experience in primary school teaching, and their professed goal was to deepen the teachers’ understanding of the mathematics they teach, while teachers were expecting the course to be pedagogically relevant for their teaching. Surprisingly, despite this mismatch in expectations, the course was considered a success by teachers and instructors alike. In our study, we analyzed a lesson on division with remainder for teachers of grades 3–6, taught by the professor. The framework used for the data analysis was mathematical discourse for teaching, a discursive adaptation of the well-known mathematical knowledge for teaching framework. Our analysis focuses on the nature of the interactions between the parties and the learning opportunities they afforded. We show how different concerns, which might have hindered communication, in fact fueled discussions, leading to understandings of the topic and its teaching that were new to all the parties involved. The findings point to a feasible model for professional development where mathematicians may contribute to the education of practicing teachers, while they are gaining new insights themselves.  相似文献   

6.
In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, Journal for Research in Mathematics Education 39:431–459, 2008) and the philosophical literature on mathematical proof, we identify three general strategies that mathematicians employ when reading proofs: appealing to the authority of other mathematicians who read the proof, line-by-line reading, and modular reading. We argue that non-deductive reasoning plays an important role in each of these three strategies.  相似文献   

7.
In this paper, we examine the support given for the ‘theory of formal discipline’ by Inglis and Simpson (Educational Studies Mathematics 67:187–204, 2008). This theory, which is widely accepted by mathematicians and curriculum bodies, suggests that the study of advanced mathematics develops general thinking skills and, in particular, conditional reasoning skills. We further examine the idea that the differences between the conditional reasoning behaviour of mathematics and arts undergraduates reported by Inglis and Simpson may be put down to different levels of general intelligence in the two groups. The studies reported in this paper call into question this suggestion, but they also cast doubt on a straightforward version of the theory of formal discipline itself (at least with respect to university study). The paper concludes by suggesting that either a pre-university formal discipline effect or a filtering effect on ‘thinking dispositions’ may give a better account for the findings.  相似文献   

8.
This study examined standard 6 and 8 (Standards 6 and 8 are the sixth and eighth years, respectively, of primary level schooling in Kenya.) students’ perceptions of how they use mathematics and science outside the classroom in an attempt to learn more about students’ everyday mathematics and science practice. The knowledge of students’ everyday mathematics and science practice may assist teachers in helping students be more powerful mathematically and scientifically both in doing mathematics and science in school and out of school. Thirty-six students at an urban school and a rural school in Kenya were interviewed before and after keeping a log for a week where they recorded their everyday mathematics and science usage. Through the interviews and log sheets, we found that the mathematics that these students perceived they used outside the classroom could be classified as 1 of the 6 activities that Bishop (Educ Stud Math 19:179–191, 1988) has called the 6 fundamental mathematical activities and was also connected to their perception of whether they learned mathematics outside school. Five categories of students’ perceptions of their out-of-school science usage emerged from the data, and we found that 4 of our codes coincided with 2 activities identified by Lederman & Lederman (Sci Child 43(2):53, 2005) as part of the nature of science and 2 of Bishop’s categories. We found that the science these students perceived that they used was connected to their views of what science is.  相似文献   

9.
The National Council of Teachers of Mathematics (NCTM) states that “Communication is an essential part of mathematics and mathematics education” (2000, p. 60). In fact, communication is one of the five process standards emphasized by NCTM. The communication standard highlights the importance of young children communicating their mathematical thinking coherently to peers and teachers. This standard also states that young children should use math language to express mathematical ideas (Baroody, 2000; Ginsburg, Inoue, & Seo, 1999; NCTM, 2000; Rubenstein & Thompson, 2002; Whitin & Whitin, 2003). Teachers must create a link between mathematics and language (Rubenstein & Thompson, 2002; Stigler & Hiebert, 2004). This article focuses on the informal strategies used by Melissa (a kindergarten teacher) that promoted the use of math language. The strategies were identified during a 3-month observational period in her classroom at Clinton Elementary (pseudonym). Clinton Elementary is located in a low-income neighborhood of a southern city that has a population of approximately 450,000. The neighborhood population is predominantly African-American (Davis, 1994).  相似文献   

10.
The aim of the paper is to advance understanding of in-service learning and skills sector trainee teachers’ learning and propose ways of improving their learning. A conceptual framework is developed by extending Billett’s (International Journal of Educational Research 47:232–240, 2008) conceptualisation of workplace learning, as a relationally interdependent process between the opportunities workplaces afford for activities and interactions and how individuals engage with these, to a third base of participation, the affordances of the initial teacher education course. Hager and Hodkinson’s (British Educational Research Journal 35:619–638, 2009) metaphor of ‘learning as becoming’ is used to conceptualise the ways trainees reconstruct learning in a continuous transactional process of boundary crossing between course and workplace. The findings of six longitudinal case studies of trainees’ development, and evidence from other studies, illustrate the complex interrelationships between LSS workplace affordances, course affordances and trainee characteristics and the ways in which trainees reconstruct learning in each setting. The experience of teaching and interacting with learners, interactions with colleagues, and access to workplace resources and training are important workplace affordances for learning. However, some trainees have limited access to these affordances. Teaching observations, course activities and experiences as a learner are significant course affordances. Trainees’ beliefs, prior experiences and dispositions vary and significantly influence their engagement with course and workplace affordances. It is proposed that better integration of course and workplace learning through guided participation in an intentional workplace curriculum and attention to the ways trainees choose to engage with this, together with the use of practical theorising has the potential to improve trainee learning.  相似文献   

11.
This article reports on a case study of a college class for pre-service teachers on the US–Mexico border in which students participated in in-depth discussion around mathematical problems every day. This pedagogical approach promotes the socialization of students into and through the specialized discourse of mathematics. The focus of this paper is on the experience of transfronterizo students in that course. Transfronterizos are Mexican residents who periodically cross the border to attend school. For these students, whose educational background in Mexico allowed them to develop proficiency in elementary mathematical discourse in Spanish, their socialization experience includes ways in which they draw on language, and other social and learning experiences in Mexico. The focus of this paper is an assignment called Thinking Logs, a genre that required the use of mathematical discourse for teaching. Drawing on data gathered from participant observation of the course, interviews, analysis of study session discourse, and genre analysis, I highlight agentive ways that each participant used in their own socialization process. I show how participants improvised writing of models, asked for clarification in the first language, and even resisted the discourse. Students who resisted the demands might incur negative effects. Furthermore, I argue that the role of the guidance from an expert (such as a professor) is imperative in a socialization process, and I offer implications for ways that teachers can guide second language writers to develop mathematics discourse.  相似文献   

12.
The present study researches the implementation of a web based intelligent tutoring system for mathematics at lower secondary schools. In recent years, there is growing concern about the worrying situation at German lower secondary schools. Data from large scale educational assessments in the county of North Rhine-Westphalia (NRW) show that children at lower secondary schools have an embarrassing paucity of basic mathematical skills (Leutner et al., Lernstandserhebungen 9. Klasse 2004 in NRW: Erster Kurzbericht zur wissenschaftlichen Begleitung, 2004). In order to improve these basic mathematical skills in lower secondary school children, several schools implemented the web based intelligent tutoring system eFit. The aim of the present research was to investigate whether eFit constitutes an effective intervention of this target group. The results show that compared to a non-treatment control group, children in the eFit group significantly improved their arithmetic performance over a period of 9 months. As will be discussed, the findings have to be treated with cautions because eFit was specifically designed to alleviate mathematical difficulties and therefore “trained for the test” whereas traditional mathematics instruction followed the regular curriculum. The implications of this will be considered in the light of existing theory and research.  相似文献   

13.
There is broad acceptance that mathematics teachers’ beliefs about the nature of mathematics influence the ways in which they teach the subject. It is also recognised that mathematics as practised in typical school classrooms is different from the mathematical activity of mathematicians. This paper presents case studies of two secondary mathematics teachers, one experienced and the other relatively new to teaching, and considers their beliefs about the nature of mathematics, as a discipline and as a school subject. Possible origins and future developments of the structures of their belief systems are discussed along with implications of such structures for their practice. It is suggested that beliefs about mathematics can usefully be considered in terms of a matrix that accommodates the possibility of differing views of school mathematics and the discipline.  相似文献   

14.
Mathematics teachers face a myriad of instructional obstacles. Since the early 1990s, mathematics education researchers have proposed the use of constructivist practices to counteract these ever-prevalent obstacles. While we do give credit to the choices of instructional activities the constructivist paradigm promotes, there are problems with its use as the foundation of mathematics pedagogy (e.g., Phillips, Educational Researcher 24: 5–12 1995; Simon, Journal for Research in Mathematics Education 26: 114–145 1995). In this paper, we will analyze and review the literature pertaining to the conceptual tenets and operational practices of constructivism, and the viability of these practices for meeting the professional teaching standards proposed by the National Council of Teachers of Mathematics (NCTM; 2000). We will then review the literature pertaining to a paradigm of teaching that may be more applicable, that of persuasive pedagogical practices, and the ways in which these practices can differentially meet the goals of the mathematics standards. The differences between constructivism and persuasive pedagogy lead us to believe that the adoption of the theory of teaching as persuasion, or persuasive pedagogy, may be more appropriate for learning mathematics and the identification and correction of misconceptions. Further, these pedagogical practices correspond with suggestions for mathematical discourse provided by NCTM (2000).  相似文献   

15.
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.  相似文献   

16.
Content area literacy has an important role in helping students understand content in specific disciplines, such as mathematics. Although the strategies are not unique to each individual content area, they are often adapted for use in a specific discipline. For example, mathematicians use mathematical language to make sense of new ideas and information and to organize that information in a specialized way. Content literacy strategies can help mathematics students accomplish these goals. In this article, we will discuss six practical strategies to help build students' content skills in the mathematics classroom and they are: the Frayer model, question generation, visual supports, think-alouds, writing to learn, and text reading.  相似文献   

17.
Elin Farnell 《PRIMUS》2017,27(2):202-211
Abstract

In this article, I present a collection of puzzles appropriate for use in a variety of undergraduate courses, along with suggestions for relevant discussion. Logic puzzles and riddles have long been sources of amusement for mathematicians and the general public alike. I describe the use of puzzles in a classroom setting, and argue for their use as a basis for discussion of the nature of mathematics, for development of problem-solving skills and confidence, and as a means of engaging students from a broad range of mathematical backgrounds. I suggest a consistent use of puzzles in a low-pressure setting as a possible means of fostering inquiry and a positive learning mindset among students.  相似文献   

18.
We report on an international study about mathematics students’ ideas of how they will use mathematics in their future study and careers. This builds on our previous research into students’ conceptions of mathematics. In this paper, we use data from two groups of students studying mathematics: those who participated in an in-depth interview and those who completed an open-ended questionnaire. We found that their responses could be grouped into four categories: don’t know; procedural skills; conceptual skills; and professional skills. Although some students held clear ideas about the role of mathematics, many were not able to articulate how it would be used in their future. This has implications for their approach to learning and our approach to teaching.  相似文献   

19.
This paper explores the possibilities of working with White, working-class teacher education students to explore the “complex social trajectory” (Reay in Women’s Stud Int Forum 20(2):225–233, 1997a, p. 19) of class border crossing as they progress through college. Through analysis of a course that I have developed, Education and the American Dream, I explore political and pedagogical issues in teaching the thousands of teacher education students who are the first in their families to attend college about social class. Arguing that faculty in teacher education too often disregard the significance of deep class differences between themselves and many of their students, I propose that teacher education include coursework in which upwardly-mobile students (a) draw upon their distinctive perspectives as class border-crossers to elucidate their “complex social positioning as a complicated amalgam of current privilege interlaced with historic disadvantage” (Reay in Women’s Stud Int Forum 20(2):225–233, 1997a, p. 25) and (b) complicate what Adair and Dahlberg (Pedagogy 1:173–175, 2001, p. 174) have termed a cultural “impulse to frame class mobility as a narrative of moral progress”. Such coursework, I suggest, has implications for the development of teacher leaders in stratified schools. The paper draws upon the literatures on social class and educational attainment, on the construction of classed identities in spite of silence about class in public and academic discourse, and on pedagogies for teaching across class differences.  相似文献   

20.
We report a study of repairs in communication between workers and visiting outsiders (students, researchers or teachers). We show how cultural models such as metaphors and mathematical models facilitated explanations and repair work in inquiry and pedagogical dialogues. We extend previous theorisations of metaphor by Black; Lakoff and Johnson; Lakoff & Nunes; and Schon, to formulate a perspective on mathematical models and modelling and show how dialogues can manifest (i) application of ‘dead’ models to new contexts, and (ii) generative modelling. In particular, we draw in some depth on one case study of the use of a double number line model of the ‘gas day’ and its mediation of communication within two dialogues, characterised by inquiry and pedagogical discourse genres respectively. In addition to spatial and gestural affordances due to its blend of grounding metaphors, the model translates between workplace objects on the one side and spreadsheet-mathematical symbols on the other. The model is found to afford generative constructions that mediate the emergence of new understandings in the dialogues. Finally we discuss the significance of this metaphorical perspective on modelling for mathematics education.  相似文献   

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